Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|
Axiom of Choice: Definition (Formal) | $$~$
f: X \rightarrow \bigcup_{Y \in X} Y
$~$$ |
---|
Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$Y \in X$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$Y$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$f$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$Y$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$f(Y) \in Y$~$ |
---|
Axiom of Choice: Definition (Formal) | $$~$
\forall_X
\left(
\left[\forall_{Y \in X} Y \not= \emptyset \right]
\Rightarrow
\left[\exists
\left( f: X \rightarrow \bigcup_{Y \in X} Y \right)
\left(\forall_{Y \in X}
\exists_{y \in Y} f(Y) = y \right) \right]
\right)
$~$$ |
---|
Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$Y_1, Y_2, Y_3$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$y_1 \in Y_1, y_2 \in Y_2, y_3 \in Y_3$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$f$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$f(Y_1) = y_1$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$f(Y_2) = y_2$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$f(Y_3) = y_3$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$X$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$Y_1, Y_2, Y_3, \ldots$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$f$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$Y$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$n$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$n$~$ |
---|
Axiom of Choice: Definition (Formal) | $~$f$~$ |
---|
""$ax2+bx+c=0$ will be displ..." | $~$ax2+bx+c=0$~$ |
---|
""$ax2+bx+c=0$ will be displ..." | $~$ax2+bx+c=0$~$ |
---|
""$ax2+bx+c=0$ will be displ..." | $~$ax2+bx+c=0$~$ |
---|
""Extreme credences" here should likely be "infi..." | $~$-\infty$~$ |
---|
""Extreme credences" here should likely be "infi..." | $~$+\infty,$~$ |
---|
""Extreme credences" here should likely be "infi..." | $~$0$~$ |
---|
""Extreme credences" here should likely be "infi..." | $~$1$~$ |
---|
""Extreme credences" here should likely be "infi..." | $~$0$~$ |
---|
""Extreme credences" here should likely be "infi..." | $~$1$~$ |
---|
""Extreme credences" here should likely be "infi..." | $~$\mathbb P(X) + \mathbb P(\lnot X)$~$ |
---|
""Extreme credences" here should likely be "infi..." | $~$\lnot X$~$ |
---|
""Extreme credences" here should likely be "infi..." | $~$X$~$ |
---|
""Extreme credences" here should likely be "infi..." | $~$\aleph_0$~$ |
---|
""Formula" and "Statement" can be interchanged f..." | $~$\{+,\dot,0,1\}$~$ |
---|
""That's because we're considering results like ..." | $~$2^6 = 64$~$ |
---|
""That's because we're considering results like ..." | $~$p<0.05$~$ |
---|
""We only ran the 2012 US Presidential Election ..." | $~$10 bet that paid out $~$ |
---|
"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$8$~$ |
---|
"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$4$~$ |
---|
"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$\log_4 8$~$ |
---|
"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$1.5$~$ |
---|
"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$3$~$ |
---|
"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$2$~$ |
---|
"$8$ is not a power of $4$, but $\log_4 8$ is $1..." | $~$log_2 3$~$ |
---|
"(5) was intended to assume that $n \in \mathbb ..." | $~$n \in \mathbb R^{\ge 1},$~$ |
---|
"(5) was intended to assume that $n \in \mathbb ..." | $~$\in \mathbb R^{\ge 0}$~$ |
---|
"(5) was intended to assume that $n \in \mathbb ..." | $~$f(x^y)=yf(x)$~$ |
---|
"(5) was intended to assume that $n \in \mathbb ..." | $~$f(b^n)=nf(b)$~$ |
---|
"(5) was intended to assume that $n \in \mathbb ..." | $~$f(b)=1 \implies f(b^n)=n,$~$ |
---|
"(8) doesn't follow from (5). The assumption in ..." | $~$n$~$ |
---|
"(8) doesn't follow from (5). The assumption in ..." | $~$f$~$ |
---|
"(8) doesn't follow from (5). The assumption in ..." | $~$(\mathbb{R}^{>0},\cdot)$~$ |
---|
"(8) doesn't follow from (5). The assumption in ..." | $~$(\mathbb{R},+)$~$ |
---|
"(8) doesn't follow from (5). The assumption in ..." | $~$log$~$ |
---|
"(8) doesn't follow from (5). The assumption in ..." | $~$\mathbb{R}$~$ |
---|
"1. I propose that this concept be called "unex..." | $$~$ s(d) = \textrm{surprise}(d \mid H) = - \log \Pr (d \mid H) $~$$ |
---|
"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$s$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$s$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$(d \mid H)$~$ |
---|
"1. I propose that this concept be called "unex..." | $$~$\textrm{log-likelihood} = -\textrm{surprise}$~$$ |
---|
"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$t(d)$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$t$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$t$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$t$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$\Pr(d \mid H)$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|
"1. I propose that this concept be called "unex..." | $$~$\Pr(H \mid d) = \Pr(H \mid t(d))$~$$ |
---|
"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$s$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$t$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$s$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$H$~$ |
---|
"1. I propose that this concept be called "unex..." | $~$d$~$ |
---|
"> "you're allowed to increase P(BadDriver) a li..." | $~$\mathbb P(e \mid GoodDriver)$~$ |
---|
"> "you're allowed to increase P(BadDriver) a li..." | $~$\mathbb P(e \mid BadDriver)$~$ |
---|
"> "you're allowed to increase P(BadDriver) a li..." | $~$\mathbb P(BadDriver)$~$ |
---|
"A summary of the relevant cardinal arithmetic, ..." | $$~$\aleph_{\alpha} + \aleph_{\alpha} = \aleph_{\alpha} = \aleph_{\alpha} \aleph_{\alpha}$~$$ |
---|
"A summary of the relevant cardinal arithmetic, ..." | $$~$2^{\aleph_{\alpha}} > \aleph_{\alpha}$~$$ |
---|
"Actually, there should be diagonal matrices ins..." | $~$\mathbf H$~$ |
---|
"Actually, there should be diagonal matrices ins..." | $~$H_1, H_2, \ldots$~$ |
---|
"Actually, there should be diagonal matrices ins..." | $~$\mathbf H,$~$ |
---|
"Actually, there should be diagonal matrices ins..." | $~$C = AB; c_{ii} = a_{ii} * b_{ii}; ∀ i ≠ j, c_{ij} = 0$~$ |
---|
"Ah, one additional thing I'm confused about -- ..." | $~$X_i$~$ |
---|
"Ah, one additional thing I'm confused about -- ..." | $~$x_i$~$ |
---|
"Ah, one additional thing I'm confused about -- ..." | $~$X_i$~$ |
---|
"Ah, one additional thing I'm confused about -- ..." | $~$X_0$~$ |
---|
"Ah, one additional thing I'm confused about -- ..." | $~$X_1$~$ |
---|
"Ah, one additional thing I'm confused about -- ..." | $~$X_2$~$ |
---|
"Ah, one additional thing I'm confused about -- ..." | $~$X_3$~$ |
---|
"Ah, one additional thing I'm confused about -- ..." | $~$x_i$~$ |
---|
"Another, speculative point:
If $V$ and $U$ we..." | $~$V$~$ |
---|
"Another, speculative point:
If $V$ and $U$ we..." | $~$U$~$ |
---|
"Any relation satisfying 1-3 is a partial order,..." | $~$S$~$ |
---|
"Any relation satisfying 1-3 is a partial order,..." | $~$\le$~$ |
---|
"Are all the words in the free group, or just th..." | $~$X$~$ |
---|
"Are all the words in the free group, or just th..." | $~$X \cup X^{-1}$~$ |
---|
"Are all the words in the free group, or just th..." | $~$r r^{-1}$~$ |
---|
"Are all the words in the free group, or just th..." | $~$r^{-1} r$~$ |
---|
"Are all the words in the free group, or just th..." | $~$r r^{-1}$~$ |
---|
"Are all the words in the free group, or just th..." | $~$r \in X$~$ |
---|
"Are all the words in the free group, or just th..." | $~$r^{-1} r$~$ |
---|
"Are all the words in the free group, or just th..." | $~$X \cup X^{-1}$~$ |
---|
"Be wary here.
We see on the next (log probabil..." | $~$(1 : 10^{100})$~$ |
---|
"Be wary here.
We see on the next (log probabil..." | $~$(1 : 10^6)$~$ |
---|
"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$n$~$ |
---|
"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x$~$ |
---|
"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x \cdot x \le n$~$ |
---|
"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x$~$ |
---|
"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x$~$ |
---|
"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x$~$ |
---|
"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x=316$~$ |
---|
"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x$~$ |
---|
"Because 5 digits is equal to 2, 2.5 digits.
(B..." | $~$x^2 \le 100000.$~$ |
---|
"Broken link :(" | $~$M$~$ |
---|
"Broken link :(" | $~$N$~$ |
---|
"Consider using [3jp] for the proof?" | $~$x!$~$ |
---|
"Consider using [3jp] for the proof?" | $$~$x! = \Gamma (x+1),$~$$ |
---|
"Consider using [3jp] for the proof?" | $~$\Gamma $~$ |
---|
"Consider using [3jp] for the proof?" | $$~$\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\mathrm{d} t$~$$ |
---|
"Consider using [3jp] for the proof?" | $~$x$~$ |
---|
"Consider using [3jp] for the proof?" | $$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|
"Consider using [3jp] for the proof?" | $~$x=1$~$ |
---|
"Consider using [3jp] for the proof?" | $$~$\prod_{i=1}^{1}i = \int_{0}^{\infty}t^{1}e^{-t}\mathrm{d} t$~$$ |
---|
"Consider using [3jp] for the proof?" | $$~$1=1$~$$ |
---|
"Consider using [3jp] for the proof?" | $~$x$~$ |
---|
"Consider using [3jp] for the proof?" | $$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|
"Consider using [3jp] for the proof?" | $~$x + 1$~$ |
---|
"Consider using [3jp] for the proof?" | $$~$\prod_{i=1}^{x+1}i = \int_{0}^{\infty}t^{x+1}e^{-t}\mathrm{d} t$~$$ |
---|
"Consider using [3jp] for the proof?" | $~$x+1$~$ |
---|
"Consider using [3jp] for the proof?" | $$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|
"Consider using [3jp] for the proof?" | $$~$(x+1)\prod_{i=1}^{x}i = (x+1)\int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|
"Consider using [3jp] for the proof?" | $$~$\prod_{i=1}^{x+1}i = (x+1)\int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
---|
"Consider using [3jp] for the proof?" | $$~$= 0+\int_{0}^{\infty}(x+1)t^{x}e^{-t}\mathrm{d} t$~$$ |
---|
"Consider using [3jp] for the proof?" | $$~$= \left (-t^{x+1}e^{-t}) \right]_{0}^{\infty}+\int_{0}^{\infty}(x+1)t^{x}e^{-t}\mathrm{d} t$~$$ |
---|
"Consider using [3jp] for the proof?" | $$~$= \left (-t^{x+1}e^{-t}) \right]_{0}^{\infty}-\int_{0}^{\infty}(x+1)t^{x}(-e^{-t})\mathrm{d} t$~$$ |
---|
"Consider using [3jp] for the proof?" | $$~$=\int_{0}^{\infty}t^{x+1}e^{-t}\mathrm{d} t$~$$ |
---|
"Correct me if I'm wrong, but isn't it idiosyncr..." | $~$(S, \le)$~$ |
---|
"Correct me if I'm wrong, but isn't it idiosyncr..." | $~$S$~$ |
---|
"Correct me if I'm wrong, but isn't it idiosyncr..." | $~$\le$~$ |
---|
"Correct me if I'm wrong, but isn't it idiosyncr..." | $~$S$~$ |
---|
"Correct me if I'm wrong, but isn't it idiosyncr..." | $~$\leq$~$ |
---|
"Correct me if I'm wrong, but isn't it idiosyncr..." | $~$\leq$~$ |
---|
"Darn it, I wanted to use th..." | $~$Y$~$ |
---|
"Darn it, I wanted to use th..." | $~$X$~$ |
---|
"Darn it, I wanted to use th..." | $~$X$~$ |
---|
"Darn it, I wanted to use th..." | $~$Y$~$ |
---|
"Darn it, I wanted to use th..." | $~$X$~$ |
---|
"Darn it, I wanted to use th..." | $~$Y$~$ |
---|
"Darn it, I wanted to use th..." | $~$X$~$ |
---|
"Darn it, I wanted to use th..." | $~$X.$~$ |
---|
"Do the different biases of coin correspond to d..." | $~$H_{0.55},$~$ |
---|
"Do the different biases of coin correspond to d..." | $~$H_{0.6}$~$ |
---|
"Do the different biases of coin correspond to d..." | $~$H_{0.8}.$~$ |
---|
"Do the different biases of coin correspond to d..." | $~$H_{0.5},$~$ |
---|
"Does this actually work for..." | $~$A$~$ |
---|
"Does this actually work for..." | $~$B$~$ |
---|
"Does this actually work for..." | $~$\bP$~$ |
---|
"Does this actually work for..." | $~$\bP$~$ |
---|
"Does this make the definiti..." | $~$Y$~$ |
---|
"Does this make the definiti..." | $~$f$~$ |
---|
"Does this make the definiti..." | $~$Y$~$ |
---|
"Does this make the definiti..." | $~$\operatorname{square} : \mathbb R \to \mathbb R$~$ |
---|
"Does this make the definiti..." | $~$\operatorname{square}(x)=x^2$~$ |
---|
"Does this make the definiti..." | $~$\mathbb R$~$ |
---|
"Does this make the definiti..." | $~$\mathbb R$~$ |
---|
"Does this make the definiti..." | $~$\mathbb R$~$ |
---|
"Does this make the definiti..." | $~$\mathbb C$~$ |
---|
"Does x correspond to a *statement* (as used in ..." | $~$Prv(x)$~$ |
---|
"Does x correspond to a *statement* (as used in ..." | $~$x$~$ |
---|
"For readers who just skimme..." | $~$n$~$ |
---|
"For readers who just skimme..." | $~$2^n$~$ |
---|
"For readers who just skimme..." | $~$2^{3,000,000,000,000}$~$ |
---|
"For readers who just skimme..." | $~$2^{3,000,000,000,000}$~$ |
---|
"For readers who just skimme..." | $~$2^\text{3 trillion}$~$ |
---|
"Had to re-read this twice. ..." | $~$a$~$ |
---|
"Had to re-read this twice. ..." | $~$b$~$ |
---|
"Had to re-read this twice. ..." | $~$31a + b$~$ |
---|
"Had to re-read this twice. ..." | $~$31\cdot 30 + 30 = 960$~$ |
---|
"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$\mathcal L(H \mid e) < 0.05$~$ |
---|
"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$H$~$ |
---|
"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$e$~$ |
---|
"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$H$~$ |
---|
"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$H$~$ |
---|
"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$e$~$ |
---|
"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$\mathcal L(H \mid e)$~$ |
---|
"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$e$~$ |
---|
"Have I gone mad, or do you mean "L(H|e) is simp..." | $~$H$~$ |
---|
"Having a long redlink which does not point anyw..." | $~$b$~$ |
---|
"Having a long redlink which does not point anyw..." | $~$n,$~$ |
---|
"Having a long redlink which does not point anyw..." | $~$\log_b(n),$~$ |
---|
"Having a long redlink which does not point anyw..." | $~$b$~$ |
---|
"Having a long redlink which does not point anyw..." | $~$n$~$ |
---|
"Having a long redlink which does not point anyw..." | $~$\log_{10}(100)=2,$~$ |
---|
"Having a long redlink which does not point anyw..." | $~$\log_{10}(316) \approx 2.5,$~$ |
---|
"Having a long redlink which does not point anyw..." | $~$316 \approx$~$ |
---|
"Having a long redlink which does not point anyw..." | $~$10 \cdot 10 \cdot \sqrt{10},$~$ |
---|
"Having a long redlink which does not point anyw..." | $~$\sqrt{10}$~$ |
---|
"How about, "because I'm goi..." | $~$\log_{10}(\text{2,310,426})$~$ |
---|
"Huh... Not sure I understand this. I have BS in..." | $~$f$~$ |
---|
"Huh... Not sure I understand this. I have BS in..." | $~$x$~$ |
---|
"Huh... Not sure I understand this. I have BS in..." | $~$f(x)$~$ |
---|
"Huh... Not sure I understand this. I have BS in..." | $~$1/2$~$ |
---|
"Huh... Not sure I understand this. I have BS in..." | $~$f$~$ |
---|
"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$H_{fair},$~$ |
---|
"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$H_{heads}$~$ |
---|
"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$H_{tails}$~$ |
---|
"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$(1/2 : 1/3 : 1/6).$~$ |
---|
"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$(3 : 2 : 1)$~$ |
---|
"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$(2 : 1 : 3).$~$ |
---|
"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$(2 : 3 : 1)$~$ |
---|
"I believe that this should be $(2 : 3 : 1)$ rat..." | $~$(3 : 2 : 1)$~$ |
---|
"I can't figure out what this paragraph means --..." | $~$A$~$ |
---|
"I can't figure out what this paragraph means --..." | $~$B$~$ |
---|
"I can't figure out what this paragraph means --..." | $~$C$~$ |
---|
"I can't figure out what this paragraph means --..." | $~$C$~$ |
---|
"I can't figure out what this paragraph means --..." | $~$\mathcal T$~$ |
---|
"I can't figure out what this paragraph means --..." | $~$B$~$ |
---|
"I can't figure out what this paragraph means --..." | $~$D$~$ |
---|
"I don't think this is what you mean, is it?" | $~$X$~$ |
---|
"I don't think this is what you mean, is it?" | $~$Y$~$ |
---|
"I don't think this is what you mean, is it?" | $~$X$~$ |
---|
"I don't think this is what you mean, is it?" | $~$Y$~$ |
---|
"I don't think this is what you mean, is it?" | $~$X \to Y$~$ |
---|
"I don't think this is what you mean, is it?" | $~$Y^X$~$ |
---|
"I don't think this is what you mean, is it?" | $~$Y^2$~$ |
---|
"I don't think this is what you mean, is it?" | $~$Y$~$ |
---|
"I don't understand this sen..." | $$~$1$~$$ |
---|
"I fail to see how this setup is not fair - but ..." | $~$99\cdot 2=198$~$ |
---|
"I fail to see how this setup is not fair - but ..." | $~$100$~$ |
---|
"I fail to see how this setup is not fair - but ..." | $~$LDT$~$ |
---|
"I fail to see how this setup is not fair - but ..." | $~$198$~$ |
---|
"I fail to see how this setup is not fair - but ..." | $~$100$~$ |
---|
"I fail to see how this setup is not fair - but ..." | $~$1$~$ |
---|
"I fail to see how this setup is not fair - but ..." | $~$0$~$ |
---|
"I got lost here (and in the following equations..." | $~$\mathbb P(X_i | \mathbf{pa}_i)$~$ |
---|
"I got lost here (and in the following equations..." | $~$X_i$~$ |
---|
"I got lost here (and in the following equations..." | $~$x_i$~$ |
---|
"I got lost here (and in the following equations..." | $~$\mathbf {pa}_i$~$ |
---|
"I got lost here (and in the following equations..." | $~$x_i$~$ |
---|
"I got lost here (and in the following equations..." | $~$\mathbf x$~$ |
---|
"I got lost here -- I feel l..." | $~$\bullet$~$ |
---|
"I got lost here -- I feel l..." | $~$G$~$ |
---|
"I got lost here -- I feel l..." | $~$G$~$ |
---|
"I love the effect, but I wo..." | $~$t = 0$~$ |
---|
"I love the effect, but I wo..." | $~$4.7 t^2$~$ |
---|
"I love the effect, but I wo..." | $~$t$~$ |
---|
"I might write this as, "whe..." | $~$x$~$ |
---|
"I might write this as, "whe..." | $~$n$~$ |
---|
"I might write this as, "whe..." | $~$n-1$~$ |
---|
"I might write this as, "whe..." | $~$n$~$ |
---|
"I might write this as, "whe..." | $~$\log_{10}(x)$~$ |
---|
"I might write this as, "whe..." | $~$x;$~$ |
---|
"I might write this as, "whe..." | $~$x$~$ |
---|
"I might write this as, "whe..." | $~$x$~$ |
---|
"I really like this domino analogy.
Also, I'd e..." | $~$P(n)$~$ |
---|
"I really like this domino analogy.
Also, I'd e..." | $~$n$~$ |
---|
"I really like this domino analogy.
Also, I'd e..." | $~$P(n)$~$ |
---|
"I really like this domino analogy.
Also, I'd e..." | $~$n$~$ |
---|
"I really like this domino analogy.
Also, I'd e..." | $~$P(m)$~$ |
---|
"I really like this domino analogy.
Also, I'd e..." | $~$k \geq m$~$ |
---|
"I really like this domino analogy.
Also, I'd e..." | $~$P(k)$~$ |
---|
"I really like this domino analogy.
Also, I'd e..." | $~$P(k+1)$~$ |
---|
"I really like this domino analogy.
Also, I'd e..." | $~$P(m)$~$ |
---|
"I really like this domino analogy.
Also, I'd e..." | $~$P(m+1)$~$ |
---|
"I really like this domino analogy.
Also, I'd e..." | $~$P(m+1)$~$ |
---|
"I really like this domino analogy.
Also, I'd e..." | $~$P(m+2)$~$ |
---|
"I see that there is a description of double sca..." | $~$-1$~$ |
---|
"I suggest making it explici..." | $~$P$~$ |
---|
"I suggest making it explici..." | $~$P(x)$~$ |
---|
"I suggest making it explici..." | $~$P(X=x)$~$ |
---|
"I suggest making it explici..." | $~$X$~$ |
---|
"I suggest making it explici..." | $~$P$~$ |
---|
"I suggest we can assume tha..." | $~$s$~$ |
---|
"I think it would be worthwhile to explicitly ca..." | $$~$ 1 + 2 + \cdots + n = \frac{n(n+1)}{2}$~$$ |
---|
"I think it would be worthwhile to explicitly ca..." | $~$n \ge 1$~$ |
---|
"I think it would be worthwhile to explicitly ca..." | $~$n=1$~$ |
---|
"I think it would be worthwhile to explicitly ca..." | $$~$ 1 = \frac{1(1+1)}{2} = \frac{2}{2} = 1.$~$$ |
---|
"I think it would be worthwhile to explicitly ca..." | $~$k$~$ |
---|
"I think it would be worthwhile to explicitly ca..." | $~$k\ge1$~$ |
---|
"I think it would be worthwhile to explicitly ca..." | $$~$1 + 2 + \cdots + k = \frac{k(k+1)}{2}$~$$ |
---|
"I think it would be worthwhile to explicitly ca..." | $$~$ 1 + 2 + \cdots + k + (k+1) = \frac{(k+1)([k+1]+1)}{2}.$~$$ |
---|
"I think it would be worthwhile to explicitly ca..." | $~$k+1$~$ |
---|
"I think it would be worthwhile to explicitly ca..." | $$~$1+2+\cdots + k + (k+1) = \frac{k(k+1)}{2} + k + 1.$~$$ |
---|
"I think it would be worthwhile to explicitly ca..." | $$~$\frac{k(k+1)}{2} + \frac{2(k+1)}{2} = \frac{(k+2)(k+1)}{2} = \frac{(k+1)([k+1]+1)}{2}.$~$$ |
---|
"I think it would be worthwhile to explicitly ca..." | $$~$ 1 + 2 + \cdots + k + (k+1) = \frac{(k+1)([k+1]+1)}{2}$~$$ |
---|
"I think it would be worthwhile to explicitly ca..." | $~$n$~$ |
---|
"I think it would be worthwhile to explicitly ca..." | $~$k+1$~$ |
---|
"I think it's confusing to introduce multi-argum..." | $~$\lambda$~$ |
---|
"I think it's confusing to introduce multi-argum..." | $~$\lambda x.f(x)$~$ |
---|
"I think it's confusing to introduce multi-argum..." | $~$x$~$ |
---|
"I think it's confusing to introduce multi-argum..." | $~$f(x)$~$ |
---|
"I think it's confusing to introduce multi-argum..." | $~$\lambda x.x+1$~$ |
---|
"I think it's confusing to introduce multi-argum..." | $~$\lambda$~$ |
---|
"I think it's confusing to introduce multi-argum..." | $~$\lambda x.\lambda y.x+y$~$ |
---|
"I think it's confusing to introduce multi-argum..." | $~$\lambda xy.x+y$~$ |
---|
"I think it's confusing to introduce multi-argum..." | $~$\lambda xy$~$ |
---|
"I think it's confusing to introduce multi-argum..." | $~$\lambda x.\lambda y$~$ |
---|
"I think that every metric space is dense in its..." | $~$\newcommand{\rats}{\mathbb{Q}} \newcommand{\Ql}{\rats^\le} \newcommand{\Qr}{\rats^\ge} \newcommand{\Qls}{\rats^<} \newcommand{\Qrs}{\rats^>}$~$ |
---|
"I think that every metric space is dense in its..." | $~$\newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\sothat}{\ |\ }$~$ |
---|
"I think the answer is no. Indeed, there are unc..." | $~$S$~$ |
---|
"I think this paragraph and ..." | $~$2^6 < 101 < 2^7$~$ |
---|
"I think this sentence would be easier to read w..." | $~$\lambda x.(\lambda y.(x+y))$~$ |
---|
"I think this sentence would be easier to read w..." | $~$(\lambda x.(\lambda y.(x+y)))$~$ |
---|
"I think this sentence would be easier to read w..." | $~$f\ x\ y$~$ |
---|
"I think this sentence would be easier to read w..." | $~$f$~$ |
---|
"I think this sentence would be easier to read w..." | $~$x$~$ |
---|
"I think this sentence would be easier to read w..." | $~$y$~$ |
---|
"I think this sentence would be easier to read w..." | $~$(f\ x)\ y$~$ |
---|
"I think this sentence would be easier to read w..." | $~$f\ (x\ y)$~$ |
---|
"I think this sentence would be easier to read w..." | $~$\lambda$~$ |
---|
"I think this sentence would be easier to read w..." | $~$\lambda x.\lambda y.x+y$~$ |
---|
"I think this sentence would be easier to read w..." | $~$\lambda x.(\lambda y.(x+y))$~$ |
---|
"I think this sentence would be easier to read w..." | $~$(\lambda x.\lambda y.x)+y$~$ |
---|
"I think this sentence would be easier to read w..." | $~$\lambda x.(\lambda y.x)+y$~$ |
---|
"I think this sentence would be easier to read w..." | $~$\lambda$~$ |
---|
"I think this sentence would be easier to read w..." | $~$\lambda xy.x+y$~$ |
---|
"I think this sentence would be easier to read w..." | $~$\lambda x.\lambda y.x+y$~$ |
---|
"I think you may need to spe..." | $~$x$~$ |
---|
"I think you may need to spe..." | $~$n$~$ |
---|
"I think you may need to spe..." | $~$n-1$~$ |
---|
"I think you may need to spe..." | $~$n$~$ |
---|
"I think you may need to spe..." | $~$\log_{10}(x)$~$ |
---|
"I think you may need to spe..." | $~$x;$~$ |
---|
"I think you may need to spe..." | $~$x$~$ |
---|
"I think you may need to spe..." | $~$x$~$ |
---|
"I would consider leading wi..." | $~$\log_{10}(12) \approx 1.08$~$ |
---|
"I would consider leading wi..." | $~$\log_2(10) \approx 3.32$~$ |
---|
"I would expect this sentence only after another..." | $~$2 : 1$~$ |
---|
"I would expect this sentence only after another..." | $~$8 : 1,$~$ |
---|
"I would expect this sentence only after another..." | $~$2 : 1$~$ |
---|
"I would expect this sentence only after another..." | $~$4 : 1.$~$ |
---|
"I'm curious if the inverse ..." | $~$(a_1 a_2 \dots a_k)$~$ |
---|
"I'm curious if the inverse ..." | $~$(a_k a_{k-1} \dots a_1)$~$ |
---|
"If these are included I think it would be good ..." | $~$0.999\dotsc=1$~$ |
---|
"If you look on Wikipedia's ..." | $~$A \cdot B = A + A + A$~$ |
---|
"If you look on Wikipedia's ..." | $~$B$~$ |
---|
"If you're going to start us..." | $~$\mathbb P$~$ |
---|
"If you're going to start us..." | $~$\operatorname{d}\!f$~$ |
---|
"If you're going to start us..." | $~$\operatorname{d}\!f$~$ |
---|
"In this page, the terms "probability" and "odds..." | $~$X$~$ |
---|
"In this page, the terms "probability" and "odds..." | $~$\mathbb P(X)$~$ |
---|
"In this page, the terms "probability" and "odds..." | $~$X.$~$ |
---|
"In this sentence I think yo..." | $~$f$~$ |
---|
"In this sentence I think yo..." | $~$X$~$ |
---|
"In this sentence I think yo..." | $~$I$~$ |
---|
"In this sentence I think yo..." | $~$Y$~$ |
---|
"In this sentence I think yo..." | $~$I$~$ |
---|
"Intro should be re-written ..." | $~$(X, \bullet)$~$ |
---|
"Intro should be re-written ..." | $~$X$~$ |
---|
"Intro should be re-written ..." | $~$\bullet$~$ |
---|
"Intro should be re-written ..." | $~$X$~$ |
---|
"Is "-1 against" the same as "+1 for"?
Expressi..." | $~${^-3}$~$ |
---|
"Is "-1 against" the same as "+1 for"?
Expressi..." | $~${^-1}$~$ |
---|
"Is "-1 against" the same as "+1 for"?
Expressi..." | $~${^-4}$~$ |
---|
"Is "-1 against" the same as "+1 for"?
Expressi..." | $~$(1 : 16)$~$ |
---|
"Is $\mathbb{N}$ itself called $\omega$, or just..." | $~$\mathbb{N}$~$ |
---|
"Is $\mathbb{N}$ itself called $\omega$, or just..." | $~$\omega$~$ |
---|
"Is [0, inf) same as R+?" | $~$d$~$ |
---|
"Is [0, inf) same as R+?" | $~$d$~$ |
---|
"Is [0, inf) same as R+?" | $~$S$~$ |
---|
"Is [0, inf) same as R+?" | $$~$d: S \times S \to [0, \infty)$~$$ |
---|
"Is this a typo? Shouldn't you buy coins if they..." | $~$10^{10} < 2^{35}.$~$ |
---|
"Is this paragraph needed? ..." | $~$x$~$ |
---|
"Is this paragraph needed? ..." | $~$n$~$ |
---|
"Is this paragraph needed? ..." | $~$n-1$~$ |
---|
"Is this paragraph needed? ..." | $~$n$~$ |
---|
"Is this paragraph needed? ..." | $~$\log_{10}(x)$~$ |
---|
"Is this paragraph needed? ..." | $~$x;$~$ |
---|
"Is this paragraph needed? ..." | $~$x$~$ |
---|
"Is this paragraph needed? ..." | $~$x$~$ |
---|
"Is this paragraph needed? ..." | $~$x$~$ |
---|
"Is this what is meant by transitive and nontran..." | $~$A = \{ \{ 1,2 \}, \{ 3,4 \}, 1, 2, 3, 4 \}$~$ |
---|
"Is this what is meant by transitive and nontran..." | $~$x = \{1,2\}$~$ |
---|
"Is this what is meant by transitive and nontran..." | $~$a = 2$~$ |
---|
"Is this what is meant by transitive and nontran..." | $~$a \in x$~$ |
---|
"Is this what is meant by transitive and nontran..." | $~$x \in A$~$ |
---|
"Is this what is meant by transitive and nontran..." | $~$a \in A$~$ |
---|
"Is this what is meant by transitive and nontran..." | $~$B = \{ \{ 1,2 \}, \{ 3,4 \} \}$~$ |
---|
"Is this what is meant by transitive and nontran..." | $~$y = \{1,2\}$~$ |
---|
"Is this what is meant by transitive and nontran..." | $~$b = 2$~$ |
---|
"Is this what is meant by transitive and nontran..." | $~$b \in y$~$ |
---|
"Is this what is meant by transitive and nontran..." | $~$y \in B$~$ |
---|
"Is this what is meant by transitive and nontran..." | $~$b \notin B$~$ |
---|
"Is what follows the colon m..." | $~$3^{10}$~$ |
---|
"Is what follows the colon m..." | $~$n^k$~$ |
---|
"Isn't one coin and three di..." | $~$\log_2(6) + \log_2(10) + 3\log_2(2) \approx 8.9$~$ |
---|
"Isn't one coin and three di..." | $~$2*3^6 = 432,$~$ |
---|
"Isn't one coin and three di..." | $~$\log_2(2) + 3*\log_2(6) \approx 8.75$~$ |
---|
"It is really confusing to apply one of the init..." | $~$\mathbb P({positive}\mid {HIV}) = .997$~$ |
---|
"It is really confusing to apply one of the init..." | $~$\mathbb P({negative}\mid \neg {HIV}) = .998$~$ |
---|
"It is really confusing to apply one of the init..." | $~$\mathbb P({positive} \mid \neg {HIV}) = .002.$~$ |
---|
"It would be nice to show how to go from 99.8% t..." | $~$1 : 100,000$~$ |
---|
"It would be nice to show how to go from 99.8% t..." | $~$500 : 1.$~$ |
---|
"Just reiterating that it's 18% of **all** stude..." | $~$\mathbb P(sick \mid blackened)$~$ |
---|
"Just reiterating that it's 18% of **all** stude..." | $~$\mathbb P(sick \wedge blackened)$~$ |
---|
"Just reiterating that it's 18% of **all** stude..." | $~$\mathbb P(blackened)$~$ |
---|
"Looks like a mathjax error?" | $~$PA$~$ |
---|
"Looks like a mathjax error?" | $~$\square_{PA}$~$ |
---|
"Looks like a mathjax error?" | $~$PA$~$ |
---|
"Looks like a mathjax error?" | $~$PA$~$ |
---|
"Looks like a mathjax error?" | $~$A$~$ |
---|
"Looks like a mathjax error?" | $~$\square_{PA}(\ulcorener A\urcorner$~$ |
---|
"Looks like a mathjax error?" | $~$A$~$ |
---|
"Looks like a mathjax error?" | $~$PA$~$ |
---|
"May need to build the intuition that knowing ho..." | $~$x$~$ |
---|
"May need to build the intuition that knowing ho..." | $~$x$~$ |
---|
"May need to build the intuition that knowing ho..." | $~$n$~$ |
---|
"May need to build the intuition that knowing ho..." | $~$c$~$ |
---|
"May need to build the intuition that knowing ho..." | $~$n$~$ |
---|
"May need to build the intuition that knowing ho..." | $~$c.$~$ |
---|
"Maybe insert an equation style definition of th..." | $~${\bf \hat v}$~$ |
---|
"Maybe insert an equation style definition of th..." | $$~$|\mathbf{\hat v}| = \left|\frac{\mathbf{v}}{|\mathbf{v}|}\right| = \left|\frac{1}{|\mathbf{v}|}\right||\mathbf{v}| = \frac{|\mathbf{v}|}{|\mathbf{v}|}=1$~$$ |
---|
"Maybe insert an equation style definition of th..." | $~$\hat{\mathbf v} = \frac{1}{| \mathbf v |}\mathbf v = \frac{\mathbf v}{| \mathbf v |}$~$ |
---|
"Might one of the following ..." | $~$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$~$ |
---|
"Might one of the following ..." | $~$\frac{1}{2}$~$ |
---|
"Might one of the following ..." | $~$G_0 \xrightarrow{f_1} G_1 \xrightarrow{f_2} G_2 \xrightarrow{f_3} \cdots \xrightarrow{f_n} G_n$~$ |
---|
"Might one of the following ..." | $~$\text{im}(f_k) = \text{ker}(f_{k+1})$~$ |
---|
"Might one of the following ..." | $~$0 \le k < n$~$ |
---|
"Might one of the following ..." | $~$n\times n$~$ |
---|
"Might one of the following ..." | $~$A$~$ |
---|
"Might one of the following ..." | $~$a_{i,j}$~$ |
---|
"Might one of the following ..." | $~$\det(A) = \sum_{\sigma\in S_n}\text{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma_i}$~$ |
---|
"Might one of the following ..." | $~$S_n$~$ |
---|
"Might one of the following ..." | $~$n$~$ |
---|
"Nice!" | $~$\log_b(x)$~$ |
---|
"Nice!" | $~$b$~$ |
---|
"Nice!" | $~$x$~$ |
---|
"No, the difference between the two sentences li..." | $~$K$~$ |
---|
"No, the difference between the two sentences li..." | $~$O$~$ |
---|
"No, this kind of factorization is used for *any..." | $~$\mathbb P(X_i | \mathbf{pa}_i)$~$ |
---|
"No, this kind of factorization is used for *any..." | $~$X_i$~$ |
---|
"No, this kind of factorization is used for *any..." | $~$x_i$~$ |
---|
"No, this kind of factorization is used for *any..." | $~$\mathbf {pa}_i$~$ |
---|
"No, this kind of factorization is used for *any..." | $~$x_i$~$ |
---|
"No, this kind of factorization is used for *any..." | $~$\mathbf x$~$ |
---|
"Not 2^100?" | $~$2^{101}$~$ |
---|
"Not clear what this means?" | $~$\prec$~$ |
---|
"Not clear what this means?" | $~$\langle \mathbb R, \leq \rangle$~$ |
---|
"Not clear what this means?" | $~$\leq$~$ |
---|
"Not clear what this means?" | $~$0 < 1$~$ |
---|
"Not clear what this means?" | $~$0$~$ |
---|
"Not clear what this means?" | $~$\mathbb R$~$ |
---|
"Not clear what this means?" | $~$x \in \mathbb R$~$ |
---|
"Not clear what this means?" | $~$x > 0$~$ |
---|
"Not clear what this means?" | $~$y \in \mathbb R$~$ |
---|
"Not clear what this means?" | $~$0 < y < x$~$ |
---|
"Not clear what this means?" | $~$\mathbb R$~$ |
---|
"Okay now I'm also confused...." | $~$f(x)=1$~$ |
---|
"Okay now I'm also confused...." | $~$1$~$ |
---|
"Okay now I'm also confused...." | $~$\{1\}$~$ |
---|
"On "Conditions for Goodhart's curse": It seems ..." | $~$V:s \mapsto V(s)$~$ |
---|
"On "Conditions for Goodhart's curse": It seems ..." | $~$s$~$ |
---|
"On "Conditions for Goodhart's curse": It seems ..." | $~$n$~$ |
---|
"One of these does log( prob/ 1 - prob) the othe..." | $~${^-2}$~$ |
---|
"One of these does log( prob/ 1 - prob) the othe..." | $~${^-6}$~$ |
---|
"One of these does log( prob/ 1 - prob) the othe..." | $~$\log_{10}(10^{-6}) - \log_{10}(10^{-2})$~$ |
---|
"One of these does log( prob/ 1 - prob) the othe..." | $~${^-4}$~$ |
---|
"One of these does log( prob/ 1 - prob) the othe..." | $~${^-13.3}$~$ |
---|
"One of these does log( prob/ 1 - prob) the othe..." | $~$\log_{10}(\frac{0.10}{0.90}) - \log_{10}(\frac{0.11}{0.89}) \approx {^-0.954}-{^-0.907} \approx {^-0.046}$~$ |
---|
"One of these does log( prob/ 1 - prob) the othe..." | $~${^-0.153}$~$ |
---|
"Pedantic remark: Aren't you missing the identit..." | $~$x^{-1}$~$ |
---|
"Pedantic remark: Aren't you missing the identit..." | $~$\rho_{x^{-1}}$~$ |
---|
"Pedantic remark: Aren't you missing the identit..." | $~$\rho_x$~$ |
---|
"Pedantic remark: Aren't you missing the identit..." | $~$\rho_{x^{-1}}$~$ |
---|
"Pedantic remark: Aren't you missing the identit..." | $~$\rho_\epsilon$~$ |
---|
"Seven tenths?" | $~$\log_{10}(500)$~$ |
---|
"Should the p's and q's in o..." | $~$p \prec q$~$ |
---|
"Should the p's and q's in o..." | $~$q$~$ |
---|
"Should the p's and q's in o..." | $~$P$~$ |
---|
"Should the p's and q's in o..." | $~$p$~$ |
---|
"Should the p's and q's in o..." | $~$p \prec q$~$ |
---|
"Should the p's and q's in o..." | $~$p$~$ |
---|
"Should the p's and q's in o..." | $~$q$~$ |
---|
"Should the p's and q's in o..." | $~$p$~$ |
---|
"Should the p's and q's in o..." | $~$q$~$ |
---|
"Should the p's and q's in o..." | $~$q$~$ |
---|
"Should the p's and q's in o..." | $~$p$~$ |
---|
"Smallest?" | $~$x,$~$ |
---|
"Smallest?" | $~$\lceil x \rceil$~$ |
---|
"Smallest?" | $~$\operatorname{ceil}(x),$~$ |
---|
"Smallest?" | $~$n \ge x.$~$ |
---|
"Smallest?" | $~$\lceil 3.72 \rceil = 4, \lceil 4 \rceil = 4,$~$ |
---|
"Smallest?" | $~$\lceil -3.72 \rceil = -3.$~$ |
---|
"Surely they are equivalent. Given a Rice-decidi..." | $~$[n]$~$ |
---|
"Surely they are equivalent. Given a Rice-decidi..." | $~$k$~$ |
---|
"Surely they are equivalent. Given a Rice-decidi..." | $~$[n]$~$ |
---|
"Surely they are equivalent. Given a Rice-decidi..." | $~$k$~$ |
---|
"Thanks for this analysis and congratulations on..." | $~$\pi_5$~$ |
---|
"Thanks for this analysis and congratulations on..." | $~$V$~$ |
---|
"Thanks for this analysis and congratulations on..." | $~$V$~$ |
---|
"Thanks for this analysis and congratulations on..." | $~$V$~$ |
---|
"The $x/y$ notation is confusing - these ratios ..." | $~$(x : y)$~$ |
---|
"The $x/y$ notation is confusing - these ratios ..." | $~$\alpha$~$ |
---|
"The $x/y$ notation is confusing - these ratios ..." | $~$(\alpha x : \alpha y).$~$ |
---|
"The $x/y$ notation is confusing - these ratios ..." | $~$x$~$ |
---|
"The $x/y$ notation is confusing - these ratios ..." | $~$y$~$ |
---|
"The $x/y$ notation is confusing - these ratios ..." | $~$\frac{x}{y}.$~$ |
---|
"The $x/y$ notation is confusing - these ratios ..." | $~$\frac{x}{y}$~$ |
---|
"The $x/y$ notation is confusing - these ratios ..." | $~$(x : y),$~$ |
---|
"The $x/y$ notation is confusing - these ratios ..." | $~$\left(\frac{x}{y} : 1\right).$~$ |
---|
"The $x/y$ notation is confusing - these ratios ..." | $~$x/y$~$ |
---|
"The expression P(a_x [ ]-> o_i) is meaningless...." | $~$\ \mathbb P(a_x \ \square \! \! \rightarrow o_i).$~$ |
---|
"The following would be simpler and more consist..." | $~$\mathbb P(X \wedge Y) = \mathbb P(Y) \cdot \mathbb P(X|Y).$~$ |
---|
"The inverse of multiplication is division. To t..." | $~$1 : 4$~$ |
---|
"The inverse of multiplication is division. To t..." | $~$3 : 1$~$ |
---|
"The inverse of multiplication is division. To t..." | $~$(1 \cdot 3) : (4 \cdot 1) = 3 : 4$~$ |
---|
"The log used to determine number of bits should..." | $~$H$~$ |
---|
"The log used to determine number of bits should..." | $~$\frac{1}{8}$~$ |
---|
"The log used to determine number of bits should..." | $~$\lnot H$~$ |
---|
"The log used to determine number of bits should..." | $~$\frac{1}{4}$~$ |
---|
"The log used to determine number of bits should..." | $~$\lnot H$~$ |
---|
"The log used to determine number of bits should..." | $~$H,$~$ |
---|
"The log used to determine number of bits should..." | $~$\mathbb P(e \mid H)$~$ |
---|
"The log used to determine number of bits should..." | $~$\mathbb P(e \mid \lnot H)$~$ |
---|
"The log used to determine number of bits should..." | $~$\left(\frac{1}{8} : \frac{1}{4}\right)$~$ |
---|
"The log used to determine number of bits should..." | $~$=$~$ |
---|
"The log used to determine number of bits should..." | $~$(1 : 2),$~$ |
---|
"The log used to determine number of bits should..." | $~$H.$~$ |
---|
"The non-existence of a total order on $\mathbb{..." | $~$\mathbb{C}$~$ |
---|
"The problem I have in mind is deciding whether ..." | $~$S$~$ |
---|
"The problem I have in mind is deciding whether ..." | $~$S$~$ |
---|
"The problem I have in mind is deciding whether ..." | $~$S$~$ |
---|
"The problem I have in mind is deciding whether ..." | $~$S$~$ |
---|
"The proof of (5) only goes through for $n\in\ma..." | $~$n\in\mathbb{N}$~$ |
---|
"The proof of (5) only goes through for $n\in\ma..." | $~$f(b)=1\Rightarrow f(b^q)=q$~$ |
---|
"The proof of (5) only goes through for $n\in\ma..." | $~$q\in\mathbb{Q}$~$ |
---|
"The proof of (5) only goes through for $n\in\ma..." | $~$f$~$ |
---|
"The urls are displaying as:
https://arbital.com..." | $~$bayes_rule_details,$~$ |
---|
"This "do" notation may seem mysterious, as it i..." | $~$\mathbb P(\mathbf x | \operatorname{do}(X_j=x_j))$~$ |
---|
"This confused me at first because I didn't real..." | $~$\mathbb P(X \mid Y)$~$ |
---|
"This confused me at first because I didn't real..." | $~$X$~$ |
---|
"This confused me at first because I didn't real..." | $~$Y$~$ |
---|
"This definition of the real numbers has a bigge..." | $~$\mathbb{N} \setminus \{1, 2, 3, 4, 5\}$~$ |
---|
"This definition of the real numbers has a bigge..." | $~${5}$~$ |
---|
"This definition of the real numbers has a bigge..." | $~$1/8$~$ |
---|
"This does not seem like it'd be transparent, es..." | $~$1$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$a$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$b$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$b$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$a$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$a$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$c$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$a$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$b$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$b$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$c$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$a$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$b$~$ |
---|
"This is a clear explanation, but I think some f..." | $~$c$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$A$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$B$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$A$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$1$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$B$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$0$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$A$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$B$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$A$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$1$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$B$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$A$~$ |
---|
"This is not universally agreed-upon, but I use ..." | $~$B$~$ |
---|
"This is slightly confusing,..." | $~$\log_{10}(\text{2,310,426})$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$\frac{a}{m}$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$a$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$\frac{1}{m}$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$\frac{1}{m}$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$n$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$a$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$\frac{1}{m}$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$n$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$n$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$\frac{1}{m}$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$\frac{1}{m} \times \frac{1}{n}$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$\frac{1}{m \times n}$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$\frac{n}{m} = n \times \frac{1}{m}$~$ |
---|
"This relies on a principle "other way" introduc..." | $~$\frac{n}{m} = n \times \frac{1}{m}$~$ |
---|
"This seems like a straw alt..." | $~$V_i$~$ |
---|
"This seems like a straw alt..." | $~$v_i.$~$ |
---|
"This seems like a straw alt..." | $~$v_i$~$ |
---|
"This seems like a straw alt..." | $~$v_i^*$~$ |
---|
"This seems like a straw alt..." | $~$V_i$~$ |
---|
"This wording suggests the group contains only s..." | $~$X = \{ a, b \}$~$ |
---|
"Underline." | $~$\log_b(316) \approx \frac{5\log_b(10)}{2}$~$ |
---|
"Underline." | $~$n$~$ |
---|
"Underline." | $~$\sqrt{n}$~$ |
---|
"Underline." | $~$n$~$ |
---|
"Underline." | $~$x$~$ |
---|
"Underline." | $~$x$~$ |
---|
"Underline." | $~$x \cdot x$~$ |
---|
"Underline." | $~$n$~$ |
---|
"Underline." | $~$n$~$ |
---|
"Underline." | $~$\sqrt{n}$~$ |
---|
"Wait, really? Is this a joke or does being tran..." | $~$\log$~$ |
---|
"Wait, really? Is this a joke or does being tran..." | $~$\log_2(3)$~$ |
---|
"Wait, really? Is this a joke or does being tran..." | $~$1$~$ |
---|
"Wait, really? Is this a joke or does being tran..." | $~$\log_2(6),$~$ |
---|
"Wait, really? Is this a joke or does being tran..." | $~$\log_2(9)$~$ |
---|
"Wait, really? Is this a joke or does being tran..." | $~$\log_2(3^{10}),$~$ |
---|
"Wait, really? Is this a joke or does being tran..." | $~$\log_2(3^9)$~$ |
---|
"Wait, really? Is this a joke or does being tran..." | $~$\log_2(3^{10}).$~$ |
---|
"Wait, really? Is this a joke or does being tran..." | $~$\log_2(3)$~$ |
---|
"What's $n$ exactly?" | $~$x$~$ |
---|
"What's $n$ exactly?" | $~$x$~$ |
---|
"What's $n$ exactly?" | $~$n$~$ |
---|
"What's $n$ exactly?" | $~$x$~$ |
---|
"What's $n$ exactly?" | $~$n$~$ |
---|
"Where did the '16' come fro..." | $~$(5 : 3 : 2) \cdot (2 : 1 : 5) \cdot (12 : 10 : 1) = (120 : 30 : 10) \cong (12/16 : 3/16 : 1/16)$~$ |
---|
"Why is it called a *decision problem*? As a rea..." | $~$D$~$ |
---|
"Why is it called a *decision problem*? As a rea..." | $~$A$~$ |
---|
"Why is it called a *decision problem*? As a rea..." | $~$A$~$ |
---|
"Why is it called a *decision problem*? As a rea..." | $~$\{0,1\}^*$~$ |
---|
"Would be cool to have an im..." | $~$C_2$~$ |
---|
"Would be cool to have an im..." | $~$2$~$ |
---|
"Would be cool to have an im..." | $~$1$~$ |
---|
"Would be cool to have an im..." | $~$-1$~$ |
---|
"Would be cool to have an im..." | $~$1$~$ |
---|
"Would be cool to have an im..." | $~$-1$~$ |
---|
"Would be cool to have an im..." | $~$f(x)$~$ |
---|
"Would be cool to have an im..." | $~$f(-x)$~$ |
---|
"Would be cool to have an im..." | $~$f(x)$~$ |
---|
"Would be cool to have an im..." | $~$(-1) \times (-1) = 1$~$ |
---|
"Would be cool to have an im..." | $~$f(-(-x)) = f(x)$~$ |
---|
"Would it be appropriate to ..." | $~$P$~$ |
---|
"Would it be appropriate to ..." | $~$\leq$~$ |
---|
"Wrong, they are exactly the same distances. I r..." | $~${+1}$~$ |
---|
"Wrong, they are exactly the same distances. I r..." | $~${^+1}$~$ |
---|
"Wrong, they are exactly the same distances. I r..." | $~$0.01$~$ |
---|
"Wrong, they are exactly the same distances. I r..." | $~$0.000001$~$ |
---|
"Wrong, they are exactly the same distances. I r..." | $~$0.11$~$ |
---|
"Wrong, they are exactly the same distances. I r..." | $~$0.100001.$~$ |
---|
"[@2] I think there should b..." | $~$\mathbb P(f\mid e\!=\!\textbf {THT}) = \dfrac{\mathcal L(e\!=\!\textbf{THT}\mid f) \cdot \mathbb P(f)}{\mathbb P(e\!=\!\textbf {THT})} = **\dfrac{(1 - x) \cdot x \cdot (1 - x) \cdot 1}{\int_0^1 (1 - x) \cdot x \cdot (1 - x) \cdot 1 \** \operatorname{d}\!f} = 12 \cdot f(1 - f)^2$~$ |
---|
"[@5hc] Thanks for the edit! I made a couple of ..." | $~$\emptyset$~$ |
---|
"[@5hc]: I've made the appropriate changes to th..." | $~$57$~$ |
---|
"[@5hc]: I've made the appropriate changes to th..." | $~$\mathrm{sin}$~$ |
---|
"in X, **such that**..." | $~$f : X \times X \to X$~$ |
---|
"in X, **such that**..." | $~$x, y, z$~$ |
---|
"in X, **such that**..." | $~$X$~$ |
---|
"in X, **such that**..." | $~$f(x, f(y, z)) = f(f(x, y), z)$~$ |
---|
"in X, **such that**..." | $~$+$~$ |
---|
"in X, **such that**..." | $~$(x + y) + z = x + (y + z)$~$ |
---|
"in X, **such that**..." | $~$x, y,$~$ |
---|
"in X, **such that**..." | $~$z$~$ |
---|
"odd + odd doesn't equal even?" | $~$0 + 2\mathbb Z$~$ |
---|
"odd + odd doesn't equal even?" | $~$1 + 2\mathbb Z$~$ |
---|
"odd + odd doesn't equal even?" | $~$+$~$ |
---|
"odd + odd doesn't equal even?" | $~$\text{even}$~$ |
---|
"odd + odd doesn't equal even?" | $~$\text{odd}$~$ |
---|
"odd + odd doesn't equal even?" | $~$\text{even}+ \text{even} = \text{even}$~$ |
---|
"odd + odd doesn't equal even?" | $~$\text{even} + \text{odd} = \text{odd}$~$ |
---|
"odd + odd doesn't equal even?" | $~$\text{odd} + \text{odd} = \text{odd}$~$ |
---|
"output?" | $~$x$~$ |
---|
"output?" | $~$x$~$ |
---|
"output?" | $~$n$~$ |
---|
"output?" | $~$c$~$ |
---|
"output?" | $~$n$~$ |
---|
"output?" | $~$c.$~$ |
---|
"test" | $~$\mathbb P(X \wedge Y) = \mathbb P(Y) \cdot \mathbb P(X|Y).$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$f(x\cdot y)=f(x)+f(y)$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$g$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$g$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$h$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$h(x+y)=h(x)+h(y)$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $$~$h(g(x\cdot y))=h(g(x))+h(g(y))$~$$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$h$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$h(x)=ch(x)$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$c$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$\mathbb{R}$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$\mathbb{Q}$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$\mathbb{R}$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$f$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$f$~$ |
---|
"tl;dr: I did some reading on related topics, an..." | $~$f$~$ |
---|
"use colon instead?" | $~$\mathsf{Fairbot}$~$ |
---|
"use colon instead?" | $~$\mathsf {Fairbot}$~$ |
---|
"use colon instead?" | $~$\mathsf {Fairbot}$~$ |
---|
"use colon instead?" | $~$\mathsf {Fairbot}$~$ |
---|
"use colon instead?" | $~$\mathsf {Fairbot}$~$ |
---|
"use colon instead?" | $~$\mathsf {CooperateBot},$~$ |
---|
"use colon instead?" | $~$\mathsf {Fairbot}$~$ |
---|
"use colon instead?" | $~$\mathsf {CooperateBot},$~$ |
---|
"use colon instead?" | $~$\mathsf {Fairbot}$~$ |
---|
"“got” would be clearer." | $~$\mathbb P(X \wedge Y) = \mathbb P(Y) \cdot \mathbb P(X|Y).$~$ |
---|
0.999...=1 | $~$0.999\dotsc$~$ |
---|
0.999...=1 | $~$1$~$ |
---|
0.999...=1 | $~$1+2+4+8+\dotsc=-1$~$ |
---|
0.999...=1 | $~$0.999\dotsc$~$ |
---|
0.999...=1 | $~$0.999\dots\neq1$~$ |
---|
0.999...=1 | $~$0.999\dots$~$ |
---|
0.999...=1 | $~$1$~$ |
---|
0.999...=1 | $~$0.999\dots$~$ |
---|
0.999...=1 | $~$9$~$ |
---|
0.999...=1 | $~$\sum_{k=1}^\infty 9 \cdot 10^{-k}$~$ |
---|
0.999...=1 | $~$(\sum_{k=1}^n 9 \cdot 10^{-k})_{n\in\mathbb N}$~$ |
---|
0.999...=1 | $~$a_n$~$ |
---|
0.999...=1 | $~$n$~$ |
---|
0.999...=1 | $~$1$~$ |
---|
0.999...=1 | $~$\varepsilon>0$~$ |
---|
0.999...=1 | $~$N\in\mathbb N$~$ |
---|
0.999...=1 | $~$n>N$~$ |
---|
0.999...=1 | $~$|1-a_n|<\varepsilon$~$ |
---|
0.999...=1 | $~$1-a_n=10^{-n}$~$ |
---|
0.999...=1 | $~$a_0$~$ |
---|
0.999...=1 | $~$0$~$ |
---|
0.999...=1 | $~$a_0=0$~$ |
---|
0.999...=1 | $~$1-a_0=1=10^0$~$ |
---|
0.999...=1 | $~$1-a_i=10^{-i}$~$ |
---|
0.999...=1 | $~$1-a_n=10^{-n}$~$ |
---|
0.999...=1 | $~$n$~$ |
---|
0.999...=1 | $~$10^{-n}$~$ |
---|
0.999...=1 | $~$10^{-n}$~$ |
---|
0.999...=1 | $~$0.999\dotsc=1$~$ |
---|
0.999...=1 | $~$0.999\dotsc=1$~$ |
---|
0.999...=1 | $~$0.999\dotsc$~$ |
---|
0.999...=1 | $~$1$~$ |
---|
0.999...=1 | $~$0.999\dotsc$~$ |
---|
0.999...=1 | $~$0.$~$ |
---|
0.999...=1 | $~$0$~$ |
---|
0.999...=1 | $~$1-0.999\dotsc=0.000\dotsc001\neq0$~$ |
---|
0.999...=1 | $~$0.000\dotsc001$~$ |
---|
0.999...=1 | $~$1$~$ |
---|
0.999...=1 | $~$0$~$ |
---|
0.999...=1 | $~$0.000\dotsc001$~$ |
---|
0.999...=1 | $~$0$~$ |
---|
0.999...=1 | $~$0.999\dotsc$~$ |
---|
0.999...=1 | $~$0.9, 0.99, 0.999, \dotsc$~$ |
---|
0.999...=1 | $~$1$~$ |
---|
0.999...=1 | $~$1$~$ |
---|
0.999...=1 | $~$1$~$ |
---|
0.999...=1 | $~$1$~$ |
---|
0.999...=1 | $~$1$~$ |
---|
0.999...=1 | $~$1$~$ |
---|
0.999...=1 | $~$0.999\dotsc$~$ |
---|
0.999...=1 | $~$9.999\dotsc$~$ |
---|
0.999...=1 | $~$9$~$ |
---|
0.999...=1 | $~$9.99-0.999=8.991$~$ |
---|
0.999...=1 | $~$9.999\dotsc-0.999\dotsc=8.999\dotsc991$~$ |
---|
0.999...=1 | $~$9$~$ |
---|
0.999...=1 | $~$0.999\dotsc$~$ |
---|
0.999...=1 | $~$8.999\dotsc991$~$ |
---|
0.999...=1 | $~$1$~$ |
---|
A googol | $~$10^{100},$~$ |
---|
A googolplex | $~$10^{10^{100}}$~$ |
---|
A googolplex | $~$10^{googol}$~$ |
---|
A googolplex | $~$ 10^{10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}.$~$ |
---|
A quick econ FAQ for AI/ML folks concerned about technological unemployment | $~$1 to be effectively +$~$ |
---|
A quick econ FAQ for AI/ML folks concerned about technological unemployment | $~$E = -mc^2,$~$ |
---|
A reply to Francois Chollet on intelligence explosion | $~$\theta$~$ |
---|
A reply to Francois Chollet on intelligence explosion | $~$\theta$~$ |
---|
A reply to Francois Chollet on intelligence explosion | $~$0$~$ |
---|
A reply to Francois Chollet on intelligence explosion | $~$1.$~$ |
---|
A reply to Francois Chollet on intelligence explosion | $~$M$~$ |
---|
A reply to Francois Chollet on intelligence explosion | $~$N$~$ |
---|
A reply to Francois Chollet on intelligence explosion | $$~$\frac{M + 1}{M + N + 2} : \frac{N + 1}{M + N + 2}$~$$ |
---|
A reply to Francois Chollet on intelligence explosion | $$~$HTHTHTHTHTHTHTHT…$~$$ |
---|
A reply to Francois Chollet on intelligence explosion | $~$H.$~$ |
---|
A reply to Francois Chollet on intelligence explosion | $~$HTTHTTHTTHTT$~$ |
---|
AI control on the cheap | $~$\mathbb{E}$~$ |
---|
AI control on the cheap | $~$\mathbb{E}$~$ |
---|
AI safety mindset | $~$\Sigma_1$~$ |
---|
AI safety mindset | $~$\Sigma_2$~$ |
---|
AIXI | $~$tl$~$ |
---|
AIXI | $~$l$~$ |
---|
AIXI | $~$t$~$ |
---|
AIXI-tl | $~$\text{AIXI}^{tl}$~$ |
---|
AIXI-tl | $~$l$~$ |
---|
AIXI-tl | $~$t$~$ |
---|
AIXI-tl | $~$tl$~$ |
---|
Abelian group | $~$G$~$ |
---|
Abelian group | $~$(X, \bullet)$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$\bullet$~$ |
---|
Abelian group | $~$x, y$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$x \bullet y$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$x \bullet y$~$ |
---|
Abelian group | $~$xy$~$ |
---|
Abelian group | $~$x(yz) = (xy)z$~$ |
---|
Abelian group | $~$x, y, z$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$e$~$ |
---|
Abelian group | $~$x$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$xe=ex=x$~$ |
---|
Abelian group | $~$x$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$x^{-1}$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$xx^{-1}=x^{-1}x=e$~$ |
---|
Abelian group | $~$x, y$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$xy=yx$~$ |
---|
Abelian group | $~$G=(X, \bullet)$~$ |
---|
Abelian group | $~$\bullet$~$ |
---|
Abelian group | $~$x, y$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$x \bullet y$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$x \bullet y$~$ |
---|
Abelian group | $~$xy$~$ |
---|
Abelian group | $~$x(yz) = (xy)z$~$ |
---|
Abelian group | $~$x, y, z$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$e$~$ |
---|
Abelian group | $~$x$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$xe=ex=x$~$ |
---|
Abelian group | $~$x$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$x^{-1}$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$xx^{-1}=x^{-1}x=e$~$ |
---|
Abelian group | $~$x, y$~$ |
---|
Abelian group | $~$X$~$ |
---|
Abelian group | $~$xy=yx$~$ |
---|
Abelian group | $~$\{1, a, a^{-1}, b, b^{-1}, c, c^{-1}, d\}$~$ |
---|
Abelian group | $~$aba^{-1}db^{-1}=d^{-1}$~$ |
---|
Abelian group | $~$aa^{-1}bb^{-1}d=d^{-1}$~$ |
---|
Abelian group | $~$d=d^{-1}$~$ |
---|
Abelian group | $~$aba^{-1}$~$ |
---|
Abelian group | $~$aa^{-1}b$~$ |
---|
Ability to read logic | $~$(\exists v: \forall w > v: \forall x>0, y>0, z>0: x^w + y^w \neq z^w) \rightarrow ((1 = 0) \vee (1 + 0 = 0 + 1))$~$ |
---|
Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $~$1 - p$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $~$1 - p$~$ |
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Absent-Minded Driver dilemma | $~$p^2$~$ |
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Absent-Minded Driver dilemma | $~$0(1-p) + 4(1-p)p + 1p^2$~$ |
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Absent-Minded Driver dilemma | $~$4 -6p$~$ |
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Absent-Minded Driver dilemma | $~$p = \frac{2}{3}$~$ |
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Absent-Minded Driver dilemma | $~$\$0\cdot\frac{1}{3} + \$4\cdot\frac{2}{3}\frac{1}{3} + \$1\cdot\frac{2}{3}\frac{2}{3} = \$\frac{4}{3} \approx \$1.33.$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $~$q.$~$ |
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Absent-Minded Driver dilemma | $~$1 : q,$~$ |
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Absent-Minded Driver dilemma | $~$\frac{1}{1+q}$~$ |
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Absent-Minded Driver dilemma | $~$\frac{q}{1+q}$~$ |
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Absent-Minded Driver dilemma | $~$p,$~$ |
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Absent-Minded Driver dilemma | $~$4p(1-p) + 1p^2.$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $~$4(1-p) + 1p.$~$ |
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Absent-Minded Driver dilemma | $~$\frac{1}{1+q}(4p(1-p) + p^2) + \frac{q}{1+q}(4(1-p) + p)$~$ |
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Absent-Minded Driver dilemma | $~$\frac{-6p - 3q + 4}{q+1}$~$ |
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Absent-Minded Driver dilemma | $~$p=\frac{4-3q}{6}.$~$ |
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Absent-Minded Driver dilemma | $~$q$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $~$q,$~$ |
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Absent-Minded Driver dilemma | $~$p=q=\frac{4}{9}.$~$ |
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Absent-Minded Driver dilemma | $~$\$4\cdot\frac{4}{9}\frac{5}{9} + \$1\cdot\frac{4}{9}\frac{4}{9} \approx \$1.19.$~$ |
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Absent-Minded Driver dilemma | $~$q$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $~$q$~$ |
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Absent-Minded Driver dilemma | $~$q,$~$ |
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Absent-Minded Driver dilemma | $~$1 : q \cong \frac{1}{1+q} : \frac{q}{1+q}$~$ |
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Absent-Minded Driver dilemma | $~$p,$~$ |
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Absent-Minded Driver dilemma | $~$q$~$ |
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Absent-Minded Driver dilemma | $~$4p(1-q) + 1pq.$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $~$4(1-p) + 1p.$~$ |
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Absent-Minded Driver dilemma | $~$q,$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $$~$\frac{1}{1+q}(4p(1-q) + pq) + \frac{q}{1+q}(4(1-p) + p)$~$$ |
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Absent-Minded Driver dilemma | $~$\frac{4 - 6q}{1+q}$~$ |
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Absent-Minded Driver dilemma | $~$p.$~$ |
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Absent-Minded Driver dilemma | $~$q$~$ |
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Absent-Minded Driver dilemma | $~$q$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $~$4-6q = 0 \implies q=\frac{2}{3}.$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $~$q$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absent-Minded Driver dilemma | $~$p$~$ |
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Absolute Complement | $~$A^\complement$~$ |
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Absolute Complement | $~$A$~$ |
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Absolute Complement | $~$A$~$ |
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Absolute Complement | $~$U$~$ |
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Absolute Complement | $~$A^\complement = U \setminus A$~$ |
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Absolute Complement | $~$A^\complement$~$ |
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Absolute Complement | $~$U$~$ |
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Absolute Complement | $~$A$~$ |
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Ackermann function | $~$A \cdot B = \underbrace{A + A + \ldots A}_{B \text{ copies of } A}$~$ |
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Ackermann function | $~$A^B = \underbrace{A \times A \times \ldots A}_{B \text{ copies of } A}$~$ |
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Ackermann function | $~$A ^ B$~$ |
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Ackermann function | $~$A \uparrow B$~$ |
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Ackermann function | $~$A \uparrow\uparrow B = \underbrace{A^{A^{\ldots^A}}}_{B \text{ copies of } A}$~$ |
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Ackermann function | $~$\uparrow^n$~$ |
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Ackermann function | $~$n$~$ |
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Ackermann function | $~$A \uparrow^2 B = \underbrace{A \uparrow^1 (A \uparrow^1 (\ldots A))}_{B \text{ copies of } A}$~$ |
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Ackermann function | $~$A \uparrow^n B = \underbrace{A \uparrow^{n-1} (A \uparrow^{n-1} (\ldots A))}_{B \text{ copies of } A}$~$ |
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Ackermann function | $~$A(n) = n \uparrow^n n$~$ |
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Ackermann function | $~$A(6)$~$ |
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Ackermann function | $~$A(1)=1$~$ |
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Ackermann function | $~$A(2)=4$~$ |
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Ackermann function | $~$A(3)$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{\text{number}}$~$ |
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Addition of rational numbers (Math 0) | $~$5$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{5}{\text{number}}$~$ |
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Addition of rational numbers (Math 0) | $~$a+b$~$ |
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Addition of rational numbers (Math 0) | $~$a$~$ |
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Addition of rational numbers (Math 0) | $~$b$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{2}{2} + \frac{3}{3} = 2$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{n}{n}$~$ |
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Addition of rational numbers (Math 0) | $~$n$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{5}{3} + \frac{8}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$5+8=13$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{5}{3} + \frac{8}{3} = \frac{13}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{5}{3} + \frac{5}{4}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{5}{3} + \frac{5}{4}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3} = \frac{4}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{4} = \frac{3}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3} = \frac{4}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{5}{3} = \frac{20}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{5}{4} = \frac{15}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$5 \times 3 = 15$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{5}{3} + \frac{5}{4}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{20}{12} + \frac{15}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{35}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{2}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{5}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{2}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{5}$~$ |
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Addition of rational numbers (Math 0) | $~$2 \times 5 = 10$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{10}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{2}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{10}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{5}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
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Addition of rational numbers (Math 0) | $~$m$~$ |
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Addition of rational numbers (Math 0) | $~$2$~$ |
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Addition of rational numbers (Math 0) | $~$n$~$ |
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Addition of rational numbers (Math 0) | $~$5$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{n} = \frac{m}{m \times n}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
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Addition of rational numbers (Math 0) | $~$m$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{m} = \frac{n}{m \times n}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
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Addition of rational numbers (Math 0) | $~$n$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{\text{thing}}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{\text{thing}}$~$ |
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Addition of rational numbers (Math 0) | $$~$\frac{1}{m} + \frac{1}{n} = \frac{n}{m \times n} + \frac{m}{m \times n}$~$$ |
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Addition of rational numbers (Math 0) | $$~$\frac{a}{m} + \frac{b}{m} = \frac{a+b}{m}$~$$ |
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Addition of rational numbers (Math 0) | $~$a$~$ |
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Addition of rational numbers (Math 0) | $~$b$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{5}{4} + \frac{5}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$3 \times 4 = 12$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{5}{4}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{15}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{4}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$5 \times 3$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{5}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{20}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$5 \times 4 = 20$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
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Addition of rational numbers (Math 0) | $~$\frac{15}{12} + \frac{20}{12} = \frac{35}{12}$~$ |
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Addition of rational numbers (Math 0) | $$~$\frac{a}{m} + \frac{b}{n} = \frac{a \times n}{m \times n} + \frac{b \times m}{m \times n} = \frac{a \times n + b \times m}{m \times n}$~$$ |
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Addition of rational numbers (Math 0) | $~$a \times n + b \times m$~$ |
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Addition of rational numbers (Math 0) | $~$a \times n$~$ |
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Addition of rational numbers (Math 0) | $~$b \times m$~$ |
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Addition of rational numbers (Math 0) | $~$a, b, m, n$~$ |
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Addition of rational numbers (Math 0) | $~$m$~$ |
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Addition of rational numbers (Math 0) | $~$n$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{10} + \frac{1}{5}$~$ |
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Addition of rational numbers exercises | $$~$\frac{1}{10} + \frac{1}{5} = \frac{1 \times 5 + 10 \times 1}{10 \times 5} = \frac{5+10}{50} = \frac{15}{50}$~$$ |
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Addition of rational numbers exercises | $~$\frac{3}{10}$~$ |
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Addition of rational numbers exercises | $~$\frac{3}{10}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{10}$~$ |
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Addition of rational numbers exercises | $~$15$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{50}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{10}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{5}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{5} = \frac{2}{10}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{10} + \frac{2}{10}$~$ |
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Addition of rational numbers exercises | $~$\frac{3}{10}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{15} + \frac{1}{10}$~$ |
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Addition of rational numbers exercises | $$~$\frac{1}{10} + \frac{1}{15} = \frac{1 \times 15 + 10 \times 1}{10 \times 15} = \frac{25}{150} = \frac{1}{6}$~$$ |
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Addition of rational numbers exercises | $~$\frac{1}{30}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{10}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{15}$~$ |
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Addition of rational numbers exercises | $~$\frac{3}{30} + \frac{2}{30} = \frac{5}{30}$~$ |
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Addition of rational numbers exercises | $~$\frac{5}{30} = \frac{1}{6}$~$ |
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Addition of rational numbers exercises | $~$\frac{25}{150} = \frac{1}{6}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{10} + \frac{1}{15}$~$ |
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Addition of rational numbers exercises | $$~$\frac{1}{15} + \frac{1}{10} = \frac{1 \times 10 + 15 \times 1}{15 \times 10} = \frac{25}{150} = \frac{1}{6}$~$$ |
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Addition of rational numbers exercises | $~$\frac{1}{10} + \frac{1}{15} = \frac{1}{15} + \frac{1}{10}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{6}$~$ |
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Addition of rational numbers exercises | $~$\frac{0}{5} + \frac{2}{5}$~$ |
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Addition of rational numbers exercises | $~$5$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{5}$~$ |
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Addition of rational numbers exercises | $~$0$~$ |
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Addition of rational numbers exercises | $~$2$~$ |
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Addition of rational numbers exercises | $~$2$~$ |
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Addition of rational numbers exercises | $~$\frac{2}{5}$~$ |
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Addition of rational numbers exercises | $~$\frac{0}{7} + \frac{2}{5}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{7}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{5}$~$ |
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Addition of rational numbers exercises | $~$\frac{2}{5}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{7}$~$ |
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Addition of rational numbers exercises | $$~$\frac{0}{7} + \frac{2}{5} = \frac{0 \times 5 + 2 \times 7}{5 \times 7} = \frac{0 + 14}{35} = \frac{14}{35}$~$$ |
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Addition of rational numbers exercises | $~$\frac{2}{5}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{5}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{5} + \frac{-1}{10}$~$ |
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Addition of rational numbers exercises | $$~$\frac{1}{15} + \frac{-1}{10} = \frac{1 \times 10 + 15 \times (-1)}{15 \times 10} = \frac{10 - 15}{150} = \frac{-5}{150} = \frac{-1}{30}$~$$ |
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Addition of rational numbers exercises | $~$\frac{7}{8}$~$ |
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Addition of rational numbers exercises | $~$\frac{13}{8}$~$ |
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Addition of rational numbers exercises | $~$\frac{a}{b}$~$ |
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Addition of rational numbers exercises | $~$a$~$ |
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Addition of rational numbers exercises | $~$b$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{8}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{8}$~$ |
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Addition of rational numbers exercises | $~$7$~$ |
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Addition of rational numbers exercises | $~$13$~$ |
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Addition of rational numbers exercises | $~$6$~$ |
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Addition of rational numbers exercises | $~$\frac{6}{8}$~$ |
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Addition of rational numbers exercises | $~$\frac{3}{4}$~$ |
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Addition of rational numbers exercises | $~$\frac{7}{8}$~$ |
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Addition of rational numbers exercises | $~$\frac{13}{7}$~$ |
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Addition of rational numbers exercises | $~$\frac{a}{b}$~$ |
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Addition of rational numbers exercises | $~$a$~$ |
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Addition of rational numbers exercises | $~$b$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{8 \times 7} = \frac{1}{56}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{8}$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{7}$~$ |
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Addition of rational numbers exercises | $~$\frac{7 \times 7}{7 \times 8} = \frac{49}{56}$~$ |
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Addition of rational numbers exercises | $~$\frac{8 \times 13}{8 \times 7} = \frac{104}{56}$~$ |
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Addition of rational numbers exercises | $~$49$~$ |
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Addition of rational numbers exercises | $~$104$~$ |
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Addition of rational numbers exercises | $~$55$~$ |
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Addition of rational numbers exercises | $~$\frac{1}{56}$~$ |
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Addition of rational numbers exercises | $~$\frac{55}{56}$~$ |
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Advanced agent properties | $~$\mathbb P(Y|X)$~$ |
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Advanced agent properties | $~$X$~$ |
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Advanced agent properties | $~$Y$~$ |
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Advanced agent properties | $~$Y,$~$ |
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Advanced agent properties | $~$X.$~$ |
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Advanced agent properties | $~$X$~$ |
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Advanced agent properties | $~$Y.$~$ |
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Advanced nonagent | $~$\pi_0$~$ |
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Advanced nonagent | $~$\mathbb E [U | \operatorname{do}(\pi_0), HumansObeyPlan]$~$ |
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Advanced nonagent | $~$\mathbb E [U | \operatorname{do}(\pi_0)],$~$ |
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Algebraic field | $~$(R, +, \times)$~$ |
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Algebraic field | $~$R$~$ |
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Algebraic field | $~$1$~$ |
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Algebraic field | $~$0$~$ |
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Algebraic field | $~$r \in R$~$ |
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Algebraic field | $~$x \in R$~$ |
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Algebraic field | $~$xr = rx = 1$~$ |
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Algebraic field | $~$0 \not = 1$~$ |
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Algebraic structure | $~$X$~$ |
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Algebraic structure tree | $~$*$~$ |
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Algebraic structure tree | $~$\circ$~$ |
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Algebraic structure tree | $~$*$~$ |
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Algebraic structure tree | $~$\circ$~$ |
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Algebraic structure tree | $~$\circ$~$ |
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Algebraic structure tree | $~$*$~$ |
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Algebraic structure tree | $~$a \circ (b * c) = (a \circ b) * (a \circ c)$~$ |
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Algebraic structure tree | $~$(a * b) \circ c = (a \circ c) * (b \circ c)$~$ |
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Algebraic structure tree | $~$*$~$ |
---|
Algebraic structure tree | $~$\circ$~$ |
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Algebraic structure tree | $~$*$~$ |
---|
Algebraic structure tree | $~$*$~$ |
---|
Algebraic structure tree | $~$*$~$ |
---|
Algebraic structure tree | $~$\circ$~$ |
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Algebraic structure tree | $~$\circ$~$ |
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Algebraic structure tree | $~$*$~$ |
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Algebraic structure tree | $~$\circ$~$ |
---|
Algebraic structure tree | $~$\circ$~$ |
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Algebraic structure tree | $~$*$~$ |
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Algebraic structure tree | $~$\circ$~$ |
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Algebraic structure tree | $~$a \circ (a * b) = a * (a \circ b) = a$~$ |
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Algebraic structure tree | $~$*$~$ |
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Algebraic structure tree | $~$\circ$~$ |
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Algebraic structure tree | $~$\wedge$~$ |
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Algebraic structure tree | $~$\vee$~$ |
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Algorithmic complexity | $~$3\uparrow\uparrow\uparrow3$~$ |
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All you need for SAT Math Here! | $~$\frac{y_2-y_1}{x_2-x_1}=\frac{rise}{run}=tan\theta$~$ |
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All you need for SAT Math Here! | $~$y=mx+b\rightarrow slope=m$~$ |
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All you need for SAT Math Here! | $~$ax+by=c\rightarrow slope=\frac{-a}{b}$~$ |
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All you need for SAT Math Here! | $~$\rightarrow$~$ |
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All you need for SAT Math Here! | $~$\rightarrow$~$ |
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All you need for SAT Math Here! | $~$\rightarrow$~$ |
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All you need for SAT Math Here! | $~$\rightarrow$~$ |
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All you need for SAT Math Here! | $~$y=mx+{b_1}, y=mx+{b_2}, {b_1}\neq {b_2}$~$ |
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All you need for SAT Math Here! | $~$y=mx+{b_1}, y=\frac{-1}{m}x+{b_2}$~$ |
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All you need for SAT Math Here! | $~${a_1}x+{b_1}y={c_1}$~$ |
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All you need for SAT Math Here! | $~${a_2}x+{b_2}y={c_2}$~$ |
---|
All you need for SAT Math Here! | $~$\frac{a_1}{a_2}\neq \frac{b_1}{b_2}$~$ |
---|
All you need for SAT Math Here! | $~$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq \frac{c_1}{c_2}$~$ |
---|
All you need for SAT Math Here! | $~$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$~$ |
---|
All you need for SAT Math Here! | $~$\big(x-h)^2+\big(y-k)^2=r^2$~$ |
---|
All you need for SAT Math Here! | $~$\big(h,k)$~$ |
---|
All you need for SAT Math Here! | $~$r=\sqrt{r^2}$~$ |
---|
All you need for SAT Math Here! | $~${x^2}+{y^2}+{ax}+{by}+c=0$~$ |
---|
All you need for SAT Math Here! | $~$\big(\frac{-a}{2},\frac{-b}{2})$~$ |
---|
All you need for SAT Math Here! | $~$\sqrt{\big(\frac{a}{2})^2+(\frac{b}{2})^2-c}$~$ |
---|
All you need for SAT Math Here! | $~$\big({x_1},{y_1})$~$ |
---|
All you need for SAT Math Here! | $~$\big(x_1-h)^2+\big(y_1-k)^2<r^2$~$ |
---|
All you need for SAT Math Here! | $~$\big(x_1-h)^2+\big(y_1-k)^2=r^2$~$ |
---|
All you need for SAT Math Here! | $~$\big(x_1-h)^2+\big(y_1-k)^2>r^2$~$ |
---|
Alternating group | $~$A_n$~$ |
---|
Alternating group | $~$S_n$~$ |
---|
Alternating group | $~$A_n$~$ |
---|
Alternating group | $~$S_n$~$ |
---|
Alternating group | $~$S_n$~$ |
---|
Alternating group | $~$(132)$~$ |
---|
Alternating group | $~$(13)(23)$~$ |
---|
Alternating group | $~$(1354)$~$ |
---|
Alternating group | $~$(54)(34)(14)$~$ |
---|
Alternating group | $~$A_4$~$ |
---|
Alternating group | $~$(12)(34)$~$ |
---|
Alternating group | $~$(13)(24)$~$ |
---|
Alternating group | $~$(14)(23)$~$ |
---|
Alternating group | $~$(123)$~$ |
---|
Alternating group | $~$(124)$~$ |
---|
Alternating group | $~$(134)$~$ |
---|
Alternating group | $~$(234)$~$ |
---|
Alternating group | $~$(132)$~$ |
---|
Alternating group | $~$(143)$~$ |
---|
Alternating group | $~$(142)$~$ |
---|
Alternating group | $~$(243)$~$ |
---|
Alternating group | $~$A_n$~$ |
---|
Alternating group | $~$2$~$ |
---|
Alternating group | $~$S_n$~$ |
---|
Alternating group | $~$A_n$~$ |
---|
Alternating group | $~$S_n$~$ |
---|
Alternating group | $~$A_n$~$ |
---|
Alternating group | $~$A_n$~$ |
---|
Alternating group | $~$3$~$ |
---|
Alternating group | $~$A_n$~$ |
---|
Alternating group | $~$A_n$~$ |
---|
Alternating group is generated by its three-cycles | $~$A_n$~$ |
---|
Alternating group is generated by its three-cycles | $~$3$~$ |
---|
Alternating group is generated by its three-cycles | $~$A_n$~$ |
---|
Alternating group is generated by its three-cycles | $~$3$~$ |
---|
Alternating group is generated by its three-cycles | $~$3$~$ |
---|
Alternating group is generated by its three-cycles | $~$(ij)(kl) = (ijk)(jkl)$~$ |
---|
Alternating group is generated by its three-cycles | $~$(ij)(jk) = (ijk)$~$ |
---|
Alternating group is generated by its three-cycles | $~$(ij)(ij) = e$~$ |
---|
Alternating group is generated by its three-cycles | $~$A_n$~$ |
---|
Alternating group is generated by its three-cycles | $~$3$~$ |
---|
Alternating group is generated by its three-cycles | $~$3$~$ |
---|
Alternating group is generated by its three-cycles | $~$A_n$~$ |
---|
Alternating group is generated by its three-cycles | $~$(ijk) = (ij)(jk)$~$ |
---|
An early stage prioritisation model | $$~$ \textbf{ Expected Value of Project } = \textbf{Decision Relevant Info} + \textbf{Rare Signals} + \textbf{Cross-Domain Skills} $~$$ |
---|
An early stage prioritisation model | $$~$ \textbf{ Expected Value of Project } = \textbf{Decision Relevant Info} + \textbf{Rare Signals} + \textbf{Cross-Domain Skills} $~$$ |
---|
An introductory guide to modern logic | $~$\phi$~$ |
---|
An introductory guide to modern logic | $~$\phi$~$ |
---|
An introductory guide to modern logic | $~$=, \wedge, \implies$~$ |
---|
An introductory guide to modern logic | $~$0$~$ |
---|
An introductory guide to modern logic | $~$n+1$~$ |
---|
An introductory guide to modern logic | $~$n$~$ |
---|
An introductory guide to modern logic | $~$\forall n. 0 \not = n+1$~$ |
---|
An introductory guide to modern logic | $~$\forall$~$ |
---|
An introductory guide to modern logic | $~$A\implies B$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$B$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$w$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$w$~$ |
---|
An introductory guide to modern logic | $~$w$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$\phi$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$\phi$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash \phi$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$\phi$~$ |
---|
An introductory guide to modern logic | $~$1$~$ |
---|
An introductory guide to modern logic | $~$=$~$ |
---|
An introductory guide to modern logic | $~$1$~$ |
---|
An introductory guide to modern logic | $~$a$~$ |
---|
An introductory guide to modern logic | $~$0$~$ |
---|
An introductory guide to modern logic | $~$n$~$ |
---|
An introductory guide to modern logic | $~$2^{a_1}3^{a_2}5^{a_3}\cdots p(n)^{a_n}$~$ |
---|
An introductory guide to modern logic | $~$n$~$ |
---|
An introductory guide to modern logic | $~$Axiom(x)$~$ |
---|
An introductory guide to modern logic | $~$IsEqualTo42(x)$~$ |
---|
An introductory guide to modern logic | $~$x = 42$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash IsEqualTo42(42)$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash \exists x IsEqualTo42(x)$~$ |
---|
An introductory guide to modern logic | $~$PA\not\vdash IsEqualTo42(7)$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash Axiom(\textbf{n})$~$ |
---|
An introductory guide to modern logic | $~$n$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$n$~$ |
---|
An introductory guide to modern logic | $~$n+1$~$ |
---|
An introductory guide to modern logic | $~$Rule(p_1, p_2,…, p_n, r)$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$p_1, …., p_n$~$ |
---|
An introductory guide to modern logic | $~$r$~$ |
---|
An introductory guide to modern logic | $~$Proof(x,y)$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$x$~$ |
---|
An introductory guide to modern logic | $~$y$~$ |
---|
An introductory guide to modern logic | $~$\exists x. Proof(x,y)$~$ |
---|
An introductory guide to modern logic | $~$\square_{PA}(y)$~$ |
---|
An introductory guide to modern logic | $~$\exists$~$ |
---|
An introductory guide to modern logic | $~$\square_{PA}(x)$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$x$~$ |
---|
An introductory guide to modern logic | $~$\ulcorner 1+1=2 \urcorner$~$ |
---|
An introductory guide to modern logic | $~$1+1=2$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash \square_{PA}(\ulcorner 1+1=2 \urcorner)$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$1+1=2$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$Proof(x,y)$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$\square_{PA}$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash A$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash \square_{PA}(\ulcorner A\urcorner)$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash \square_{PA}(\ulcorner A\rightarrow B\urcorner) \rightarrow [\square_{PA}(\ulcorner A \urcorner)\rightarrow \square_{PA}(\ulcorner B \urcorner)]$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash \square_{PA}(\ulcorner A\urcorner) \rightarrow \square_{PA} \square_{PA} (\ulcorner A\urcorner)$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$B$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$B$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$A\rightarrow B$~$ |
---|
An introductory guide to modern logic | $~$B$~$ |
---|
An introductory guide to modern logic | $~$\square_{PA}(\ulcorner A \urcorner)$~$ |
---|
An introductory guide to modern logic | $~$\phi(x)$~$ |
---|
An introductory guide to modern logic | $~$\psi$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash \psi \leftrightarrow \phi(\ulcorner \psi \urcorner)$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash \square_{PA}(\ulcorner A\urcorner) \rightarrow A$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash A$~$ |
---|
An introductory guide to modern logic | $~$PA\not\vdash A$~$ |
---|
An introductory guide to modern logic | $~$PA\not\vdash \square_{PA}(\ulcorner A\urcorner) \rightarrow A$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$n$~$ |
---|
An introductory guide to modern logic | $~$PA\vdash Proof(\textbf n, \ulcorner A\urcorner)$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$n$~$ |
---|
An introductory guide to modern logic | $~$Proof(\textbf n,\ulcorner A\urcorner)$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$n$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$P\wedge \neg P$~$ |
---|
An introductory guide to modern logic | $~$P\wedge \neg P$~$ |
---|
An introductory guide to modern logic | $~$P$~$ |
---|
An introductory guide to modern logic | $~$\bot$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$PA\not \vdash \neg \square_{PA}(\bot)$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$\square_{PA}$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$\square_{PA}(\ulcorner A\urcorner$~$ |
---|
An introductory guide to modern logic | $~$A$~$ |
---|
An introductory guide to modern logic | $~$PA$~$ |
---|
Antisymmetric relation | $~$R$~$ |
---|
Antisymmetric relation | $~$(aRb ∧ bRa) → a = b$~$ |
---|
Antisymmetric relation | $~$a ≠ b → (¬aRb ∨ ¬bRa)$~$ |
---|
Antisymmetric relation | $~$aRa$~$ |
---|
Antisymmetric relation | $~$\{(0,0), (1,1), (2,2)…\}$~$ |
---|
Antisymmetric relation | $~$\{(0,1), (1,2), (2,3), (3,4)…\}$~$ |
---|
Antisymmetric relation | $~$\{…(9,3),(10,5),(10,2),(14,7),(14,2)…)\}$~$ |
---|
Arbital Markdown | $~$ax^2 + bx + c = 0$~$ |
---|
Arbital Markdown | $~$ax^2 + bx + c = 0$~$ |
---|
Arbital Markdown | $$~$\lim_{N \to \infty} \sum_{k=1}^N f(t_k) \Delta t$~$$ |
---|
Arbital Markdown | $$~$\lim_{N \to \infty} \sum_{k=1}^N f(t_k) \Delta t$~$$ |
---|
Arbital examplar pages | $~$n^\text{th}$~$ |
---|
Arithmetical hierarchy | $~$\Pi_1$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_1$~$ |
---|
Arithmetical hierarchy | $~$\Pi_n$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_{n+1}$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_n$~$ |
---|
Arithmetical hierarchy | $~$\Pi_{n+1}$~$ |
---|
Arithmetical hierarchy | $~$\Pi_n$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_n$~$ |
---|
Arithmetical hierarchy | $~$\Delta_n$~$ |
---|
Arithmetical hierarchy | $~$\Pi_1$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_1$~$ |
---|
Arithmetical hierarchy | $~$\Delta_0$~$ |
---|
Arithmetical hierarchy | $~$\Pi_0$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_0$~$ |
---|
Arithmetical hierarchy | $~$\forall x < 10: \exists y < x: x + y < 10$~$ |
---|
Arithmetical hierarchy | $~$x, y, z…$~$ |
---|
Arithmetical hierarchy | $~$\phi(x, y, z…)$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_n,$~$ |
---|
Arithmetical hierarchy | $~$\forall x: \forall y: \forall z: … \phi(x, y, z…)$~$ |
---|
Arithmetical hierarchy | $~$\Pi_{n+1}$~$ |
---|
Arithmetical hierarchy | $~$\Pi_n$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_{n+1}$~$ |
---|
Arithmetical hierarchy | $~$\Pi_n$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_n$~$ |
---|
Arithmetical hierarchy | $~$\Delta_n$~$ |
---|
Arithmetical hierarchy | $~$\Pi_1$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_1$~$ |
---|
Arithmetical hierarchy | $~$\forall x$~$ |
---|
Arithmetical hierarchy | $~$\exists y$~$ |
---|
Arithmetical hierarchy | $~$\phi(x, y) \leftrightarrow [(x + y) = (y + x)],$~$ |
---|
Arithmetical hierarchy | $~$x$~$ |
---|
Arithmetical hierarchy | $~$y$~$ |
---|
Arithmetical hierarchy | $~$\Delta_0 = \Pi_0 = \Sigma_0.$~$ |
---|
Arithmetical hierarchy | $~$+$~$ |
---|
Arithmetical hierarchy | $~$=$~$ |
---|
Arithmetical hierarchy | $~$\Delta_0$~$ |
---|
Arithmetical hierarchy | $~$c$~$ |
---|
Arithmetical hierarchy | $~$d$~$ |
---|
Arithmetical hierarchy | $~$c + d = d + c$~$ |
---|
Arithmetical hierarchy | $~$\forall x_1: \forall x_2: …$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_n$~$ |
---|
Arithmetical hierarchy | $~$x_i$~$ |
---|
Arithmetical hierarchy | $~$\Pi_{n+1}.$~$ |
---|
Arithmetical hierarchy | $~$\forall x: (x + 3) = (3 + x)$~$ |
---|
Arithmetical hierarchy | $~$\Pi_1.$~$ |
---|
Arithmetical hierarchy | $~$\exists x_1: \exists x_2: …$~$ |
---|
Arithmetical hierarchy | $~$\Pi_n$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_{n+1}.$~$ |
---|
Arithmetical hierarchy | $~$\exists y: \forall x: (x + y) = (y + x)$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_2$~$ |
---|
Arithmetical hierarchy | $~$\exists y: \exists x: (x + y) = (y + x)$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_1.$~$ |
---|
Arithmetical hierarchy | $~$\Pi_n$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_n$~$ |
---|
Arithmetical hierarchy | $~$\Delta_n.$~$ |
---|
Arithmetical hierarchy | $~$\Delta_0$~$ |
---|
Arithmetical hierarchy | $~$\forall x: \exists y < x: (x + y) = (y + x)$~$ |
---|
Arithmetical hierarchy | $~$\Pi_1$~$ |
---|
Arithmetical hierarchy | $~$\Pi_2$~$ |
---|
Arithmetical hierarchy | $~$c,$~$ |
---|
Arithmetical hierarchy | $~$\forall x < c: \phi(x)$~$ |
---|
Arithmetical hierarchy | $~$\phi(0) \wedge \phi(1) … \wedge \phi(c)$~$ |
---|
Arithmetical hierarchy | $~$\exists x < c: \phi(x)$~$ |
---|
Arithmetical hierarchy | $~$\phi(0) \vee \phi(1) \vee …$~$ |
---|
Arithmetical hierarchy | $~$z = 2^x \cdot 3^y$~$ |
---|
Arithmetical hierarchy | $~$\Delta_{n+1}$~$ |
---|
Arithmetical hierarchy | $~$\Pi_n$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_n$~$ |
---|
Arithmetical hierarchy | $~$\Pi_{n}$~$ |
---|
Arithmetical hierarchy | $~$\Pi_{n+1}$~$ |
---|
Arithmetical hierarchy | $~$\exists$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_{n+1}$~$ |
---|
Arithmetical hierarchy | $~$\Pi_{n}$~$ |
---|
Arithmetical hierarchy | $~$\forall$~$ |
---|
Arithmetical hierarchy | $~$\phi \in \Pi_n$~$ |
---|
Arithmetical hierarchy | $~$\Pi_n$~$ |
---|
Arithmetical hierarchy | $~$\phi$~$ |
---|
Arithmetical hierarchy | $~$\Pi_n$~$ |
---|
Arithmetical hierarchy | $~$\Pi_n.$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_1$~$ |
---|
Arithmetical hierarchy | $~$\phi \in \Delta_0$~$ |
---|
Arithmetical hierarchy | $~$\exists x: \phi(x)$~$ |
---|
Arithmetical hierarchy | $~$\Pi_1$~$ |
---|
Arithmetical hierarchy | $~$\phi$~$ |
---|
Arithmetical hierarchy | $~$\forall x: \phi(x)$~$ |
---|
Arithmetical hierarchy | $~$\phi$~$ |
---|
Arithmetical hierarchy | $~$\Sigma_1$~$ |
---|
Arithmetical hierarchy | $~$\Pi_1.$~$ |
---|
Arithmetical hierarchy | $~$\Pi_2$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Delta_0,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_0,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Sigma_0$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_1.$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y^9 = 9^y.$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y^9 = 9^y.$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Delta_0,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Sigma_1.$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$c$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$c$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Sigma_1.$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$c,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Sigma_1$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$c,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_2.$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$(x + y) > 10^9$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Sigma_2,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_1$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x.$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_n$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Sigma_{n+1}$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Sigma_n$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_{n+1}$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Sigma_n$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_n$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Delta_n.$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Sigma_1$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y^9 = 9^y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y^9 = 9^y,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y^9 = 9^y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_1$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Delta_1$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Sigma_2$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x^x$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_1$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x^x$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x^x$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x = 2,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$2^2$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x^x$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$c,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$c^c,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$c=1.$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$z = 2^x \cdot 3^y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x^3 + y^3 = z^3$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$w$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$w = 2^x \cdot 3^y \cdot 5^z$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x^3 + y^3 = z^3.$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_1,$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x^w + y^w = z^w.$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_1$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$X \rightarrow Y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$Y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$X$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$X$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$Y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_1$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$x$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$y = f(x) = 4x+1$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_1$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$4x+1$~$ |
---|
Arithmetical hierarchy: If you don't read logic | $~$\Pi_2$~$ |
---|
Arity (of a function) | $~$f(a, b, c, d) = ac - bd$~$ |
---|
Arity (of a function) | $~$+$~$ |
---|
Arity (of a function) | $~$(\mathrm{People} \times \mathrm{Ages})$~$ |
---|
Associative operation | $~$\bullet : X \times X \to X$~$ |
---|
Associative operation | $~$x, y, z$~$ |
---|
Associative operation | $~$X$~$ |
---|
Associative operation | $~$x \bullet (y \bullet z) = (x \bullet y) \bullet z$~$ |
---|
Associative operation | $~$+$~$ |
---|
Associative operation | $~$(x + y) + z = x + (y + z)$~$ |
---|
Associative operation | $~$x, y,$~$ |
---|
Associative operation | $~$z$~$ |
---|
Associative operation | $~$f$~$ |
---|
Associative operation | $~$x, y,$~$ |
---|
Associative operation | $~$z$~$ |
---|
Associative operation | $~$f$~$ |
---|
Associative operation | $~$f$~$ |
---|
Associative operation | $~$f(f(x, y), z) = f(x, f(y, z)),$~$ |
---|
Associative operation | $~$f$~$ |
---|
Associative operation | $~$x$~$ |
---|
Associative operation | $~$y$~$ |
---|
Associative operation | $~$z$~$ |
---|
Associative operation | $~$f$~$ |
---|
Associative operation | $~$y$~$ |
---|
Associative operation | $~$z$~$ |
---|
Associative operation | $~$x$~$ |
---|
Associative operation | $~$f$~$ |
---|
Associative operation | $~$f$~$ |
---|
Associative operation | $~$f_3 : X \times X \times X \to X,$~$ |
---|
Associative operation | $~$f$~$ |
---|
Associative operation | $~$f$~$ |
---|
Associative operation | $~$f$~$ |
---|
Associative operation | $~$f_4, f_5, \ldots,$~$ |
---|
Associative operation | $~$\bullet$~$ |
---|
Associative operation | $~$2 \cdot 3 \cdot 4 \cdot 5$~$ |
---|
Associativity vs commutativity | $~$x$~$ |
---|
Associativity vs commutativity | $~$y,$~$ |
---|
Associativity vs commutativity | $~$y$~$ |
---|
Associativity vs commutativity | $~$x.$~$ |
---|
Associativity vs commutativity | $~$a \cdot (b \cdot (c \cdot d)),$~$ |
---|
Associativity vs commutativity | $~$((a \cdot b) \cdot c) \cdot d.$~$ |
---|
Associativity vs commutativity | $~$\cdot$~$ |
---|
Associativity vs commutativity | $~$3 + 2 + (-7) + 5 + (-2) + (-3) + 7,$~$ |
---|
Associativity vs commutativity | $~$3 - 3 + 2 - 2 + 7 - 7 + 5 = 5,$~$ |
---|
Associativity: Examples | $~$(x + y) + z = x + (y + z)$~$ |
---|
Associativity: Examples | $~$x, y,$~$ |
---|
Associativity: Examples | $~$z.$~$ |
---|
Associativity: Examples | $~$n$~$ |
---|
Associativity: Examples | $~$n$~$ |
---|
Associativity: Examples | $~$(x \times y) \times z = x \times (y \times z)$~$ |
---|
Associativity: Examples | $~$x, y,$~$ |
---|
Associativity: Examples | $~$z.$~$ |
---|
Associativity: Examples | $~$n$~$ |
---|
Associativity: Examples | $~$n$~$ |
---|
Associativity: Examples | $~$x$~$ |
---|
Associativity: Examples | $~$y$~$ |
---|
Associativity: Examples | $~$z$~$ |
---|
Associativity: Examples | $~$(x \times y) \times z$~$ |
---|
Associativity: Examples | $~$x \times (y \times z).$~$ |
---|
Associativity: Examples | $~$x$~$ |
---|
Associativity: Examples | $~$y$~$ |
---|
Associativity: Examples | $~$z$~$ |
---|
Associativity: Examples | $~$z$~$ |
---|
Associativity: Examples | $~$(5-3)-2=0$~$ |
---|
Associativity: Examples | $~$5-(3-2)=4.$~$ |
---|
Associativity: Examples | $~$\uparrow$~$ |
---|
Associativity: Examples | $~$\uparrow$~$ |
---|
Associativity: Examples | $~$\uparrow\downarrow.$~$ |
---|
Associativity: Examples | $~$\uparrow\downarrow$~$ |
---|
Associativity: Examples | $~$\uparrow,$~$ |
---|
Associativity: Examples | $~$\uparrow\downarrow\downarrow,$~$ |
---|
Associativity: Examples | $~$\uparrow$~$ |
---|
Associativity: Examples | $~$\uparrow\downarrow,$~$ |
---|
Associativity: Examples | $~$\uparrow\downarrow\uparrow,$~$ |
---|
Associativity: Examples | $~$?$~$ |
---|
Associativity: Examples | $~$(red\ ?\ green)\ ?\ blue = blue$~$ |
---|
Associativity: Examples | $~$red\ ?\ (green\ ?\ blue)=red.$~$ |
---|
Associativity: Intuition | $~$f : X \times X \to X$~$ |
---|
Associativity: Intuition | $~$X$~$ |
---|
Associativity: Intuition | $~$3 + 4 + 5 + 6,$~$ |
---|
Associativity: Intuition | $~$+$~$ |
---|
Associativity: Intuition | $~$[a, b, c, d, \ldots]$~$ |
---|
Associativity: Intuition | $~$a$~$ |
---|
Associativity: Intuition | $~$b$~$ |
---|
Associativity: Intuition | $~$[a, b]$~$ |
---|
Associativity: Intuition | $~$c$~$ |
---|
Associativity: Intuition | $~$b$~$ |
---|
Associativity: Intuition | $~$c$~$ |
---|
Associativity: Intuition | $~$[b, c]$~$ |
---|
Associativity: Intuition | $~$a$~$ |
---|
Associativity: Intuition | $~$[a, b, c]$~$ |
---|
Associativity: Intuition | $~$f : X \times X \to Y$~$ |
---|
Associativity: Intuition | $~$f$~$ |
---|
Associativity: Intuition | $~$f$~$ |
---|
Associativity: Intuition | $~$f$~$ |
---|
Associativity: Intuition | $~$f$~$ |
---|
Associativity: Intuition | $~$f$~$ |
---|
Associativity: Intuition | $~$f$~$ |
---|
Associativity: Intuition | $~$f$~$ |
---|
Associativity: Intuition | $~$f$~$ |
---|
Associativity: Intuition | $~$+$~$ |
---|
Associativity: Intuition | $~$n$~$ |
---|
Associativity: Intuition | $~$n$~$ |
---|
Associativity: Intuition | $~$+$~$ |
---|
Associativity: Intuition | $~$x$~$ |
---|
Associativity: Intuition | $~$y$~$ |
---|
Associativity: Intuition | $~$z$~$ |
---|
Associativity: Intuition | $~$x$~$ |
---|
Associativity: Intuition | $~$y$~$ |
---|
Associativity: Intuition | $~$z$~$ |
---|
Associativity: Intuition | $~$f$~$ |
---|
Associativity: Intuition | $~$f(red,blue)=red,$~$ |
---|
Associativity: Intuition | $~$f(red,green)=green,$~$ |
---|
Associativity: Intuition | $~$f(blue,blue)=blue,$~$ |
---|
Associativity: Intuition | $~$f(blue,green=blue).$~$ |
---|
Associativity: Intuition | $~$f$~$ |
---|
Associativity: Intuition | $~$f(f(red, green), blue))=blue,$~$ |
---|
Associativity: Intuition | $~$f(red, f(green, blue))=red.$~$ |
---|
Associativity: Intuition | $~$f(green, blue)$~$ |
---|
Associativity: Intuition | $~$f$~$ |
---|
Associativity: Intuition | $~$f$~$ |
---|
Asymptotic Notation | $$~$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = 0$~$$ |
---|
Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|
Asymptotic Notation | $~$g(x)$~$ |
---|
Asymptotic Notation | $~$f(x)$~$ |
---|
Asymptotic Notation | $~$x$~$ |
---|
Asymptotic Notation | $~$f(x) = x$~$ |
---|
Asymptotic Notation | $~$g(x) = x^2$~$ |
---|
Asymptotic Notation | $~$\lim_{x \rightarrow \infty} \frac{x}{x^2} = 0$~$ |
---|
Asymptotic Notation | $~$x = o(x^2)$~$ |
---|
Asymptotic Notation | $~$x^2$~$ |
---|
Asymptotic Notation | $~$x$~$ |
---|
Asymptotic Notation | $~$x$~$ |
---|
Asymptotic Notation | $~$\frac{g(x)}{f(x)}$~$ |
---|
Asymptotic Notation | $~$g(x) - f(x)$~$ |
---|
Asymptotic Notation | $~$x$~$ |
---|
Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|
Asymptotic Notation | $~$f(x) \in o(g(x))$~$ |
---|
Asymptotic Notation | $~$o(g(x))$~$ |
---|
Asymptotic Notation | $~$g(x)$~$ |
---|
Asymptotic Notation | $~$f(x) = 200x + 10000$~$ |
---|
Asymptotic Notation | $~$g(x) = x^2$~$ |
---|
Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|
Asymptotic Notation | $~$x$~$ |
---|
Asymptotic Notation | $~$g(x) > f(x)$~$ |
---|
Asymptotic Notation | $$~$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim{x \rightarrow \infty} \frac{200x + 10000}{x^2} = 0$~$$ |
---|
Asymptotic Notation | $$~$\lim_{x \rightarrow \infty} \frac{200x + 10000}{x^2} = \lim_{x \rightarrow \infty} \frac{200}{2x}$~$$ |
---|
Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = 20x^2 - 10x + 5$~$ |
---|
Asymptotic Notation | $~$g(x) = 2x^2 - x + 10$~$ |
---|
Asymptotic Notation | $~$g(x) = o(f(x))$~$ |
---|
Asymptotic Notation | $$~$\lim_{x \rightarrow \infty} \frac{g(x)}{f(x)} = \lim_{x \rightarrow \infty} \frac{2x^2 - x + 10}{20x^2 - 10x + 5} = \lim_{x \rightarrow \infty} \frac{4x - 1}{40x - 10}$~$$ |
---|
Asymptotic Notation | $$~$= \lim_{x \rightarrow \infty} \frac{4}{40} = \frac{1}{10}$~$$ |
---|
Asymptotic Notation | $~$f(x)$~$ |
---|
Asymptotic Notation | $~$g(x)$~$ |
---|
Asymptotic Notation | $~$f(x)$~$ |
---|
Asymptotic Notation | $~$g(x)$~$ |
---|
Asymptotic Notation | $~$g(x) \neq o(f(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|
Asymptotic Notation | $$~$\forall_{c>0} \exists_{n>0} \text{ such that } \forall_{x>n} c \cdot f(x) \leq g(x)$~$$ |
---|
Asymptotic Notation | $~$g(x)$~$ |
---|
Asymptotic Notation | $~$f(x)$~$ |
---|
Asymptotic Notation | $~$f(x)$~$ |
---|
Asymptotic Notation | $~$200 x + 10000 = o(x^2)$~$ |
---|
Asymptotic Notation | $~$c$~$ |
---|
Asymptotic Notation | $~$c(200x + 10000)$~$ |
---|
Asymptotic Notation | $~$x^2$~$ |
---|
Asymptotic Notation | $~$n$~$ |
---|
Asymptotic Notation | $~$f(x) = o(f(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = o(g(x))\ \ \implies\ \ g(x) \neq o(f(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = o(g(x)) \text{ and } g(x) = o(h(x))\ \ \implies\ \ f(x)= o(h(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = o(g(x))\ \ \implies\ \ c + f(x) = o(g(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = o(g(x))\ \ \implies\ \ c \cdot f(x) = o(g(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = 1$~$ |
---|
Asymptotic Notation | $~$f(x) = log(log(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = log(x)$~$ |
---|
Asymptotic Notation | $~$f(x) = x$~$ |
---|
Asymptotic Notation | $~$f(x) = x \cdot log(x)$~$ |
---|
Asymptotic Notation | $~$f(x) = x^{1+\epsilon}$~$ |
---|
Asymptotic Notation | $~$0 < \epsilon < 1$~$ |
---|
Asymptotic Notation | $~$f(x) = x^2$~$ |
---|
Asymptotic Notation | $~$f(x) = x^3$~$ |
---|
Asymptotic Notation | $~$f(x) = x^4$~$ |
---|
Asymptotic Notation | $~$f(x) = e^{cx}$~$ |
---|
Asymptotic Notation | $~$f(x) = x!$~$ |
---|
Asymptotic Notation | $~$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = 0$~$ |
---|
Asymptotic Notation | $~$0 < \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} < \infty$~$ |
---|
Asymptotic Notation | $~$f(x) = \Theta(g(x))$~$ |
---|
Asymptotic Notation | $~$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \infty$~$ |
---|
Asymptotic Notation | $~$f(x) = \omega(g(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|
Asymptotic Notation | $~$g(x) = \omega(f(x))$~$ |
---|
Asymptotic Notation | $~$g(x)$~$ |
---|
Asymptotic Notation | $~$f(x)$~$ |
---|
Asymptotic Notation | $~$o(g(x))$~$ |
---|
Asymptotic Notation | $~$\Theta(g(x))$~$ |
---|
Asymptotic Notation | $~$\omega(g(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = O(g(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = o(g(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = \Theta(g(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = \Omega(g(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = \omega(g(x))$~$ |
---|
Asymptotic Notation | $~$f(x) = \Theta(g(x))$~$ |
---|
Asymptotic Notation | $~$\Theta(n\ lg(n))$~$ |
---|
Asymptotic Notation | $~$\Theta(n^2)$~$ |
---|
Asymptotic Notation | $~$n\ lg(n)$~$ |
---|
Asymptotic Notation | $~$n^2$~$ |
---|
Asymptotic Notation | $~$n lg(n) = o(n^2)$~$ |
---|
Asymptotic Notation | $~$[6,5,4,3,2,1]$~$ |
---|
Asymptotic Notation | $~$[1,2,3,4,6,5]$~$ |
---|
Asymptotic Notation | $~$n$~$ |
---|
Asymptotic Notation | $~$n^2$~$ |
---|
Asymptotic Notation | $~$O(n^2)$~$ |
---|
Author's guide to Arbital | $~$e$~$ |
---|
Author's guide to Arbital | $~$\approx 2.718…$~$ |
---|
Axiom | $~$T$~$ |
---|
Axiom | $~$\forall w. weight(w)\rightarrow 0<w \wedge w < 1$~$ |
---|
Axiom | $~$0$~$ |
---|
Axiom | $~$1$~$ |
---|
Axiom | $~$[P(0) \wedge \forall n. P(n)\rightarrow P(n+1)]\rightarrow \forall n. P(n)$~$ |
---|
Axiom | $~$PA$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $$~$
f: X \rightarrow \bigcup_{Y \in X} Y
$~$$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$Y \in X$~$ |
---|
Axiom of Choice | $~$Y$~$ |
---|
Axiom of Choice | $~$f$~$ |
---|
Axiom of Choice | $~$Y$~$ |
---|
Axiom of Choice | $~$f(Y) \in Y$~$ |
---|
Axiom of Choice | $$~$
\forall_X
\left(
\left[\forall_{Y \in X} Y \not= \emptyset \right]
\Rightarrow
\left[\exists
\left( f: X \rightarrow \bigcup_{Y \in X} Y \right)
\left(\forall_{Y \in X}
\exists_{y \in Y} f(Y) = y \right) \right]
\right)
$~$$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$Y_1, Y_2, Y_3$~$ |
---|
Axiom of Choice | $~$y_1 \in Y_1, y_2 \in Y_2, y_3 \in Y_3$~$ |
---|
Axiom of Choice | $~$f$~$ |
---|
Axiom of Choice | $~$f(Y_1) = y_1$~$ |
---|
Axiom of Choice | $~$f(Y_2) = y_2$~$ |
---|
Axiom of Choice | $~$f(Y_3) = y_3$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$Y_1, Y_2, Y_3, \ldots$~$ |
---|
Axiom of Choice | $~$f$~$ |
---|
Axiom of Choice | $~$Y$~$ |
---|
Axiom of Choice | $~$n$~$ |
---|
Axiom of Choice | $~$n$~$ |
---|
Axiom of Choice | $~$f$~$ |
---|
Axiom of Choice | $~$X_1, X_2, X_3, \ldots$~$ |
---|
Axiom of Choice | $~$\prod_{i \in \mathbb{N}} X_i$~$ |
---|
Axiom of Choice | $~$(x_1, x_2, x_3, \ldots )$~$ |
---|
Axiom of Choice | $~$x_1 \in X_1$~$ |
---|
Axiom of Choice | $~$x_2 \in X_2$~$ |
---|
Axiom of Choice | $~$X_1$~$ |
---|
Axiom of Choice | $~$X_2$~$ |
---|
Axiom of Choice | $~$X_3$~$ |
---|
Axiom of Choice | $~$f: C \rightarrow C$~$ |
---|
Axiom of Choice | $~$C$~$ |
---|
Axiom of Choice | $~$x_0$~$ |
---|
Axiom of Choice | $~$C$~$ |
---|
Axiom of Choice | $~$x_0 \in C$~$ |
---|
Axiom of Choice | $~$f(x_0) = x_0$~$ |
---|
Axiom of Choice | $~$(x , y)$~$ |
---|
Axiom of Choice | $~$x$~$ |
---|
Axiom of Choice | $~$y$~$ |
---|
Axiom of Choice | $~$I$~$ |
---|
Axiom of Choice | $~$(A_i)_{i \in I}$~$ |
---|
Axiom of Choice | $~$I$~$ |
---|
Axiom of Choice | $~$I$~$ |
---|
Axiom of Choice | $~$\mathbb{N}$~$ |
---|
Axiom of Choice | $~$A_n$~$ |
---|
Axiom of Choice | $~$\mathcal{U}$~$ |
---|
Axiom of Choice | $~$I$~$ |
---|
Axiom of Choice | $~$I$~$ |
---|
Axiom of Choice | $~$I$~$ |
---|
Axiom of Choice | $~$I$~$ |
---|
Axiom of Choice | $~$\mathcal{U}$~$ |
---|
Axiom of Choice | $~$\mathcal{U}$~$ |
---|
Axiom of Choice | $~$X \subseteq I$~$ |
---|
Axiom of Choice | $~$X \in \mathcal{U}$~$ |
---|
Axiom of Choice | $~$(A_i)_{i \in X}$~$ |
---|
Axiom of Choice | $~$(A_i)_{i \in I}$~$ |
---|
Axiom of Choice | $~$A$~$ |
---|
Axiom of Choice | $~$A_i$~$ |
---|
Axiom of Choice | $~$A$~$ |
---|
Axiom of Choice | $~$A_i$~$ |
---|
Axiom of Choice | $~$A_i$~$ |
---|
Axiom of Choice | $~$A_i$~$ |
---|
Axiom of Choice | $~$A$~$ |
---|
Axiom of Choice | $~$A$~$ |
---|
Axiom of Choice | $~$\in$~$ |
---|
Axiom of Choice | $~$x \in X$~$ |
---|
Axiom of Choice | $~$x$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$\in$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$\phi$~$ |
---|
Axiom of Choice | $~$\in$~$ |
---|
Axiom of Choice | $~$\{x \in X : \phi(x) \}$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$\phi$~$ |
---|
Axiom of Choice | $~$\mathbb{N}$~$ |
---|
Axiom of Choice | $~$x$~$ |
---|
Axiom of Choice | $~$\phi(x)$~$ |
---|
Axiom of Choice | $~$A, B, C, \ldots$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$xy = yx$~$ |
---|
Axiom of Choice | $~$x$~$ |
---|
Axiom of Choice | $~$y$~$ |
---|
Axiom of Choice | $~$xy \not= yx$~$ |
---|
Axiom of Choice | $~$S_3$~$ |
---|
Axiom of Choice | $~$C$~$ |
---|
Axiom of Choice | $~$C$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$C$~$ |
---|
Axiom of Choice | $~$A$~$ |
---|
Axiom of Choice | $~$A \times A$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$Y$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$C$~$ |
---|
Axiom of Choice | $~$u \in X$~$ |
---|
Axiom of Choice | $~$C$~$ |
---|
Axiom of Choice | $~$u \geq c$~$ |
---|
Axiom of Choice | $~$c \in C$~$ |
---|
Axiom of Choice | $~$m \in X$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$x \in X$~$ |
---|
Axiom of Choice | $~$m \not< x$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$m$~$ |
---|
Axiom of Choice | $~$V$~$ |
---|
Axiom of Choice | $~$V$~$ |
---|
Axiom of Choice | $~$v_1 \in V$~$ |
---|
Axiom of Choice | $~$v$~$ |
---|
Axiom of Choice | $~$\{v_1\}$~$ |
---|
Axiom of Choice | $~$\{v_1\} \subseteq \{v, v_2\} \subseteq \{v_1, v_2, v_3 \} \subseteq \cdots$~$ |
---|
Axiom of Choice | $~$ \{v_1\} \cup \{v_1, v_2\} \cup \{v_1, v_2, v_3 \} \cdots = \{v_1, v_2, v_3, \ldots \}$~$ |
---|
Axiom of Choice | $~$B$~$ |
---|
Axiom of Choice | $~$B$~$ |
---|
Axiom of Choice | $~$v_i$~$ |
---|
Axiom of Choice | $~$B$~$ |
---|
Axiom of Choice | $~$M$~$ |
---|
Axiom of Choice | $~$V$~$ |
---|
Axiom of Choice | $~$V$~$ |
---|
Axiom of Choice | $~$M$~$ |
---|
Axiom of Choice | $~$v \in V$~$ |
---|
Axiom of Choice | $~$M$~$ |
---|
Axiom of Choice | $~$M$~$ |
---|
Axiom of Choice | $~$M \cup \{v\}$~$ |
---|
Axiom of Choice | $~$M$~$ |
---|
Axiom of Choice | $~$v$~$ |
---|
Axiom of Choice | $~$M$~$ |
---|
Axiom of Choice | $~$M$~$ |
---|
Axiom of Choice | $~$V$~$ |
---|
Axiom of Choice | $~$\{v_1\}$~$ |
---|
Axiom of Choice | $~$\{v_1, v_2\}$~$ |
---|
Axiom of Choice | $~$\mathbb{N}$~$ |
---|
Axiom of Choice | $~$\mathbb{N}$~$ |
---|
Axiom of Choice | $~$\mathbb{N}$~$ |
---|
Axiom of Choice | $~$\{42, 48, 64, \ldots\}$~$ |
---|
Axiom of Choice | $~$42$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$R$~$ |
---|
Axiom of Choice | $~$(x_n)_{n \in \mathbb{N}}$~$ |
---|
Axiom of Choice | $~$x_n$~$ |
---|
Axiom of Choice | $~$R$~$ |
---|
Axiom of Choice | $~$x_{n+1}$~$ |
---|
Axiom of Choice | $~$\mathbb{N}$~$ |
---|
Axiom of Choice | $~$\mathbb{R}$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$X$~$ |
---|
Axiom of Choice | $~$P(X)$~$ |
---|
Axiom of Choice | $~$\mathbb{R}$~$ |
---|
Axiom of Choice | $~$P(\mathbb{N})$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|
Axiom of Choice Definition (Intuitive) | $$~$
f: X \rightarrow \bigcup_{Y \in X} Y
$~$$ |
---|
Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$Y \in X$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$Y$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$f$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$Y$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$f(Y) \in Y$~$ |
---|
Axiom of Choice Definition (Intuitive) | $$~$
\forall_X
\left(
\left[\forall_{Y \in X} Y \not= \emptyset \right]
\Rightarrow
\left[\exists
\left( f: X \rightarrow \bigcup_{Y \in X} Y \right)
\left(\forall_{Y \in X}
\exists_{y \in Y} f(Y) = y \right) \right]
\right)
$~$$ |
---|
Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$Y_1, Y_2, Y_3$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$y_1 \in Y_1, y_2 \in Y_2, y_3 \in Y_3$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$f$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$f(Y_1) = y_1$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$f(Y_2) = y_2$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$f(Y_3) = y_3$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$X$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$Y_1, Y_2, Y_3, \ldots$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$f$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$Y$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$n$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$n$~$ |
---|
Axiom of Choice Definition (Intuitive) | $~$f$~$ |
---|
Bag | $~$\operatorname{Bag}(1, 1, 2, 3) = \operatorname{Bag}(2, 1, 3, 1) \neq \operatorname{Bag}(1, 2, 3).$~$ |
---|
Bayes' rule | $~$2 \times \dfrac{1}{4} = \dfrac{1}{2}.$~$ |
---|
Bayes' rule | $~$h_1$~$ |
---|
Bayes' rule | $~$\mathbb {P}(h_1)$~$ |
---|
Bayes' rule | $~$\mathbb {P}(h_2)$~$ |
---|
Bayes' rule | $~$e_0$~$ |
---|
Bayes' rule | $~$e_0$~$ |
---|
Bayes' rule | $~$h_1$~$ |
---|
Bayes' rule | $~$\mathbb {P}(e_0\mid h_1)$~$ |
---|
Bayes' rule | $~$\mathbb {P}(e_0\mid h_2)$~$ |
---|
Bayes' rule | $~$e_0$~$ |
---|
Bayes' rule | $~$h_2$~$ |
---|
Bayes' rule | $~$e_0$~$ |
---|
Bayes' rule | $~$h_1$~$ |
---|
Bayes' rule | $~$h_2$~$ |
---|
Bayes' rule | $$~$\frac{\mathbb {P}(h_1\mid e_0)}{\mathbb {P}(h_2\mid e_0)} = \frac{\mathbb {P}(h_1)}{\mathbb {P}(h_2)} \cdot \frac{\mathbb {P}(e_0\mid h_1)}{\mathbb {P}(e_0\mid h_2)}$~$$ |
---|
Bayes' rule | $~$\mathbb P(\mathbf{H}\mid e) \propto \operatorname{\mathbb {P}}(e\mid \mathbf{H}) \cdot \operatorname{\mathbb {P}}(\mathbf{H}).$~$ |
---|
Bayes' rule: Definition | $~$H_1$~$ |
---|
Bayes' rule: Definition | $~$H_2$~$ |
---|
Bayes' rule: Definition | $~$e_0.$~$ |
---|
Bayes' rule: Definition | $~$\mathbb P(H_i)$~$ |
---|
Bayes' rule: Definition | $~$H_i$~$ |
---|
Bayes' rule: Definition | $~$\mathbb P(e_0\mid H_i)$~$ |
---|
Bayes' rule: Definition | $~$e_0$~$ |
---|
Bayes' rule: Definition | $~$H_i$~$ |
---|
Bayes' rule: Definition | $~$\mathbb P(H_i\mid e_0)$~$ |
---|
Bayes' rule: Definition | $~$H_i$~$ |
---|
Bayes' rule: Definition | $~$e_0.$~$ |
---|
Bayes' rule: Definition | $$~$\dfrac{\mathbb P(H_1)}{\mathbb P(H_2)} \times \dfrac{\mathbb P(e_0\mid H_1)}{\mathbb P(e_0\mid H_2)} = \dfrac{\mathbb P(H_1\mid e_0)}{\mathbb P(H_2\mid e_0)}$~$$ |
---|
Bayes' rule: Definition | $~$h_i$~$ |
---|
Bayes' rule: Definition | $~$\alpha$~$ |
---|
Bayes' rule: Definition | $~$h_k$~$ |
---|
Bayes' rule: Definition | $~$\beta$~$ |
---|
Bayes' rule: Definition | $~$h_i$~$ |
---|
Bayes' rule: Definition | $~$h_k$~$ |
---|
Bayes' rule: Definition | $~$h_i$~$ |
---|
Bayes' rule: Definition | $~$\alpha \cdot \beta$~$ |
---|
Bayes' rule: Definition | $~$h_k.$~$ |
---|
Bayes' rule: Definition | $~$2 \times \dfrac{1}{4} = \dfrac{1}{2}.$~$ |
---|
Bayes' rule: Definition | $~$\mathbb P(X\wedge Y) = \mathbb P(X\mid Y) \cdot \mathbb P(Y):$~$ |
---|
Bayes' rule: Definition | $$~$
\dfrac{\mathbb P(H_i)}{\mathbb P(H_j)} \times \dfrac{\mathbb P(e\mid H_i)}{\mathbb P(e\mid H_j)}
= \dfrac{\mathbb P(e \wedge H_i)}{\mathbb P(e \wedge H_j)}
= \dfrac{\mathbb P(e \wedge H_i) / \mathbb P(e)}{\mathbb P(e \wedge H_j) / \mathbb P(e)}
= \dfrac{\mathbb P(H_i\mid e)}{\mathbb P(H_j\mid e)}
$~$$ |
---|
Bayes' rule: Definition | $$~$\log \left ( \dfrac
{\mathbb P(H_i)}
{\mathbb P(H_j)}
\right )
+
\log \left ( \dfrac
{\mathbb P(e\mid H_i)}
{\mathbb P(e\mid H_j)}
\right )
=
\log \left ( \dfrac
{\mathbb P(H_i\mid e)}
{\mathbb P(H_j\mid e)}
\right )
$~$$ |
---|
Bayes' rule: Definition | $$~$\begin{array}{rll}
(1/2 : 1/3 : 1/6) \cong & (3 : 2 : 1) & \\
\times & (2 : 1 : 3) & \\
\times & (2 : 3 : 1) & \\
\times & (2 : 1 : 3) & \\
= & (24 : 6 : 9) & \cong (8 : 2 : 3)
\end{array}$~$$ |
---|
Bayes' rule: Definition | $~$\mathbb P(H_i\mid e),$~$ |
---|
Bayes' rule: Definition | $$~$\mathbb P(H_i\mid e) = \dfrac{\mathbb P(e\mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P(e\mid H_k) \cdot \mathbb P(H_k)}$~$$ |
---|
Bayes' rule: Definition | $$~$\mathbb P(\mathbf{H}\mid e) \propto \mathbb P(e\mid \mathbf{H}) \cdot \mathbb P(\mathbf{H}).$~$$ |
---|
Bayes' rule: Definition | $~$1,$~$ |
---|
Bayes' rule: Functional form | $$~$\mathbb P(H_x\mid e) \propto \mathcal L_e(H_x) \cdot \mathbb P(H_x)$~$$ |
---|
Bayes' rule: Functional form | $$~$\mathbb P(H_x\mid e) \propto \mathcal L_e(H_x) \cdot \mathbb P(H_x)$~$$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$\mathbb P(b),$~$ |
---|
Bayes' rule: Functional form | $~$\mathbb P(b)\cdot \mathrm{d}b$~$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$[b + \mathrm{d}b]$~$ |
---|
Bayes' rule: Functional form | $~$\mathrm db$~$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$[a, b]$~$ |
---|
Bayes' rule: Functional form | $~$\int_a^b \mathbb P(b) \, \mathrm db.$~$ |
---|
Bayes' rule: Functional form | $~$b,$~$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$\mathbb P(b) = 1$~$ |
---|
Bayes' rule: Functional form | $~$b,$~$ |
---|
Bayes' rule: Functional form | $~$\mathbb P(b)\, \mathrm db = \mathrm db$~$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$\mathcal L_{t_1}(b)$~$ |
---|
Bayes' rule: Functional form | $~$t_1$~$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$b = 0.6,$~$ |
---|
Bayes' rule: Functional form | $~$b = 0.33,$~$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$\mathcal L_{t_1}(b)$~$ |
---|
Bayes' rule: Functional form | $~$t_1$~$ |
---|
Bayes' rule: Functional form | $~$b,$~$ |
---|
Bayes' rule: Functional form | $~$\mathcal L_{t_1}(b) = 1 - b.$~$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$\mathbb O(b\mid t_1) = \mathcal L_{t_1}(b) \cdot \mathbb P(b) = 1 - b,$~$ |
---|
Bayes' rule: Functional form | $~$\int_0^1 (1 - b) \, \mathrm db = \frac{1}{2}.$~$ |
---|
Bayes' rule: Functional form | $~$\mathbb P(b \mid t_1) = \dfrac{\mathbb O(b \mid t_1)}{\int_0^1 \mathbb O(b \mid t_1) \, \mathrm db} = 2 \cdot (1 - f)$~$ |
---|
Bayes' rule: Functional form | $~$h_2t_3.$~$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$b$~$ |
---|
Bayes' rule: Functional form | $~$b.$~$ |
---|
Bayes' rule: Functional form | $$~$\mathbb P(b \mid t_1h_2t_3) = \frac{\mathcal L_{t_1h_2t_3}(b) \cdot \mathbb P(b)}{\mathbb P(t_1h_2t_3)} = \frac{(1 - b) \cdot b \cdot (1 - b) \cdot 1}{\int_0^1 (1 - b) \cdot b \cdot (1 - b) \cdot 1 \, \mathrm{d}b} = {12\cdot b(1 - b)^2}$~$$ |
---|
Bayes' rule: Log-odds form | $~$H_i$~$ |
---|
Bayes' rule: Log-odds form | $~$H_j$~$ |
---|
Bayes' rule: Log-odds form | $~$e$~$ |
---|
Bayes' rule: Log-odds form | $$~$
\log \left ( \dfrac
{\mathbb P(H_i\mid e)}
{\mathbb P(H_j\mid e)}
\right )
=
\log \left ( \dfrac
{\mathbb P(H_i)}
{\mathbb P(H_j)}
\right )
+
\log \left ( \dfrac
{\mathbb P(e\mid H_i)}
{\mathbb P(e\mid H_j)}
\right ).
$~$$ |
---|
Bayes' rule: Log-odds form | $~$H_i$~$ |
---|
Bayes' rule: Log-odds form | $~$H_j$~$ |
---|
Bayes' rule: Log-odds form | $~$e$~$ |
---|
Bayes' rule: Log-odds form | $$~$
\log \left ( \dfrac
{\mathbb P(H_i\mid e)}
{\mathbb P(H_j\mid e)}
\right )
=
\log \left ( \dfrac
{\mathbb P(H_i)}
{\mathbb P(H_j)}
\right )
+
\log \left ( \dfrac
{\mathbb P(e\mid H_i)}
{\mathbb P(e\mid H_j)}
\right ).
$~$$ |
---|
Bayes' rule: Log-odds form | $~$(1 : 1)$~$ |
---|
Bayes' rule: Log-odds form | $~$(1 : 2) \times (4 : 1) \times (2 : 1),$~$ |
---|
Bayes' rule: Log-odds form | $~$(1 \times 4 \times 2 : 2 \times 1 \times 1) = (8 : 2) = (4 : 1)$~$ |
---|
Bayes' rule: Log-odds form | $~$2$~$ |
---|
Bayes' rule: Log-odds form | $~$\log_2 (\frac{1}{1}) = 0$~$ |
---|
Bayes' rule: Log-odds form | $~$\log_2 (\frac{1}{2}) = {-1}$~$ |
---|
Bayes' rule: Log-odds form | $~$\log_2 (\frac{4}{1}) = {+2}$~$ |
---|
Bayes' rule: Log-odds form | $~$\log_2 (\frac{2}{1}) = {+1}$~$ |
---|
Bayes' rule: Log-odds form | $~$0 + {^-1} + {^+2} + {^+1} = {^+2}$~$ |
---|
Bayes' rule: Log-odds form | $~$(2^{+2} : 1) = (4 : 1),$~$ |
---|
Bayes' rule: Log-odds form | $~$H$~$ |
---|
Bayes' rule: Log-odds form | $~$\lnot H,$~$ |
---|
Bayes' rule: Log-odds form | $~$2 : 1$~$ |
---|
Bayes' rule: Log-odds form | $~$H.$~$ |
---|
Bayes' rule: Log-odds form | $~$H$~$ |
---|
Bayes' rule: Log-odds form | $~$(1 : 1)$~$ |
---|
Bayes' rule: Log-odds form | $~$(2 : 1)$~$ |
---|
Bayes' rule: Log-odds form | $~$(4 : 1)$~$ |
---|
Bayes' rule: Log-odds form | $~$(8 : 1)$~$ |
---|
Bayes' rule: Log-odds form | $~$(16 : 1)$~$ |
---|
Bayes' rule: Log-odds form | $~$(32 : 1).$~$ |
---|
Bayes' rule: Log-odds form | $~$\frac{1}{2} = 50\%$~$ |
---|
Bayes' rule: Log-odds form | $~$\frac{2}{3} \approx 67\%$~$ |
---|
Bayes' rule: Log-odds form | $~$\frac{4}{5} = 80\%$~$ |
---|
Bayes' rule: Log-odds form | $~$\frac{8}{9} \approx 89\%$~$ |
---|
Bayes' rule: Log-odds form | $~$\frac{16}{17} \approx 94\%$~$ |
---|
Bayes' rule: Log-odds form | $~$\frac{32}{33} \approx 97\%.$~$ |
---|
Bayes' rule: Log-odds form | $~$(2 : 1)$~$ |
---|
Bayes' rule: Log-odds form | $~$H$~$ |
---|
Bayes' rule: Log-odds form | $~$-\infty$~$ |
---|
Bayes' rule: Log-odds form | $~$+\infty$~$ |
---|
Bayes' rule: Log-odds form | $~$(0,1)$~$ |
---|
Bayes' rule: Log-odds form | $~${+1}$~$ |
---|
Bayes' rule: Log-odds form | $~${^+1}$~$ |
---|
Bayes' rule: Log-odds form | $~$0.01$~$ |
---|
Bayes' rule: Log-odds form | $~$0.000001$~$ |
---|
Bayes' rule: Log-odds form | $~$0.11$~$ |
---|
Bayes' rule: Log-odds form | $~$0.100001.$~$ |
---|
Bayes' rule: Log-odds form | $~${^-2}$~$ |
---|
Bayes' rule: Log-odds form | $~${^-6}$~$ |
---|
Bayes' rule: Log-odds form | $~$\log_{10}(10^{-6}) - \log_{10}(10^{-2})$~$ |
---|
Bayes' rule: Log-odds form | $~${^-4}$~$ |
---|
Bayes' rule: Log-odds form | $~${^-13.3}$~$ |
---|
Bayes' rule: Log-odds form | $~$\log_{10}(\frac{0.10}{0.90}) - \log_{10}(\frac{0.11}{0.89}) \approx {^-0.954}-{^-0.907} \approx {^-0.046}$~$ |
---|
Bayes' rule: Log-odds form | $~${^-0.153}$~$ |
---|
Bayes' rule: Log-odds form | $~$2 : 1,$~$ |
---|
Bayes' rule: Log-odds form | $~$H$~$ |
---|
Bayes' rule: Log-odds form | $~$H$~$ |
---|
Bayes' rule: Log-odds form | $~$H$~$ |
---|
Bayes' rule: Log-odds form | $~$1 : 2$~$ |
---|
Bayes' rule: Log-odds form | $~${^-3}$~$ |
---|
Bayes' rule: Log-odds form | $~${^-1}$~$ |
---|
Bayes' rule: Log-odds form | $~${^-4}$~$ |
---|
Bayes' rule: Log-odds form | $~$(1 : 16)$~$ |
---|
Bayes' rule: Log-odds form | $~$\mathbb P({positive}\mid {HIV}) = .997$~$ |
---|
Bayes' rule: Log-odds form | $~$\mathbb P({negative}\mid \neg {HIV}) = .998$~$ |
---|
Bayes' rule: Log-odds form | $~$\mathbb P({positive} \mid \neg {HIV}) = .002.$~$ |
---|
Bayes' rule: Log-odds form | $~$1 : 100,000$~$ |
---|
Bayes' rule: Log-odds form | $~$500 : 1.$~$ |
---|
Bayes' rule: Log-odds form | $~$\log_{10}(500) \approx 2.7$~$ |
---|
Bayes' rule: Log-odds form | $~$500 : 1$~$ |
---|
Bayes' rule: Log-odds form | $~$0$~$ |
---|
Bayes' rule: Log-odds form | $~$1$~$ |
---|
Bayes' rule: Log-odds form | $~$-\infty$~$ |
---|
Bayes' rule: Log-odds form | $~$+\infty,$~$ |
---|
Bayes' rule: Log-odds form | $~$0$~$ |
---|
Bayes' rule: Log-odds form | $~$1$~$ |
---|
Bayes' rule: Log-odds form | $~$0$~$ |
---|
Bayes' rule: Log-odds form | $~$1$~$ |
---|
Bayes' rule: Log-odds form | $~$\mathbb P(X) + \mathbb P(\lnot X)$~$ |
---|
Bayes' rule: Log-odds form | $~$\lnot X$~$ |
---|
Bayes' rule: Log-odds form | $~$X$~$ |
---|
Bayes' rule: Log-odds form | $~$\aleph_0$~$ |
---|
Bayes' rule: Log-odds form | $~$o$~$ |
---|
Bayes' rule: Log-odds form | $~$e = 10\log_{10}(o)$~$ |
---|
Bayes' rule: Odds form | $~$(1 : 2) \times (10 : 1) = (10 : 2) = (5 : 1)$~$ |
---|
Bayes' rule: Odds form | $~$e,$~$ |
---|
Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H \mid e)$~$ |
---|
Bayes' rule: Odds form | $~$\boldsymbol H$~$ |
---|
Bayes' rule: Odds form | $~$e$~$ |
---|
Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H)$~$ |
---|
Bayes' rule: Odds form | $~$\boldsymbol H$~$ |
---|
Bayes' rule: Odds form | $~$\mathcal L_e(\boldsymbol H).$~$ |
---|
Bayes' rule: Odds form | $~$(1 : 2) \times (10 : 1) = (10 : 2) = (5 : 1)$~$ |
---|
Bayes' rule: Odds form | $~$\boldsymbol H$~$ |
---|
Bayes' rule: Odds form | $~$\mathbb O$~$ |
---|
Bayes' rule: Odds form | $~$\boldsymbol H$~$ |
---|
Bayes' rule: Odds form | $~$\boldsymbol H = (H_1, H_2, H_3),$~$ |
---|
Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H)$~$ |
---|
Bayes' rule: Odds form | $~$(6 : 2 : 1),$~$ |
---|
Bayes' rule: Odds form | $~$H_1$~$ |
---|
Bayes' rule: Odds form | $~$H_2$~$ |
---|
Bayes' rule: Odds form | $~$H_3.$~$ |
---|
Bayes' rule: Odds form | $~$\boldsymbol H;$~$ |
---|
Bayes' rule: Odds form | $~$H_i$~$ |
---|
Bayes' rule: Odds form | $$~$\mathbb O(\boldsymbol H) \propto \mathbb P(\boldsymbol H).$~$$ |
---|
Bayes' rule: Odds form | $~$H_1$~$ |
---|
Bayes' rule: Odds form | $~$H_2$~$ |
---|
Bayes' rule: Odds form | $~$H_3$~$ |
---|
Bayes' rule: Odds form | $~$\boldsymbol H$~$ |
---|
Bayes' rule: Odds form | $~$(H_1, H_2, H_3).$~$ |
---|
Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H) = (80 : 8 : 4) = (20 : 2 : 1)$~$ |
---|
Bayes' rule: Odds form | $~$e_w$~$ |
---|
Bayes' rule: Odds form | $~$\mathbb P(e_w\mid \boldsymbol H) = (0.6, 0.9, 0.3).$~$ |
---|
Bayes' rule: Odds form | $~$\mathcal L_{e_w}(\boldsymbol H) = P(e_w \mid \boldsymbol H).$~$ |
---|
Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H\mid e)$~$ |
---|
Bayes' rule: Odds form | $~$\boldsymbol H$~$ |
---|
Bayes' rule: Odds form | $~$e.$~$ |
---|
Bayes' rule: Odds form | $$~$\mathbb O(\boldsymbol H) \times \mathcal L_{e}(\boldsymbol H) = \mathbb O(\boldsymbol H\mid e)$~$$ |
---|
Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H)$~$ |
---|
Bayes' rule: Odds form | $~$\mathcal L_{e}(\boldsymbol H)$~$ |
---|
Bayes' rule: Odds form | $~$\mathbb O(\boldsymbol H\mid e).$~$ |
---|
Bayes' rule: Odds form | $~$\mathcal L_e(\boldsymbol H) = (0.6, 0.9, 0.3)$~$ |
---|
Bayes' rule: Odds form | $~$(2 : 3 : 1).$~$ |
---|
Bayes' rule: Odds form | $~$(20 : 2 : 1).$~$ |
---|
Bayes' rule: Odds form | $~$(0.6 : 0.9 : 0.3)$~$ |
---|
Bayes' rule: Odds form | $~$(2 : 3 : 1).$~$ |
---|
Bayes' rule: Odds form | $~$e_w$~$ |
---|
Bayes' rule: Odds form | $~$(20 : 2 : 1) \times (2 : 3 : 1) = (40 : 6 : 1).$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$(2 : 8) \times (9 : 3) \ = \ (1 : 4) \times (3 : 1) \ = \ (3 : 4),$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$(x : y)$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$(x : y)$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$\alpha$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$(\alpha x : \alpha y).$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$(1 : 2 : 1)$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$\frac{1}{4} : \frac{2}{4} : \frac{1}{4}.$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$(a : b : c)$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$(\frac{a}{a + b + c} : \frac{b}{a + b + c} : \frac{c}{a + b + c}).$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$A, B, C$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$\mathbb P(A), \mathbb P(B), \mathbb P(C)$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$1.$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$\textbf{Prior odds} \times \textbf{Likelihood ratio} = \textbf{Posterior odds}$~$ |
---|
Bayes' rule: Odds form (Intro, Math 1) | $~$(1 : 9 ) \times (3 : 1) \ = \ (3 : 9) \ \cong \ (1 : 3)$~$ |
---|
Bayes' rule: Probability form | $$~$\mathbb P(H_i\mid e) = \dfrac{\mathbb P(e\mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P(e\mid H_k) \cdot \mathbb P(H_k)}$~$$ |
---|
Bayes' rule: Probability form | $~$\mathbb P(X \mid Y) = \frac{\mathbb P(X \wedge Y)}{\mathbb P (Y)}$~$ |
---|
Bayes' rule: Probability form | $~$\mathbb P(Y) = \sum_k \mathbb P(Y \wedge X_k)$~$ |
---|
Bayes' rule: Probability form | $~$H$~$ |
---|
Bayes' rule: Probability form | $~$e$~$ |
---|
Bayes' rule: Probability form | $~$e$~$ |
---|
Bayes' rule: Probability form | $~$H_k$~$ |
---|
Bayes' rule: Probability form | $~$H_k$~$ |
---|
Bayes' rule: Probability form | $~$H_i$~$ |
---|
Bayes' rule: Probability form | $~$e,$~$ |
---|
Bayes' rule: Probability form | $~$H_i$~$ |
---|
Bayes' rule: Probability form | $~$e,$~$ |
---|
Bayes' rule: Probability form | $~$e$~$ |
---|
Bayes' rule: Probability form | $~$H.$~$ |
---|
Bayes' rule: Probability form | $$~$\mathbb P(H_i\mid e) = \dfrac{\mathbb P(e\mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P(e\mid H_k) \cdot \mathbb P(H_k)}$~$$ |
---|
Bayes' rule: Probability form | $~$H_i$~$ |
---|
Bayes' rule: Probability form | $~$e$~$ |
---|
Bayes' rule: Probability form | $~$\sum_k (\text {expression containing } k)$~$ |
---|
Bayes' rule: Probability form | $~$k$~$ |
---|
Bayes' rule: Probability form | $~$k$~$ |
---|
Bayes' rule: Probability form | $~$\mathbf H$~$ |
---|
Bayes' rule: Probability form | $~$H_i$~$ |
---|
Bayes' rule: Probability form | $~$H_k$~$ |
---|
Bayes' rule: Probability form | $~$\mathbf H$~$ |
---|
Bayes' rule: Probability form | $~$H_1, H_2, H_3$~$ |
---|
Bayes' rule: Probability form | $~$\mathbb P(H_2 \mid heads).$~$ |
---|
Bayes' rule: Probability form | $$~$\mathbb P(H_2 \mid heads) = \frac{\mathbb P(heads \mid H_2) \cdot \mathbb P(H_2)}{\sum_k \mathbb P(heads \mid H_k) \cdot \mathbb P(H_k)}$~$$ |
---|
Bayes' rule: Probability form | $$~$\mathbb P(H_2 \mid heads) = \frac{\mathbb P(heads \mid H_2) \cdot \mathbb P(H_2)}{[\mathbb P(heads \mid H_1) \cdot \mathbb P(H_1)] + [\mathbb P(heads \mid H_2) \cdot \mathbb P(H_2)] + [\mathbb P(heads \mid H_3) \cdot \mathbb P(H_3)]}$~$$ |
---|
Bayes' rule: Probability form | $$~$\mathbb P(H_2 \mid heads) = \frac{0.70 \cdot 0.35 }{[0.50 \cdot 0.40] + [0.70 \cdot 0.35] + [0.20 \cdot 0.25]} = \frac{0.245}{0.20 + 0.245 + 0.05} = 0.\overline{49}$~$$ |
---|
Bayes' rule: Probability form | $~$H$~$ |
---|
Bayes' rule: Probability form | $~$e$~$ |
---|
Bayes' rule: Probability form | $~$e$~$ |
---|
Bayes' rule: Probability form | $~$H_k$~$ |
---|
Bayes' rule: Probability form | $~$H_k$~$ |
---|
Bayes' rule: Probability form | $~$H_i$~$ |
---|
Bayes' rule: Probability form | $~$e,$~$ |
---|
Bayes' rule: Probability form | $~$H_i$~$ |
---|
Bayes' rule: Probability form | $~$e,$~$ |
---|
Bayes' rule: Probability form | $~$e$~$ |
---|
Bayes' rule: Probability form | $~$H.$~$ |
---|
Bayes' rule: Probability form | $~$H_1,H_2,H_3\ldots$~$ |
---|
Bayes' rule: Probability form | $~$1$~$ |
---|
Bayes' rule: Probability form | $~$H_k$~$ |
---|
Bayes' rule: Probability form | $~$\mathbb P(H_k)$~$ |
---|
Bayes' rule: Probability form | $~$\mathbb P(H_4)=\frac{1}{5}$~$ |
---|
Bayes' rule: Probability form | $~$E,$~$ |
---|
Bayes' rule: Probability form | $~$e_1, e_2, \ldots.$~$ |
---|
Bayes' rule: Probability form | $~$E = e_j,$~$ |
---|
Bayes' rule: Probability form | $~$e_j.$~$ |
---|
Bayes' rule: Probability form | $~$H_4$~$ |
---|
Bayes' rule: Probability form | $~$e_3,$~$ |
---|
Bayes' rule: Probability form | $~$H_4$~$ |
---|
Bayes' rule: Probability form | $~$e_3,$~$ |
---|
Bayes' rule: Probability form | $~$H_4$~$ |
---|
Bayes' rule: Probability form | $~$e_3.$~$ |
---|
Bayes' rule: Probability form | $~$H_4$~$ |
---|
Bayes' rule: Probability form | $~$H_4$~$ |
---|
Bayes' rule: Probability form | $~$e_3.$~$ |
---|
Bayes' rule: Probability form | $~$e_3,$~$ |
---|
Bayes' rule: Probability form | $~$H_k$~$ |
---|
Bayes' rule: Probability form | $~$H_k$~$ |
---|
Bayes' rule: Probability form | $~$e_3.$~$ |
---|
Bayes' rule: Probability form | $$~$\mathbb P(H_4 \mid e_3) = \frac{\mathbb P(e_3 \mid H_4) \cdot \mathbb P(H_4)}{\sum_k \mathbb P(e_3 \mid H_k) \cdot \mathbb P(H_k)}$~$$ |
---|
Bayes' rule: Probability form | $~$e_j,$~$ |
---|
Bayes' rule: Probability form | $~$e_3$~$ |
---|
Bayes' rule: Probability form | $~$e_3.$~$ |
---|
Bayes' rule: Probability form | $~$e_3$~$ |
---|
Bayes' rule: Probability form | $~$e_5$~$ |
---|
Bayes' rule: Probability form | $~$e_5$~$ |
---|
Bayes' rule: Probability form | $~$e_5$~$ |
---|
Bayes' rule: Probability form | $~$e_5,$~$ |
---|
Bayes' rule: Probability form | $~$e_j$~$ |
---|
Bayes' rule: Probability form | $~$e_3,$~$ |
---|
Bayes' rule: Probability form | $~$e_5.$~$ |
---|
Bayes' rule: Probability form | $~$e_3$~$ |
---|
Bayes' rule: Probability form | $~$e_3$~$ |
---|
Bayes' rule: Probability form | $~$e_5$~$ |
---|
Bayes' rule: Probability form | $~$e_3$~$ |
---|
Bayes' rule: Probability form | $~$H_4$~$ |
---|
Bayes' rule: Probability form | $~$H_4$~$ |
---|
Bayes' rule: Probability form | $~$e_3$~$ |
---|
Bayes' rule: Probability form | $~$H_4$~$ |
---|
Bayes' rule: Probability form | $~$e_3$~$ |
---|
Bayes' rule: Probability form | $~$e_3$~$ |
---|
Bayes' rule: Probability form | $~$H_k$~$ |
---|
Bayes' rule: Probability form | $~$e_j$~$ |
---|
Bayes' rule: Probability form | $~$\mathbb P(e \mid GoodDriver)$~$ |
---|
Bayes' rule: Probability form | $~$\mathbb P(e \mid BadDriver)$~$ |
---|
Bayes' rule: Probability form | $~$\mathbb P(BadDriver)$~$ |
---|
Bayes' rule: Probability form | $$~$\mathbb P(X \mid Y) = \frac{\mathbb P(X \wedge Y)}{\mathbb P (Y)}$~$$ |
---|
Bayes' rule: Probability form | $$~$\mathbb P(Y) = \sum_k \mathbb P(Y \wedge X_k)$~$$ |
---|
Bayes' rule: Probability form | $$~$
\mathbb P(H_i \mid e) = \frac{\mathbb P(H_i \wedge e)}{\mathbb P (e)} \tag{defn. conditional prob.}
$~$$ |
---|
Bayes' rule: Probability form | $$~$
\mathbb P(H_i \mid e) = \frac{\mathbb P(e \wedge H_i)}{\sum_k \mathbb P (e \wedge H_k)} \tag {law of marginal prob.}
$~$$ |
---|
Bayes' rule: Probability form | $$~$
\mathbb P(H_i \mid e) = \frac{\mathbb P(e \mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P (e \mid H_k) \cdot \mathbb P(H_k)} \tag {defn. conditional prob.}
$~$$ |
---|
Bayes' rule: Proportional form | $~$2 \times \dfrac{1}{4} = \dfrac{1}{2}.$~$ |
---|
Bayes' rule: Proportional form | $~$H_i$~$ |
---|
Bayes' rule: Proportional form | $~$H_j$~$ |
---|
Bayes' rule: Proportional form | $~$e$~$ |
---|
Bayes' rule: Proportional form | $$~$\dfrac{\mathbb P(H_i)}{\mathbb P(H_j)} \times \dfrac{\mathbb P(e\mid H_i)}{\mathbb P(e\mid H_j)} = \dfrac{\mathbb P(H_i\mid e)}{\mathbb P(H_j\mid e)}$~$$ |
---|
Bayes' rule: Proportional form | $~$(1 : 4) \times (3 : 1) = (3 : 4).$~$ |
---|
Bayes' rule: Proportional form | $~$(1 : 4) \times (3 : 1) = (3 : 4).$~$ |
---|
Bayes' rule: Proportional form | $~$\frac{1}{4} \times \frac{3}{1} = \frac{3}{4},$~$ |
---|
Bayes' rule: Proportional form | $~$0.25 \times 3 = 0.75.$~$ |
---|
Bayes' rule: Proportional form | $~$(0.25 : 1) \cdot (3 : 1) = (0.75 : 1),$~$ |
---|
Bayes' rule: Vector form | $$~$\begin{array}{rll}
(1/2 : 1/3 : 1/6) = & (3 : 2 : 1) & \\
\times & (2 : 1 : 3) & \\
\times & (2 : 3 : 1) & \\
\times & (2 : 1 : 3) & \\
= & (24 : 6 : 9) & = (8 : 2 : 3)
\end{array}$~$$ |
---|
Bayes' rule: Vector form | $~$\mathbf H$~$ |
---|
Bayes' rule: Vector form | $~$H_1, H_2, \ldots$~$ |
---|
Bayes' rule: Vector form | $~$\mathbf H,$~$ |
---|
Bayes' rule: Vector form | $$~$\mathbb O(\mathbf H) \times \mathcal L_e(\mathbf H) = \mathbb O(\mathbf H \mid e)$~$$ |
---|
Bayes' rule: Vector form | $~$\mathbb O(\mathbf H)$~$ |
---|
Bayes' rule: Vector form | $~$H_i$~$ |
---|
Bayes' rule: Vector form | $~$\mathcal L_e(\mathbf H)$~$ |
---|
Bayes' rule: Vector form | $~$H_i$~$ |
---|
Bayes' rule: Vector form | $~$e,$~$ |
---|
Bayes' rule: Vector form | $~$\mathbb O(\mathbf H \mid e)$~$ |
---|
Bayes' rule: Vector form | $~$H_i.$~$ |
---|
Bayes' rule: Vector form | $$~$\begin{array}{r}
\mathbb O(\mathbf H) \\
\times\ \mathcal L_{e_1}(\mathbf H) \\
\times\ \mathcal L_{e_2}(\mathbf H \wedge e_1) \\
\times\ \mathcal L_{e_3}(\mathbf H \wedge e_1 \wedge e_2) \\
= \mathbb O(\mathbf H \mid e_1 \wedge e_2 \wedge e_3)
\end{array}$~$$ |
---|
Bayes' rule: Vector form | $~$H_{fair},$~$ |
---|
Bayes' rule: Vector form | $~$H_{heads}$~$ |
---|
Bayes' rule: Vector form | $~$H_{tails}$~$ |
---|
Bayes' rule: Vector form | $~$(1/2 : 1/3 : 1/6).$~$ |
---|
Bayes' rule: Vector form | $~$(2 : 3 : 1)$~$ |
---|
Bayes' rule: Vector form | $~$(2 : 1 : 3).$~$ |
---|
Bayes' rule: Vector form | $$~$\begin{array}{rll}
(1/2 : 1/3 : 1/6) = & (3 : 2 : 1) & \\
\times & (2 : 1 : 3) & \\
\times & (2 : 3 : 1) & \\
\times & (2 : 1 : 3) & \\
= & (24 : 6 : 9) & = (8 : 2 : 3) = (8/13 : 2/13 : 3/13)
\end{array}$~$$ |
---|
Bayes' rule: Vector form | $$~$\mathbb P(H_i\mid e) = \dfrac{\mathbb P(e\mid H_i)P(H_i)}{\sum_k \mathbb P(e\mid H_k)P(H_k)}$~$$ |
---|
Bayes' rule: Vector form | $~$(5 : 3 : 2)$~$ |
---|
Bayes' rule: Vector form | $$~$\left(\frac{10}{50} : \frac{3}{30} : \frac{10}{20}\right) = \left(\frac{1}{5} : \frac{1}{10} : \frac{1}{2}\right) = (2 : 1 : 5)$~$$ |
---|
Bayes' rule: Vector form | $$~$\left(\frac{30}{50} : \frac{15}{30} : \frac{1}{20}\right) = \left(\frac{3}{5} : \frac{1}{2} : \frac{1}{20}\right) = (12 : 10 : 1)$~$$ |
---|
Bayes' rule: Vector form | $$~$(5 : 3 : 2) \times (2 : 1 : 5) \times (12 : 10 : 1) = (120 : 30 : 10) = \left(\frac{12}{16} : \frac{3}{16} : \frac{1}{16}\right)$~$$ |
---|
Bayes' rule: Vector form | $$~$\mathbb P({workplace}\mid \neg {romance} \wedge {museum}) \neq \mathbb P({workplace}\mid \neg {romance})$~$$ |
---|
Bayes' rule: Vector form | $~$\mathbb P({museum} \wedge {workplace} \mid \neg {romance})$~$ |
---|
Bayes' rule: Vector form | $~$\mathbb P({museum}\mid \neg {romance}) \cdot \mathbb P({workplace}\mid \neg {romance}).$~$ |
---|
Bayesian view of scientific virtues | $~$Grek$~$ |
---|
Bayesian view of scientific virtues | $~$up, down,$~$ |
---|
Bayesian view of scientific virtues | $~$other.$~$ |
---|
Bayesian view of scientific virtues | $~$Thag$~$ |
---|
Bayesian view of scientific virtues | $~$up, down,$~$ |
---|
Bayesian view of scientific virtues | $~$other$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(\cdot\mid Thag)$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(up\mid Thag) + \mathbb P(down\mid Thag) + \mathbb P(other\mid Thag) = 1.$~$ |
---|
Bayesian view of scientific virtues | $~$1/3$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(up\mid Thag), \mathbb P(down\mid Thag),$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(other\mid Thag)$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(down\mid Grek)!$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(up\mid Grek)$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(other\mid Grek)$~$ |
---|
Bayesian view of scientific virtues | $~$up,$~$ |
---|
Bayesian view of scientific virtues | $~$up$~$ |
---|
Bayesian view of scientific virtues | $~$other,$~$ |
---|
Bayesian view of scientific virtues | $~$down$~$ |
---|
Bayesian view of scientific virtues | $~$down$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(down\mid Thag)$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(up\mid Thag) = 1.$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(up\mid Thag) = 1$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(down\mid Thag) = 1$~$ |
---|
Bayesian view of scientific virtues | $~$1$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(down\mid Grek) = 0.95$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(down\mid Grek) = 0$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(down\mid Grek) = 0.95$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(down\mid Thag) = 0.95$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(blue\mid Thag) = 0.90$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(blue\mid \neg Thag) < 0.01$~$ |
---|
Bayesian view of scientific virtues | $~$\dfrac{\mathbb P(Thag\mid blue)}{\mathbb P(\neg Thag\mid blue)} > 90 \cdot \dfrac{\mathbb P(Thag)}{\mathbb P(\neg Thag)}$~$ |
---|
Bayesian view of scientific virtues | $~$H \rightarrow E,$~$ |
---|
Bayesian view of scientific virtues | $~$\neg E$~$ |
---|
Bayesian view of scientific virtues | $~$\neg H$~$ |
---|
Bayesian view of scientific virtues | $~$E,$~$ |
---|
Bayesian view of scientific virtues | $~$H.$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(UranusLocation\mid currentNewton)$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(UranusLocation\mid newModel)$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(UranusLocation\mid Neptune \wedge Newton),$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(UranusLocation\mid Neptune \wedge Other).$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(MercuryLocation\mid Einstein)$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(MercuryLocation\mid Newton),$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(MercuryLocation\mid Other)$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(newObservation\mid Other),$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(MercuryLocation\mid Newton)$~$ |
---|
Bayesian view of scientific virtues | $~$\mathbb P(observation\mid hypothesis)$~$ |
---|
Bayesian view of scientific virtues | $~$observation$~$ |
---|
Bayesian view of scientific virtues | $~$\neg observation$~$ |
---|
Belief revision as probability elimination | $~$\mathbb P$~$ |
---|
Belief revision as probability elimination | $~$\mathbb P$~$ |
---|
Belief revision as probability elimination | $$~$\begin{array}{l|r|r}
& Sick & Healthy \\
\hline
Test + & 18\% & 24\% \\
\hline
Test - & 2\% & 56\%
\end{array}$~$$ |
---|
Binary function | $~$f$~$ |
---|
Binary function | $~$+,$~$ |
---|
Binary function | $~$-,$~$ |
---|
Binary function | $~$\times,$~$ |
---|
Binary function | $~$\div$~$ |
---|
Binary notation | $~$8207$~$ |
---|
Binary notation | $~$(7 \times 10^0) + (0 \times 10^1) + (2 \times 10^2) + (8 \times 10^3)$~$ |
---|
Binary notation | $~$0$~$ |
---|
Binary notation | $~$1$~$ |
---|
Binary notation | $~$11010$~$ |
---|
Binary notation | $~$(0 \times 2^0) + (1 \times 2^1) + (0 \times 2^2) + (1 \times 2^3) + (1 \times 2^4)$~$ |
---|
Binary notation | $~$26$~$ |
---|
Bit | $~$\log_2$~$ |
---|
Bit | $~$\mathbb B$~$ |
---|
Bit | $~$2 : 1$~$ |
---|
Bit | $~$\mathbb B$~$ |
---|
Bit | $~$2 : 1$~$ |
---|
Bit | $~$\log_2$~$ |
---|
Bit | $~$\log_2$~$ |
---|
Bit | $~$\log_2$~$ |
---|
Bit | $~$\log_2$~$ |
---|
Bit (abstract) | $~$\mathbb B$~$ |
---|
Bit (abstract) | $~$\mathbb B$~$ |
---|
Bit (abstract) | $~$\mathbb N$~$ |
---|
Bit (abstract) | $~$\mathbb N$~$ |
---|
Bit (abstract) | $~$\mathbb B$~$ |
---|
Bit (abstract) | $~$\mathbb B$~$ |
---|
Bit (of data) | $~$n$~$ |
---|
Bit (of data) | $~$\log_2(n)$~$ |
---|
Bit (of data) | $~$n$~$ |
---|
Bit (of data) | $~$\log_2(n)$~$ |
---|
Bit (of data) | $~$\log_2(10) \approx 3.32$~$ |
---|
Bit (of data) | $~$2^{10}=1024.$~$ |
---|
Bit (of data) | $~$2^{20}=1048576.$~$ |
---|
Bit (of data) | $~$n$~$ |
---|
Bit (of data) | $~$n$~$ |
---|
Bit (of data) | $~$\log_2(n)$~$ |
---|
Boolean | $~$\land$~$ |
---|
Boolean | $~$\lor$~$ |
---|
Boolean | $~$\neg$~$ |
---|
Boolean | $~$\rightarrow$~$ |
---|
Bézout's theorem | $~$a$~$ |
---|
Bézout's theorem | $~$b$~$ |
---|
Bézout's theorem | $~$c$~$ |
---|
Bézout's theorem | $~$ax+by = c$~$ |
---|
Bézout's theorem | $~$x$~$ |
---|
Bézout's theorem | $~$y$~$ |
---|
Bézout's theorem | $~$a$~$ |
---|
Bézout's theorem | $~$b$~$ |
---|
Bézout's theorem | $~$c$~$ |
---|
Bézout's theorem | $~$a$~$ |
---|
Bézout's theorem | $~$b$~$ |
---|
Bézout's theorem | $~$c$~$ |
---|
Bézout's theorem | $~$ax+by = c$~$ |
---|
Bézout's theorem | $~$x$~$ |
---|
Bézout's theorem | $~$y$~$ |
---|
Bézout's theorem | $~$a$~$ |
---|
Bézout's theorem | $~$b$~$ |
---|
Bézout's theorem | $~$c$~$ |
---|
Bézout's theorem | $~$ax+by=c$~$ |
---|
Bézout's theorem | $~$ax+by=c$~$ |
---|
Bézout's theorem | $~$x$~$ |
---|
Bézout's theorem | $~$y$~$ |
---|
Bézout's theorem | $~$a$~$ |
---|
Bézout's theorem | $~$b$~$ |
---|
Bézout's theorem | $~$a$~$ |
---|
Bézout's theorem | $~$b$~$ |
---|
Bézout's theorem | $~$ax$~$ |
---|
Bézout's theorem | $~$by$~$ |
---|
Bézout's theorem | $~$c$~$ |
---|
Bézout's theorem | $~$c$~$ |
---|
Bézout's theorem | $~$\mathrm{hcf}(a,b) \mid c$~$ |
---|
Bézout's theorem | $~$d$~$ |
---|
Bézout's theorem | $~$d \times \mathrm{hcf}(a,b) = c$~$ |
---|
Bézout's theorem | $~$a, b$~$ |
---|
Bézout's theorem | $~$x$~$ |
---|
Bézout's theorem | $~$y$~$ |
---|
Bézout's theorem | $~$ax + by = \mathrm{hcf}(a,b)$~$ |
---|
Bézout's theorem | $~$a (xd) + b (yd) = d \mathrm{hcf}(a, b) = c$~$ |
---|
Bézout's theorem | $~$d \times \mathrm{hcf}(a,b) = c$~$ |
---|
Bézout's theorem | $~$ax+by$~$ |
---|
Bézout's theorem | $~$a$~$ |
---|
Bézout's theorem | $~$b$~$ |
---|
Bézout's theorem | $~$\mathrm{hcf}(a,b)$~$ |
---|
Bézout's theorem | $~$ax+by=c$~$ |
---|
Bézout's theorem | $~$d$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$1 < 2$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$2<1$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$a < b$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$b < a$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f: A \to B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$g: B \to A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$h: A \to B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$b$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$a \in A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f(a) = b$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f^{-1}(b)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$a \in A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f(a) = b$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$g$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f^{-1}(a)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f^{-1}(a)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$a \in A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $$~$\dots, f^{-1}(g^{-1}(a)), g^{-1}(a), a, f(a), g(f(a)), \dots$~$$ |
---|
Cantor-Schröder-Bernstein theorem | $~$a$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$g^{-1}(a)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$gfgf(a) = a$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$b \in B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $$~$\dots g^{-1} f^{-1}(b), f^{-1}(b), b, g(b), f(g(b)), \dots$~$$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$a \in A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$g^{-1} f^{-1}(b)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$b$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$h(a) = f(a)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$h(a) = f(a)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$h(a) = g^{-1}(a)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$h(a) = f(a)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$b \in B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$a$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$h$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$a$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$b$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$b \in B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$h$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$g(b)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$b$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$h$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$h$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$b \in B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$X$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f: X \to X$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$x$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f(x) = x$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f: A \to B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$g: B \to A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$P \cup Q$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$R \cup S$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$P$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$R$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$g$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$S$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$Q$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$A \to B$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$f$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$P$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$g^{-1}$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$Q$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$P \mapsto A \setminus g(B \setminus f(P))$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$\mathcal{P}(A)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$\mathcal{P}(A)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$\mathcal{P}(A)$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$P$~$ |
---|
Cantor-Schröder-Bernstein theorem | $~$P = A \setminus g(B \setminus f(P))$~$ |
---|
Cardinality | $~$A$~$ |
---|
Cardinality | $~$A$~$ |
---|
Cardinality | $~$|A|$~$ |
---|
Cardinality | $~$A$~$ |
---|
Cardinality | $~$|A| = n$~$ |
---|
Cardinality | $~$A$~$ |
---|
Cardinality | $~$n$~$ |
---|
Cardinality | $~$n$~$ |
---|
Cardinality | $~$\{0, …, (n-1)\}$~$ |
---|
Cardinality | $~$n$~$ |
---|
Cardinality | $~$\mathbb N$~$ |
---|
Cardinality | $~$\mathbb N$~$ |
---|
Cardinality | $~$|X|$~$ |
---|
Cardinality | $~$X$~$ |
---|
Cardinality | $~$X.$~$ |
---|
Cardinality | $~$X = \{a, b, c, d\}, |X|=4.$~$ |
---|
Cardinality | $~$S$~$ |
---|
Cardinality | $~$n$~$ |
---|
Cardinality | $~$S$~$ |
---|
Cardinality | $~$1$~$ |
---|
Cardinality | $~$n$~$ |
---|
Cardinality | $~$\{9, 15, 12, 20\}$~$ |
---|
Cardinality | $~$\{1, 2, 3, 4\}$~$ |
---|
Cardinality | $~$m$~$ |
---|
Cardinality | $~$m$~$ |
---|
Cardinality | $~$4$~$ |
---|
Cardinality | $~$S$~$ |
---|
Cardinality | $~$T$~$ |
---|
Cardinality | $~$f : S \to \{1, 2, 3, \ldots, n\}$~$ |
---|
Cardinality | $~$g : \{1, 2, 3, \ldots, n\} \to T$~$ |
---|
Cardinality | $~$g \circ f$~$ |
---|
Cardinality | $~$S$~$ |
---|
Cardinality | $~$T$~$ |
---|
Cardinality | $~$n$~$ |
---|
Cardinality | $~$\aleph_0$~$ |
---|
Cardinality | $~$\aleph_1, \aleph_2, \aleph_3,$~$ |
---|
Cartesian product | $~$A$~$ |
---|
Cartesian product | $~$B,$~$ |
---|
Cartesian product | $~$A \times B,$~$ |
---|
Cartesian product | $~$(a, b)$~$ |
---|
Cartesian product | $~$a \in A$~$ |
---|
Cartesian product | $~$b \in B.$~$ |
---|
Cartesian product | $~$\mathbb B \times \mathbb N$~$ |
---|
Cartesian product | $~$\mathbb B^3 = \mathbb B \times \mathbb B \times \mathbb B$~$ |
---|
Cartesian product | $~$\times$~$ |
---|
Cartesian product | $~$n$~$ |
---|
Cartesian product | $~$n$~$ |
---|
Category (mathematics) | $~$f$~$ |
---|
Category (mathematics) | $~$X$~$ |
---|
Category (mathematics) | $~$Y$~$ |
---|
Category (mathematics) | $~$X$~$ |
---|
Category (mathematics) | $~$Y$~$ |
---|
Category (mathematics) | $~$X$~$ |
---|
Category (mathematics) | $~$Y$~$ |
---|
Category (mathematics) | $~$f$~$ |
---|
Category (mathematics) | $~$X$~$ |
---|
Category (mathematics) | $~$f$~$ |
---|
Category (mathematics) | $~$Y$~$ |
---|
Category (mathematics) | $~$f$~$ |
---|
Category (mathematics) | $~$f$~$ |
---|
Category (mathematics) | $~$X$~$ |
---|
Category (mathematics) | $~$Y$~$ |
---|
Category (mathematics) | $~$f: X \rightarrow Y$~$ |
---|
Category (mathematics) | $~$f: X \rightarrow Y$~$ |
---|
Category (mathematics) | $~$g: Y \rightarrow Z$~$ |
---|
Category (mathematics) | $~$X \rightarrow Z$~$ |
---|
Category (mathematics) | $~$g \circ f$~$ |
---|
Category (mathematics) | $~$gf$~$ |
---|
Category (mathematics) | $~$f: X \rightarrow Y$~$ |
---|
Category (mathematics) | $~$g: Y \rightarrow Z$~$ |
---|
Category (mathematics) | $~$h:Z \rightarrow W$~$ |
---|
Category (mathematics) | $~$h(gf) = (hg)f$~$ |
---|
Category (mathematics) | $~$X$~$ |
---|
Category (mathematics) | $~$1_X : X \rightarrow X$~$ |
---|
Category (mathematics) | $~$f:W \rightarrow X$~$ |
---|
Category (mathematics) | $~$g:X \rightarrow Y$~$ |
---|
Category (mathematics) | $~$1_X f = f$~$ |
---|
Category (mathematics) | $~$g 1_X = g$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$\text{dom}(f)$~$ |
---|
Category theory | $~$\text{cod}(f)$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$\text{dom}(f) = X$~$ |
---|
Category theory | $~$\text{cod}(f) = Y$~$ |
---|
Category theory | $~$f: X \rightarrow Y$~$ |
---|
Category theory | $~$f: X \rightarrow Y$~$ |
---|
Category theory | $~$g: Y \rightarrow Z$~$ |
---|
Category theory | $~$X \rightarrow Z$~$ |
---|
Category theory | $~$g \circ f$~$ |
---|
Category theory | $~$gf$~$ |
---|
Category theory | $~$f: X \rightarrow Y$~$ |
---|
Category theory | $~$g: Y \rightarrow Z$~$ |
---|
Category theory | $~$h:Z \rightarrow W$~$ |
---|
Category theory | $~$h(gf) = (hg)f$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$1_X : X \rightarrow X$~$ |
---|
Category theory | $~$f:W \rightarrow X$~$ |
---|
Category theory | $~$g:X \rightarrow Y$~$ |
---|
Category theory | $~$1_X f = f$~$ |
---|
Category theory | $~$g 1_X = g$~$ |
---|
Category theory | $~$(P, \leq)$~$ |
---|
Category theory | $~$x \rightarrow y$~$ |
---|
Category theory | $~$x$~$ |
---|
Category theory | $~$y$~$ |
---|
Category theory | $~$x \leq y$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$Y$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$Y$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$Y$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$Y$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$f: X \rightarrow Y$~$ |
---|
Category theory | $~$f: X \rightarrow Y$~$ |
---|
Category theory | $~$g: Y \rightarrow Z$~$ |
---|
Category theory | $~$X \rightarrow Z$~$ |
---|
Category theory | $~$g \circ f$~$ |
---|
Category theory | $~$gf$~$ |
---|
Category theory | $~$f: X \rightarrow Y$~$ |
---|
Category theory | $~$g: Y \rightarrow Z$~$ |
---|
Category theory | $~$h:Z \rightarrow W$~$ |
---|
Category theory | $~$h(gf) = (hg)f$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$1_X : X \rightarrow X$~$ |
---|
Category theory | $~$f:W \rightarrow X$~$ |
---|
Category theory | $~$g:X \rightarrow Y$~$ |
---|
Category theory | $~$1_X f = f$~$ |
---|
Category theory | $~$g 1_X = g$~$ |
---|
Category theory | $~$x \in X$~$ |
---|
Category theory | $~$f: X \rightarrow Y$~$ |
---|
Category theory | $~$g: Y \rightarrow Z$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$g$~$ |
---|
Category theory | $~$g(f(x))$~$ |
---|
Category theory | $~$(g \circ f)(x)$~$ |
---|
Category theory | $~$\mathbb{A}, \mathbb{B}, \mathbb{C}$~$ |
---|
Category theory | $~$A, B, C, W, X, Y, Z$~$ |
---|
Category theory | $~$e, f, g, h, u, v, w$~$ |
---|
Category theory | $~$a, b, c, x, y, z$~$ |
---|
Category theory | $~$E, F, G, H$~$ |
---|
Category theory | $~$\alpha, \beta, \gamma, \delta$~$ |
---|
Category theory | $~$\kappa$~$ |
---|
Category theory | $~$\lambda$~$ |
---|
Category theory | $~$\mathbb{C}$~$ |
---|
Category theory | $~$T$~$ |
---|
Category theory | $~$\mathbb{C}$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$\mathbb{C}$~$ |
---|
Category theory | $~$f: X \rightarrow T$~$ |
---|
Category theory | $~$f: X \rightarrow T$~$ |
---|
Category theory | $~$g: X \rightarrow T$~$ |
---|
Category theory | $~$f=g$~$ |
---|
Category theory | $~$\{a\}$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$f: X \rightarrow \{a\}$~$ |
---|
Category theory | $~$x$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$a$~$ |
---|
Category theory | $~$T$~$ |
---|
Category theory | $~$T$~$ |
---|
Category theory | $~$T$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$Y$~$ |
---|
Category theory | $~$P$~$ |
---|
Category theory | $~$f: P \rightarrow X$~$ |
---|
Category theory | $~$g: P \rightarrow Y$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$Y$~$ |
---|
Category theory | $~$W$~$ |
---|
Category theory | $~$u: W \rightarrow X$~$ |
---|
Category theory | $~$v:W \rightarrow Y$~$ |
---|
Category theory | $~$h: W \rightarrow P$~$ |
---|
Category theory | $~$fh = u$~$ |
---|
Category theory | $~$gh = v$~$ |
---|
Category theory | $~$T$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$f: X \rightarrow T$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$f: X \leftarrow T$~$ |
---|
Category theory | $~$T'$~$ |
---|
Category theory | $~$T'$~$ |
---|
Category theory | $~$T$~$ |
---|
Category theory | $~$f: T \rightarrow T'$~$ |
---|
Category theory | $~$g: T' \rightarrow T$~$ |
---|
Category theory | $~$gf = 1_T$~$ |
---|
Category theory | $~$fg = 1_{T'}$~$ |
---|
Category theory | $~$f: T \leftarrow T'$~$ |
---|
Category theory | $~$g: T' \leftarrow T$~$ |
---|
Category theory | $~$fg = 1_T$~$ |
---|
Category theory | $~$gf = 1_{T'}$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$g$~$ |
---|
Category theory | $~$\mathbb{A}$~$ |
---|
Category theory | $~$\mathbb{B}$~$ |
---|
Category theory | $~$F$~$ |
---|
Category theory | $~$\mathbb{A}$~$ |
---|
Category theory | $~$\mathbb{B}$~$ |
---|
Category theory | $~$F: \mathbb{A} \rightarrow \mathbb{B}$~$ |
---|
Category theory | $~$F_0:$~$ |
---|
Category theory | $~$\mathbb{A}$~$ |
---|
Category theory | $~$\rightarrow$~$ |
---|
Category theory | $~$\mathbb{B}$~$ |
---|
Category theory | $~$F_1:$~$ |
---|
Category theory | $~$\mathbb{A}$~$ |
---|
Category theory | $~$\rightarrow$~$ |
---|
Category theory | $~$\mathbb{B}$~$ |
---|
Category theory | $~$f: X \rightarrow Y$~$ |
---|
Category theory | $~$F_1(f): F_0(X) \rightarrow F_1(Y)$~$ |
---|
Category theory | $~$F_1(f)$~$ |
---|
Category theory | $~$F_0$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$F_1(f)$~$ |
---|
Category theory | $~$F_0$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$1_X: X \rightarrow X$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$F_1(1_X): F_0(X) \rightarrow F_0(X)$~$ |
---|
Category theory | $~$F_0(X)$~$ |
---|
Category theory | $~$f: X \rightarrow Y$~$ |
---|
Category theory | $~$g: Y \rightarrow Z$~$ |
---|
Category theory | $~$F_1(g) \circ F_1(f): F_0(X) \rightarrow F_0(Z)$~$ |
---|
Category theory | $~$F_1(g \circ f): F_0(X) \rightarrow F_0(Z)$~$ |
---|
Category theory | $~$F_0$~$ |
---|
Category theory | $~$F_1$~$ |
---|
Category theory | $~$F$~$ |
---|
Category theory | $~$F(f): F(X) \rightarrow F(Y)$~$ |
---|
Category theory | $~$f: X \rightarrow Y$~$ |
---|
Category theory | $~$g: Y \rightarrow X$~$ |
---|
Category theory | $~$gf = 1_X$~$ |
---|
Category theory | $~$fg = 1_Y$~$ |
---|
Category theory | $~$W$~$ |
---|
Category theory | $~$g,h: W \rightarrow X$~$ |
---|
Category theory | $~$fg = fh$~$ |
---|
Category theory | $~$g = h$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$X$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$Z$~$ |
---|
Category theory | $~$g,h: X \rightarrow Z$~$ |
---|
Category theory | $~$gf = hf$~$ |
---|
Category theory | $~$g = h$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$Y$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$X = Y$~$ |
---|
Category theory | $~$f: X \rightarrow X$~$ |
---|
Category theory | $~$f$~$ |
---|
Category theory | $~$g: Y \rightarrow X$~$ |
---|
Category theory | $~$gf = 1_X$~$ |
---|
Category theory | $~$g: Y \rightarrow X$~$ |
---|
Category theory | $~$fg = 1_Y$~$ |
---|
Cauchy sequence | $~$X$~$ |
---|
Cauchy sequence | $~$d$~$ |
---|
Cauchy sequence | $~$(x_n)_{n=0}^\infty$~$ |
---|
Cauchy sequence | $~$\varepsilon > 0$~$ |
---|
Cauchy sequence | $~$N$~$ |
---|
Cauchy sequence | $~$m, n > N$~$ |
---|
Cauchy sequence | $~$d(x_m, x_n) < \varepsilon$~$ |
---|
Cauchy sequence | $~$|x_m - x_n|$~$ |
---|
Cauchy's theorem on subgroup existence | $~$G$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$|G|$~$ |
---|
Cauchy's theorem on subgroup existence | $~$G$~$ |
---|
Cauchy's theorem on subgroup existence | $~$G$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $$~$X = \{ (x_1, x_2, \dots, x_p) : x_1 x_2 \dots x_p = e \}$~$$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence | $~$(e, e, \dots, e)$~$ |
---|
Cauchy's theorem on subgroup existence | $~$C_p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence | $$~$(h, (x_1, \dots, x_p)) \mapsto (x_2, x_3, \dots, x_p, x_1)$~$$ |
---|
Cauchy's theorem on subgroup existence | $~$h$~$ |
---|
Cauchy's theorem on subgroup existence | $~$C_p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$h^i$~$ |
---|
Cauchy's theorem on subgroup existence | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence | $~$(x_1, \dots, x_p)$~$ |
---|
Cauchy's theorem on subgroup existence | $~$(x_{i+1}, x_{i+2} , \dots, x_p, x_1, \dots, x_i)$~$ |
---|
Cauchy's theorem on subgroup existence | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence | $~$x_1 x_2 \dots x_p = e$~$ |
---|
Cauchy's theorem on subgroup existence | $$~$x_{i+1} x_{i+2} \dots x_p x_1 \dots x_i = (x_1 \dots x_i)^{-1} (x_1 \dots x_p) (x_1 \dots x_i) = (x_1 \dots x_i)^{-1} e (x_1 \dots x_i) = e$~$$ |
---|
Cauchy's theorem on subgroup existence | $~$0$~$ |
---|
Cauchy's theorem on subgroup existence | $~$(h^i h^j)(x_1, x_2, \dots, x_p) = h^i(h^j(x_1, x_2, \dots, x_p))$~$ |
---|
Cauchy's theorem on subgroup existence | $~$h^{i+j}$~$ |
---|
Cauchy's theorem on subgroup existence | $~$i+j$~$ |
---|
Cauchy's theorem on subgroup existence | $~$j$~$ |
---|
Cauchy's theorem on subgroup existence | $~$i$~$ |
---|
Cauchy's theorem on subgroup existence | $~$i+j$~$ |
---|
Cauchy's theorem on subgroup existence | $~$\bar{x} = (x_1, \dots, x_p) \in X$~$ |
---|
Cauchy's theorem on subgroup existence | $~$\mathrm{Orb}_{C_p}(\bar{x})$~$ |
---|
Cauchy's theorem on subgroup existence | $~$\bar{x}$~$ |
---|
Cauchy's theorem on subgroup existence | $~$|C_p| = p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$\bar{x} \in X$~$ |
---|
Cauchy's theorem on subgroup existence | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence | $~$|G|^{p-1}$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$x_p = (x_1 \dots x_{p-1})^{-1}$~$ |
---|
Cauchy's theorem on subgroup existence | $~$C_p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$|G|$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$|G|^{p-1} = |X|$~$ |
---|
Cauchy's theorem on subgroup existence | $~$|\mathrm{Orb}_{C_p}((e, e, \dots, e))| = 1$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p-1$~$ |
---|
Cauchy's theorem on subgroup existence | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p-1$~$ |
---|
Cauchy's theorem on subgroup existence | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$p \mid |X|$~$ |
---|
Cauchy's theorem on subgroup existence | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence | $~$\{ \bar{x} \}$~$ |
---|
Cauchy's theorem on subgroup existence | $~$\bar{x} = (x_1, \dots, x_p)$~$ |
---|
Cauchy's theorem on subgroup existence | $~$C_p$~$ |
---|
Cauchy's theorem on subgroup existence | $~$\bar{x}$~$ |
---|
Cauchy's theorem on subgroup existence | $~$\bar{x}$~$ |
---|
Cauchy's theorem on subgroup existence | $~$x_i$~$ |
---|
Cauchy's theorem on subgroup existence | $~$(x, x, \dots, x) \in X$~$ |
---|
Cauchy's theorem on subgroup existence | $~$x^p = e$~$ |
---|
Cauchy's theorem on subgroup existence | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$G$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$G$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$G$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$G$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$x \not = e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$x^p = e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$x^i$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$i < p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p=5$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$\{ a, b, c, d, e\}$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(e, e, a, b, a)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(e,a,b,a,e)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$x \not = e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(x, x, \dots, x)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$x$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(x, x, \dots, x)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(e, e, a, b, a)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$eeaba$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$aba = e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(a,b,c,b,b)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$x$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$x^p = e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$abcbb = e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1}$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p-1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p-1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p-1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p=5$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(a, a, b, e, \cdot)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$b^{-1} a^{-2}$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$aabe(a^{-1} a^{-2}) = e$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1}$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$|G|$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$|X|$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(e,e,\dots,e)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(a_1, a_2, \dots, a_p)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(a_2, a_3, \dots, a_p, a_1)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(a, a, \dots, a)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$A$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $$~$(a_1, a_2, \dots, a_p), (a_2, a_3, \dots, a_p, a_1), \dots, (a_{p-1}, a_p, a_1, \dots, a_{p-2}), (a_p, a_1, a_2, \dots, a_{p-1})$~$$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p=8$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(1,1,2,2,1,1,2,2)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$n$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$n$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$n$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$n=1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$T$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$n=p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$X$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1}$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(e,e,\dots,e)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1}$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1} - 1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p=2$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1} - 1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$|G|^{p-1}$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$1$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$(a,a,\dots,a)$~$ |
---|
Cauchy's theorem on subgroup existence: intuitive version | $~$p$~$ |
---|
Causal decision theories | $~$\mathcal U$~$ |
---|
Causal decision theories | $~$\mathcal O$~$ |
---|
Causal decision theories | $~$a_x$~$ |
---|
Causal decision theories | $$~$\mathbb E[\mathcal U|a_x] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(a_x \ \square \!\! \rightarrow o_i)$~$$ |
---|
Causal decision theories | $~$operatorname{do}()$~$ |
---|
Causal decision theories | $$~$\mathbb E[\mathcal U| \operatorname{do}(a_x)] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(o_i | \operatorname{do}(a_x))$~$$ |
---|
Causal decision theories | $~$a_0$~$ |
---|
Causal decision theories | $~$o_i$~$ |
---|
Causal decision theories | $~$\mathbb P(o_i|a_0).$~$ |
---|
Causal decision theories | $~$a_0,$~$ |
---|
Causal decision theories | $~$a_0.$~$ |
---|
Causal decision theories | $~$O$~$ |
---|
Causal decision theories | $~$\neg O$~$ |
---|
Causal decision theories | $~$O$~$ |
---|
Causal decision theories | $~$K$~$ |
---|
Causal decision theories | $~$O$~$ |
---|
Causal decision theories | $~$\mathbb P(K|\neg O),$~$ |
---|
Causal decision theories | $~$\mathbb P(\neg O \ \square \!\! \rightarrow K).$~$ |
---|
Causal decision theories | $~$\mathbb P(\neg O \ \square \!\! \rightarrow K),$~$ |
---|
Causal decision theories | $~$\mathbb P(K|\neg O).$~$ |
---|
Causal decision theories | $~$\mathbb P(\bullet \ || \ \bullet)$~$ |
---|
Causal decision theories | $~$X_1$~$ |
---|
Causal decision theories | $~$X_2$~$ |
---|
Causal decision theories | $~$X_3$~$ |
---|
Causal decision theories | $~$X_4$~$ |
---|
Causal decision theories | $~$X_5$~$ |
---|
Causal decision theories | $~$\mathbb P(X_i | \mathbf{pa}_i)$~$ |
---|
Causal decision theories | $~$X_i$~$ |
---|
Causal decision theories | $~$x_i$~$ |
---|
Causal decision theories | $~$\mathbf {pa}_i$~$ |
---|
Causal decision theories | $~$x_i$~$ |
---|
Causal decision theories | $~$\mathbf x$~$ |
---|
Causal decision theories | $$~$\mathbb P(\mathbf x) = \prod_i \mathbb P(x_i | \mathbf{pa}_i)$~$$ |
---|
Causal decision theories | $~$\mathbb P(\mathbf x | \operatorname{do}(X_j=x_j))$~$ |
---|
Causal decision theories | $$~$\mathbb P(\mathbf x | \operatorname{do}(X_j=x_j)) = \prod_{i \neq j} \mathbb P(x_i | \mathbf{pa}_i)$~$$ |
---|
Causal decision theories | $~$\mathbf x$~$ |
---|
Causal decision theories | $~$x_j$~$ |
---|
Causal decision theories | $~$\operatorname{do}$~$ |
---|
Causal decision theories | $~$X_j$~$ |
---|
Causal decision theories | $~$0$~$ |
---|
Causal decision theories | $~$\operatorname{do}(X_j=x_j)$~$ |
---|
Causal decision theories | $~$X_j$~$ |
---|
Causal decision theories | $~$\mathbf{pa}_j,$~$ |
---|
Causal decision theories | $~$X_j = x_j$~$ |
---|
Causal decision theories | $~$\operatorname{do}(X_j=x_j)$~$ |
---|
Causal decision theories | $~$X_k$~$ |
---|
Causal decision theories | $~$X_j$~$ |
---|
Causal decision theories | $$~$\mathbb E[\mathcal U| \operatorname{do}(a_x)] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(o_i | \operatorname{do}(a_x))$~$$ |
---|
Causal decision theories | $~$\operatorname{do}()$~$ |
---|
Causal decision theories | $~$W, X, Y, Z$~$ |
---|
Causal decision theories | $$~$\begin{array}{r|c|c}
& \text{One-boxing predicted} & \text{Two-boxing predicted} \\
\hline
\text{W: Take both boxes, no fee:} & \$500,500 & \$500 \\ \hline
\text{X: Take only Box B, no fee:} & \$500,000 & \$0 \\ \hline
\text{Y: Take both boxes, pay fee:} & \$1,000,100 & \$100 \\ \hline
\text{Z: Take only Box B, pay fee:} & \$999,100 & -\$900
\end{array}$~$$ |
---|
Causal decision theories | $~$\operatorname{do}()$~$ |
---|
Cayley's Theorem on symmetric groups | $~$G$~$ |
---|
Cayley's Theorem on symmetric groups | $~$\mathrm{Sym}(G)$~$ |
---|
Cayley's Theorem on symmetric groups | $~$G$~$ |
---|
Cayley's Theorem on symmetric groups | $~$G$~$ |
---|
Cayley's Theorem on symmetric groups | $~$G$~$ |
---|
Cayley's Theorem on symmetric groups | $~$\mathrm{Sym}(G)$~$ |
---|
Cayley's Theorem on symmetric groups | $~$G$~$ |
---|
Cayley's Theorem on symmetric groups | $~$G$~$ |
---|
Cayley's Theorem on symmetric groups | $~$G \times G \to G$~$ |
---|
Cayley's Theorem on symmetric groups | $~$(g, h) \mapsto gh$~$ |
---|
Cayley's Theorem on symmetric groups | $~$\Phi: G \to \mathrm{Sym}(G)$~$ |
---|
Cayley's Theorem on symmetric groups | $~$g \mapsto (h \mapsto gh)$~$ |
---|
Cayley's Theorem on symmetric groups | $~$g \in \mathrm{ker}(\Phi)$~$ |
---|
Cayley's Theorem on symmetric groups | $~$\Phi$~$ |
---|
Cayley's Theorem on symmetric groups | $~$(h \mapsto gh)$~$ |
---|
Cayley's Theorem on symmetric groups | $~$gh = h$~$ |
---|
Cayley's Theorem on symmetric groups | $~$h$~$ |
---|
Cayley's Theorem on symmetric groups | $~$g$~$ |
---|
Cayley's Theorem on symmetric groups | $~$G$~$ |
---|
Cayley's Theorem on symmetric groups | $~$G$~$ |
---|
Cayley's Theorem on symmetric groups | $~$\mathrm{Sym}(G)$~$ |
---|
Ceiling | $~$x,$~$ |
---|
Ceiling | $~$\lceil x \rceil$~$ |
---|
Ceiling | $~$\operatorname{ceil}(x),$~$ |
---|
Ceiling | $~$n \ge x.$~$ |
---|
Ceiling | $~$\lceil 3.72 \rceil = 4, \lceil 4 \rceil = 4,$~$ |
---|
Ceiling | $~$\lceil -3.72 \rceil = -3.$~$ |
---|
Ceiling | $~$\mathbb R \to \mathbb Z.$~$ |
---|
Church encoding | $~$\lambda$~$ |
---|
Church encoding | $~$\lambda$~$ |
---|
Church encoding | $~$0,1,2,\dots$~$ |
---|
Church encoding | $~$\lambda$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$\lambda$~$ |
---|
Church encoding | $~$\lambda$~$ |
---|
Church encoding | $~$\lambda x.M$~$ |
---|
Church encoding | $~$M$~$ |
---|
Church encoding | $~$\lambda$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$M$~$ |
---|
Church encoding | $~$x\ (x\ (x\ x))$~$ |
---|
Church encoding | $~$((x\ x)\ x)\ x$~$ |
---|
Church encoding | $~$0$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$3$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$\lambda$~$ |
---|
Church encoding | $~$\lambda f.\lambda x.M$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$0$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$0=\lambda f.\lambda x.x$~$ |
---|
Church encoding | $~$1$~$ |
---|
Church encoding | $~$1=\lambda f.\lambda x.f\ x$~$ |
---|
Church encoding | $~$2=\lambda f.\lambda x.f\ (f\ x)$~$ |
---|
Church encoding | $~$3=\lambda f.\lambda x.f\ (f\ (f\ x))$~$ |
---|
Church encoding | $~$4=\lambda f.\lambda x.f\ (f\ (f\ (f\ x)))$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$n=\lambda f.\lambda x.f^n(x)$~$ |
---|
Church encoding | $~$\lambda$~$ |
---|
Church encoding | $~$S(n)=n+1$~$ |
---|
Church encoding | $~$S$~$ |
---|
Church encoding | $~$\lambda n$~$ |
---|
Church encoding | $~$\lambda f.\lambda x$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$n+1$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$n+1$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $$~$S=\lambda n.\lambda f.\lambda x.f\ (n\ f\ x).$~$$ |
---|
Church encoding | $~$(n\ f\ x)$~$ |
---|
Church encoding | $~$f^n(x)$~$ |
---|
Church encoding | $~$f\ (n\ f\ x)$~$ |
---|
Church encoding | $~$f(f^n(x))=f^{n+1}(x)$~$ |
---|
Church encoding | $~$f\ x$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $$~$S^\prime=\lambda n.\lambda y.\lambda x.n\ f\ (f\ x).$~$$ |
---|
Church encoding | $~$S$~$ |
---|
Church encoding | $~$S^\prime$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$\lambda a.\lambda b.a$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$S$~$ |
---|
Church encoding | $~$S\ 3=4$~$ |
---|
Church encoding | $~$1$~$ |
---|
Church encoding | $~$m$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$m$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$m$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$m+n$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$m+n$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$m$~$ |
---|
Church encoding | $~$\lambda$~$ |
---|
Church encoding | $$~$+=\lambda m.\lambda n.\lambda f.\lambda x.m\ f\ (n\ f\ x)$~$$ |
---|
Church encoding | $~$n\ f\ x$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$m\ f$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$m$~$ |
---|
Church encoding | $~$m$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$2+3=5$~$ |
---|
Church encoding | $~$2+3$~$ |
---|
Church encoding | $~$+\ 2\ 3$~$ |
---|
Church encoding | $~$\lambda$~$ |
---|
Church encoding | $~$m+n$~$ |
---|
Church encoding | $~$+\ m\ n$~$ |
---|
Church encoding | $~$m$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$x$~$ |
---|
Church encoding | $~$m\times n$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$m$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$m\times n$~$ |
---|
Church encoding | $~$(f^n)^m(x)=f^{m\times n}(x)$~$ |
---|
Church encoding | $~$f$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$\lambda x.n\ f\ x$~$ |
---|
Church encoding | $~$\eta$~$ |
---|
Church encoding | $~$n\ f$~$ |
---|
Church encoding | $~$n\ f$~$ |
---|
Church encoding | $~$m$~$ |
---|
Church encoding | $$~$\times=\lambda m.\lambda n.\lambda f.\lambda x.m\ (n\ f) x$~$$ |
---|
Church encoding | $~$\eta$~$ |
---|
Church encoding | $$~$\times=\lambda m.\lambda n.\lambda f.m\ (n\ f).$~$$ |
---|
Church encoding | $~$m$~$ |
---|
Church encoding | $~$n$~$ |
---|
Church encoding | $~$\times\ 2\ 3=6$~$ |
---|
Church-Turing thesis: Evidence for the Church-Turing thesis | $~$f$~$ |
---|
Church-Turing thesis: Evidence for the Church-Turing thesis | $~$x$~$ |
---|
Church-Turing thesis: Evidence for the Church-Turing thesis | $~$f(x)$~$ |
---|
Church-Turing thesis: Evidence for the Church-Turing thesis | $~$1/2$~$ |
---|
Church-Turing thesis: Evidence for the Church-Turing thesis | $~$f$~$ |
---|
Closure | $~$S$~$ |
---|
Closure | $~$f$~$ |
---|
Closure | $~$f$~$ |
---|
Closure | $~$S$~$ |
---|
Closure | $~$S$~$ |
---|
Closure | $~$f$~$ |
---|
Closure | $~$S$~$ |
---|
Closure | $~$f$~$ |
---|
Closure | $~$x, y, z \in S$~$ |
---|
Closure | $~$f(x, y, z) \in S$~$ |
---|
Closure | $~$\mathbb Z$~$ |
---|
Closure | $~$\mathbb Z_5 = \{0, 1, 2, 3, 4, 5\}$~$ |
---|
Closure | $~$1 + 5$~$ |
---|
Closure | $~$\mathbb Z_5$~$ |
---|
Codomain (of a function) | $~$\operatorname{cod}(f)$~$ |
---|
Codomain (of a function) | $~$f : X \to Y$~$ |
---|
Codomain (of a function) | $~$Y$~$ |
---|
Codomain (of a function) | $~$+$~$ |
---|
Codomain (of a function) | $~$Y$~$ |
---|
Codomain (of a function) | $~$f$~$ |
---|
Codomain (of a function) | $~$Y$~$ |
---|
Codomain (of a function) | $~$\operatorname{square} : \mathbb R \to \mathbb R$~$ |
---|
Codomain (of a function) | $~$+$~$ |
---|
Codomain (of a function) | $~$\mathbb N$~$ |
---|
Codomain (of a function) | $~$\mathbb Z$~$ |
---|
Codomain vs image | $~$X$~$ |
---|
Codomain vs image | $~$Y$~$ |
---|
Codomain vs image | $~$Y$~$ |
---|
Codomain vs image | $~$f : X \to Y$~$ |
---|
Codomain vs image | $~$X$~$ |
---|
Codomain vs image | $~$Y$~$ |
---|
Codomain vs image | $~$Y$~$ |
---|
Codomain vs image | $~$\mathbb R$~$ |
---|
Codomain vs image | $~$f$~$ |
---|
Codomain vs image | $~$X$~$ |
---|
Codomain vs image | $~$I$~$ |
---|
Codomain vs image | $~$Y$~$ |
---|
Codomain vs image | $~$I$~$ |
---|
Codomain vs image | $~$\mathbb N$~$ |
---|
Codomain vs image | $~$2^{65536} − 3$~$ |
---|
Codomain vs image | $~$\{0, 1\},$~$ |
---|
Codomain vs image | $~$\{0, 1\}$~$ |
---|
Codomain vs image | $~$\{0, 1\}$~$ |
---|
Coherent decisions imply consistent utilities | $~$\mathbb P(X),$~$ |
---|
Coherent decisions imply consistent utilities | $~$\mathbb P(\neg X),$~$ |
---|
Coherent decisions imply consistent utilities | $~$\mathbb P(X) + \mathbb P(\neg X) = 1.$~$ |
---|
Coherent decisions imply consistent utilities | $~$>_P$~$ |
---|
Coherent decisions imply consistent utilities | $~$X >_P Y$~$ |
---|
Coherent decisions imply consistent utilities | $$~$\text{onions} >_P \text{pineapple} >_P \text{mushrooms} >_P \text{onions}$~$$ |
---|
Coherent decisions imply consistent utilities | $~$>$~$ |
---|
Coherent decisions imply consistent utilities | $~$>_P$~$ |
---|
Coherent decisions imply consistent utilities | $~$x > y, y > z \implies x > z$~$ |
---|
Coherent decisions imply consistent utilities | $~$>_P$~$ |
---|
Coherent decisions imply consistent utilities | $~$x, y, z$~$ |
---|
Coherent decisions imply consistent utilities | $~$x > y > z > x.$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$0.01$~$ |
---|
Coherent decisions imply consistent utilities | $~$\text{mushroom} >_P \text{pineapple} >_P \text{onion}$~$ |
---|
Coherent decisions imply consistent utilities | $~$>_P$~$ |
---|
Coherent decisions imply consistent utilities | $~$\text{onions} >_P \text{pineapple}.$~$ |
---|
Coherent decisions imply consistent utilities | $~$0.5$~$ |
---|
Coherent decisions imply consistent utilities | $~$0.5$~$ |
---|
Coherent decisions imply consistent utilities | $$~$\mathbb P(heads) \cdot U(\text{1 orange}) + \mathbb P(tails) \cdot U(\text{3 plums}) \\
= 0.50 \cdot €2 + 0.50 \cdot €1.5 = €1.75$~$$ |
---|
Coherent decisions imply consistent utilities | $~$1 \cdot U(\text{1 apple}) = €1.$~$ |
---|
Coherent decisions imply consistent utilities | $~$0.5$~$ |
---|
Coherent decisions imply consistent utilities | $~$-0.2$~$ |
---|
Coherent decisions imply consistent utilities | $~$3$~$ |
---|
Coherent decisions imply consistent utilities | $~$0$~$ |
---|
Coherent decisions imply consistent utilities | $~$1$~$ |
---|
Coherent decisions imply consistent utilities | $~$-0.3$~$ |
---|
Coherent decisions imply consistent utilities | $~$27.$~$ |
---|
Coherent decisions imply consistent utilities | $~$0.6$~$ |
---|
Coherent decisions imply consistent utilities | $~$0.7$~$ |
---|
Coherent decisions imply consistent utilities | $~$1.3$~$ |
---|
Coherent decisions imply consistent utilities | $~$1!$~$ |
---|
Coherent decisions imply consistent utilities | $~$1,$~$ |
---|
Coherent decisions imply consistent utilities | $$~$\mathbb P(\text{heads}) \cdot U(\text{0.8 apples}) + \mathbb P(\text{tails}) \cdot U(\text{0.8 apples}) \\
= 0.6 \cdot €0.8 + 0.7 \cdot €0.8 = €1.04.$~$$ |
---|
Coherent decisions imply consistent utilities | $~$X.$~$ |
---|
Coherent decisions imply consistent utilities | $~$X$~$ |
---|
Coherent decisions imply consistent utilities | $~$x$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$x$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$1$~$ |
---|
Coherent decisions imply consistent utilities | $~$X$~$ |
---|
Coherent decisions imply consistent utilities | $~$X$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$x.$~$ |
---|
Coherent decisions imply consistent utilities | $~$N \cdot \$x$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$N$~$ |
---|
Coherent decisions imply consistent utilities | $~$X$~$ |
---|
Coherent decisions imply consistent utilities | $~$X$~$ |
---|
Coherent decisions imply consistent utilities | $~$Y$~$ |
---|
Coherent decisions imply consistent utilities | $~$X$~$ |
---|
Coherent decisions imply consistent utilities | $~$Y$~$ |
---|
Coherent decisions imply consistent utilities | $~$X$~$ |
---|
Coherent decisions imply consistent utilities | $~$Y$~$ |
---|
Coherent decisions imply consistent utilities | $~$x$~$ |
---|
Coherent decisions imply consistent utilities | $~$y$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$1.$~$ |
---|
Coherent decisions imply consistent utilities | $~$x + y < \$1,$~$ |
---|
Coherent decisions imply consistent utilities | $~$X$~$ |
---|
Coherent decisions imply consistent utilities | $~$Y$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$1$~$ |
---|
Coherent decisions imply consistent utilities | $~$x + y.$~$ |
---|
Coherent decisions imply consistent utilities | $~$x + y > \$1,$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$1$~$ |
---|
Coherent decisions imply consistent utilities | $~$x + y.$~$ |
---|
Coherent decisions imply consistent utilities | $~$x + y - \$1 > \$0.$~$ |
---|
Coherent decisions imply consistent utilities | $~$X$~$ |
---|
Coherent decisions imply consistent utilities | $~$X$~$ |
---|
Coherent decisions imply consistent utilities | $~$Q$~$ |
---|
Coherent decisions imply consistent utilities | $~$R$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$x$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$1$~$ |
---|
Coherent decisions imply consistent utilities | $~$Q$~$ |
---|
Coherent decisions imply consistent utilities | $~$Q$~$ |
---|
Coherent decisions imply consistent utilities | $~$Q$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$y,$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$1$~$ |
---|
Coherent decisions imply consistent utilities | $~$R$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$z$~$ |
---|
Coherent decisions imply consistent utilities | $~$\$1$~$ |
---|
Coherent decisions imply consistent utilities | $~$Q$~$ |
---|
Coherent decisions imply consistent utilities | $~$R$~$ |
---|
Coherent decisions imply consistent utilities | $~$Q$~$ |
---|
Coherent decisions imply consistent utilities | $~$R$~$ |
---|
Coherent decisions imply consistent utilities | $~$Q$~$ |
---|
Coherent decisions imply consistent utilities | $~$R$~$ |
---|
Coherent decisions imply consistent utilities | $~$Q$~$ |
---|
Coherent decisions imply consistent utilities | $$~$\mathbb P(Q \wedge R) = \mathbb P(Q) \cdot \mathbb P(R \mid Q)$~$$ |
---|
Coherent decisions imply consistent utilities | $~$z = x \cdot y.$~$ |
---|
Coherent decisions imply consistent utilities | $~$\mathbb P(Q)$~$ |
---|
Coherent decisions imply consistent utilities | $~$\mathbb P(R \mid Q)$~$ |
---|
Coherent decisions imply consistent utilities | $~$\mathbb P(Q \wedge R),$~$ |
---|
Coherent decisions imply consistent utilities | $~$Q$~$ |
---|
Coherent decisions imply consistent utilities | $~$Q$~$ |
---|
Coherent decisions imply consistent utilities | $~$R$~$ |
---|
Coherent decisions imply consistent utilities | $~$Q$~$ |
---|
Coherent decisions imply consistent utilities | $~$R$~$ |
---|
Coherent decisions imply consistent utilities | $~$A, B, C$~$ |
---|
Coherent decisions imply consistent utilities | $~$X, Y, Z$~$ |
---|
Coherent decisions imply consistent utilities | $~$x, y, z$~$ |
---|
Coherent decisions imply consistent utilities | $$~$\begin{array}{rrrl}
-Ax & + 0 & - Cz & \geqq 0 \\
A(1-x) & - By & - Cz & \geqq 0 \\
A(1-x) & + B(1-y) & + C(1-z) & \geqq 0
\end{array}$~$$ |
---|
Coherent decisions imply consistent utilities | $~$x, y, z \in (0..1)$~$ |
---|
Coherent decisions imply consistent utilities | $~$z = x * y.$~$ |
---|
Coherent decisions imply consistent utilities | $$~$\begin{array}{rcl}
U(\text{gain \$1 million}) & > & 0.9 \cdot U(\text{gain \$5 million}) + 0.1 \cdot U(\text{gain \$0}) \\
0.5 \cdot U(\text{gain \$0}) + 0.5 \cdot U(\text{gain \$1 million}) & > & 0.45 \cdot U(\text{gain \$5 million}) + 0.55 \cdot U(\text{gain \$0})
\end{array}$~$$ |
---|
Coherent decisions imply consistent utilities | $~$L$~$ |
---|
Coherent decisions imply consistent utilities | $~$M$~$ |
---|
Coherent decisions imply consistent utilities | $~$L > M$~$ |
---|
Coherent decisions imply consistent utilities | $~$p > 0$~$ |
---|
Coherent decisions imply consistent utilities | $~$N$~$ |
---|
Coherent decisions imply consistent utilities | $~$p \cdot L + (1-p)\cdot N > p \cdot M + (1-p) \cdot N.$~$ |
---|
Coherent decisions imply consistent utilities | $~$N,$~$ |
---|
Coherent decisions imply consistent utilities | $~$L$~$ |
---|
Coherent decisions imply consistent utilities | $~$M,$~$ |
---|
Coherent decisions imply consistent utilities | $~$L$~$ |
---|
Coherent decisions imply consistent utilities | $~$M$~$ |
---|
Coherent decisions imply consistent utilities | $~$L$~$ |
---|
Coherent decisions imply consistent utilities | $~$M$~$ |
---|
Coherent decisions imply consistent utilities | $~$L$~$ |
---|
Coherent decisions imply consistent utilities | $~$M,$~$ |
---|
Colon-to notation | $~$f : X \to Y$~$ |
---|
Colon-to notation | $~$\to$~$ |
---|
Colon-to notation | $~$f$~$ |
---|
Colon-to notation | $~$X$~$ |
---|
Colon-to notation | $~$Y$~$ |
---|
Colon-to notation | $~$f$~$ |
---|
Colon-to notation | $~$X$~$ |
---|
Colon-to notation | $~$Y$~$ |
---|
Colon-to notation | $~$f$~$ |
---|
Colon-to notation | $~$f : \mathbb{R} \to \mathbb{R}$~$ |
---|
Colon-to notation | $~$f$~$ |
---|
Colon-to notation | $~$x \mapsto x^2$~$ |
---|
Colon-to notation | $~$f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$~$ |
---|
Colon-to notation | $~$f$~$ |
---|
Colon-to notation | $~$\times$~$ |
---|
Combining vectors | $~$\mathbf u$~$ |
---|
Combining vectors | $~$\mathbf v$~$ |
---|
Combining vectors | $~$\mathbf w$~$ |
---|
Combining vectors | $~$\mathbf s$~$ |
---|
Combining vectors | $~$\mathbf u$~$ |
---|
Combining vectors | $~$\mathbf v$~$ |
---|
Combining vectors | $~$\mathbf w$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf d$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf d$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf v$~$ |
---|
Combining vectors | $~$\mathbf v$~$ |
---|
Combining vectors | $~$\mathbf v$~$ |
---|
Combining vectors | $~$\mathbf v$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf z$~$ |
---|
Combining vectors | $~$\mathbf r$~$ |
---|
Combining vectors | $~$\mathbf s$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf v$~$ |
---|
Combining vectors | $~$\mathbf v = 3\mathbf {x} + 4 \mathbf {y}$~$ |
---|
Combining vectors | $~$v$~$ |
---|
Combining vectors | $~$v$~$ |
---|
Combining vectors | $~$\mathbf x$~$ |
---|
Combining vectors | $~$\mathbf v$~$ |
---|
Combining vectors | $~$3$~$ |
---|
Combining vectors | $~$\mathbf y$~$ |
---|
Combining vectors | $~$\mathbf v$~$ |
---|
Combining vectors | $~$-1$~$ |
---|
Combining vectors | $~$\mathbf v$~$ |
---|
Combining vectors | $~$(3,4)$~$ |
---|
Combining vectors | $~$O$~$ |
---|
Combining vectors | $~$p = O + 2\mathbf x + 3\mathbf y$~$ |
---|
Combining vectors | $~$q = O - 3\mathbf x + \mathbf y$~$ |
---|
Combining vectors | $~$p = (2, 3)$~$ |
---|
Combining vectors | $~$q = (-3,1)$~$ |
---|
Combining vectors | $~$\mathbf s, \mathbf t$~$ |
---|
Combining vectors | $~$p$~$ |
---|
Combining vectors | $~$(2,\frac{1}{2})$~$ |
---|
Combining vectors | $~$q = (-3,2)$~$ |
---|
Communication: magician example | $~$\log_2(2 \times 6 \times 6) \approx 6.17$~$ |
---|
Communication: magician example | $~$A♠$~$ |
---|
Communication: magician example | $~$K♡.$~$ |
---|
Communication: magician example | $~$2 \cdot 6 \cdot 6 = 72$~$ |
---|
Commutative operation | $~$f$~$ |
---|
Commutative operation | $~$X$~$ |
---|
Commutative operation | $~$+$~$ |
---|
Commutative operation | $~$3 + 4 = 4 + 3.$~$ |
---|
Commutativity: Examples | $~$x+y = y+x$~$ |
---|
Commutativity: Examples | $~$x$~$ |
---|
Commutativity: Examples | $~$y,$~$ |
---|
Commutativity: Examples | $~$x \times y = y \times x$~$ |
---|
Commutativity: Examples | $~$x$~$ |
---|
Commutativity: Examples | $~$y,$~$ |
---|
Commutativity: Examples | $~$x \times y$~$ |
---|
Commutativity: Examples | $~$x$~$ |
---|
Commutativity: Examples | $~$y$~$ |
---|
Commutativity: Examples | $~$x$~$ |
---|
Commutativity: Examples | $~$y$~$ |
---|
Commutativity: Examples | $~$y$~$ |
---|
Commutativity: Examples | $~$x.$~$ |
---|
Commutativity: Examples | $~$x$~$ |
---|
Commutativity: Examples | $~$y$~$ |
---|
Commutativity: Examples | $~$x \times y$~$ |
---|
Commutativity: Examples | $~$x$~$ |
---|
Commutativity: Examples | $~$y$~$ |
---|
Commutativity: Examples | $~$x$~$ |
---|
Commutativity: Examples | $~$y$~$ |
---|
Commutativity: Examples | $~$y$~$ |
---|
Commutativity: Examples | $~$x$~$ |
---|
Commutativity: Examples | $~$r$~$ |
---|
Commutativity: Examples | $~$p$~$ |
---|
Commutativity: Examples | $~$s$~$ |
---|
Commutativity: Examples | $~$?$~$ |
---|
Commutativity: Examples | $~$r ? p = p,$~$ |
---|
Commutativity: Examples | $~$r ? s = r,$~$ |
---|
Commutativity: Examples | $~$p ? s = s,$~$ |
---|
Commutativity: Examples | $~$r?p=p?r$~$ |
---|
Commutativity: Examples | $~$(r?p)?s=s$~$ |
---|
Commutativity: Examples | $~$r?(p?s)=r.$~$ |
---|
Commutativity: Examples | $~$x / y$~$ |
---|
Commutativity: Examples | $~$y / x$~$ |
---|
Commutativity: Examples | $~$x$~$ |
---|
Commutativity: Examples | $~$y$~$ |
---|
Commutativity: Examples | $~$2 \times 3$~$ |
---|
Commutativity: Examples | $~$3 \times 5$~$ |
---|
Commutativity: Examples | $~$2 \times 3$~$ |
---|
Commutativity: Intuition | $~$f(x, y)$~$ |
---|
Commutativity: Intuition | $~$f$~$ |
---|
Commutativity: Intuition | $~$f(x, y)$~$ |
---|
Commutativity: Intuition | $~$f$~$ |
---|
Commutativity: Intuition | $~$x$~$ |
---|
Commutativity: Intuition | $~$y$~$ |
---|
Commutativity: Intuition | $~$\{b, d, e, l, u, r\}$~$ |
---|
Commutativity: Intuition | $~$X^2$~$ |
---|
Commutativity: Intuition | $~$X;$~$ |
---|
Commutativity: Intuition | $~$X^2$~$ |
---|
Commutativity: Intuition | $~$(x_1, x_2).$~$ |
---|
Commutativity: Intuition | $~$X^2$~$ |
---|
Commutativity: Intuition | $~$|X|$~$ |
---|
Commutativity: Intuition | $~$|X|$~$ |
---|
Commutativity: Intuition | $~$f : X^2 \to Y$~$ |
---|
Commutativity: Intuition | $~$X^2$~$ |
---|
Commutativity: Intuition | $~$f(x_1, x_2)$~$ |
---|
Commutativity: Intuition | $~$(x_1, x_2);$~$ |
---|
Commutativity: Intuition | $~$X^2$~$ |
---|
Commutativity: Intuition | $~$f$~$ |
---|
Commutativity: Intuition | $~$\operatorname{swap} : X^2 \to X^2$~$ |
---|
Commutativity: Intuition | $~$(x_1, x_2)$~$ |
---|
Commutativity: Intuition | $~$(x_2, x_1),$~$ |
---|
Commutativity: Intuition | $~$\operatorname{swap}(X^2)$~$ |
---|
Commutativity: Intuition | $~$\operatorname{swap}$~$ |
---|
Commutativity: Intuition | $~$X^2$~$ |
---|
Commutativity: Intuition | $~$X^2$~$ |
---|
Commutativity: Intuition | $~$f$~$ |
---|
Commutativity: Intuition | $~$\operatorname{swap}(X^2).$~$ |
---|
Commutativity: Intuition | $~$f$~$ |
---|
Commutativity: Intuition | $~$X^2$~$ |
---|
Commutativity: Intuition | $~$f$~$ |
---|
Commutativity: Intuition | $~$\operatorname{swap}(X^2),$~$ |
---|
Commutativity: Intuition | $~$f$~$ |
---|
Commutativity: Intuition | $~$X^2$~$ |
---|
Commutativity: Intuition | $~$\operatorname{swap}$~$ |
---|
Commutativity: Intuition | $~$f(x_1, x_2)=f(x_2, x_1)$~$ |
---|
Commutativity: Intuition | $~$(x_1, x_2)$~$ |
---|
Complete lattice | $~$L$~$ |
---|
Complete lattice | $~$\bigvee \emptyset$~$ |
---|
Complete lattice | $~$\bigvee L$~$ |
---|
Complete lattice | $~$\bigvee \emptyset$~$ |
---|
Complete lattice | $~$L$~$ |
---|
Complete lattice | $~$\bigvee L$~$ |
---|
Complete lattice | $~$L$~$ |
---|
Complete lattice | $~$P$~$ |
---|
Complete lattice | $~$A \subseteq P$~$ |
---|
Complete lattice | $~$A^L$~$ |
---|
Complete lattice | $~$A$~$ |
---|
Complete lattice | $~$\{ p \in P \mid \forall a \in A. p \leq a \}$~$ |
---|
Complete lattice | $~$P$~$ |
---|
Complete lattice | $~$\bigvee A^L$~$ |
---|
Complete lattice | $~$P$~$ |
---|
Complete lattice | $~$\bigvee A^L$~$ |
---|
Complete lattice | $~$A$~$ |
---|
Complete lattice | $~$\bigvee A^L$~$ |
---|
Complete lattice | $~$A$~$ |
---|
Complete lattice | $~$a \in A$~$ |
---|
Complete lattice | $~$A^L$~$ |
---|
Complete lattice | $~$a$~$ |
---|
Complete lattice | $~$A^L$~$ |
---|
Complete lattice | $~$\bigvee A^L$~$ |
---|
Complete lattice | $~$A^L$~$ |
---|
Complete lattice | $~$\bigvee A^L \leq a$~$ |
---|
Complete lattice | $~$\bigvee A^L$~$ |
---|
Complete lattice | $~$A$~$ |
---|
Complete lattice | $~$\bigvee A^L$~$ |
---|
Complete lattice | $~$A$~$ |
---|
Complete lattice | $~$p \in P$~$ |
---|
Complete lattice | $~$A$~$ |
---|
Complete lattice | $~$p \in A^L$~$ |
---|
Complete lattice | $~$\bigvee A^L$~$ |
---|
Complete lattice | $~$A^L$~$ |
---|
Complete lattice | $~$p \leq \bigvee A^L$~$ |
---|
Complete lattice | $~$L$~$ |
---|
Complete lattice | $~$\bigvee \emptyset$~$ |
---|
Complete lattice | $~$\bigvee L$~$ |
---|
Complete lattice | $~$L$~$ |
---|
Complete lattice | $~$L$~$ |
---|
Complete lattice | $~$L$~$ |
---|
Complete lattice | $~$L$~$ |
---|
Complete lattice | $~$X$~$ |
---|
Complete lattice | $~$\langle \mathcal P(X), \subseteq \rangle$~$ |
---|
Complete lattice | $~$X$~$ |
---|
Complete lattice | $~$Y \subset \mathcal P(X)$~$ |
---|
Complete lattice | $~$\bigvee Y = \bigcup Y$~$ |
---|
Complete lattice | $~$\bigvee Y = \bigcup Y$~$ |
---|
Complete lattice | $~$A \in Y$~$ |
---|
Complete lattice | $~$A \subseteq \bigcup Y$~$ |
---|
Complete lattice | $~$\bigcup Y$~$ |
---|
Complete lattice | $~$Y$~$ |
---|
Complete lattice | $~$B \in \mathcal P(X)$~$ |
---|
Complete lattice | $~$Y$~$ |
---|
Complete lattice | $~$A \in Y$~$ |
---|
Complete lattice | $~$A \subseteq B$~$ |
---|
Complete lattice | $~$x \in \bigcup Y$~$ |
---|
Complete lattice | $~$x \in A$~$ |
---|
Complete lattice | $~$A \in Y$~$ |
---|
Complete lattice | $~$A \subseteq B$~$ |
---|
Complete lattice | $~$x \in B$~$ |
---|
Complete lattice | $~$\bigcup Y \subseteq B$~$ |
---|
Complete lattice | $~$\bigcup Y$~$ |
---|
Complete lattice | $~$Y$~$ |
---|
Complete lattice | $~$X$~$ |
---|
Complete lattice | $~$F : X \to X$~$ |
---|
Complete lattice | $~$x \in X$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$x \leq F(x)$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$F(x) \leq x$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$X$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$A \subseteq X$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$A$~$ |
---|
Complete lattice | $~$A$~$ |
---|
Complete lattice | $~$\mu F$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$\mu F$~$ |
---|
Complete lattice | $~$\mu F$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$\nu F$~$ |
---|
Complete lattice | $~$L$~$ |
---|
Complete lattice | $~$F : L \to L$~$ |
---|
Complete lattice | $~$\mu F$~$ |
---|
Complete lattice | $~$\nu F$~$ |
---|
Complete lattice | $~$L = \langle \mathbb R, \leq \rangle$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$F(x) = x$~$ |
---|
Complete lattice | $~$x \leq y \implies F(x) = x \leq y = F(y)$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$\mathbb R$~$ |
---|
Complete lattice | $~$\mathbb R$~$ |
---|
Complete lattice | $~$\mu F$~$ |
---|
Complete lattice | $~$\nu F$~$ |
---|
Complete lattice | $~$L$~$ |
---|
Complete lattice | $~$F : L \to L$~$ |
---|
Complete lattice | $~$\mu F$~$ |
---|
Complete lattice | $~$\nu F$~$ |
---|
Complete lattice | $~$L$~$ |
---|
Complete lattice | $~$F : L \to L$~$ |
---|
Complete lattice | $~$L$~$ |
---|
Complete lattice | $~$\mu F$~$ |
---|
Complete lattice | $~$\bigwedge \{x \in L \mid F(x) \leq x\}$~$ |
---|
Complete lattice | $~$\nu F$~$ |
---|
Complete lattice | $~$\bigvee \{x \in L \mid x \leq F(x) \}$~$ |
---|
Complete lattice | $~$\bigwedge \{x \in L \mid F(x) \leq x\}$~$ |
---|
Complete lattice | $~$\bigvee \{x \in L \mid F(x) \leq x \}$~$ |
---|
Complete lattice | $~$\bigwedge \{x \in L \mid F(x) \leq x\}$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$U = \{x \in L \mid F(x) \leq x\}$~$ |
---|
Complete lattice | $~$y = \bigwedge U$~$ |
---|
Complete lattice | $~$F(y) = y$~$ |
---|
Complete lattice | $~$V$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$V \subseteq U$~$ |
---|
Complete lattice | $~$y \leq u$~$ |
---|
Complete lattice | $~$u \in U$~$ |
---|
Complete lattice | $~$y \leq v$~$ |
---|
Complete lattice | $~$v \in V$~$ |
---|
Complete lattice | $~$y$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$u \in U$~$ |
---|
Complete lattice | $~$y \leq u$~$ |
---|
Complete lattice | $~$F(y) \leq F(u) \leq u$~$ |
---|
Complete lattice | $~$F(y)$~$ |
---|
Complete lattice | $~$U$~$ |
---|
Complete lattice | $~$y$~$ |
---|
Complete lattice | $~$F(y) \leq y$~$ |
---|
Complete lattice | $~$y \in U$~$ |
---|
Complete lattice | $~$F$~$ |
---|
Complete lattice | $~$F(y) \leq y$~$ |
---|
Complete lattice | $~$F(F(y)) \leq F(y)$~$ |
---|
Complete lattice | $~$F(y) \in U$~$ |
---|
Complete lattice | $~$y$~$ |
---|
Complete lattice | $~$y \leq F(y)$~$ |
---|
Complete lattice | $~$y \leq F(y)$~$ |
---|
Complete lattice | $~$F(y) \leq y$~$ |
---|
Complete lattice | $~$F(y) = y$~$ |
---|
Complex number | $~$z = a + b\textrm{i}$~$ |
---|
Complex number | $~$\textrm{i}$~$ |
---|
Complex number | $~$\textrm{i}=\sqrt{-1}$~$ |
---|
Complex number | $~$5-3$~$ |
---|
Complex number | $~$0$~$ |
---|
Complex number | $~$\frac{1}{2}, \frac{5}{3}$~$ |
---|
Complex number | $~$-\frac{6}{7}$~$ |
---|
Complex number | $~$\sqrt{9}=3$~$ |
---|
Complex number | $~$\sqrt{2}$~$ |
---|
Complex number | $~$\sqrt{}$~$ |
---|
Complex number | $~$\textrm{i}$~$ |
---|
Complex number | $~$\textrm{i}$~$ |
---|
Complex number | $~$x^2+1=0$~$ |
---|
Complex number | $~$\textrm{i}$~$ |
---|
Complex number | $~$\sqrt{-1}$~$ |
---|
Complex number | $~$\textrm{i}$~$ |
---|
Complex number | $~$-a$~$ |
---|
Complex number | $~$\sqrt{-a}=\textrm{i}\sqrt{a}$~$ |
---|
Complexity theory | $~$P$~$ |
---|
Complexity theory | $~$NP$~$ |
---|
Complexity theory | $~$221$~$ |
---|
Complexity theory | $~$13$~$ |
---|
Complexity theory | $~$17$~$ |
---|
Complexity theory | $~$13 \cdot 17 = 221$~$ |
---|
Complexity theory: Complexity zoo | $~$P$~$ |
---|
Complexity theory: Complexity zoo | $~$x$~$ |
---|
Complexity theory: Complexity zoo | $~$1000 x^{42}+10^{100}$~$ |
---|
Complexity theory: Complexity zoo | $~$P$~$ |
---|
Complexity theory: Complexity zoo | $~$\mathcal{O}(n)$~$ |
---|
Complexity theory: Complexity zoo | $~$\mathcal{O}(n*log(n))$~$ |
---|
Complexity theory: Complexity zoo | $~$P$~$ |
---|
Complexity theory: Complexity zoo | $~$NP$~$ |
---|
Complexity theory: Complexity zoo | $~$NP$~$ |
---|
Complexity theory: Complexity zoo | $~$P$~$ |
---|
Complexity theory: Complexity zoo | $~$P$~$ |
---|
Complexity theory: Complexity zoo | $~$NP$~$ |
---|
Complexity theory: Complexity zoo | $~$P\subset NP$~$ |
---|
Complexity theory: Complexity zoo | $~$P=NP$~$ |
---|
Complexity theory: Complexity zoo | $~$P!=NP$~$ |
---|
Complexity theory: Complexity zoo | $~$P!=NP$~$ |
---|
Complexity theory: Complexity zoo | $~$P=NP$~$ |
---|
Compressing multiple messages | $~$n$~$ |
---|
Compressing multiple messages | $~$\lceil \log_2(n) \rceil$~$ |
---|
Compressing multiple messages | $~$n$~$ |
---|
Compressing multiple messages | $~$3^{10} < 2^{16}.$~$ |
---|
Compressing multiple messages | $~$3^{10}$~$ |
---|
Compressing multiple messages | $~$n$~$ |
---|
Compressing multiple messages | $~$k$~$ |
---|
Compressing multiple messages | $~$n^k$~$ |
---|
Compressing multiple messages | $~$n^k$~$ |
---|
Compressing multiple messages | $~$k$~$ |
---|
Compressing multiple messages | $~$n$~$ |
---|
Compressing multiple messages | $~$k$~$ |
---|
Compressing multiple messages | $~$n$~$ |
---|
Compressing multiple messages | $~$n$~$ |
---|
Concrete groups (Draft) | $~$1$~$ |
---|
Concrete groups (Draft) | $~$2$~$ |
---|
Concrete groups (Draft) | $~$3$~$ |
---|
Concrete groups (Draft) | $~$4$~$ |
---|
Concrete groups (Draft) | $~$90^\circ$~$ |
---|
Concrete groups (Draft) | $~$1 \mapsto 2$~$ |
---|
Concrete groups (Draft) | $~$2 \mapsto 3$~$ |
---|
Concrete groups (Draft) | $~$3 \mapsto 4$~$ |
---|
Concrete groups (Draft) | $~$4 \mapsto 1$~$ |
---|
Concrete groups (Draft) | $~$r := (1234)$~$ |
---|
Concrete groups (Draft) | $~$r^2 = (13)(24)$~$ |
---|
Concrete groups (Draft) | $~$180^\circ$~$ |
---|
Concrete groups (Draft) | $~$r^3 = (4321)$~$ |
---|
Concrete groups (Draft) | $~$270^\circ$~$ |
---|
Concrete groups (Draft) | $~$f:= (1 4)(2 3)$~$ |
---|
Concrete groups (Draft) | $~$180^\circ$~$ |
---|
Concrete groups (Draft) | $~$(13)(24)\circ(14)(23) = (1 2)(3 4)$~$ |
---|
Concrete groups (Draft) | $~$f$~$ |
---|
Concrete groups (Draft) | $~$r$~$ |
---|
Concrete groups (Draft) | $~$rf = (1234)(14)(23)$~$ |
---|
Concrete groups (Draft) | $~$(13) = r^3f$~$ |
---|
Concrete groups (Draft) | $~$90^\circ$~$ |
---|
Concrete groups (Draft) | $~$270^\circ$~$ |
---|
Concrete groups (Draft) | $~$(24)(24) = ()$~$ |
---|
Concrete groups (Draft) | $~$(4321)(1234) = ()$~$ |
---|
Concrete groups (Draft) | $~$r$~$ |
---|
Concrete groups (Draft) | $~$r^2$~$ |
---|
Concrete groups (Draft) | $~$r^3$~$ |
---|
Concrete groups (Draft) | $~$f$~$ |
---|
Concrete groups (Draft) | $~$rf$~$ |
---|
Concrete groups (Draft) | $~$r^2f$~$ |
---|
Concrete groups (Draft) | $~$r^3f$~$ |
---|
Concrete groups (Draft) | $~$e := ()$~$ |
---|
Concrete groups (Draft) | $~$(12)$~$ |
---|
Concrete groups (Draft) | $~$G$~$ |
---|
Concrete groups (Draft) | $~$\circ : G \times G \to G$~$ |
---|
Conditional probability | $~$\mathbb{P}(X\mid Y)$~$ |
---|
Conditional probability | $~$\mathbb{P}(yellow\mid banana)$~$ |
---|
Conditional probability | $~$\mathbb{P}(banana\mid yellow)$~$ |
---|
Conditional probability | $~$\mathbb{P}(X\mid Y)$~$ |
---|
Conditional probability | $~$\mathbb{P}(yellow\mid banana)$~$ |
---|
Conditional probability | $~$\mathbb{P}(banana\mid yellow)$~$ |
---|
Conditional probability | $~$\mathbb{P}(X\mid Y)$~$ |
---|
Conditional probability | $~$\mathbb{P}(blue \wedge round)$~$ |
---|
Conditional probability | $~$\mathbb{P}(blue\mid round) := \frac{\mathbb{P}(blue \wedge round)}{\mathbb{P}(round)} = \frac{\text{5% blue and round marbles}}{\text{20% round marbles}} = \frac{5}{20} = 0.25.$~$ |
---|
Conditional probability | $~$\mathbb{P}(X\mid Y) := \frac{\mathbb{P}(X \wedge Y)}{\mathbb{P}(Y)}.$~$ |
---|
Conditional probability | $~$\mathbb{P}(X\mid Y) := \frac{\mathbb{P}(X \wedge Y)}{\mathbb{P}(Y)}$~$ |
---|
Conditional probability | $~$Y$~$ |
---|
Conditional probability | $~$X$~$ |
---|
Conditional probability | $~$Y$~$ |
---|
Conditional probability | $~$X \wedge Y$~$ |
---|
Conditional probability | $~$X \wedge Y$~$ |
---|
Conditional probability | $~$\mathbb P(observation\mid hypothesis)$~$ |
---|
Conditional probability | $~$\mathbb P(hypothesis\mid observation)$~$ |
---|
Conditional probability | $~$\mathbb{P}(X\mid Y)$~$ |
---|
Conditional probability | $~$X$~$ |
---|
Conditional probability | $~$Y$~$ |
---|
Conditional probability | $~$\mathbb P(left\mid right)$~$ |
---|
Conditional probability | $~$left$~$ |
---|
Conditional probability | $~$right$~$ |
---|
Conditional probability | $~$\mathbb P(yellow\mid banana)$~$ |
---|
Conditional probability | $~$\mathbb P(banana\mid yellow)$~$ |
---|
Conditional probability | $~$yellow$~$ |
---|
Conditional probability | $~$banana$~$ |
---|
Conditional probability | $~$\mathbb P(left \mid right),$~$ |
---|
Conditional probability | $~$right$~$ |
---|
Conditional probability | $~$right$~$ |
---|
Conditional probability | $~$left$~$ |
---|
Conditional probability | $~$X \wedge Y$~$ |
---|
Conditional probability | $~$X$~$ |
---|
Conditional probability | $~$Y$~$ |
---|
Conditional probability | $~$X$~$ |
---|
Conditional probability | $~$Y$~$ |
---|
Conditional probability | $$~$\mathbb P(left \mid right) = \dfrac{\mathbb P(left \wedge right)}{\mathbb P(right)}.$~$$ |
---|
Conditional probability | $~$right$~$ |
---|
Conditional probability | $~$right$~$ |
---|
Conditional probability | $~$left$~$ |
---|
Conditional probability | $$~$\begin{array}{l\mid r\mid r}
& Red & Blue \\
\hline
Square & 1 & 2 \\
\hline
Round & 3 & 4
\end{array}$~$$ |
---|
Conditional probability | $$~$\mathbb P(red\mid round) = \dfrac{\mathbb P(red \wedge round)}{\mathbb P(round)} = \dfrac{3}{3 + 4} = \dfrac{3}{7}$~$$ |
---|
Conditional probability | $$~$\mathbb P(square\mid blue) = \dfrac{\mathbb P(square \wedge blue)}{\mathbb P(blue)} = \dfrac{2}{2 + 4} = \dfrac{1}{3}$~$$ |
---|
Conditional probability | $~$\mathbb P(red hair\mid Scarlet) = 99\%,$~$ |
---|
Conditional probability | $~$\mathbb P(redhair\mid Scarlet),$~$ |
---|
Conditional probability | $~$\mathbb P(Scarlet\mid redhair),$~$ |
---|
Conditional probability | $~$\mathbb P(redhair\mid Scarlet)$~$ |
---|
Conditional probability | $~$1$~$ |
---|
Conditional probability | $~$\mathbb P(redhair\mid Scarlet)$~$ |
---|
Conditional probability | $~$\mathbb P(Scarlet\mid redhair)$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(\text{left} \mid \text{right})$~$ |
---|
Conditional probability: Refresher | $~$\frac{\mathbb P(\text{left} \land \text{right})}{\mathbb P(\text{right})}.$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(yellow \mid banana)$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(banana \mid yellow)$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(\text{left} \mid \text{right})$~$ |
---|
Conditional probability: Refresher | $~$\frac{\mathbb P(\text{left} \land \text{right})}{\mathbb P(\text{right})}.$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(yellow \mid banana)$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(banana \mid yellow)$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(v)$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(V = v)$~$ |
---|
Conditional probability: Refresher | $~$V$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(yellow)$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P({ColorOfNextObjectInBag}=yellow)$~$ |
---|
Conditional probability: Refresher | $~$ColorOfNextObjectInBag$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P,$~$ |
---|
Conditional probability: Refresher | $~$yellow$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(x \land y)$~$ |
---|
Conditional probability: Refresher | $~$x$~$ |
---|
Conditional probability: Refresher | $~$y$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(x\mid y)$~$ |
---|
Conditional probability: Refresher | $$~$\frac{\mathbb P(x \wedge y)}{\mathbb P(y)}.$~$$ |
---|
Conditional probability: Refresher | $~$\mathbb P({sick}\mid {positive})$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P({sick}\mid {positive})$~$ |
---|
Conditional probability: Refresher | $~$=$~$ |
---|
Conditional probability: Refresher | $~$\frac{\mathbb P({sick} \land {positive})}{\mathbb P({positive})}.$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(sick \mid positive)$~$ |
---|
Conditional probability: Refresher | $~$sick$~$ |
---|
Conditional probability: Refresher | $~$positive$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(x\mid y)$~$ |
---|
Conditional probability: Refresher | $~$y$~$ |
---|
Conditional probability: Refresher | $~$y$~$ |
---|
Conditional probability: Refresher | $~$x$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(positive \mid sick)$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(sick \mid positive).$~$ |
---|
Conditional probability: Refresher | $~$\frac{18}{20} = 0.9$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(positive \mid sick) = 90\%,$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(sick \mid positive) \approx 43\%.$~$ |
---|
Conditional probability: Refresher | $~$\mathbb P(\text{left} \mid \text{right})$~$ |
---|
Conjugacy class | $~$g$~$ |
---|
Conjugacy class | $~$G$~$ |
---|
Conjugacy class | $~$g$~$ |
---|
Conjugacy class | $~$G$~$ |
---|
Conjugacy class | $~$\{ x g x^{-1} : x \in G \}$~$ |
---|
Conjugacy class | $~$g$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$S_n$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$S_n$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$\sigma$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$n_1, \dots, n_k$~$ |
---|
Conjugacy class is cycle type in symmetric group | $$~$\sigma = (a_{11} a_{12} \dots a_{1 n_1})(a_{21} \dots a_{2 n_2}) \dots (a_{k 1} a_{k 2} \dots a_{k n_k})$~$$ |
---|
Conjugacy class is cycle type in symmetric group | $~$\tau \in S_n$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$\tau \sigma \tau^{-1}(\tau(a_{ij})) = \tau \sigma(a_{ij}) = a_{i (j+1)}$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$a_{i (n_i+1)}$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$a_{i 1}$~$ |
---|
Conjugacy class is cycle type in symmetric group | $$~$\tau \sigma \tau^{-1} = (\tau(a_{11}) \tau(a_{12}) \dots \tau(a_{1 n_1}))(\tau(a_{21}) \dots \tau(a_{2 n_2})) \dots (\tau(a_{k 1}) \tau(a_{k 2}) \dots \tau(a_{k n_k}))$~$$ |
---|
Conjugacy class is cycle type in symmetric group | $~$\sigma$~$ |
---|
Conjugacy class is cycle type in symmetric group | $$~$\pi = (b_{11} b_{12} \dots b_{1 n_1})(b_{21} \dots b_{2 n_2}) \dots (b_{k 1} b_{k 2} \dots b_{k n_k})$~$$ |
---|
Conjugacy class is cycle type in symmetric group | $~$\pi$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$\sigma$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$\tau(a_{ij}) = b_{ij}$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$\tau$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$\tau \sigma \tau^{-1} = \pi$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$\sigma$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$\pi$~$ |
---|
Conjugacy class is cycle type in symmetric group | $~$S_5$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$A_5$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$A_5$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$5!/2 = 60$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$S_5$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$A_5$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$S_5$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$S_5$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$A_5$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$(5)$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$(3, 1, 1)$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$(2, 2, 1)$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$(1,1,1,1,1)$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$(5)$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$(12345)$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$(12345)$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$S_5$~$ |
---|
Conjugacy classes of the alternating group on five elements | $~$(12)(12345)(12)^{-1} = (21345)$~$ |
---|
Conjugacy classes of the alternating group on five elements | $$~$\begin{array}{|c|c|c|c|}
\hline
\text{Representative}& \text{Size of class} & \text{Cycle type} & \text{Order of element} \\ \hline
(12345) & 12 & 5 & 5 \\ \hline
(21345) & 12 & 5 & 5 \\ \hline
(123) & 20 & 3,1,1 & 3 \\ \hline
(12)(34) & 15 & 2,2,1 & 2 \\ \hline
e & 1 & 1,1,1,1,1 & 1 \\ \hline
\end{array}$~$$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_5$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$S_5$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_5$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$60$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$S_5$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$5! = 120$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $$~$\begin{array}{|c|c|c|c|}
\hline
\text{Representative}& \text{Size of class} & \text{Cycle type} & \text{Order of element} \\ \hline
(12345) & 12 & 5 & 5 \\ \hline
(21345) & 12 & 5 & 5 \\ \hline
(123) & 20 & 3,1,1 & 3 \\ \hline
(12)(34) & 15 & 2,2,1 & 2 \\ \hline
e & 1 & 1,1,1,1,1 & 1 \\ \hline
\end{array}$~$$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\tau e \tau^{-1} = \tau \tau^{-1} = e$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\tau$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$S_n$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_n$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(5)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(3,1,1)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(2,2,1)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(1,1,1,1,1)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(2,2,1)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ab)(cd)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ab)(ce)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ab)(de)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ab)(cd)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ac)(bd)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(cba)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$e$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ab)(cd)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ac)(be)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(bc)(de)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$e$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(3,1,1)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(abc)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(acb)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(bc)(de)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(abc)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(abd)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(cde)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(abc)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(ade)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(bd)(ce)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(12345)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(21345)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\{ \rho (12345) \rho^{-1}: \rho \ \text{even} \}$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_5$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\{ \rho (12345) \rho^{-1}: \rho \ \text{odd} \}$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_5$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(12345)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_5$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(21345) = (12)(12345)(12)^{-1}$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\tau (12345) \tau^{-1} = (\tau(1), \tau(2), \tau(3), \tau(4), \tau(5))$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\tau$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$\tau$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$1$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$2$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$2$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$1$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$3$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$3$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$4$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$4$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$5$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$5$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(12)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$A_5$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(12345)$~$ |
---|
Conjugacy classes of the alternating group on five elements: Simpler proof | $~$(21345)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$S_5$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$5! = 120$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$S_5$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $$~$\begin{array}{|c|c|c|c|}
\hline
\text{Representative}& \text{Size of class} & \text{Cycle type} & \text{Order of element} \\ \hline
(12345) & 24 & 5 & 5 \\ \hline
(1234) & 30 & 4,1 & 4 \\ \hline
(123) & 20 & 3,1,1 & 3 \\ \hline
(123)(45) & 20 & 3,2 & 6 \\ \hline
(12)(34) & 15 & 2,2,1 & 2 \\ \hline
(12) & 10 & 2,1,1,1 & 2 \\ \hline
e & 1 & 1,1,1,1,1 & 1 \\ \hline
\end{array}$~$$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$6$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$5$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$5$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$5$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(12345)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$5$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$5$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(12345)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(23451)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(34512)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4!$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$24$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4,1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(1234)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$a$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$b$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$a$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$c$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$b$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$c$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4 \times 3 \times 2 = 24$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$a$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$b$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$a$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$c$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$b$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$c$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3 \times 2 \times 1 = 6$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$30$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3,1,1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3,2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3,1,1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(123)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$4,1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$\binom{5}{3} = 10$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$\{1,2,3\}$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(123)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(231)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(312)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(132)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(321)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(213)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2 \times 10 = 20$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3,2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(123)(45)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$\binom{5}{3} = 10$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(12)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(21)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$3$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2 \times 10 = 20$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2,2,1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2,1,1,1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2,2,1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(12)(34)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$\binom{5}{2}$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$\binom{3}{2}$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(12)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(21)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(12)(34)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(34)(12)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$\binom{5}{2} \times \binom{3}{2} / 2 = 15$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2,1,1,1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(12)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$\binom{5}{2}$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$2$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(12)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$(21)$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$\binom{5}{2} = 10$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|
Conjugacy classes of the symmetric group on five elements | $~$1$~$ |
---|
Conjunctions and disjunctions | $~$P \land Q$~$ |
---|
Conjunctions and disjunctions | $~$P \lor Q$~$ |
---|
Conjunctions and disjunctions | $~$R$~$ |
---|
Conjunctions and disjunctions | $~$P$~$ |
---|
Conjunctions and disjunctions | $~$Q$~$ |
---|
Conjunctions and disjunctions | $~$R \equiv P \land Q $~$ |
---|
Conjunctions and disjunctions | $~$S$~$ |
---|
Conjunctions and disjunctions | $~$P$~$ |
---|
Conjunctions and disjunctions | $~$Q$~$ |
---|
Conjunctions and disjunctions | $~$S$~$ |
---|
Conjunctions and disjunctions | $~$P$~$ |
---|
Conjunctions and disjunctions | $~$Q$~$ |
---|
Conjunctions and disjunctions | $~$S \equiv P \lor Q$~$ |
---|
Consequentialist cognition | $~$X$~$ |
---|
Consequentialist cognition | $~$X$~$ |
---|
Consequentialist cognition | $~$Y$~$ |
---|
Consequentialist cognition | $~$Y$~$ |
---|
Consequentialist cognition | $~$Y'$~$ |
---|
Consequentialist cognition | $~$X$~$ |
---|
Consequentialist cognition | $~$X',$~$ |
---|
Consequentialist cognition | $~$X$~$ |
---|
Consequentialist cognition | $~$Y$~$ |
---|
Consistency | $~$X$~$ |
---|
Consistency | $~$T\vdash X$~$ |
---|
Consistency | $~$T\vdash \neg X$~$ |
---|
Consistency | $~$\square_{PA}$~$ |
---|
Consistency | $~$\neg\square_{PA}(\ulcorner 0=1\urcorner)$~$ |
---|
Consistency | $~$PA$~$ |
---|
Consistency | $~$PA$~$ |
---|
Context disaster | $~$V$~$ |
---|
Context disaster | $~$V$~$ |
---|
Context disaster | $~$0$~$ |
---|
Context disaster | $~$0,$~$ |
---|
Context disaster | $~$V$~$ |
---|
Context disaster | $~$0$~$ |
---|
Context disaster | $~$U$~$ |
---|
Context disaster | $~$\mathbb P_t(X)$~$ |
---|
Context disaster | $~$X$~$ |
---|
Context disaster | $~$t,$~$ |
---|
Context disaster | $~$\mathbb Q_t(X)$~$ |
---|
Context disaster | $~$X$~$ |
---|
Context disaster | $~$\pi \in \Pi$~$ |
---|
Context disaster | $~$\pi$~$ |
---|
Context disaster | $~$\Pi$~$ |
---|
Context disaster | $~$\mathbb E_{\mathbb P, t} [W \mid \pi]$~$ |
---|
Context disaster | $~$\mathbb P_t$~$ |
---|
Context disaster | $~$W$~$ |
---|
Context disaster | $~$\pi$~$ |
---|
Context disaster | $$~$\underset{\pi \in \Pi}{\operatorname {optimum}} F(\pi)$~$$ |
---|
Context disaster | $~$\pi$~$ |
---|
Context disaster | $~$\Pi$~$ |
---|
Context disaster | $~$F$~$ |
---|
Context disaster | $~$\Pi_1$~$ |
---|
Context disaster | $~$t,$~$ |
---|
Context disaster | $~$\Pi_2$~$ |
---|
Context disaster | $~$u$~$ |
---|
Context disaster | $$~$\mathbb E_{\mathbb Q, t} [V \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U \mid \pi] \big ] > 0 \\
\mathbb E_{\mathbb P, t} [V \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U \mid \pi] \big ] > 0 \\
\mathbb E_{\mathbb P, u} [V \mid \big [ \underset{\pi \in \Pi_2}{\operatorname {optimum}} \mathbb E_{\mathbb P, u} [U \mid \pi] \big ] < 0$~$$ |
---|
Context disaster | $~$t$~$ |
---|
Context disaster | $~$\Pi_1$~$ |
---|
Context disaster | $~$V$~$ |
---|
Context disaster | $~$u$~$ |
---|
Context disaster | $~$\Pi_2,$~$ |
---|
Context disaster | $~$V.$~$ |
---|
Context disaster | $$~$\mathbb E_{\mathbb Q, t} [V \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U \mid \pi] \big ] > 0 \\
\mathbb E_{\mathbb P, t} [V \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U \mid \pi] \big ] < 0 \\
\mathbb E_{\mathbb P, u} [V \mid \big [ \underset{\pi \in \Pi_2}{\operatorname {optimum}} \mathbb E_{\mathbb P, u} [U \mid \pi] \big ] < 0$~$$ |
---|
Context disaster | $~$V.$~$ |
---|
Context disaster | $~$W_{t}$~$ |
---|
Context disaster | $~$W$~$ |
---|
Context disaster | $~$t,$~$ |
---|
Context disaster | $$~$\mathbb E_{\mathbb Q, t} [V_\infty \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U_\infty \mid \pi] \big ] > 0 \\
\mathbb E_{\mathbb P, t} [V_{u} \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U_\infty \mid \pi] \big ] > 0 \\
\mathbb E_{\mathbb P, t} [V_\infty \mid \big [ \underset{\pi \in \Pi_1}{\operatorname {optimum}} \mathbb E_{\mathbb P, t} [U_\infty \mid \pi] \big ] < 0 \\
\mathbb E_{\mathbb P, u} [V_\infty \mid \big [ \underset{\pi \in \Pi_2}{\operatorname {optimum}} \mathbb E_{\mathbb P, u} [U_\infty \mid \pi] \big ] < 0$~$$ |
---|
Context disaster | $~$t$~$ |
---|
Context disaster | $~$u$~$ |
---|
Context disaster | $~$V,$~$ |
---|
Context disaster | $~$t$~$ |
---|
Context disaster | $~$\mathbb Q_t$~$ |
---|
Context disaster | $~$V$~$ |
---|
Context disaster | $~$U,$~$ |
---|
Context disaster | $~$U$~$ |
---|
Context disaster | $~$U$~$ |
---|
Context disaster | $~$V,$~$ |
---|
Context disaster | $~$U$~$ |
---|
Context disaster | $~$V.$~$ |
---|
Convergent instrumental strategies | $~$X$~$ |
---|
Convergent instrumental strategies | $~$X$~$ |
---|
Convergent instrumental strategies | $~$X,$~$ |
---|
Convergent instrumental strategies | $~$X'$~$ |
---|
Convergent instrumental strategies | $~$X$~$ |
---|
Convergent instrumental strategies | $~$X'$~$ |
---|
Convergent instrumental strategies | $~$X^*$~$ |
---|
Convergent instrumental strategies | $~$\pi_1$~$ |
---|
Convergent instrumental strategies | $~$\pi_2$~$ |
---|
Convergent strategies of self-modification | $~$X$~$ |
---|
Convergent strategies of self-modification | $~$Y.$~$ |
---|
Convergent strategies of self-modification | $~$Y$~$ |
---|
Convergent strategies of self-modification | $~$X$~$ |
---|
Convergent strategies of self-modification | $~$Y.$~$ |
---|
Convergent strategies of self-modification | $~$Y$~$ |
---|
Convergent strategies of self-modification | $~$X$~$ |
---|
Convergent strategies of self-modification | $~$X$~$ |
---|
Convergent strategies of self-modification | $~$Y.$~$ |
---|
Convex set | $~$x$~$ |
---|
Convex set | $~$y$~$ |
---|
Convex set | $~$x$~$ |
---|
Convex set | $~$y$~$ |
---|
Convex set | $~$S$~$ |
---|
Convex set | $$~$\forall x, y \in S, \theta \in [0, 1]: \theta x + (1 - \theta) y \in S$~$$ |
---|
Cosmic endowment | $~$\approx 4 \times 10^{20}$~$ |
---|
Cosmic endowment | $~$\approx 10^{42}$~$ |
---|
Cosmic endowment | $~$\approx 10^{25}$~$ |
---|
Cosmic endowment | $~$\approx 10^{54}$~$ |
---|
Countability | $~$\mathbb{Z}^+ = \{1, 2, 3, 4, \ldots\}$~$ |
---|
Countability | $~$S$~$ |
---|
Countability | $~$S$~$ |
---|
Countability | $~$\mathbb Q$~$ |
---|
Countability | $~$\frac{p}{q}$~$ |
---|
Countability | $~$p$~$ |
---|
Countability | $~$q$~$ |
---|
Countability | $~$q > 0$~$ |
---|
Countability | $~$\mathbb Z^+ \times \mathbb Z^+$~$ |
---|
Countability | $~$\mathbb Z$~$ |
---|
Countability | $~$\frac{a}{b}$~$ |
---|
Countability | $~$|a| + |b|$~$ |
---|
Countability | $~$a$~$ |
---|
Countability | $~$b$~$ |
---|
Countability | $~$0 / 1$~$ |
---|
Countability | $~$-1 / 1$~$ |
---|
Countability | $~$1 / 1$~$ |
---|
Countability | $~$-2 / 1$~$ |
---|
Countability | $~$-1 / 2$~$ |
---|
Countability | $~$1 / 2$~$ |
---|
Countability | $~$2 / 1$~$ |
---|
Countability | $~$\ldots$~$ |
---|
Countability | $~$(2d+1)^2$~$ |
---|
Countability | $~$d$~$ |
---|
Countability | $~$d$~$ |
---|
Countability | $~$(2d+1)^2$~$ |
---|
Countability | $~$\square$~$ |
---|
Countability | $~$(\mathbb Z^+)^n$~$ |
---|
Countability | $~$n$~$ |
---|
Countability | $~$f$~$ |
---|
Countability | $~$A$~$ |
---|
Countability | $~$B$~$ |
---|
Countability | $~$B$~$ |
---|
Countability | $~$E$~$ |
---|
Countability | $~$A$~$ |
---|
Countability | $~$E\circ f$~$ |
---|
Countability | $~$B$~$ |
---|
Countability | $~$B$~$ |
---|
Countability | $~$\Sigma^*$~$ |
---|
Countability | $~$\mathbb N^n$~$ |
---|
Countability | $~$n$~$ |
---|
Countability | $~$n\in \mathbb N$~$ |
---|
Countability | $~$E_n: \mathbb N \to \mathbb N^n$~$ |
---|
Countability | $~$\mathbb N ^n$~$ |
---|
Countability | $~$(J_1,J_2)(n)$~$ |
---|
Countability | $~$\mathbb N^2$~$ |
---|
Countability | $~$E: \mathbb N \to \Sigma^* , n\hookrightarrow E_{J_1(n)}(J_2(n))$~$ |
---|
Countability | $~$\Sigma^*$~$ |
---|
Countability | $~$E$~$ |
---|
Countability | $~$\Sigma^*$~$ |
---|
Countability | $~$\square$~$ |
---|
Countability | $~$P_\omega(A)$~$ |
---|
Countability | $~$A$~$ |
---|
Countability | $~$E$~$ |
---|
Countability | $~$A$~$ |
---|
Countability | $~$E': \mathbb N^* \to P_\omega(A)$~$ |
---|
Countability | $~$n_0 n_1 n_2 … n_r$~$ |
---|
Countability | $~$\{a\in A:\exists m\le k E(n_m)=a\}\subseteq A$~$ |
---|
Countability | $~$E'$~$ |
---|
Countability | $~$P_\omega(A)$~$ |
---|
Creating a /learn/ link | $~$bayes_rule_details,$~$ |
---|
Currying | $~$F:(X,Y,Z,N)→R$~$ |
---|
Currying | $~$curry(F)$~$ |
---|
Currying | $~$X→(Y→(Z→(N→R)))$~$ |
---|
Currying | $~$curry(F)(4)(3)(2)(1)$~$ |
---|
Currying | $~$F(4,3,2,1)$~$ |
---|
Cycle notation in symmetric groups | $~$k$~$ |
---|
Cycle notation in symmetric groups | $~$k$~$ |
---|
Cycle notation in symmetric groups | $~$S_n$~$ |
---|
Cycle notation in symmetric groups | $~$k$~$ |
---|
Cycle notation in symmetric groups | $~$a_1, \dots, a_k$~$ |
---|
Cycle notation in symmetric groups | $~$\{1,2,\dots,n\}$~$ |
---|
Cycle notation in symmetric groups | $~$k$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma(a_i) = a_{i+1}$~$ |
---|
Cycle notation in symmetric groups | $~$1 \leq i < k$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma(a_k) = a_1$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma(x) = x$~$ |
---|
Cycle notation in symmetric groups | $~$x \not \in \{a_1, \dots, a_k \}$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma = (a_1 a_2 \dots a_k)$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma = (a_1, a_2, \dots, a_k)$~$ |
---|
Cycle notation in symmetric groups | $~$(a_1 a_2 \dots a_k) = (a_2 a_3 \dots a_k a_1)$~$ |
---|
Cycle notation in symmetric groups | $~$a_i$~$ |
---|
Cycle notation in symmetric groups | $~$(a_1 a_2 \dots a_k)$~$ |
---|
Cycle notation in symmetric groups | $~$(a_k a_{k-1} \dots a_1)$~$ |
---|
Cycle notation in symmetric groups | $$~$\begin{pmatrix}1 & 2 & 3 \\ 2 & 3 & 1 \\ \end{pmatrix}$~$$ |
---|
Cycle notation in symmetric groups | $~$(123)$~$ |
---|
Cycle notation in symmetric groups | $~$(231)$~$ |
---|
Cycle notation in symmetric groups | $~$(312)$~$ |
---|
Cycle notation in symmetric groups | $~$(123)$~$ |
---|
Cycle notation in symmetric groups | $~$S_n$~$ |
---|
Cycle notation in symmetric groups | $~$n \geq 3$~$ |
---|
Cycle notation in symmetric groups | $~$(145)$~$ |
---|
Cycle notation in symmetric groups | $~$S_n$~$ |
---|
Cycle notation in symmetric groups | $~$n \geq 5$~$ |
---|
Cycle notation in symmetric groups | $~$S_n$~$ |
---|
Cycle notation in symmetric groups | $~$S_4$~$ |
---|
Cycle notation in symmetric groups | $~$2$~$ |
---|
Cycle notation in symmetric groups | $~$2$~$ |
---|
Cycle notation in symmetric groups | $$~$\begin{pmatrix}1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \\ \end{pmatrix}$~$$ |
---|
Cycle notation in symmetric groups | $~$(12)$~$ |
---|
Cycle notation in symmetric groups | $~$(34)$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma$~$ |
---|
Cycle notation in symmetric groups | $~$c_1 = (a_1 a_2 \dots a_k)$~$ |
---|
Cycle notation in symmetric groups | $~$c_2$~$ |
---|
Cycle notation in symmetric groups | $~$c_3$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma = c_3 c_2 c_1$~$ |
---|
Cycle notation in symmetric groups | $~$(a_1 a_2 \dots a_k)$~$ |
---|
Cycle notation in symmetric groups | $~$a_1 \mapsto a_2 \mapsto a_3 \dots \mapsto a_k \mapsto a_1$~$ |
---|
Cycle notation in symmetric groups | $~$k$~$ |
---|
Cycle notation in symmetric groups | $~$i$~$ |
---|
Cycle notation in symmetric groups | $~$a_1 \mapsto a_{i+1}$~$ |
---|
Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)(a_4 a_5)$~$ |
---|
Cycle notation in symmetric groups | $~$a_i$~$ |
---|
Cycle notation in symmetric groups | $~$3 \times 2 = 6$~$ |
---|
Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)$~$ |
---|
Cycle notation in symmetric groups | $~$(a_4 a_5)$~$ |
---|
Cycle notation in symmetric groups | $~$[(a_1 a_2 a_3)(a_4 a_5)]^n = (a_1 a_2 a_3)^n (a_4 a_5)^n$~$ |
---|
Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)^n (a_4 a_5)^n$~$ |
---|
Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)^n = (a_4 a_5)^n = e$~$ |
---|
Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)^n$~$ |
---|
Cycle notation in symmetric groups | $~$a_1$~$ |
---|
Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)^n$~$ |
---|
Cycle notation in symmetric groups | $~$n$~$ |
---|
Cycle notation in symmetric groups | $~$3$~$ |
---|
Cycle notation in symmetric groups | $~$(a_1 a_2 a_3)$~$ |
---|
Cycle notation in symmetric groups | $~$3$~$ |
---|
Cycle notation in symmetric groups | $~$(a_4 a_5)^n$~$ |
---|
Cycle notation in symmetric groups | $~$n$~$ |
---|
Cycle notation in symmetric groups | $~$2$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma$~$ |
---|
Cycle notation in symmetric groups | $~$S_5$~$ |
---|
Cycle notation in symmetric groups | $~$(123)$~$ |
---|
Cycle notation in symmetric groups | $~$(345)$~$ |
---|
Cycle notation in symmetric groups | $~$(345)(123) = (12453)$~$ |
---|
Cycle notation in symmetric groups | $~$1$~$ |
---|
Cycle notation in symmetric groups | $~$2$~$ |
---|
Cycle notation in symmetric groups | $~$2$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma$~$ |
---|
Cycle notation in symmetric groups | $~$1$~$ |
---|
Cycle notation in symmetric groups | $~$2$~$ |
---|
Cycle notation in symmetric groups | $~$2$~$ |
---|
Cycle notation in symmetric groups | $~$3$~$ |
---|
Cycle notation in symmetric groups | $~$3$~$ |
---|
Cycle notation in symmetric groups | $~$4$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma$~$ |
---|
Cycle notation in symmetric groups | $~$2$~$ |
---|
Cycle notation in symmetric groups | $~$4$~$ |
---|
Cycle notation in symmetric groups | $~$4$~$ |
---|
Cycle notation in symmetric groups | $~$4$~$ |
---|
Cycle notation in symmetric groups | $~$5$~$ |
---|
Cycle notation in symmetric groups | $~$\sigma$~$ |
---|
Cycle notation in symmetric groups | $~$4$~$ |
---|
Cycle notation in symmetric groups | $~$5$~$ |
---|
Cycle type of a permutation | $~$\sigma$~$ |
---|
Cycle type of a permutation | $~$S_n$~$ |
---|
Cycle type of a permutation | $~$\sigma$~$ |
---|
Cycle type of a permutation | $~$\sigma$~$ |
---|
Cycle type of a permutation | $~$\sigma$~$ |
---|
Cycle type of a permutation | $~$1$~$ |
---|
Cycle type of a permutation | $~$(123)(45)$~$ |
---|
Cycle type of a permutation | $~$S_7$~$ |
---|
Cycle type of a permutation | $~$3,2$~$ |
---|
Cycle type of a permutation | $~$(6)$~$ |
---|
Cycle type of a permutation | $~$(7)$~$ |
---|
Cycle type of a permutation | $~$3,2,1,1$~$ |
---|
Cycle type of a permutation | $~$k$~$ |
---|
Cycle type of a permutation | $~$k$~$ |
---|
Cycle type of a permutation | $~$k$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5$~$ |
---|
Cyclic Group Intro (Math 0) | $~$6$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5$~$ |
---|
Cyclic Group Intro (Math 0) | $~$6$~$ |
---|
Cyclic Group Intro (Math 0) | $~$11$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7$~$ |
---|
Cyclic Group Intro (Math 0) | $~$9$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7$~$ |
---|
Cyclic Group Intro (Math 0) | $~$9$~$ |
---|
Cyclic Group Intro (Math 0) | $~$4$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7+9 = 16$~$ |
---|
Cyclic Group Intro (Math 0) | $~$16- 12 = 4$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$4$~$ |
---|
Cyclic Group Intro (Math 0) | $~$4 + 12 = 16$~$ |
---|
Cyclic Group Intro (Math 0) | $~$16 - 12 = 4$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12 - 5 = 7$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$4$~$ |
---|
Cyclic Group Intro (Math 0) | $~$2$~$ |
---|
Cyclic Group Intro (Math 0) | $~$6$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7$~$ |
---|
Cyclic Group Intro (Math 0) | $~$9$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7+9 = 16$~$ |
---|
Cyclic Group Intro (Math 0) | $~$16-12 = 4$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7 +5 = 12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12 - 12 = 0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$\bullet$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7 \bullet 9 = 4$~$ |
---|
Cyclic Group Intro (Math 0) | $~$12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$15$~$ |
---|
Cyclic Group Intro (Math 0) | $~$15$~$ |
---|
Cyclic Group Intro (Math 0) | $~$15$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5 \bullet 7 = 12$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7 \bullet 9 = 1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7 + 9 = 16$~$ |
---|
Cyclic Group Intro (Math 0) | $~$16 - 15 = 1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7$~$ |
---|
Cyclic Group Intro (Math 0) | $~$15 - 5 = 10$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5$~$ |
---|
Cyclic Group Intro (Math 0) | $~$10$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5 + 10 = 15$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5 \bullet 10 = 0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$15$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5$~$ |
---|
Cyclic Group Intro (Math 0) | $~$-5$~$ |
---|
Cyclic Group Intro (Math 0) | $~$-5 = 10$~$ |
---|
Cyclic Group Intro (Math 0) | $~$5$~$ |
---|
Cyclic Group Intro (Math 0) | $~$7$~$ |
---|
Cyclic Group Intro (Math 0) | $~$-5 = 7$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1 \bullet 1 \bullet 1 \bullet \cdots \bullet 1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$-1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$-1 = 11$~$ |
---|
Cyclic Group Intro (Math 0) | $~$15$~$ |
---|
Cyclic Group Intro (Math 0) | $~$-1 = 14$~$ |
---|
Cyclic Group Intro (Math 0) | $~$-1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$-1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$h$~$ |
---|
Cyclic Group Intro (Math 0) | $~$t$~$ |
---|
Cyclic Group Intro (Math 0) | $~$\bullet$~$ |
---|
Cyclic Group Intro (Math 0) | $~$h \bullet h = t$~$ |
---|
Cyclic Group Intro (Math 0) | $~$h \bullet t = h$~$ |
---|
Cyclic Group Intro (Math 0) | $~$t \bullet h = h$~$ |
---|
Cyclic Group Intro (Math 0) | $~$t \bullet t = t$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1 \bullet 1 = 0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1 \bullet 0 = 1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0 \bullet 1 = 1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$0 \bullet 0 = 0$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1$~$ |
---|
Cyclic Group Intro (Math 0) | $~$1$~$ |
---|
Cyclic group | $~$G$~$ |
---|
Cyclic group | $~$g$~$ |
---|
Cyclic group | $~$g$~$ |
---|
Cyclic group | $~$(G, +)$~$ |
---|
Cyclic group | $~$G$~$ |
---|
Cyclic group | $~$g \in G$~$ |
---|
Cyclic group | $~$h \in G$~$ |
---|
Cyclic group | $~$n \in \mathbb{Z}$~$ |
---|
Cyclic group | $~$h = g^n$~$ |
---|
Cyclic group | $~$g^n$~$ |
---|
Cyclic group | $~$g + g + \dots + g$~$ |
---|
Cyclic group | $~$n$~$ |
---|
Cyclic group | $~$G = \langle g \rangle$~$ |
---|
Cyclic group | $~$g$~$ |
---|
Cyclic group | $~$G$~$ |
---|
Cyclic group | $~$(\mathbb{Z}, +) = \langle 1 \rangle = \langle -1 \rangle$~$ |
---|
Cyclic group | $~$\{ e, g \}$~$ |
---|
Cyclic group | $~$e$~$ |
---|
Cyclic group | $~$g^2 = e$~$ |
---|
Cyclic group | $~$g$~$ |
---|
Cyclic group | $~$g^2 = g^0 = e$~$ |
---|
Cyclic group | $~$n$~$ |
---|
Cyclic group | $~$n$~$ |
---|
Cyclic group | $~$1$~$ |
---|
Cyclic group | $~$n-1$~$ |
---|
Cyclic group | $~$S_n$~$ |
---|
Cyclic group | $~$n > 2$~$ |
---|
Cyclic group | $~$a, b \in G$~$ |
---|
Cyclic group | $~$g$~$ |
---|
Cyclic group | $~$G$~$ |
---|
Cyclic group | $~$a = g^i, b = g^j$~$ |
---|
Cyclic group | $~$ab = g^i g^j = g^{i+j} = g^{j+i} = g^j g^i = ba$~$ |
---|
Cyclic group | $~$\{ g^0, g^1, g^{-1}, g^2, g^{-2}, \dots \}$~$ |
---|
Data capacity | $~$\log(2)$~$ |
---|
Data capacity | $~$\log_2(2)=1$~$ |
---|
Data capacity | $~$\log_2(36) \approx 5.17$~$ |
---|
Data capacity | $~$\log_2(8) = 3$~$ |
---|
Data capacity | $~$n$~$ |
---|
Data capacity | $~$b$~$ |
---|
Data capacity | $~$b^n$~$ |
---|
Data capacity | $~$5 \cdot 8 = 40$~$ |
---|
Death in Damascus | $~$\operatorname {do}()$~$ |
---|
Death in Damascus | $~$D$~$ |
---|
Death in Damascus | $~$A$~$ |
---|
Death in Damascus | $~$Y$~$ |
---|
Death in Damascus | $~$N$~$ |
---|
Death in Damascus | $~$DY, AY, DN, AN$~$ |
---|
Death in Damascus | $$~$
\begin{array}{r|c|c}
& \text {Damascus fatal} & \text {Aleppo fatal} \\ \hline
\ {DN} & \text {Die} & \text{Live} \\ \hline
\ {AN} & \text {Live} & \text {Die} \\ \hline
\ {DY} & \text {Die, \$-1} & \text{Live, \$+10} \\ \hline
\ {AY} & \text {Live, \$+10} & \text {Die, \$-1}
\end{array}
$~$$ |
---|
Death in Damascus | $~$AY$~$ |
---|
Death in Damascus | $~$AN.$~$ |
---|
Decimal notation | $~$e$~$ |
---|
Decimal notation | $~$(2 \cdot 100) + (4 \cdot 10) + (6 \cdot 1),$~$ |
---|
Decision problem | $~$w$~$ |
---|
Decision problem | $~$p$~$ |
---|
Decision problem | $~$D$~$ |
---|
Decision problem | $~$A$~$ |
---|
Decision problem | $~$A$~$ |
---|
Decision problem | $~$\{0,1\}^*$~$ |
---|
Decision problem | $~$w$~$ |
---|
Decision problem | $~$p$~$ |
---|
Decision problem | $~$w$~$ |
---|
Decision problem | $~$A$~$ |
---|
Decision problem | $~$w$~$ |
---|
Decision problem | $~$D$~$ |
---|
Decision problem | $~$D$~$ |
---|
Decision problem | $~$A$~$ |
---|
Decision problem | $~$D$~$ |
---|
Decision problem | $~$D$~$ |
---|
Decision problem | $$~$
CONNECTED = \{s\in\{0,1\}^*:\text{$s$ represents a connected graph}\}
$~$$ |
---|
Decision problem | $~$TAUTOLOGY$~$ |
---|
Decision problem | $~$TAUTOLOGY$~$ |
---|
Decision problem | $$~$
PRIMES = \{ x\in \mathbb{N}:\text{$x$ is prime}\}
$~$$ |
---|
Decision problem | $~$PRIMES$~$ |
---|
Decision problem | $$~$
PRIMES = \{s\in\{0,1\}^*:\text{$s$ represent a prime number in base $2$}\}
$~$$ |
---|
Decit | $~$\log_2(10)\approx 3.32$~$ |
---|
Dependent messages can be encoded cheaply | $~$m_1, m_2, m_3$~$ |
---|
Dependent messages can be encoded cheaply | $~$E$~$ |
---|
Dependent messages can be encoded cheaply | $~$E(m_1)E(m_2)E(m_3)$~$ |
---|
Dependent messages can be encoded cheaply | $~$(m_1, m_2, m_3)$~$ |
---|
Derivative | $~$y$~$ |
---|
Derivative | $~$x$~$ |
---|
Derivative | $~$y$~$ |
---|
Derivative | $~$x$~$ |
---|
Derivative | $~$f(x)$~$ |
---|
Derivative | $~$x$~$ |
---|
Derivative | $~$f(x)$~$ |
---|
Derivative | $~$(x, f(x))$~$ |
---|
Derivative | $~$t = 0$~$ |
---|
Derivative | $~$4.7 t^2$~$ |
---|
Derivative | $~$t$~$ |
---|
Derivative | $$~$\frac{\mathrm{d}}{\mathrm{d} t} mileage = speed$~$$ |
---|
Derivative | $~$t$~$ |
---|
Derivative | $~$t$~$ |
---|
Derivative | $~$t$~$ |
---|
Derivative | $~$4.7 t^2$~$ |
---|
Derivative | $$~$\frac{\mathrm{d}}{\mathrm{d} t} 4.7 t^2 = speed$~$$ |
---|
Derivative | $$~$distance\ traveled = 2t$~$$ |
---|
Derivative | $~$distance\ traveled = 2t$~$ |
---|
Derivative | $~$distance\ traveled = t^2$~$ |
---|
Derivative | $~$t=1$~$ |
---|
Derivative | $~$d = t^2$~$ |
---|
Derivative | $~$d$~$ |
---|
Derivative | $~$t$~$ |
---|
Derivative | $~$t$~$ |
---|
Derivative | $~$\frac{\Delta d}{\Delta t}$~$ |
---|
Derivative | $~$(t,t^2)$~$ |
---|
Derivative | $~$h$~$ |
---|
Derivative | $~$((t+h),(t+h)^2)$~$ |
---|
Derivative | $$~$∆d=(t+h)^2-t^2$~$$ |
---|
Derivative | $$~$∆t=(t+h) - t$~$$ |
---|
Derivative | $$~$∆d=2ht + h^2$~$$ |
---|
Derivative | $$~$∆t=h$~$$ |
---|
Derivative | $$~$\frac{\Delta d}{\Delta t}=\frac{2ht + h^2}{h}=2t+h$~$$ |
---|
Derivative | $~$h$~$ |
---|
Derivative | $~$2t$~$ |
---|
Derivative | $~$t$~$ |
---|
Derivative | $~$1$~$ |
---|
Derivative | $~$2$~$ |
---|
Derivative | $~$t$~$ |
---|
Derivative | $~$5$~$ |
---|
Derivative | $~$10$~$ |
---|
Derivative | $~$t^2$~$ |
---|
Derivative | $~$2t$~$ |
---|
Derivative | $~$4.7t^2$~$ |
---|
Derivative | $~$9.4t$~$ |
---|
Derivative | $~$t=0$~$ |
---|
Derivative | $~$t$~$ |
---|
Derivative | $~$9.4t$~$ |
---|
Derivative | $~$t^2$~$ |
---|
Derivative | $~$2t$~$ |
---|
Derivative | $~$t$~$ |
---|
Derivative | $~$t$~$ |
---|
Derivative | $~$c$~$ |
---|
Derivative | $~$n$~$ |
---|
Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}c=0$~$$ |
---|
Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}ct=c$~$$ |
---|
Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}ct^2=2ct$~$$ |
---|
Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}ct^2=3ct^2$~$$ |
---|
Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}ct^n=nct^{n-1}$~$$ |
---|
Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}e^t=e^t$~$$ |
---|
Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}sin(t)=cos(t)$~$$ |
---|
Derivative | $$~$\frac{\mathrm{d} }{\mathrm{d} t}cos(t)=-sin(t)$~$$ |
---|
Diagonal lemma | $~$T$~$ |
---|
Diagonal lemma | $~$S$~$ |
---|
Diagonal lemma | $~$T\vdash S\iff F(\ulcorner S \urcorner)$~$ |
---|
Diagonal lemma | $~$\phi(x)$~$ |
---|
Diagonal lemma | $~$T$~$ |
---|
Diagonal lemma | $~$\phi(x)$~$ |
---|
Diagonal lemma | $~$x$~$ |
---|
Diagonal lemma | $~$S$~$ |
---|
Diagonal lemma | $~$T\vdash S\leftrightarrow \phi(\ulcorner S\urcorner)$~$ |
---|
Diagonal lemma | $~$\neg \square_{PA} (x)$~$ |
---|
Diagonal lemma | $~$PA$~$ |
---|
Diagonal lemma | $~$x$~$ |
---|
Diagonal lemma | $~$G$~$ |
---|
Diagonal lemma | $~$PA\vdash G\leftrightarrow \neg \square_{PA} (\ulcorner G\urcorner)$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
\vec{y_{n}}=\mathbf{W_n}^T \times \vec{y_{n-1}} + \vec{b_n}
$~$$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$n$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$\vec{y_n}$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$n^{th}$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$l_n \times 1$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$l_n$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$n^th$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$\mathbf{W_n}$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$l_{n-1} \times l_{n}$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$n$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$n-1$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$\vec{b_n}$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$n^th$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$(n-1)^th$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$l_n\times1$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$w$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
f(x)=w\times x
$~$$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$f(x)=m\times x$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $~$y=mx+b$~$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
\vec{y_{n}}=\mathbf{W_n}^T \times \vec{y_{n-1}} + 1 \times \vec{b_n}
$~$$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
\vec{y_{n}}=
\left[ \begin{array}{c}
x, \\ 1
\end{array} \right]^T
\cdot
\left[ \begin{array}{c}
\mathbf{W_n},
\\ \vec{b_n}
\end{array} \right]
$~$$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
\vec{y_{n}} = \vec{y_{new_{n-1}}}^T \times \vec{W_{new}}
$~$$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
\vec{W_{new}} =\vec{W_{new}}-\frac{\delta W_{new}}{\delta Error}
$~$$ |
---|
Difference Between Weights and Biases: Another way of Looking at Forward Propagation | $$~$
\vec{W_{new}} =Activation(\vec{W_{new}}-\frac{\delta W_{new}}{\delta Error})
$~$$ |
---|
Dihedral group | $~$D_{2n}$~$ |
---|
Dihedral group | $~$n$~$ |
---|
Dihedral group | $$~$D_{2n} \cong \langle a, b \mid a^n, b^2, b a b^{-1} = a^{-1} \rangle$~$$ |
---|
Dihedral group | $~$a$~$ |
---|
Dihedral group | $~$b$~$ |
---|
Dihedral group | $~$D_{2n}$~$ |
---|
Dihedral group | $~$n > 2$~$ |
---|
Dihedral group | $~$D_{2n}$~$ |
---|
Dihedral group | $~$S_n$~$ |
---|
Dihedral group | $~$a = (123 \dots n)$~$ |
---|
Dihedral group | $~$b = (2, n)(3, n-1) \dots (\frac{n}{2}+1, \frac{n}{2}+3)$~$ |
---|
Dihedral group | $~$n$~$ |
---|
Dihedral group | $~$b = (2, n)(3, n-1)\dots(\frac{n-1}{2}, \frac{n+1}{2})$~$ |
---|
Dihedral group | $~$n$~$ |
---|
Dihedral group | $~$D_6$~$ |
---|
Dihedral group | $~$\langle a, b \mid b^2, b a b^{-1} = a^{-1} \rangle$~$ |
---|
Dihedral group | $~$D_{2n}$~$ |
---|
Dihedral group | $~$\mathbb{R}^2$~$ |
---|
Dihedral group | $~$x=0$~$ |
---|
Dihedral group | $~$D_{2n}$~$ |
---|
Dihedral groups are non-abelian | $~$n \geq 3$~$ |
---|
Dihedral groups are non-abelian | $~$n$~$ |
---|
Dihedral groups are non-abelian | $~$D_{2n}$~$ |
---|
Dihedral groups are non-abelian | $~$\langle a, b \mid a^n, b^2, bab^{-1} = a^{-1} \rangle$~$ |
---|
Dihedral groups are non-abelian | $~$ba = a^{-1} b = a^{-2} a b$~$ |
---|
Dihedral groups are non-abelian | $~$ab = ba$~$ |
---|
Dihedral groups are non-abelian | $~$a^2$~$ |
---|
Dihedral groups are non-abelian | $~$a$~$ |
---|
Dihedral groups are non-abelian | $~$n > 2$~$ |
---|
Dihedral groups are non-abelian | $~$ab$~$ |
---|
Dihedral groups are non-abelian | $~$ba$~$ |
---|
Direct sum of vector spaces | $~$U$~$ |
---|
Direct sum of vector spaces | $~$W,$~$ |
---|
Direct sum of vector spaces | $~$U \oplus W,$~$ |
---|
Direct sum of vector spaces | $~$U$~$ |
---|
Direct sum of vector spaces | $~$W,$~$ |
---|
Direct sum of vector spaces | $~$U$~$ |
---|
Direct sum of vector spaces | $~$W$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$(a_1 a_2 \dots a_k)$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$(b_1 b_2 \dots b_m)$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$S_n$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$a_i, b_j$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$S_n$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$\sigma$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$(a_1 a_2 \dots a_k)$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$(b_1 b_2 \dots b_m)$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$\tau$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$(b_1 b_2 \dots b_m)$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$(a_1 a_2 \dots a_k)$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$\sigma(a_i) = (b_1 b_2 \dots b_m)[(a_1 a_2 \dots a_k)(a_i)] = (b_1 b_2 \dots b_m)(a_{i+1}) = a_{i+1}$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$a_{k+1}$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$a_1$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$\tau(a_i) = (a_1 a_2 \dots a_k)[(b_1 b_2 \dots b_m)(a_i)] = (a_1 a_2 \dots a_k)(a_i) = a_{i+1}$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$(a_1 a_2 \dots a_k)$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$(b_1 b_2 \dots b_m)$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$a_i$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$b_j$~$ |
---|
Disjoint cycles commute in symmetric groups | $~$\{1,2,\dots, n\}$~$ |
---|
Disjoint union of sets | $~$\sqcup$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$A \sqcup B$~$ |
---|
Disjoint union of sets | $~$A = \{6,7\}$~$ |
---|
Disjoint union of sets | $~$B = \{8, 9\}$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$\{6,7,8,9\}$~$ |
---|
Disjoint union of sets | $~$A \sqcup B = \{6,7,8,9\}$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$A \cup B$~$ |
---|
Disjoint union of sets | $~$\sqcup$~$ |
---|
Disjoint union of sets | $~$\cup$~$ |
---|
Disjoint union of sets | $~$\{1,2\} \sqcup \{1,3\} = \{1,2,3\}$~$ |
---|
Disjoint union of sets | $~$\{1,2\} \cup \{1,3\} = \{1,2,3\}$~$ |
---|
Disjoint union of sets | $~$1$~$ |
---|
Disjoint union of sets | $~$A = \{6,7\}$~$ |
---|
Disjoint union of sets | $~$B = \{6,8\}$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$a$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$1$~$ |
---|
Disjoint union of sets | $~$(a, 1)$~$ |
---|
Disjoint union of sets | $~$a$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$A'$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $$~$A' = \{ (a, 1) : a \in A \}$~$$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$1$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$2$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$B'$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $$~$B' = \{ (b,2) : b \in B \}$~$$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$A'$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$A'$~$ |
---|
Disjoint union of sets | $~$a \mapsto (a,1)$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$B'$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$A'$~$ |
---|
Disjoint union of sets | $~$B'$~$ |
---|
Disjoint union of sets | $~$A'$~$ |
---|
Disjoint union of sets | $~$1$~$ |
---|
Disjoint union of sets | $~$B'$~$ |
---|
Disjoint union of sets | $~$2$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$A' \sqcup B'$~$ |
---|
Disjoint union of sets | $~$\sqcup$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$A = \{6,7\}$~$ |
---|
Disjoint union of sets | $~$B=\{6,8\}$~$ |
---|
Disjoint union of sets | $~$A = \{6,7\}$~$ |
---|
Disjoint union of sets | $~$B=\{6,8\}$~$ |
---|
Disjoint union of sets | $~$\sqcup$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$6$~$ |
---|
Disjoint union of sets | $~$A' = \{ (6, 1), (7, 1) \}$~$ |
---|
Disjoint union of sets | $~$B' = \{ (6, 2), (8, 2) \}$~$ |
---|
Disjoint union of sets | $$~$A \sqcup B = \{ (6,1), (7,1), (6,2), (8,2) \}$~$$ |
---|
Disjoint union of sets | $~$A \cup B = \{ 6, 7, 8 \}$~$ |
---|
Disjoint union of sets | $~$6$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$A \sqcup B$~$ |
---|
Disjoint union of sets | $~$6$~$ |
---|
Disjoint union of sets | $~$(6,1)$~$ |
---|
Disjoint union of sets | $~$(6,2)$~$ |
---|
Disjoint union of sets | $~$A = \{1,2\}$~$ |
---|
Disjoint union of sets | $~$B = \{3,4\}$~$ |
---|
Disjoint union of sets | $~$A \sqcup B$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$A \cup B = \{1,2,3,4\}$~$ |
---|
Disjoint union of sets | $~$A' \cup B' = \{(1,1), (2,1), (3,2), (4,2) \}$~$ |
---|
Disjoint union of sets | $~$A' = \{(1,1), (2,1)\}$~$ |
---|
Disjoint union of sets | $~$B' = \{(3,2), (4,2) \}$~$ |
---|
Disjoint union of sets | $~$A = B = \{6,7\}$~$ |
---|
Disjoint union of sets | $~$A' = \{(6,1), (7,1)\}$~$ |
---|
Disjoint union of sets | $~$B' = \{(6,2), (7,2)\}$~$ |
---|
Disjoint union of sets | $$~$A \sqcup B = \{(6,1),(7,1),(6,2),(7,2)\}$~$$ |
---|
Disjoint union of sets | $~$A = \mathbb{N}$~$ |
---|
Disjoint union of sets | $~$B = \{ 1, 2, x \}$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$\mathbb{N}$~$ |
---|
Disjoint union of sets | $~$0$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$\{1,2,x\}$~$ |
---|
Disjoint union of sets | $~$x$~$ |
---|
Disjoint union of sets | $~$A \sqcup B$~$ |
---|
Disjoint union of sets | $~$A' = \{ (0,1), (1,1), (2,1), (3,1), \dots\}$~$ |
---|
Disjoint union of sets | $~$B' = \{(1,2), (2,2), (x,2)\}$~$ |
---|
Disjoint union of sets | $$~$\{(0,1), (1,1),(2,1),(3,1), \dots, (1,2),(2,2),(x,2)\}$~$$ |
---|
Disjoint union of sets | $~$A = \mathbb{N}$~$ |
---|
Disjoint union of sets | $~$B = \{x, y\}$~$ |
---|
Disjoint union of sets | $~$A \sqcup B$~$ |
---|
Disjoint union of sets | $~$\{ 0,1,2,\dots, x, y \}$~$ |
---|
Disjoint union of sets | $~$\{(0,1), (1,1), (2,1), \dots, (x,2), (y,2)\}$~$ |
---|
Disjoint union of sets | $~$A \sqcup B \sqcup C$~$ |
---|
Disjoint union of sets | $~$A \sqcup B$~$ |
---|
Disjoint union of sets | $~$A \cup B \cup C$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$C$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$B$~$ |
---|
Disjoint union of sets | $~$C$~$ |
---|
Disjoint union of sets | $~$A$~$ |
---|
Disjoint union of sets | $~$C$~$ |
---|
Disjoint union of sets | $~$A' = \{(a, 1) : a \in A \}$~$ |
---|
Disjoint union of sets | $~$B' = \{ (b, 2) : b \in B \}$~$ |
---|
Disjoint union of sets | $~$C' = \{ (c, 3) : c \in C \}$~$ |
---|
Disjoint union of sets | $~$A \sqcup B \sqcup C$~$ |
---|
Disjoint union of sets | $~$A' \cup B' \cup C'$~$ |
---|
Disjoint union of sets | $$~$\bigsqcup_{i \in I} A_i = \bigcup_{i \in I} A_i$~$$ |
---|
Disjoint union of sets | $~$A_i$~$ |
---|
Disjoint union of sets | $$~$\bigsqcup_{i \in I} A_i = \bigcup_{i \in I} A'_i$~$$ |
---|
Disjoint union of sets | $~$A'_i = \{ (a, i) : a \in A_i \}$~$ |
---|
Disjoint union of sets | $$~$\bigsqcup_{n \in \mathbb{N}} \{0, 1,2,\dots,n\} = \{(0,0)\} \cup \{(0,1), (1,1) \} \cup \{ (0,2), (1,2), (2,2)\} \cup \dots = \{ (n, m) : n \leq m \}$~$$ |
---|
Disjoint union of sets | $~$A \sqcup B$~$ |
---|
Disjoint union of sets | $~$A' \cup B'$~$ |
---|
Disjoint union of sets | $~$A' = \{ (a, 2) : a \in A \}$~$ |
---|
Disjoint union of sets | $~$B' = \{ (b,1) : b \in B \}$~$ |
---|
Division of rational numbers (Math 0) | $~$1$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{4}{3}$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|
Division of rational numbers (Math 0) | $~$1$~$ |
---|
Division of rational numbers (Math 0) | $$~$1 + \frac{1}{3} = \frac{1}{1} + \frac{1}{3} = \frac{3 \times 1 + 1 \times 1}{3 \times 1} = \frac{3+1}{3} = \frac{4}{3}$~$$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{4}{3}$~$ |
---|
Division of rational numbers (Math 0) | $~$x$~$ |
---|
Division of rational numbers (Math 0) | $~$y$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{x}{y}$~$ |
---|
Division of rational numbers (Math 0) | $~$x$~$ |
---|
Division of rational numbers (Math 0) | $~$y$~$ |
---|
Division of rational numbers (Math 0) | $~$a/n$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|
Division of rational numbers (Math 0) | $~$1$~$ |
---|
Division of rational numbers (Math 0) | $~$m$~$ |
---|
Division of rational numbers (Math 0) | $~$1$~$ |
---|
Division of rational numbers (Math 0) | $~$m$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|
Division of rational numbers (Math 0) | $~$n$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|
Division of rational numbers (Math 0) | $~$n$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|
Division of rational numbers (Math 0) | $~$a$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|
Division of rational numbers (Math 0) | $~$n$~$ |
---|
Division of rational numbers (Math 0) | $~$a$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|
Division of rational numbers (Math 0) | $~$n$~$ |
---|
Division of rational numbers (Math 0) | $~$n$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{m} \times \frac{1}{n}$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
---|
Division of rational numbers (Math 0) | $$~$\frac{a}{m} / n = \frac{a}{m \times n}$~$$ |
---|
Division of rational numbers (Math 0) | $~$x$~$ |
---|
Division of rational numbers (Math 0) | $~$x$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{-1}$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{1} = 1$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{1}{1} = 1$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{-1}{-1}$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{-1}{-1}$~$ |
---|
Division of rational numbers (Math 0) | $~$1$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{-1}{-1} = 1$~$ |
---|
Division of rational numbers (Math 0) | $~$\frac{a}{m} \times \frac{b}{n} = \frac{a \times b}{m \times n}$~$ |
---|
Division of rational numbers (Math 0) | $$~$\frac{1}{-m} = \frac{1}{-m} \times 1 = \frac{1}{-m} \times \frac{-1}{-1} = \frac{-1 \times 1}{-m \times -1} = \frac{-1}{m}$~$$ |
---|
Division of rational numbers (Math 0) | $~$\frac{a}{-b} = \frac{-a}{b}$~$ |
---|
Domain (of a function) | $~$\operatorname{dom}(f)$~$ |
---|
Domain (of a function) | $~$f : X \to Y$~$ |
---|
Domain (of a function) | $~$X$~$ |
---|
Domain (of a function) | $~$+$~$ |
---|
Domain (of a function) | $~$(x, y)$~$ |
---|
Domain (of a function) | $~$y$~$ |
---|
Effective number of political parties | $~$1, 2, \ldots, n$~$ |
---|
Effective number of political parties | $~$p_n$~$ |
---|
Effective number of political parties | $~$n$~$ |
---|
Effective number of political parties | $~$0$~$ |
---|
Effective number of political parties | $~$1$~$ |
---|
Effective number of political parties | $~$\displaystyle \frac{1}{\sum_{i=1}^n p_i^2}$~$ |
---|
Effective number of political parties | $~$x$~$ |
---|
Effective number of political parties | $~$n$~$ |
---|
Effective number of political parties | $~$n$~$ |
---|
Effective number of political parties | $~$n$~$ |
---|
Effective number of political parties | $~$k$~$ |
---|
Effective number of political parties | $~$k$~$ |
---|
Effective number of political parties | $~$k = 1$~$ |
---|
Effective number of political parties | $~$n$~$ |
---|
Effective number of political parties | $~$p_i$~$ |
---|
Effective number of political parties | $~$n$~$ |
---|
Effective number of political parties | $~$1/n$~$ |
---|
Effective number of political parties | $~$p_i$~$ |
---|
Effective number of political parties | $~$p_i$~$ |
---|
Effective number of political parties | $~$p_i$~$ |
---|
Effective number of political parties | $~$(p_1 \cdot p_1) + (p_2 \cdot p_2) + \ldots + (p_n \cdot p_n) = \sum_{i=1}^n p_i^2$~$ |
---|
Eigenvalues and eigenvectors | $~$A$~$ |
---|
Eigenvalues and eigenvectors | $~$v$~$ |
---|
Eigenvalues and eigenvectors | $~$Av = \lambda v$~$ |
---|
Eigenvalues and eigenvectors | $~$v$~$ |
---|
Eigenvalues and eigenvectors | $~$A$~$ |
---|
Eigenvalues and eigenvectors | $~$\lambda$~$ |
---|
Eigenvalues and eigenvectors | $~$A$~$ |
---|
Eigenvalues and eigenvectors | $~$v$~$ |
---|
Eigenvalues and eigenvectors | $~$|\lambda| > 1$~$ |
---|
Eigenvalues and eigenvectors | $~$|\lambda| < 1$~$ |
---|
Eigenvalues and eigenvectors | $~$\lambda < 0$~$ |
---|
Elementary Algebra | $$~$2 + 2 = 4$~$$ |
---|
Elementary Algebra | $~$2 < 4$~$ |
---|
Elementary Algebra | $~$5 > 1$~$ |
---|
Elementary Algebra | $$~$2 + (3 \times 4) = 14$~$$ |
---|
Elementary Algebra | $$~$(2 + 3) \times 4 = 20$~$$ |
---|
Elementary Algebra | $~$2 + 3 \times 4$~$ |
---|
Elementary Algebra | $~$2 + (3 \times 4)$~$ |
---|
Elementary Algebra | $~$2+2=4$~$ |
---|
Elementary Algebra | $~$(2 + 2) + 3 = 4 + 3$~$ |
---|
Elementary Algebra | $~$2^3 \times 2^4$~$ |
---|
Elementary Algebra | $~$2^3 = 2 \times 2 \times 2$~$ |
---|
Elementary Algebra | $~$2 \times 2 \times 2 = 8$~$ |
---|
Elementary Algebra | $~$2^3 = 8$~$ |
---|
Elementary Algebra | $~$2^4 = 2 \times 2 \times 2 \times 2 = 16$~$ |
---|
Elementary Algebra | $~$2^3 \times 2^4 = 8 \times 16$~$ |
---|
Elementary Algebra | $~$2^3\times 2^4 = 128$~$ |
---|
Elementary Algebra | $~$0 \times 3 = 0$~$ |
---|
Elementary Algebra | $~$0 \times -4 = 0$~$ |
---|
Elementary Algebra | $~$0 \times 1224 = 0$~$ |
---|
Elementary Algebra | $~$0 \times \text{any number} = 0$~$ |
---|
Elementary Algebra | $~$0 \times x = 0$~$ |
---|
Elementary Algebra | $~$x$~$ |
---|
Elementary Algebra | $$~$ a + b = b + a$~$$ |
---|
Elementary Algebra | $$~$ a \times b = b\times a$~$$ |
---|
Elementary Algebra | $$~$ 0 + a = a$~$$ |
---|
Elementary Algebra | $$~$ 1 \times a = a$~$$ |
---|
Elementary Algebra | $$~$ (a + b) + c = a + (b + c)$~$$ |
---|
Elementary Algebra | $$~$ (a \times b ) \times c = a \times (b\times c)$~$$ |
---|
Elementary Algebra | $$~$ a \times (b + c) = a\times b + a\times c$~$$ |
---|
Elementary Algebra | $$~$ a + (-a) = a - a = 0 $~$$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$1$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$1$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$1,$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$1$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$1$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$1$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$1$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$0$~$ |
---|
Empirical probabilities are not exactly 0 or 1 | $~$1,$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $$~$\exists B \forall x : x∉B$~$$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$A$~$ |
---|
Empty set | $~$B$~$ |
---|
Empty set | $~$\forall x : x∉A$~$ |
---|
Empty set | $~$\forall x: x∉B$~$ |
---|
Empty set | $~$\forall x : (x ∈ A \Leftrightarrow x ∈ B)$~$ |
---|
Empty set | $~$A=B$~$ |
---|
Empty set | $~$x$~$ |
---|
Empty set | $~$(x ∈ A \Leftrightarrow x ∈ B)$~$ |
---|
Empty set | $~$x \not \in A$~$ |
---|
Empty set | $~$x \not \in B$~$ |
---|
Empty set | $~$\phi$~$ |
---|
Empty set | $~$\forall a \exists b \forall x : x \in b \Leftrightarrow (x \in a \wedge \phi(x))$~$ |
---|
Empty set | $~$\phi$~$ |
---|
Empty set | $~$\bot$~$ |
---|
Empty set | $~$\forall a \exists b \forall x : x \in b \Leftrightarrow (x \in a \wedge \bot)$~$ |
---|
Empty set | $~$x \in b \Leftrightarrow (x \in a \wedge \bot)$~$ |
---|
Empty set | $~$x \in b \Leftrightarrow \bot$~$ |
---|
Empty set | $~$x \notin b$~$ |
---|
Empty set | $~$\forall a \exists b \forall x : x \notin b$~$ |
---|
Empty set | $~$a$~$ |
---|
Empty set | $~$\{\emptyset\}$~$ |
---|
Empty set | $~$\{\emptyset\} \not= \emptyset$~$ |
---|
Empty set | $~$\emptyset ∈ \{\emptyset\}$~$ |
---|
Empty set | $~$\emptyset ∉ \emptyset$~$ |
---|
Empty set | $~$\{\emptyset\}$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$|\{\emptyset\}| = 1$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$|\emptyset| = 0$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$x$~$ |
---|
Empty set | $~$x$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$X$~$ |
---|
Empty set | $~$X$~$ |
---|
Empty set | $~$x$~$ |
---|
Empty set | $~$X$~$ |
---|
Empty set | $~$x$~$ |
---|
Empty set | $~$X$~$ |
---|
Empty set | $~$X$~$ |
---|
Empty set | $~$X$~$ |
---|
Empty set | $~$A$~$ |
---|
Empty set | $~$X$~$ |
---|
Empty set | $~$A$~$ |
---|
Empty set | $~$A$~$ |
---|
Empty set | $~$B$~$ |
---|
Empty set | $~$A$~$ |
---|
Empty set | $~$B$~$ |
---|
Empty set | $~$B$~$ |
---|
Empty set | $~$A$~$ |
---|
Empty set | $~$A = B$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$\{ \emptyset \}$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$\{\emptyset\}$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$P$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$P$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Empty set | $~$\emptyset$~$ |
---|
Emulating digits | $~$n$~$ |
---|
Emulating digits | $~$m$~$ |
---|
Emulating digits | $~$m, n \in$~$ |
---|
Emulating digits | $~$\mathbb N$~$ |
---|
Emulating digits | $~$m < n,$~$ |
---|
Emulating digits | $~$m$~$ |
---|
Emulating digits | $~$n$~$ |
---|
Emulating digits | $~$7$~$ |
---|
Emulating digits | $~$m > n,$~$ |
---|
Emulating digits | $~$n$~$ |
---|
Emulating digits | $~$n^2$~$ |
---|
Emulating digits | $~$n$~$ |
---|
Emulating digits | $~$(x, y)$~$ |
---|
Emulating digits | $~$0 \le x < n$~$ |
---|
Emulating digits | $~$0 \le y < n$~$ |
---|
Emulating digits | $~$(x, y)$~$ |
---|
Emulating digits | $~$xn + y.$~$ |
---|
Emulating digits | $~$x = y = 0$~$ |
---|
Emulating digits | $~$n^2 - 1$~$ |
---|
Emulating digits | $~$x = y = n-1$~$ |
---|
Emulating digits | $~$n$~$ |
---|
Emulating digits | $~$n^2$~$ |
---|
Emulating digits | $~$n^3$~$ |
---|
Emulating digits | $~$(x, y, z)$~$ |
---|
Emulating digits | $~$xn^2 + yn + z$~$ |
---|
Emulating digits | $~$n^4$~$ |
---|
Emulating digits | $~$m$~$ |
---|
Emulating digits | $~$a$~$ |
---|
Emulating digits | $~$n^a > m,$~$ |
---|
Emulating digits | $~$a$~$ |
---|
Emulating digits | $~$n$~$ |
---|
Emulating digits | $~$n$~$ |
---|
Emulating digits | $~$m$~$ |
---|
Emulating digits | $~$m$~$ |
---|
Emulating digits | $~$n$~$ |
---|
Emulating digits | $~$m$~$ |
---|
Emulating digits | $~$m$~$ |
---|
Emulating digits | $~$m$~$ |
---|
Emulating digits | $~$m$~$ |
---|
Encoding trits with GalCom bits | $~$\log_2(3) \approx 1.585$~$ |
---|
Encoding trits with GalCom bits | $~$2 - \frac{1}{3} \approx 1.67$~$ |
---|
Environmental goals | $~$E_{1,t} \ldots E_{n,t}$~$ |
---|
Environmental goals | $~$t.$~$ |
---|
Environmental goals | $~$S_t$~$ |
---|
Environmental goals | $~$E_t$~$ |
---|
Environmental goals | $~$A_t$~$ |
---|
Environmental goals | $~$t.$~$ |
---|
Environmental goals | $~$R_t$~$ |
---|
Environmental goals | $~$E_t$~$ |
---|
Environmental goals | $~$A_t$~$ |
---|
Environmental goals | $~$E_{t+1}$~$ |
---|
Environmental goals | $~$E_t$~$ |
---|
Environmental goals | $~$A_t.$~$ |
---|
Environmental goals | $~$E_{1,t}$~$ |
---|
Environmental goals | $~$t.$~$ |
---|
Environmental goals | $~$A_t$~$ |
---|
Environmental goals | $~$\theta.$~$ |
---|
Environmental goals | $~$A_t$~$ |
---|
Environmental goals | $~$\theta$~$ |
---|
Environmental goals | $~$E_1 \ldots E_m$~$ |
---|
Environmental goals | $~$E_{m+1} \ldots E_n$~$ |
---|
Environmental goals | $~$E_{m+1} \ldots E_n$~$ |
---|
Environmental goals | $~$R$~$ |
---|
Environmental goals | $~$E_1 \ldots E_m$~$ |
---|
Environmental goals | $~$E_1$~$ |
---|
Environmental goals | $~$E_{m+1, t} \ldots E_{n,t} = 0 \implies R_t=E_{1, t}.$~$ |
---|
Environmental goals | $~$E_{m+1} \ldots E_n,$~$ |
---|
Environmental goals | $~$E_1$~$ |
---|
Environmental goals | $~$E_1.$~$ |
---|
Environmental goals | $~$R$~$ |
---|
Environmental goals | $~$R$~$ |
---|
Environmental goals | $~$E_1$~$ |
---|
Environmental goals | $~$E_1$~$ |
---|
Environmental goals | $~$A_t$~$ |
---|
Environmental goals | $~$S_{t+1}$~$ |
---|
Environmental goals | $~$S_{1, t}$~$ |
---|
Environmental goals | $~$E_{1, t},$~$ |
---|
Environmental goals | $~$S_1.$~$ |
---|
Environmental goals | $~$S_1$~$ |
---|
Environmental goals | $~$R$~$ |
---|
Environmental goals | $~$Q$~$ |
---|
Environmental goals | $~$E_1$~$ |
---|
Environmental goals | $~$S_1$~$ |
---|
Environmental goals | $~$E_1.$~$ |
---|
Environmental goals | $~$S_1,$~$ |
---|
Environmental goals | $~$E_1$~$ |
---|
Environmental goals | $~$E_1$~$ |
---|
Environmental goals | $~$E_1,$~$ |
---|
Environmental goals | $~$E_1$~$ |
---|
Environmental goals | $~$R$~$ |
---|
Environmental goals | $~$Q$~$ |
---|
Environmental goals | $~$R$~$ |
---|
Environmental goals | $~$Q.$~$ |
---|
Environmental goals | $~$R$~$ |
---|
Environmental goals | $~$E_1.$~$ |
---|
Environmental goals | $~$Q$~$ |
---|
Environmental goals | $~$R$~$ |
---|
Environmental goals | $~$Q.$~$ |
---|
Environmental goals | $~$R$~$ |
---|
Environmental goals | $~$E_1$~$ |
---|
Environmental goals | $~$R$~$ |
---|
Environmental goals | $~$R.$~$ |
---|
Environmental goals | $~$E_1$~$ |
---|
Environmental goals | $~$E_1.$~$ |
---|
Environmental goals | $~$S$~$ |
---|
Environmental goals | $~$E_1.$~$ |
---|
Environmental goals | $~$Q$~$ |
---|
Environmental goals | $~$S$~$ |
---|
Environmental goals | $~$E_1.$~$ |
---|
Equaliser (category theory) | $~$f, g: A \to B$~$ |
---|
Equaliser (category theory) | $~$E$~$ |
---|
Equaliser (category theory) | $~$e: E \to A$~$ |
---|
Equaliser (category theory) | $~$ge = fe$~$ |
---|
Equaliser (category theory) | $~$ge = fe$~$ |
---|
Equaliser (category theory) | $~$X$~$ |
---|
Equaliser (category theory) | $~$x: X \to A$~$ |
---|
Equaliser (category theory) | $~$fx = gx$~$ |
---|
Equaliser (category theory) | $~$\bar{x} : X \to A$~$ |
---|
Equaliser (category theory) | $~$e \bar{x} = x$~$ |
---|
Equivalence relation | $~$\sim$~$ |
---|
Equivalence relation | $~$S$~$ |
---|
Equivalence relation | $~$S$~$ |
---|
Equivalence relation | $~$x \in S$~$ |
---|
Equivalence relation | $~$x \sim x$~$ |
---|
Equivalence relation | $~$x,y \in S$~$ |
---|
Equivalence relation | $~$x \sim y$~$ |
---|
Equivalence relation | $~$y \sim x$~$ |
---|
Equivalence relation | $~$x,y,z \in S$~$ |
---|
Equivalence relation | $~$x \sim y$~$ |
---|
Equivalence relation | $~$y \sim z$~$ |
---|
Equivalence relation | $~$x \sim z$~$ |
---|
Equivalence relation | $~$S$~$ |
---|
Equivalence relation | $~$\sim$~$ |
---|
Equivalence relation | $~$S$~$ |
---|
Equivalence relation | $~$x \in S$~$ |
---|
Equivalence relation | $~$S$~$ |
---|
Equivalence relation | $~$x$~$ |
---|
Equivalence relation | $~$[x]=\{y \in S \mid x \sim y\}$~$ |
---|
Equivalence relation | $~$x$~$ |
---|
Equivalence relation | $~$[x]$~$ |
---|
Equivalence relation | $~$S/\sim = \{[x] \mid x \in S\}$~$ |
---|
Equivalence relation | $~$x \in [x]$~$ |
---|
Equivalence relation | $~$[x]=[y]$~$ |
---|
Equivalence relation | $~$x \sim y$~$ |
---|
Equivalence relation | $~$S$~$ |
---|
Equivalence relation | $~$A$~$ |
---|
Equivalence relation | $~$x \sim y$~$ |
---|
Equivalence relation | $~$U \in A$~$ |
---|
Equivalence relation | $~$x,y \in U$~$ |
---|
Equivalence relation | $~$[x] \in A$~$ |
---|
Equivalence relation | $~$A=S/\sim$~$ |
---|
Equivalence relation | $~$f: S \to U$~$ |
---|
Equivalence relation | $~$f^*: S/\sim \to U$~$ |
---|
Equivalence relation | $~$U$~$ |
---|
Equivalence relation | $~$f^*([x])$~$ |
---|
Equivalence relation | $~$f(x)$~$ |
---|
Equivalence relation | $~$x \sim y$~$ |
---|
Equivalence relation | $~$f(x) \neq f(y)$~$ |
---|
Equivalence relation | $~$f^*([x])=f^*([y])$~$ |
---|
Equivalence relation | $~$x \sim y$~$ |
---|
Equivalence relation | $~$f(x)=f(y)$~$ |
---|
Equivalence relation | $~$f: S \to S$~$ |
---|
Equivalence relation | $~$f^*: S/\sim \to S/\sim$~$ |
---|
Equivalence relation | $~$f^*([x])=[f(x)]$~$ |
---|
Equivalence relation | $~$x \sim y$~$ |
---|
Equivalence relation | $~$[f(x)]=[f(y)]$~$ |
---|
Equivalence relation | $~$f(x) \sim f(y)$~$ |
---|
Equivalence relation | $~$x \sim y$~$ |
---|
Equivalence relation | $~$n|x-y$~$ |
---|
Equivalence relation | $~$n \in \mathbb N$~$ |
---|
Equivalence relation | $~$n$~$ |
---|
Equivalence relation | $~$n$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$ab$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$a$~$ |
---|
Euclid's Lemma on prime numbers | $~$b$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$a$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$b$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$p \mid ab$~$ |
---|
Euclid's Lemma on prime numbers | $~$p \mid a$~$ |
---|
Euclid's Lemma on prime numbers | $~$p \mid b$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$ab$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$a$~$ |
---|
Euclid's Lemma on prime numbers | $~$b$~$ |
---|
Euclid's Lemma on prime numbers | $~$p \mid ab$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$ab$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$a$~$ |
---|
Euclid's Lemma on prime numbers | $~$p \mid b$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$a$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$a$~$ |
---|
Euclid's Lemma on prime numbers | $~$1$~$ |
---|
Euclid's Lemma on prime numbers | $~$x, y$~$ |
---|
Euclid's Lemma on prime numbers | $~$ax+py = 1$~$ |
---|
Euclid's Lemma on prime numbers | $~$p \mid ab$~$ |
---|
Euclid's Lemma on prime numbers | $~$p \mid a$~$ |
---|
Euclid's Lemma on prime numbers | $~$p \mid b$~$ |
---|
Euclid's Lemma on prime numbers | $~$a$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$c$~$ |
---|
Euclid's Lemma on prime numbers | $~$c \mid a$~$ |
---|
Euclid's Lemma on prime numbers | $~$c \mid p$~$ |
---|
Euclid's Lemma on prime numbers | $~$d$~$ |
---|
Euclid's Lemma on prime numbers | $~$a$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$d \mid c$~$ |
---|
Euclid's Lemma on prime numbers | $~$a$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$c \mid p$~$ |
---|
Euclid's Lemma on prime numbers | $~$c = p$~$ |
---|
Euclid's Lemma on prime numbers | $~$c=1$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$c$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$c \mid a$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$a$~$ |
---|
Euclid's Lemma on prime numbers | $~$c = 1$~$ |
---|
Euclid's Lemma on prime numbers | $~$b$~$ |
---|
Euclid's Lemma on prime numbers | $~$abx + pby = b$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$ab$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$p \mid b$~$ |
---|
Euclid's Lemma on prime numbers | $~$\mathbb{Z}$~$ |
---|
Euclid's Lemma on prime numbers | $~$\mathbb{Z}$~$ |
---|
Euclid's Lemma on prime numbers | $~$\mathbb{Z}$~$ |
---|
Euclid's Lemma on prime numbers | $~$pq$~$ |
---|
Euclid's Lemma on prime numbers | $~$p, q$~$ |
---|
Euclid's Lemma on prime numbers | $~$1$~$ |
---|
Euclid's Lemma on prime numbers | $~$pq$~$ |
---|
Euclid's Lemma on prime numbers | $~$p$~$ |
---|
Euclid's Lemma on prime numbers | $~$q$~$ |
---|
Euclidean domains are principal ideal domains | $~$R$~$ |
---|
Euclidean domains are principal ideal domains | $~$R$~$ |
---|
Euclidean domains are principal ideal domains | $~$\mathbb{Z}$~$ |
---|
Euclidean domains are principal ideal domains | $~$\mathbb{Z}$~$ |
---|
Euclidean domains are principal ideal domains | $~$\mathbb{Z}$~$ |
---|
Euclidean domains are principal ideal domains | $~$\mathbb{Z}$~$ |
---|
Euclidean domains are principal ideal domains | $~$R$~$ |
---|
Euclidean domains are principal ideal domains | $~$R$~$ |
---|
Euclidean domains are principal ideal domains | $~$\mathbb{Z}$~$ |
---|
Euclidean domains are principal ideal domains | $~$R$~$ |
---|
Euclidean domains are principal ideal domains | $~$n > 0$~$ |
---|
Euclidean domains are principal ideal domains | $~$n$~$ |
---|
Euclidean domains are principal ideal domains | $~$n < 0$~$ |
---|
Euclidean domains are principal ideal domains | $~$-n$~$ |
---|
Euclidean domains are principal ideal domains | $~$R$~$ |
---|
Euclidean domains are principal ideal domains | $~$\phi: \mathbb{R} \setminus \{ 0 \} \to \mathbb{N}^{\geq 0}$~$ |
---|
Euclidean domains are principal ideal domains | $~$a$~$ |
---|
Euclidean domains are principal ideal domains | $~$b$~$ |
---|
Euclidean domains are principal ideal domains | $~$\phi(a) \leq \phi(b)$~$ |
---|
Euclidean domains are principal ideal domains | $~$a$~$ |
---|
Euclidean domains are principal ideal domains | $~$b$~$ |
---|
Euclidean domains are principal ideal domains | $~$a$~$ |
---|
Euclidean domains are principal ideal domains | $~$q$~$ |
---|
Euclidean domains are principal ideal domains | $~$r$~$ |
---|
Euclidean domains are principal ideal domains | $~$a = qb+r$~$ |
---|
Euclidean domains are principal ideal domains | $~$\phi(r) < \phi(b)$~$ |
---|
Euclidean domains are principal ideal domains | $~$I \subseteq R$~$ |
---|
Euclidean domains are principal ideal domains | $~$I$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha: R \to S$~$ |
---|
Euclidean domains are principal ideal domains | $~$r \in R$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha(x) = 0$~$ |
---|
Euclidean domains are principal ideal domains | $~$x$~$ |
---|
Euclidean domains are principal ideal domains | $~$r$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha$~$ |
---|
Euclidean domains are principal ideal domains | $~$0$~$ |
---|
Euclidean domains are principal ideal domains | $~$0$~$ |
---|
Euclidean domains are principal ideal domains | $~$0$~$ |
---|
Euclidean domains are principal ideal domains | $~$r = 0$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha$~$ |
---|
Euclidean domains are principal ideal domains | $~$0$~$ |
---|
Euclidean domains are principal ideal domains | $~$r$~$ |
---|
Euclidean domains are principal ideal domains | $~$\phi$~$ |
---|
Euclidean domains are principal ideal domains | $~$r$~$ |
---|
Euclidean domains are principal ideal domains | $~$x$~$ |
---|
Euclidean domains are principal ideal domains | $~$r$~$ |
---|
Euclidean domains are principal ideal domains | $~$ar$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha(ar) = \alpha(a) \alpha(r) = \alpha(a) \times 0 = 0$~$ |
---|
Euclidean domains are principal ideal domains | $~$r$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha$~$ |
---|
Euclidean domains are principal ideal domains | $~$0$~$ |
---|
Euclidean domains are principal ideal domains | $~$x$~$ |
---|
Euclidean domains are principal ideal domains | $~$r$~$ |
---|
Euclidean domains are principal ideal domains | $~$x = ar+b$~$ |
---|
Euclidean domains are principal ideal domains | $~$\phi(b) < \phi(r)$~$ |
---|
Euclidean domains are principal ideal domains | $~$b$~$ |
---|
Euclidean domains are principal ideal domains | $~$\phi$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha(x) = \alpha(ar)+\alpha(b)$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha(r) = 0$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha(x) = \alpha(b)$~$ |
---|
Euclidean domains are principal ideal domains | $~$b$~$ |
---|
Euclidean domains are principal ideal domains | $~$\phi$~$ |
---|
Euclidean domains are principal ideal domains | $~$r$~$ |
---|
Euclidean domains are principal ideal domains | $~$r$~$ |
---|
Euclidean domains are principal ideal domains | $~$\phi$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha$~$ |
---|
Euclidean domains are principal ideal domains | $~$0$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha(b)$~$ |
---|
Euclidean domains are principal ideal domains | $~$0$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha(x)$~$ |
---|
Euclidean domains are principal ideal domains | $~$\alpha(x) = 0$~$ |
---|
Euclidean domains are principal ideal domains | $~$x$~$ |
---|
Euclidean domains are principal ideal domains | $~$r$~$ |
---|
Euclidean domains are principal ideal domains | $~$\mathbb{Z}[\frac{1}{2} (1+\sqrt{-19})]$~$ |
---|
Every group is a quotient of a free group | $~$G$~$ |
---|
Every group is a quotient of a free group | $~$F(X)$~$ |
---|
Every group is a quotient of a free group | $~$X$~$ |
---|
Every group is a quotient of a free group | $~$G$~$ |
---|
Every group is a quotient of a free group | $~$F(X)$~$ |
---|
Every group is a quotient of a free group | $~$T: \mathcal{C} \to \mathcal{C}$~$ |
---|
Every group is a quotient of a free group | $~$\mathcal{C}$~$ |
---|
Every group is a quotient of a free group | $~$(A, \alpha)$~$ |
---|
Every group is a quotient of a free group | $~$T$~$ |
---|
Every group is a quotient of a free group | $~$\alpha: TA \to A$~$ |
---|
Every group is a quotient of a free group | $~$F(G)$~$ |
---|
Every group is a quotient of a free group | $~$G$~$ |
---|
Every group is a quotient of a free group | $~$G$~$ |
---|
Every group is a quotient of a free group | $~$\theta: F(G) \to G$~$ |
---|
Every group is a quotient of a free group | $~$(a_1, a_2, \dots, a_n)$~$ |
---|
Every group is a quotient of a free group | $~$a_1 a_2 \dots a_n$~$ |
---|
Every group is a quotient of a free group | $~$F(G)$~$ |
---|
Every group is a quotient of a free group | $~$G$~$ |
---|
Every group is a quotient of a free group | $~$w_1 = (a_1, \dots, a_m)$~$ |
---|
Every group is a quotient of a free group | $~$w_2 = (b_1, \dots, b_n)$~$ |
---|
Every group is a quotient of a free group | $$~$\theta(w_1 w_2) = \theta(a_1, \dots, a_m, b_1, \dots, b_m) = a_1 \dots a_m b_1 \dots b_m = \theta(w_1) \theta(w_2)$~$$ |
---|
Every group is a quotient of a free group | $~$G$~$ |
---|
Every group is a quotient of a free group | $~$F(G)$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$\sigma$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$S_n$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$\tau_1, \dots, \tau_k$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$\sigma = \tau_k \tau_{k-1} \dots \tau_1$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(123)$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(23)(13)$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$3$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$\sigma$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$\sigma$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$\sigma$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_1 a_2 \dots a_r)$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_{r-1} a_r) (a_{r-2} a_r) \dots (a_2 a_r) (a_1 a_r)$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_i$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_i$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_1 a_r)$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_2 a_r)$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_{i-1} a_r)$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_i a_r)$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_r$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_{i+1} a_r)$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_r$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_{i+1}$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$(a_{i+2} a_r), \dots, (a_{r-1} a_r)$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_{i+1}$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_i$~$ |
---|
Every member of a symmetric group on finitely many elements is a product of transpositions | $~$a_{i+1}$~$ |
---|
Examination through isomorphism | $~$(X,d)$~$ |
---|
Examination through isomorphism | $~$d(x,y)$~$ |
---|
Examination through isomorphism | $~$x,y \in X$~$ |
---|
Examination through isomorphism | $~$[0,1]$~$ |
---|
Examination through isomorphism | $~$[0,2]$~$ |
---|
Examination through isomorphism | $~$\mathbb{R}$~$ |
---|
Examination through isomorphism | $~$\mathbb{R}$~$ |
---|
Examination through isomorphism | $~$f : [0,1] \to [0,2]$~$ |
---|
Examination through isomorphism | $~$g : [0,2] \to [0,1]$~$ |
---|
Examination through isomorphism | $~$fg$~$ |
---|
Examination through isomorphism | $~$gf$~$ |
---|
Examination through isomorphism | $~$f$~$ |
---|
Examination through isomorphism | $~$2$~$ |
---|
Examination through isomorphism | $~$g$~$ |
---|
Examination through isomorphism | $~$2$~$ |
---|
Examination through isomorphism | $~$[0,1]$~$ |
---|
Examination through isomorphism | $~$1$~$ |
---|
Examination through isomorphism | $~$[0,2]$~$ |
---|
Examination through isomorphism | $~$2$~$ |
---|
Examination through isomorphism | $~$\text{Set}\times\text{Set}\to\text{Set}$~$ |
---|
Examination through isomorphism | $~$A \times (B \times C)$~$ |
---|
Examination through isomorphism | $~$(a,(b,c))$~$ |
---|
Examination through isomorphism | $~$(A \times B) \times C$~$ |
---|
Examination through isomorphism | $~$((a,b),c)$~$ |
---|
Examination through isomorphism | $~$\text{Set}\times\text{Set}\times\text{Set}\to\text{Set}$~$ |
---|
Examination through isomorphism | $~$(A,B,C) \mapsto A \times (B \times C)$~$ |
---|
Examination through isomorphism | $~$(A,B,C) \mapsto (A \times B) \times C$~$ |
---|
Examination through isomorphism | $~$\text{Set}\times\text{Set}\times\text{Set}\to\text{Set}$~$ |
---|
Example: Dragon Pox | $$~$
\newcommand{\bP}{\mathbb{P}}
$~$$ |
---|
Example: Dragon Pox | $$~$
\newcommand{\bP}{\mathbb{P}}
$~$$ |
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Example: Dragon Pox | $~$\bP(D) = 0.4$~$ |
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Example: Dragon Pox | $~$\bP(S \mid D) = 0.7$~$ |
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Example: Dragon Pox | $~$\bP(S \mid \neg D) = 0.2$~$ |
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Example: Dragon Pox | $~$(C)$~$ |
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Example: Dragon Pox | $~$(\neg C)$~$ |
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Example: Dragon Pox | $~$(L)$~$ |
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Example: Dragon Pox | $~$(\neg L)$~$ |
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Example: Dragon Pox | $$~$
\begin{align}
\bP(L \mid \;\;D,\;\;C) &= 0.4\\
\bP(L \mid \;\;D,\neg C) &= 0.1\\
\bP(L \mid \neg D,\;\;C) &= 0.7\\
\bP(L \mid \neg D,\neg C) &= 0.9
\end{align}
$~$$ |
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Example: Dragon Pox | $~$D$~$ |
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Example: Dragon Pox | $~$\bP(D) = 0.4$~$ |
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Example: Dragon Pox | $~$S$~$ |
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Example: Dragon Pox | $~$\bP(S \mid D) = 0.7$~$ |
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Example: Dragon Pox | $~$\bP(S \mid \neg D) = 0.2$~$ |
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Example: Dragon Pox | $~$(C)$~$ |
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Example: Dragon Pox | $~$(\neg C)$~$ |
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Example: Dragon Pox | $~$(L)$~$ |
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Example: Dragon Pox | $~$D$~$ |
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Example: Dragon Pox | $~$C$~$ |
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Example: Dragon Pox | $$~$
\begin{align}
\bP(L \mid \;\;D,\;\;C) &= 0.4\\
\bP(L \mid \;\;D,\neg C) &= 0.1\\
\bP(L \mid \neg D,\;\;C) &= 0.7\\
\bP(L \mid \neg D,\neg C) &= 0.9
\end{align}
$~$$ |
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Example: Dragon Pox | $~$\bP(L \mid D,C) > \bP(L \mid D,\neg C)$~$ |
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Example: Dragon Pox | $~$\neg D$~$ |
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Example: Dragon Pox | $~$\bP(L \mid \neg D,C) < \bP(L \mid \neg D,\neg C)$~$ |
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Exchange rates between digits | $~$n$~$ |
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Exchange rates between digits | $~$b$~$ |
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Exchange rates between digits | $~$\log_b(n).$~$ |
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Exchange rates between digits | $~$2^\text{3,000,000,000,000}$~$ |
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Exchange rates between digits | $~$n$~$ |
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Exchange rates between digits | $~$2^n$~$ |
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Exchange rates between digits | $~$2^4=16$~$ |
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Exchange rates between digits | $~$2^6 < 101 < 2^7$~$ |
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Exchange rates between digits | $~$2^{12} < 8000 < 2^{13}$~$ |
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Exchange rates between digits | $~$2^{13} < 15,000 < 2^{14}$~$ |
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Exchange rates between digits | $~$x$~$ |
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Exchange rates between digits | $~$n$~$ |
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Exchange rates between digits | $~$n$~$ |
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Exchange rates between digits | $~$3n$~$ |
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Exchange rates between digits | $~$10^n > 2^{3n}$~$ |
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Exchange rates between digits | $~$n$~$ |
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Exchange rates between digits | $~$x$~$ |
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Exchange rates between digits | $~$3n$~$ |
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Exchange rates between digits | $~$n$~$ |
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Exchange rates between digits | $~$10^n$~$ |
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Exchange rates between digits | $~$2^3$~$ |
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Exchange rates between digits | $~$2^{3n}$~$ |
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Exchange rates between digits | $~$n$~$ |
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Exchange rates between digits | $~$2^{3(n-1)}$~$ |
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Exchange rates between digits | $~$n \ge 11,$~$ |
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Exchange rates between digits | $~$x$~$ |
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Exchange rates between digits | $~$10^{10} < 2^{35}.$~$ |
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Exchange rates between digits | $~$2^{33} < 10^{10} < 2^{34},$~$ |
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Exchange rates between digits | $~$2^{332} < 10^{100} < 2^{333},$~$ |
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Exchange rates between digits | $~$p$~$ |
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Exchange rates between digits | $~$2^p > 10$~$ |
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Exchange rates between digits | $~$2^p < 10$~$ |
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Exchange rates between digits | $~$p$~$ |
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Exchange rates between digits | $~$2^p = 10,$~$ |
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Exchange rates between digits | $~$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$~$ |
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Exchange rates between digits | $~$2 + 2 + 2 + 2 + 2 = 10.$~$ |
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Exchange rates between digits | $~$p$~$ |
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Exchange rates between digits | $~$p$~$ |
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Exchange rates between digits | $~$2^p = 10$~$ |
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Exchange rates between digits | $~$\log_2(10),$~$ |
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Exchange rates between digits | $~$\log_b(x)$~$ |
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Exchange rates between digits | $~$x$~$ |
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Exchange rates between digits | $~$b$~$ |
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Exchange rates between digits | $~$\log_2(6) \approx 2.58$~$ |
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Exchange rates between digits | $~$2^2 < 6 < 2^3$~$ |
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Exchange rates between digits | $~$2^{25} < 6^{10} < 2^{26}$~$ |
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Exchange rates between digits | $~$2^{258} < 6^{100} < 2^{259}.$~$ |
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Exchange rates between digits | $~$\log_2(6)$~$ |
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Exchange rates between digits | $~$\log_b(x)$~$ |
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Exchange rates between digits | $~$b$~$ |
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Exchange rates between digits | $~$x$~$ |
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Exchange rates between digits | $~$b$~$ |
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Exchange rates between digits | $~$x$~$ |
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Exchange rates between digits | $~$b$~$ |
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Exchange rates between digits | $~$x$~$ |
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Exchange rates between digits | $~$x$~$ |
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Exchange rates between digits | $~$b$~$ |
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Exchange rates between digits | $~$b$~$ |
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Exchange rates between digits | $~$x$~$ |
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Exchange rates between digits | $~$\log_b(x)$~$ |
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Exchange rates between digits | $~$x$~$ |
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Exchange rates between digits | $~$b$~$ |
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Exchange rates between digits | $~$\log_x(b) = \frac{1}{\log_b(x)}$~$ |
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Exchange rates between digits | $~$x$~$ |
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Exchange rates between digits | $~$b$~$ |
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Exchange rates between digits | $~$b$~$ |
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Exchange rates between digits | $~$x$~$ |
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Exchange rates between digits | $~$\log_{1.5}(2.5)$~$ |
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Existence Proof of Logical Inductor | $~$\overline{\mathbb{P}}$~$ |
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Existence Proof of Logical Inductor | $~$\overline{D}$~$ |
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Existence Proof of Logical Inductor | $~$\overline{T}$~$ |
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Existence Proof of Logical Inductor | $~$\overline{\mathbb{P}}$~$ |
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Existence Proof of Logical Inductor | $~$\overline{D}$~$ |
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Existence Proof of Logical Inductor | $~$\overline{LIA}$~$ |
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Existence Proof of Logical Inductor | $~$\overline{D}$~$ |
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Existence Proof of Logical Inductor | $~$-b$~$ |
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Existence Proof of Logical Inductor | $~$-b$~$ |
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Existence Proof of Logical Inductor | $~$\overline{T}$~$ |
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Existence Proof of Logical Inductor | $~$n$~$ |
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Existence Proof of Logical Inductor | $~$\mathbb{P}_n$~$ |
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Existence Proof of Logical Inductor | $~$T_n(\mathbb{P}_{\leq n})$~$ |
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Existence Proof of Logical Inductor | $~$\text{fix}(\mathbb{V})(\phi) := \max{(0,\min{(1, \mathbb{V}(\phi) + T(\mathbb{P}_{\leq n-1},\mathbb{V})[\phi])})}$~$ |
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Existence Proof of Logical Inductor | $~$\mathbb{V}^{\text{fix}}$~$ |
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Existence Proof of Logical Inductor | $~$\phi$~$ |
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Existence Proof of Logical Inductor | $~$\mathbb{V}^{\text{fix}}(\phi)= \max{(0,\min{(1, \mathbb{V}^{\text{fix}}(\phi) + T(\mathbb{P}_{\leq n-1},\mathbb{V}^{\text{fix}})[\phi])})}$~$ |
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Existence Proof of Logical Inductor | $~$\mathcal{V}' \to \mathcal{V}'$~$ |
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Existence Proof of Logical Inductor | $~$\mathcal{V}'$~$ |
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Existence Proof of Logical Inductor | $~$[0,1]^{S'}$~$ |
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Existence Proof of Logical Inductor | $~$x$~$ |
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Existence Proof of Logical Inductor | $~$f(x)=x$~$ |
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Existence Proof of Logical Inductor | $~$\text{fix}\mathbb{V}(\phi)$~$ |
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Existence Proof of Logical Inductor | $~$T$~$ |
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Existence Proof of Logical Inductor | $~$\mathbb{P}$~$ |
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Existence Proof of Logical Inductor | $~$T$~$ |
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Existence Proof of Logical Inductor | $~$1-2^{-n}$~$ |
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Existence Proof of Logical Inductor | $~$2^{-n}$~$ |
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Existence Proof of Logical Inductor | $~$B(n,b, T_n, \mathbb{P}_{\leq n-1})$~$ |
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Existence Proof of Logical Inductor | $~$(n-1)$~$ |
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Existence Proof of Logical Inductor | $~$m<n$~$ |
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Expected value | $~$V = x_{1},$~$ |
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Expected value | $~$V = x_{2}, …, $~$ |
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Expected value | $~$V = x_{k}$~$ |
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Expected value | $~$P(x_{i})$~$ |
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Expected value | $~$V = x_{i}$~$ |
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Expected value | $$~$\sum_{i=1}^{k}x_{i}P(x_{i})$~$$ |
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Expected value | $~$x \in \mathbb{R}$~$ |
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Expected value | $~$P(x)$~$ |
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Expected value | $~$\lim_{dx \to 0}$~$ |
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Expected value | $~$x<V<(x+dx)$~$ |
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Expected value | $~$dx$~$ |
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Expected value | $$~$\int_{-∞}^{∞}xP(x)dx$~$$ |
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Explicit Bayes as a counter for 'worrying' | $~$\mathbb P(\text{cancel}|\text{desirable})$~$ |
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Explicit Bayes as a counter for 'worrying' | $~$\mathbb P(\text{cancel}|\text{undesirable})$~$ |
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Exponential | $~$b$~$ |
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Exponential | $~$x$~$ |
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Exponential | $~$b^x,$~$ |
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Exponential | $~$b$~$ |
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Exponential | $~$x$~$ |
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Exponential | $~$10^3$~$ |
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Exponential | $~$10 \cdot 10 \cdot 10 = 1000$~$ |
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Exponential | $~$2^4=16,$~$ |
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Exponential | $~$2 \cdot 2 \cdot 2 \cdot 2 = 16.$~$ |
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Exponential | $~$x$~$ |
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Exponential | $~$10^{1/2}$~$ |
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Exponential | $~$n$~$ |
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Exponential | $~$n$~$ |
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Exponential | $~$n \approx 3.16,$~$ |
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Exponential | $~$n \cdot n \approx 10.$~$ |
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Exponential | $~$f(x) = c \times a^x$~$ |
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Exponential | $~$c$~$ |
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Exponential | $~$a$~$ |
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Exponential | $~$1.02$~$ |
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Exponential | $~$f(x) = 100 \times 1.02^x$~$ |
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Exponential | $~$x$~$ |
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Exponential | $~$x$~$ |
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Exponential | $~$f(x) = 1 \times 2^x$~$ |
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Exponential | $~$f(x) = f(x-1) \times 1.02$~$ |
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Exponential | $~$\Delta f(x) = f(x+1) - f(x) = 0.02 \times f(x)$~$ |
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Exponential | $~$f(x) = f(x-1) + 0.02 \times f(0)$~$ |
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Exponential | $~$f(0)$~$ |
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Exponential | $~$f(x)$~$ |
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Exponential notation for function spaces | $~$X$~$ |
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Exponential notation for function spaces | $~$Y$~$ |
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Exponential notation for function spaces | $~$X$~$ |
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Exponential notation for function spaces | $~$Y$~$ |
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Exponential notation for function spaces | $~$X \to Y$~$ |
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Exponential notation for function spaces | $~$Y^X$~$ |
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Exponential notation for function spaces | $~$Y^3$~$ |
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Exponential notation for function spaces | $~$Y$~$ |
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Exponential notation for function spaces | $~$f : X \to Y$~$ |
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Exponential notation for function spaces | $~$X$~$ |
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Exponential notation for function spaces | $~$Y$~$ |
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Exponential notation for function spaces | $~$Y$~$ |
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Exponential notation for function spaces | $~$X$~$ |
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Exponential notation for function spaces | $~$Y^n$~$ |
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Exponential notation for function spaces | $~$n$~$ |
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Exponential notation for function spaces | $~$Y$~$ |
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Exponential notation for function spaces | $~$|X| = n$~$ |
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Exponential notation for function spaces | $~$Y^X \cong Y^n$~$ |
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Exponential notation for function spaces | $~$Z^{X \times Y} \cong (Z^X)^Y$~$ |
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Exponential notation for function spaces | $~$Z^{X + Y} \cong Z^X \times Z^Y$~$ |
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Exponential notation for function spaces | $~$Z^1 \cong Z$~$ |
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Exponential notation for function spaces | $~$1$~$ |
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Exponential notation for function spaces | $~$Z$~$ |
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Exponential notation for function spaces | $~$Z$~$ |
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Exponential notation for function spaces | $~$Z^0 \cong 1$~$ |
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Exponential notation for function spaces | $~$0$~$ |
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Exponential notation for function spaces | $~$Y^X$~$ |
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Exponential notation for function spaces | $~$\text{Hom}_{\mathcal{C}}(X, Y)$~$ |
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Exponential notation for function spaces | $~$\mathcal{C}$~$ |
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Extensionality Axiom | $$~$ \forall A \forall B : ( \forall x : (x \in A \iff x \in B) \Rightarrow A=B)$~$$ |
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Extensionality Axiom | $~$\{1,2\} = \{2,1\}$~$ |
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Extensionality Axiom | $~$1$~$ |
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Extensionality Axiom | $~$2$~$ |
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Extensionality Axiom | $~$5$~$ |
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Extensionality Axiom | $~$73$~$ |
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Extraordinary claims require extraordinary evidence | $~$(1 : 9 ) \times (3 : 1) \ = \ (3 : 9) \ \cong \ (1 : 3)$~$ |
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Extraordinary claims require extraordinary evidence | $~$X$~$ |
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Extraordinary claims require extraordinary evidence | $~$X$~$ |
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Extraordinary claims require extraordinary evidence | $~$X.$~$ |
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Extraordinary claims require extraordinary evidence | $$~$\text{Likelihood ratio} = \dfrac{\text{Probability of seeing the evidence, assuming the claim is true}}{\text{Probability of seeing the evidence, assuming the claim is false}}$~$$ |
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Extraordinary claims require extraordinary evidence | $~$10^{100}$~$ |
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Extraordinary claims require extraordinary evidence | $~$10^{94}$~$ |
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Extraordinary claims require extraordinary evidence | $~$(10^{94} : 1)$~$ |
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Extraordinary claims require extraordinary evidence | $~$10^{-94}$~$ |
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Extraordinary claims require extraordinary evidence | $~$(1 : 10^{100})$~$ |
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Extraordinary claims require extraordinary evidence | $~$(1 : 10^6)$~$ |
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Factorial | $~$5!$~$ |
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Factorial | $~$1*2*3*4*5$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n!=\prod_{i=1}^{n}i$~$ |
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Factorial | $~$0! = 1$~$ |
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Factorial | $~$n!$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$A$~$ |
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Factorial | $~$B$~$ |
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Factorial | $~$C$~$ |
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Factorial | $$~$ABC$~$$ |
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Factorial | $$~$ACB$~$$ |
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Factorial | $$~$BAC$~$$ |
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Factorial | $$~$BCA$~$$ |
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Factorial | $$~$CAB$~$$ |
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Factorial | $$~$CBA$~$$ |
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Factorial | $~$6$~$ |
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Factorial | $~$3$~$ |
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Factorial | $~$6 = 3*2*1 = 3!$~$ |
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Factorial | $~$1$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n+1$~$ |
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Factorial | $~$1$~$ |
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Factorial | $$~$A$~$$ |
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Factorial | $$~$1 = \prod_{i=1}^{1}i = 1!$~$$ |
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Factorial | $~$\{A_{1},A_{2},…,A_{n},A_{n+1}\}$~$ |
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Factorial | $~$n+1$~$ |
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Factorial | $~$A_{n+1}$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n!$~$ |
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Factorial | $~$A_{n+1}$~$ |
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Factorial | $~$A_{n+1}$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n!$~$ |
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Factorial | $~$A_{n+1}$~$ |
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Factorial | $~$n!$~$ |
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Factorial | $~$A_{n+1}$~$ |
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Factorial | $~$n!*(n+1)$~$ |
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Factorial | $~$(n+1)!$~$ |
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Factorial | $~$x!$~$ |
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Factorial | $$~$x! = \Gamma (x+1),$~$$ |
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Factorial | $~$\Gamma $~$ |
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Factorial | $$~$\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\mathrm{d} t$~$$ |
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Factorial | $~$x$~$ |
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Factorial | $$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
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Factorial | $~$x=1$~$ |
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Factorial | $$~$\prod_{i=1}^{1}i = \int_{0}^{\infty}t^{1}e^{-t}\mathrm{d} t$~$$ |
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Factorial | $$~$1=1$~$$ |
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Factorial | $~$x$~$ |
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Factorial | $$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
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Factorial | $~$x + 1$~$ |
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Factorial | $$~$\prod_{i=1}^{x+1}i = \int_{0}^{\infty}t^{x+1}e^{-t}\mathrm{d} t$~$$ |
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Factorial | $~$x+1$~$ |
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Factorial | $$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
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Factorial | $$~$(x+1)\prod_{i=1}^{x}i = (x+1)\int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
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Factorial | $$~$\prod_{i=1}^{x+1}i = (x+1)\int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$ |
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Factorial | $$~$= 0+\int_{0}^{\infty}(x+1)t^{x}e^{-t}\mathrm{d} t$~$$ |
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Factorial | $$~$= \left (-t^{x+1}e^{-t}) \right]_{0}^{\infty}+\int_{0}^{\infty}(x+1)t^{x}e^{-t}\mathrm{d} t$~$$ |
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Factorial | $$~$= \left (-t^{x+1}e^{-t}) \right]_{0}^{\infty}-\int_{0}^{\infty}(x+1)t^{x}(-e^{-t})\mathrm{d} t$~$$ |
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Factorial | $$~$=\int_{0}^{\infty}t^{x+1}e^{-t}\mathrm{d} t$~$$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$1,2,3$~$ |
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Factorial | $~$1,2,3$~$ |
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Factorial | $~$1,3,2$~$ |
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Factorial | $~$1$~$ |
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Factorial | $~$2$~$ |
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Factorial | $~$3$~$ |
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Factorial | $~$6$~$ |
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Factorial | $~$1,2,3$~$ |
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Factorial | $~$1,3,2$~$ |
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Factorial | $~$2,1,3$~$ |
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Factorial | $~$2,3,1$~$ |
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Factorial | $~$3,1,2$~$ |
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Factorial | $~$3,2,1$~$ |
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Factorial | $~$1$~$ |
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Factorial | $~$2$~$ |
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Factorial | $~$3$~$ |
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Factorial | $~$6$~$ |
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Factorial | $~$24$~$ |
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Factorial | $~$1,2,3,4$~$ |
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Factorial | $~$1,2,4,3$~$ |
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Factorial | $~$1,3,2,4$~$ |
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Factorial | $~$1,3,4,2$~$ |
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Factorial | $~$1,4,2,3$~$ |
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Factorial | $~$1,4,3,2$~$ |
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Factorial | $~$2,1,3,4$~$ |
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Factorial | $~$24$~$ |
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Factorial | $~$6$~$ |
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Factorial | $~$6$~$ |
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Factorial | $~$1$~$ |
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Factorial | $~$6$~$ |
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Factorial | $~$2$~$ |
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Factorial | $~$6$~$ |
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Factorial | $~$3$~$ |
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Factorial | $~$6$~$ |
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Factorial | $~$4$~$ |
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Factorial | $~$24$~$ |
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Factorial | $~$120$~$ |
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Factorial | $~$24$~$ |
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Factorial | $~$24$~$ |
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Factorial | $~$1$~$ |
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Factorial | $~$24$~$ |
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Factorial | $~$2$~$ |
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Factorial | $~$24$~$ |
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Factorial | $~$3$~$ |
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Factorial | $~$24$~$ |
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Factorial | $~$4$~$ |
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Factorial | $~$24$~$ |
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Factorial | $~$5$~$ |
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Factorial | $~$120$~$ |
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Factorial | $~$5$~$ |
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Factorial | $~$4$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n-1$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n-1$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$1$~$ |
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Factorial | $~$2$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n-1$~$ |
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Factorial | $~$n-1$~$ |
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Factorial | $~$5!$~$ |
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Factorial | $~$120$~$ |
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Factorial | $~$4!$~$ |
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Factorial | $~$n!$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$n$~$ |
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Factorial | $~$5! = 5 \times 4!$~$ |
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Factorial | $~$4! = 4 \times 3!$~$ |
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Factorial | $~$5$~$ |
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Factorial | $~$n-1$~$ |
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Factorial | $~$n \times n - 1!$~$ |
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Factorial | $~$(n \times n)-1!$~$ |
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Factorial | $$~$n! = n \times (n-1)!$~$$ |
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Factorial | $~$n! = n \times (n-1)!$~$ |
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Factorial | $~$(n-1)! = (n-1) \times (n-2)!$~$ |
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Factorial | $~$(n-2)! = (n-2) \times (n-3)!$~$ |
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Factorial | $$~$n! = n \times (n-1)! = n \times (n-1) \times (n-2)! = n \times (n-1) \times (n-2) \times (n-3)!$~$$ |
---|
Factorial | $$~$n \times (n-1) \times (n-2) \times \dots \times 5 \times 4 \times 3!$~$$ |
---|
Factorial | $~$3! = 6$~$ |
---|
Factorial | $~$3 \times 2 \times 1$~$ |
---|
Factorial | $$~$n! = n \times (n-1) \times \dots \times 4 \times 3 \times 2 \times 1$~$$ |
---|
Factorial | $~$n!$~$ |
---|
Factorial | $~$n$~$ |
---|
Factorial | $~$n$~$ |
---|
Factorial | $~$3!$~$ |
---|
Factorial | $~$2!$~$ |
---|
Factorial | $~$1!$~$ |
---|
Factorial | $~$1,2$~$ |
---|
Factorial | $~$2,1$~$ |
---|
Factorial | $~$2! = 2$~$ |
---|
Factorial | $~$1$~$ |
---|
Factorial | $~$1! = 1$~$ |
---|
Factorial | $~$1$~$ |
---|
Factorial | $~$0! = 1$~$ |
---|
Faithful simulation | $~$D$~$ |
---|
Faithful simulation | $~$S_D$~$ |
---|
Faithful simulation | $~$D$~$ |
---|
Faithful simulation | $~$D$~$ |
---|
Faithful simulation | $~$S_D$~$ |
---|
Faithful simulation | $~$D.$~$ |
---|
Field homomorphism is trivial or injective | $~$F$~$ |
---|
Field homomorphism is trivial or injective | $~$G$~$ |
---|
Field homomorphism is trivial or injective | $~$f: F \to G$~$ |
---|
Field homomorphism is trivial or injective | $~$f$~$ |
---|
Field homomorphism is trivial or injective | $~$0$~$ |
---|
Field homomorphism is trivial or injective | $~$0$~$ |
---|
Field homomorphism is trivial or injective | $~$f$~$ |
---|
Field homomorphism is trivial or injective | $~$f$~$ |
---|
Field homomorphism is trivial or injective | $~$0$~$ |
---|
Field homomorphism is trivial or injective | $~$F$~$ |
---|
Field homomorphism is trivial or injective | $~$G$~$ |
---|
Field homomorphism is trivial or injective | $~$f: F \to G$~$ |
---|
Field homomorphism is trivial or injective | $~$f$~$ |
---|
Field homomorphism is trivial or injective | $~$0$~$ |
---|
Field homomorphism is trivial or injective | $~$x \in F$~$ |
---|
Field homomorphism is trivial or injective | $~$f(x) = 0_G$~$ |
---|
Field homomorphism is trivial or injective | $~$f$~$ |
---|
Field homomorphism is trivial or injective | $~$f: F \to G$~$ |
---|
Field homomorphism is trivial or injective | $~$f$~$ |
---|
Field homomorphism is trivial or injective | $~$x,y$~$ |
---|
Field homomorphism is trivial or injective | $~$f(x) = f(y)$~$ |
---|
Field homomorphism is trivial or injective | $~$x = y$~$ |
---|
Field homomorphism is trivial or injective | $~$f(x) = f(y)$~$ |
---|
Field homomorphism is trivial or injective | $~$f(x)-f(y) = 0_G$~$ |
---|
Field homomorphism is trivial or injective | $~$f(x-y) = 0_G$~$ |
---|
Field homomorphism is trivial or injective | $~$f$~$ |
---|
Field homomorphism is trivial or injective | $~$f$~$ |
---|
Field homomorphism is trivial or injective | $~$f(z) = 0_G$~$ |
---|
Field homomorphism is trivial or injective | $~$z = 0_F$~$ |
---|
Field homomorphism is trivial or injective | $~$z = x-y$~$ |
---|
Field homomorphism is trivial or injective | $~$f(z) = 0_G$~$ |
---|
Field homomorphism is trivial or injective | $~$z$~$ |
---|
Field homomorphism is trivial or injective | $~$0_F$~$ |
---|
Field homomorphism is trivial or injective | $~$z^{-1}$~$ |
---|
Field homomorphism is trivial or injective | $~$f(z^{-1}) f(z) = f(z^{-1}) \times 0_G = 0_G$~$ |
---|
Field homomorphism is trivial or injective | $~$f$~$ |
---|
Field homomorphism is trivial or injective | $~$f(z^{-1} \times z) = 0_G$~$ |
---|
Field homomorphism is trivial or injective | $~$f(1_F) = 0_G$~$ |
---|
Field homomorphism is trivial or injective | $~$f$~$ |
---|
Field homomorphism is trivial or injective | $~$F \setminus \{ 0_F \}$~$ |
---|
Field homomorphism is trivial or injective | $~$G \setminus \{0_G\}$~$ |
---|
Field homomorphism is trivial or injective | $~$1_F$~$ |
---|
Field homomorphism is trivial or injective | $~$F \setminus \{0_F\}$~$ |
---|
Field homomorphism is trivial or injective | $~$1_G$~$ |
---|
Field homomorphism is trivial or injective | $~$F \setminus \{0_G\}$~$ |
---|
Field homomorphism is trivial or injective | $~$z$~$ |
---|
Field homomorphism is trivial or injective | $~$z \not = 0_F$~$ |
---|
Field homomorphism is trivial or injective | $~$f(z) = 0_G$~$ |
---|
Field homomorphism is trivial or injective | $~$z = 0_F$~$ |
---|
Field structure of rational numbers | $~$\frac{a}{b} + \frac{p}{q} = \frac{aq+bp}{bq}$~$ |
---|
Field structure of rational numbers | $~$\frac{a}{b} \frac{c}{d} = \frac{ac}{bd}$~$ |
---|
Field structure of rational numbers | $~$\frac{0}{1}$~$ |
---|
Field structure of rational numbers | $~$\frac{1}{1}$~$ |
---|
Field structure of rational numbers | $~$\frac{a}{b}$~$ |
---|
Field structure of rational numbers | $~$\frac{-a}{b}$~$ |
---|
Field structure of rational numbers | $~$\frac{a}{b}$~$ |
---|
Field structure of rational numbers | $~$a \not = 0$~$ |
---|
Field structure of rational numbers | $~$\frac{b}{a}$~$ |
---|
Field structure of rational numbers | $~$0 < \frac{c}{d}$~$ |
---|
Field structure of rational numbers | $~$c$~$ |
---|
Field structure of rational numbers | $~$d$~$ |
---|
Field structure of rational numbers | $~$c$~$ |
---|
Field structure of rational numbers | $~$d$~$ |
---|
Field structure of rational numbers | $~$\frac{a}{b} < \frac{c}{d}$~$ |
---|
Field structure of rational numbers | $~$0 < \frac{c}{d} - \frac{a}{b}$~$ |
---|
Finite set | $~$X$~$ |
---|
Finite set | $~$n \in \mathbb{N}$~$ |
---|
Finite set | $~$X$~$ |
---|
Finite set | $~$n$~$ |
---|
Finite set | $~$\{ 1,2 \}$~$ |
---|
Finite set | $~$\{ \mathbb{N} \}$~$ |
---|
Finite set | $~$\mathbb{N}$~$ |
---|
Finite set | $~$\mathbb{R}$~$ |
---|
First order linear equations | $$~$
u'=a(t)u+b(t)
$~$$ |
---|
First order linear equations | $~$a$~$ |
---|
First order linear equations | $~$b$~$ |
---|
First order linear equations | $~$[\alpha, \beta]$~$ |
---|
First order linear equations | $~$b$~$ |
---|
First order linear equations | $~$b=0$~$ |
---|
First order linear equations | $$~$
u'=a(t)u
$~$$ |
---|
First order linear equations | $~$C^1$~$ |
---|
First order linear equations | $~$[\alpha, \beta]$~$ |
---|
First order linear equations | $~$b$~$ |
---|
First order linear equations | $~$\Sigma_b$~$ |
---|
First order linear equations | $~$\Sigma_0$~$ |
---|
First order linear equations | $~$\Sigma_0$~$ |
---|
First order linear equations | $~$\Sigma_b$~$ |
---|
First order linear equations | $~$\Sigma_0$~$ |
---|
First order linear equations | $~$\Sigma_0$~$ |
---|
First order linear equations | $~$\Sigma_0$~$ |
---|
First order linear equations | $~$\Sigma_b$~$ |
---|
First order linear equations | $~$\Sigma_b$~$ |
---|
First order linear equations | $~$a$~$ |
---|
First order linear equations | $~$b$~$ |
---|
First order linear equations | $$~$
u' = au+b
$~$$ |
---|
First order linear equations | $~$u'=au$~$ |
---|
First order linear equations | $~$ke^{\int_{t_0}^ta}$~$ |
---|
First order linear equations | $~$k$~$ |
---|
First order linear equations | $~$t_0\in [\alpha, \beta]$~$ |
---|
First order linear equations | $~$u=h\dot v$~$ |
---|
First order linear equations | $~$h$~$ |
---|
First order linear equations | $~$e^{\int_{t_0}^ta}$~$ |
---|
First order linear equations | $~$u$~$ |
---|
First order linear equations | $$~$
u'=(hv)'=h'v+hv'=au+b=a(hv)+b
$~$$ |
---|
First order linear equations | $~$h\in\Sigma_0$~$ |
---|
First order linear equations | $~$h'=ah$~$ |
---|
First order linear equations | $$~$
v'=bh^{-1}=be^{-\int_{t_0}^ta}
$~$$ |
---|
First order linear equations | $$~$
v=\int_{t_0}^tbe^{\int_{t}^sa}ds
$~$$ |
---|
First order linear equations | $~$\Sigma_b$~$ |
---|
First order linear equations | $~$ke^{\int_{t_0}^ta}+\int_{t_0}^tbe^{\int_{t}^sa}ds$~$ |
---|
First order linear equations | $~$k$~$ |
---|
Fixed point theorem of provability logic | $~$\phi(p, q_1,…,q_n)$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$H(q_1,..,q_n)$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot[p\leftrightarrow \phi(p,q_1,…,q_n)] \leftrightarrow \boxdot[p\leftrightarrow H(q_1,..,q_n)]$~$ |
---|
Fixed point theorem of provability logic | $~$\phi(p)$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot[p\leftrightarrow \phi(p)] \leftrightarrow \boxdot[p\leftrightarrow H]$~$ |
---|
Fixed point theorem of provability logic | $~$\boxdot A = A\wedge \square A$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$\phi(p)$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$\psi(p, q_1…,q_n)$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$H(q_1,…,q_n)$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot[p\leftrightarrow\psi(p, q_1,…,q_n)] \leftrightarrow \boxdot[p_i\leftrightarrow H(q_1,…,q_n)]$~$ |
---|
Fixed point theorem of provability logic | $~$\psi$~$ |
---|
Fixed point theorem of provability logic | $~$\psi$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$\phi$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash H(q_1,…,q_n)\leftrightarrow \phi(H(q_1,…,q_n),q_1,…,q_n)$~$ |
---|
Fixed point theorem of provability logic | $~$\phi$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot[p\leftrightarrow\psi(p, q_1,…,q_n)] \leftrightarrow \boxdot[p_i\leftrightarrow H(q_1,…,q_n)]$~$ |
---|
Fixed point theorem of provability logic | $~$GL$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot[H(q_1,…,q_n)\leftrightarrow\psi(H(q_1,…,q_n), q_1,…,q_n)] \leftrightarrow \boxdot[H(q_1,…,q_n)\leftrightarrow H(q_1,…,q_n)]$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot[H(q_1,…,q_n)\leftrightarrow H(q_1,…,q_n)$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot[H(q_1,…,q_n)\leftrightarrow\psi(H(q_1,…,q_n), q_1,…,q_n)]$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$I$~$ |
---|
Fixed point theorem of provability logic | $~$\phi$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash H\leftrightarrow I$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$\phi(p)$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot(p\leftrightarrow \phi(p))\leftrightarrow (p\leftrightarrow H)$~$ |
---|
Fixed point theorem of provability logic | $~$I$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash H\leftrightarrow I$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash F(I)\leftrightarrow F(H)$~$ |
---|
Fixed point theorem of provability logic | $~$F(q)$~$ |
---|
Fixed point theorem of provability logic | $~$F(q)=\boxdot(p\leftrightarrow q)$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot(p\leftrightarrow H)\leftrightarrow \boxdot(p\leftrightarrow I)$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot(p\leftrightarrow \phi(p))\leftrightarrow (p\leftrightarrow I)$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$I$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot (p\leftrightarrow H)\leftrightarrow \boxdot (p\leftrightarrow I)$~$ |
---|
Fixed point theorem of provability logic | $~$GL$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash\boxdot (H\leftrightarrow H)\leftrightarrow \boxdot (H\leftrightarrow I)$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot (H\leftrightarrow H)$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash (H\leftrightarrow I)$~$ |
---|
Fixed point theorem of provability logic | $~$\phi(p)$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot[p\leftrightarrow \phi(p)] \leftrightarrow \boxdot[p\leftrightarrow H]$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$\square^n \bot$~$ |
---|
Fixed point theorem of provability logic | $~$\square^n A = \underbrace{\square,\square,\ldots,\square}_{n\text{-times}} A$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$A$~$ |
---|
Fixed point theorem of provability logic | $~$A$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$A$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$A$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$B$~$ |
---|
Fixed point theorem of provability logic | $~$[[B]]_A$~$ |
---|
Fixed point theorem of provability logic | $~$A$~$ |
---|
Fixed point theorem of provability logic | $~$[[\bot]]_A = \emptyset$~$ |
---|
Fixed point theorem of provability logic | $~$[[B\to C]]_A = (\mathbb{N} \setminus [[B]]_A)\cup [[C]]_A$~$ |
---|
Fixed point theorem of provability logic | $~$[[\square D]]_A=\{m:\forall i < m i\in [[D]]_A\}$~$ |
---|
Fixed point theorem of provability logic | $~$[[p]]_A=[[A]]_A$~$ |
---|
Fixed point theorem of provability logic | $~$M$~$ |
---|
Fixed point theorem of provability logic | $~$(p\leftrightarrow A) is valid, and $~$ |
---|
Fixed point theorem of provability logic | $~$ a $~$ |
---|
Fixed point theorem of provability logic | $~$-sentence. Then $~$ |
---|
Fixed point theorem of provability logic | $~$ iff $~$ |
---|
Fixed point theorem of provability logic | $~$A$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$B$~$ |
---|
Fixed point theorem of provability logic | $~$n$~$ |
---|
Fixed point theorem of provability logic | $~$n$~$ |
---|
Fixed point theorem of provability logic | $~$n$~$ |
---|
Fixed point theorem of provability logic | $~$\square$~$ |
---|
Fixed point theorem of provability logic | $~$A$~$ |
---|
Fixed point theorem of provability logic | $~$A$~$ |
---|
Fixed point theorem of provability logic | $~$p\leftrightarrow A$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$\square^{n+1}\bot\wedge \square^n \bot$~$ |
---|
Fixed point theorem of provability logic | $~$n$~$ |
---|
Fixed point theorem of provability logic | $~$p\leftrightarrow \neg\square p$~$ |
---|
Fixed point theorem of provability logic | $~$\neg\square p$~$ |
---|
Fixed point theorem of provability logic | $~$0$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$\square B$~$ |
---|
Fixed point theorem of provability logic | $~$0$~$ |
---|
Fixed point theorem of provability logic | $~$B$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$\neg\square p$~$ |
---|
Fixed point theorem of provability logic | $$~$
\begin{array}{cccc}
\text{world= } & p & \square (p) & \neg \square (p) \\
0 & \bot & \top & \bot \\
1 & \top & \bot & \top \\
2 & \top & \bot & \top \\
\end{array}
$~$$ |
---|
Fixed point theorem of provability logic | $~$\square$~$ |
---|
Fixed point theorem of provability logic | $~$2$~$ |
---|
Fixed point theorem of provability logic | $~$[[p]]_{\neg\square p} = \mathbb{N}\setminus \{0\}$~$ |
---|
Fixed point theorem of provability logic | $~$H = \square^{0+1}\bot \wedge \square^0\bot = \neg\square\bot$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \square [p\leftrightarrow \neg\square p]\leftrightarrow \square[p\leftrightarrow \neg\square \bot]$~$ |
---|
Fixed point theorem of provability logic | $~$PA$~$ |
---|
Fixed point theorem of provability logic | $~$PA\vdash \square_{PA} [G\leftrightarrow \neg\square_{PA} G]\leftrightarrow \square_{PA}[G\leftrightarrow \neg\square_{PA} \bot]$~$ |
---|
Fixed point theorem of provability logic | $~$G$~$ |
---|
Fixed point theorem of provability logic | $~$PA$~$ |
---|
Fixed point theorem of provability logic | $~$G$~$ |
---|
Fixed point theorem of provability logic | $~$PA\vdash G\leftrightarrow \neg\square_{PA} G$~$ |
---|
Fixed point theorem of provability logic | $~$G$~$ |
---|
Fixed point theorem of provability logic | $~$PA\vdash \square_PA[ G\leftrightarrow \neg\square_{PA} G]$~$ |
---|
Fixed point theorem of provability logic | $~$PA\vdash \square_{PA}[G\leftrightarrow \neg\square_{PA} \bot]$~$ |
---|
Fixed point theorem of provability logic | $~$PA$~$ |
---|
Fixed point theorem of provability logic | $~$PA\vdash G\leftrightarrow \neg\square_{PA} \bot$~$ |
---|
Fixed point theorem of provability logic | $~$G$~$ |
---|
Fixed point theorem of provability logic | $~$PA$~$ |
---|
Fixed point theorem of provability logic | $~$\omega$~$ |
---|
Fixed point theorem of provability logic | $~$H\leftrightarrow\square H$~$ |
---|
Fixed point theorem of provability logic | $$~$
\begin{array}{ccc}
\text{world= } & p & \square (p) \\
0 & \top & \top \\
1 & \top & \top \\
\end{array}
$~$$ |
---|
Fixed point theorem of provability logic | $~$\top$~$ |
---|
Fixed point theorem of provability logic | $~$\phi(p, q_1,…,q_n)$~$ |
---|
Fixed point theorem of provability logic | $~$\phi$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$\phi$~$ |
---|
Fixed point theorem of provability logic | $~$\phi$~$ |
---|
Fixed point theorem of provability logic | $~$B(\square D_1(p), …, \square D_{k}(p))$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$\square$~$ |
---|
Fixed point theorem of provability logic | $~$q_i$~$ |
---|
Fixed point theorem of provability logic | $~$B$~$ |
---|
Fixed point theorem of provability logic | $~$D_i$~$ |
---|
Fixed point theorem of provability logic | $~$k$~$ |
---|
Fixed point theorem of provability logic | $~$\phi$~$ |
---|
Fixed point theorem of provability logic | $~$\phi$~$ |
---|
Fixed point theorem of provability logic | $~$0$~$ |
---|
Fixed point theorem of provability logic | $~$p$~$ |
---|
Fixed point theorem of provability logic | $~$B_i = B(\square D_1(p), …, \square D_{i-1}(p),\top, \square D_{i+1}(p),…,\square D_k(p))$~$ |
---|
Fixed point theorem of provability logic | $~$k-1$~$ |
---|
Fixed point theorem of provability logic | $~$k-1$~$ |
---|
Fixed point theorem of provability logic | $~$H_i$~$ |
---|
Fixed point theorem of provability logic | $~$B_i$~$ |
---|
Fixed point theorem of provability logic | $~$H=B(\square D_1(H_1),…,\square D_k(H_k))$~$ |
---|
Fixed point theorem of provability logic | $~$\phi$~$ |
---|
Fixed point theorem of provability logic | $~$p\leftrightarrow \neg\square(q\to p)$~$ |
---|
Fixed point theorem of provability logic | $~$B(d)=\neg d$~$ |
---|
Fixed point theorem of provability logic | $~$D_1(p)=q\to p$~$ |
---|
Fixed point theorem of provability logic | $~$B_1(p)=\neg \top = \bot$~$ |
---|
Fixed point theorem of provability logic | $~$H=B(\square D_1(\bot))=\neg\square \neg q$~$ |
---|
Fixed point theorem of provability logic | $~$p\leftrightarrow \square [\square(p\wedge q)\wedge \square(p\wedge r)]$~$ |
---|
Fixed point theorem of provability logic | $~$B(a)=a$~$ |
---|
Fixed point theorem of provability logic | $~$D_1(p)=\square(p\wedge q)\wedge \square(p\wedge r)$~$ |
---|
Fixed point theorem of provability logic | $~$B(\top)$~$ |
---|
Fixed point theorem of provability logic | $~$\top$~$ |
---|
Fixed point theorem of provability logic | $~$B(\square D_1(p=\top))=\square[\square(\top\wedge q)\wedge \square(\top\wedge r)]=\square[\square(q)\wedge \square(r)]$~$ |
---|
Fixed point theorem of provability logic | $~$A_i(p_1,…,p_n)$~$ |
---|
Fixed point theorem of provability logic | $~$n$~$ |
---|
Fixed point theorem of provability logic | $~$A_i$~$ |
---|
Fixed point theorem of provability logic | $~$p_n$~$ |
---|
Fixed point theorem of provability logic | $~$p_js$~$ |
---|
Fixed point theorem of provability logic | $~$H_1, …,H_n$~$ |
---|
Fixed point theorem of provability logic | $~$p_j$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \wedge_{i\le n} \{\boxdot (p_i\leftrightarrow A_i(p_1,…,p_n)\}\leftrightarrow \wedge_{i\le n} \{\boxdot(p_i\leftrightarrow H_i)\}$~$ |
---|
Fixed point theorem of provability logic | $~$H$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot(p_1\leftrightarrow A_i(p_1,…,p_n)) \leftrightarrow \boxdot(p_1\leftrightarrow H(p_2,…,p_n))$~$ |
---|
Fixed point theorem of provability logic | $~$j$~$ |
---|
Fixed point theorem of provability logic | $~$H_1,…,H_j$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \wedge_{i\le j} \{\boxdot(p_i\leftrightarrow A_i(p_1,…,p_n)\}\leftrightarrow \wedge_{i\le j} \{\boxdot(p_i\leftrightarrow H_i(p_{j+1},…,p_n))\}$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot(A\leftrightarrow B)\rightarrow [F(A)\leftrightarrow F(B)]$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot(p_i\leftrightarrow H_i(p_{j+1},…,p_n)\rightarrow [\boxdot(p_{j+1}\leftrightarrow A_{j+1}(p_{1},…,p_n))\leftrightarrow \boxdot(p_{j+1}\leftrightarrow A_{j+1}(p_{1},…,p_{i-1},H_i(p_{j+1},…,p_n),p_{i+1},…,p_n))]$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \wedge_{i\le j} \{\boxdot(p_i\leftrightarrow A_i(p_1,…,p_n)\}\rightarrow \boxdot(p_{j+1}\leftrightarrow A_{j+1}(H_1,…,H_j,p_{j+1},…,p_n))$~$ |
---|
Fixed point theorem of provability logic | $~$H_{j+1}'$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot(p_{j+1}\leftrightarrow A_{j+1}(H_1,…,H_j,p_{j+1},…,p_n)) \leftrightarrow \boxdot[p_{j+1}\leftrightarrow H_{j+1}'(p_{j+2},…,p_n)]$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \boxdot[p_{j+1}\leftrightarrow H_{j+1}'(p_{j+2},…,p_n)]\rightarrow [\boxdot(p_i\leftrightarrow H_i(p_{j+1},…,p_n)) \leftrightarrow \boxdot(p_i\leftrightarrow H_i(H_{j+1}',…,p_n))$~$ |
---|
Fixed point theorem of provability logic | $~$H_{i}'$~$ |
---|
Fixed point theorem of provability logic | $~$H_i(H_{j+1}',…,p_n)$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \wedge_{i\le j+1} \{\boxdot(p_i\leftrightarrow A_i(p_1,…,p_n)\}\rightarrow \wedge_{i\le j+1} \{\boxdot(p_i\leftrightarrow H_i'(p_{j+2},…,p_n))\}$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \wedge_{i\le j+1} \{\boxdot(p_i\leftrightarrow A_i(p_1,…,p_n)\}\leftrightarrow \wedge_{i\le j+1} \{\boxdot(p_i\leftrightarrow H_i'(p_{j+2},…,p_n))\}$~$ |
---|
Fixed point theorem of provability logic | $~$\square$~$ |
---|
Fixed point theorem of provability logic | $~$H_i'$~$ |
---|
Fixed point theorem of provability logic | $~$H_i$~$ |
---|
Fixed point theorem of provability logic | $~$A_i$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash H_i\leftrightarrow A_i(H_1,…,H_n)$~$ |
---|
Fixed point theorem of provability logic | $~$GL$~$ |
---|
Fixed point theorem of provability logic | $~$p_i$~$ |
---|
Fixed point theorem of provability logic | $~$H_i$~$ |
---|
Fixed point theorem of provability logic | $~$GL\vdash \wedge_{i\le n} \{\boxdot (H_i\leftrightarrow A_i(H_1,…,H_n)\}\leftrightarrow \wedge_{i\le n} \{\boxdot(H_i\leftrightarrow H_i)\}$~$ |
---|
Fixed point theorem of provability logic | $~$GL$~$ |
---|
Flag the load-bearing premises | $~$\neg X$~$ |
---|
Formal Logic | $~$S$~$ |
---|
Formal Logic | $~$O$~$ |
---|
Formal Logic | $~$M$~$ |
---|
Formal Logic | $~$C$~$ |
---|
Formal Logic | $~$S$~$ |
---|
Formal Logic | $~$O$~$ |
---|
Formal Logic | $~$S$~$ |
---|
Formal Logic | $~$O$~$ |
---|
Formal Logic | $~$M$~$ |
---|
Formal Logic | $~$C$~$ |
---|
Formal Logic | $~$M$~$ |
---|
Formal Logic | $~$C$~$ |
---|
Formal Logic | $~$\rightarrow$~$ |
---|
Formal Logic | $~$A$~$ |
---|
Formal Logic | $~$B$~$ |
---|
Formal Logic | $~$A \rightarrow B$~$ |
---|
Formal Logic | $~$\therefore$~$ |
---|
Formal definition of the free group | $~$X^r$~$ |
---|
Formal definition of the free group | $~$X \cup X^{-1}$~$ |
---|
Formal definition of the free group | $~$aa^{-1}$~$ |
---|
Formal definition of the free group | $~$r$~$ |
---|
Formal definition of the free group | $~$F(X)$~$ |
---|
Formal definition of the free group | $~$FX$~$ |
---|
Formal definition of the free group | $~$X$~$ |
---|
Formal definition of the free group | $~$\mathrm{Sym}(X^r)$~$ |
---|
Formal definition of the free group | $~$x \in X \cup X^{-1}$~$ |
---|
Formal definition of the free group | $~$\rho_x : \mathrm{Sym}(X^r) \to \mathrm{Sym}(X^r)$~$ |
---|
Formal definition of the free group | $~$a_1 a_2 \dots a_n \mapsto a_1 a_2 \dots a_n x$~$ |
---|
Formal definition of the free group | $~$a_n \not = x^{-1}$~$ |
---|
Formal definition of the free group | $~$a_1 a_2 \dots a_{n-1} x^{-1} \mapsto a_1 a_2 \dots a_{n-1}$~$ |
---|
Formal definition of the free group | $~$\rho_{x^{-1}} : \mathrm{Sym}(X^r) \to \mathrm{Sym}(X^r)$~$ |
---|
Formal definition of the free group | $~$a_1 a_2 \dots a_n \mapsto a_1 a_2 \dots a_n x^{-1}$~$ |
---|
Formal definition of the free group | $~$a_n \not = x$~$ |
---|
Formal definition of the free group | $~$a_1 a_2 \dots a_{n-1} x \mapsto a_1 a_2 \dots a_{n-1}$~$ |
---|
Formal definition of the free group | $~$\rho_x$~$ |
---|
Formal definition of the free group | $~$\mathrm{Sym}(X^r)$~$ |
---|
Formal definition of the free group | $~$X^r$~$ |
---|
Formal definition of the free group | $~$X^r$~$ |
---|
Formal definition of the free group | $~$X^r$~$ |
---|
Formal definition of the free group | $~$X$~$ |
---|
Formal definition of the free group | $~$x^{-1}$~$ |
---|
Formal definition of the free group | $~$\rho_x$~$ |
---|
Formal definition of the free group | $~$x$~$ |
---|
Formal definition of the free group | $~$x^{-1}$~$ |
---|
Formal definition of the free group | $~$\rho_x$~$ |
---|
Formal definition of the free group | $~$x^{-1}$~$ |
---|
Formal definition of the free group | $~$\rho_x$~$ |
---|
Formal definition of the free group | $~$x$~$ |
---|
Formal definition of the free group | $~$\rho_x$~$ |
---|
Formal definition of the free group | $~$X^r \to X^r$~$ |
---|
Formal definition of the free group | $~$x^{-1}$~$ |
---|
Formal definition of the free group | $~$\rho_{x^{-1}}$~$ |
---|
Formal definition of the free group | $~$\rho_{\varepsilon}$~$ |
---|
Formal definition of the free group | $~$\rho_x$~$ |
---|
Formal definition of the free group | $~$\rho_{x^{-1}}$~$ |
---|
Formal definition of the free group | $~$\mathrm{Sym}(X^r)$~$ |
---|
Formal definition of the free group | $~$\rho_x$~$ |
---|
Formal definition of the free group | $~$\rho_{x^{-1}}$~$ |
---|
Formal definition of the free group | $~$\rho_x \cdot \rho_y = \rho_x \circ \rho_y$~$ |
---|
Formal definition of the free group | $~$\rho_x \rho_y$~$ |
---|
Formal definition of the free group | $~$\rho_{a_n} \rho_{a_{n-1}} \dots \rho_{a_1}$~$ |
---|
Formal definition of the free group | $~$\varepsilon$~$ |
---|
Formal definition of the free group | $$~$\rho_{a_n} \rho_{a_{n-1}} \dots \rho_{a_1}(\varepsilon) = \rho_{a_n} \rho_{a_{n-1}} \dots \rho_{a_3}(\rho_{a_2}(a_1)) = \rho_{a_n a_{n-1} \dots a_3}(a_1 a_2) = \dots = a_1 a_2 \dots a_n$~$$ |
---|
Formal definition of the free group | $~$a_1 a_2 \dots a_n$~$ |
---|
Formal definition of the free group | $~$\rho_{a_i}, \rho_{a_{i+1}}$~$ |
---|
Formal definition of the free group | $~$\rho_{a_i}$~$ |
---|
Formal definition of the free group | $~$w = a_1 a_2 \dots a_n$~$ |
---|
Formal definition of the free group | $~$\rho_{a_1} \rho_{a_2} \dots \rho_{a_n}$~$ |
---|
Formal definition of the free group | $~$\rho_{a_1} \circ \rho_{a_2} \circ \dots \circ \rho_{a_n}$~$ |
---|
Formal definition of the free group | $~$a_i$~$ |
---|
Formal definition of the free group | $~$X \cup X^{-1}$~$ |
---|
Formal definition of the free group | $~$\rho_{a_i}$~$ |
---|
Formal definition of the free group | $~$a_1 a_2 \dots a_n$~$ |
---|
Formal definition of the free group | $~$b_1 b_2 \dots b_m$~$ |
---|
Formal definition of the free group | $~$\rho_{a_1} \rho_{a_2} \dots \rho_{a_n} = \rho_{b_1} \rho_{b_2} \dots \rho_{b_m}$~$ |
---|
Formal definition of the free group | $~$a_1 \dots a_n = b_1 \dots b_m$~$ |
---|
Formal definition of the free group | $~$\varepsilon$~$ |
---|
Formal definition of the free group | $~$\rho_{a_1} \rho_{a_2} \dots \rho_{a_n}$~$ |
---|
Formal definition of the free group | $~$a_n a_{n-1} \dots a_2 a_1$~$ |
---|
Formal definition of the free group | $~$\rho_{b_1} \rho_{b_2} \dots \rho_{b_m}$~$ |
---|
Formal definition of the free group | $~$b_m b_{m-1} \dots b_2 b_1$~$ |
---|
Formal definition of the free group | $~$\rho_x$~$ |
---|
Formal definition of the free group | $~$\rho_{x^{-1}}$~$ |
---|
Formal definition of the free group | $~$x \in X$~$ |
---|
Formal definition of the free group | $~$\rho_{x_1} \dots \rho_{x_n}$~$ |
---|
Formal definition of the free group | $~$x_1, \dots, x_n \in X \cup X^{-1}$~$ |
---|
Formal definition of the free group | $~$x_1 \dots x_n$~$ |
---|
Formal definition of the free group | $~$x_i, x_{i+1}$~$ |
---|
Formal definition of the free group | $~$\rho_{x_1} \dots \rho_{x_n}$~$ |
---|
Formal definition of the free group | $~$\rho_{x_1} \rho_{x_1^{-1}} \rho_{x_2} = \rho_{x_2}$~$ |
---|
Fractional bits | $~$\log_2(8) = 3$~$ |
---|
Fractional bits | $~$\log_2(1024) = 10$~$ |
---|
Fractional bits | $~$\log_2(3) \approx 1.58.$~$ |
---|
Fractional bits | $~$\log_2(3),$~$ |
---|
Fractional bits | $~$n \ge 5$~$ |
---|
Fractional bits | $~$n - 5.$~$ |
---|
Fractional bits: Digit usage interpretation | $~$10 \cdot 10 \cdot \sqrt{10} \approx 316,$~$ |
---|
Fractional bits: Digit usage interpretation | $~$\sqrt{10}$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\log_2(7)$~$ |
---|
Fractional bits: Expected cost interpretation | $~$n$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\lceil \log_2(n) \rceil$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\log_2(7) \neq 2.875,$~$ |
---|
Fractional bits: Expected cost interpretation | $~$(m, n)$~$ |
---|
Fractional bits: Expected cost interpretation | $~$7m + n,$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\lceil \log_2(49) \rceil = 6$~$ |
---|
Fractional bits: Expected cost interpretation | $~$64 - 49 = 15$~$ |
---|
Fractional bits: Expected cost interpretation | $~$6 - \frac{15}{49} \approx 5.694$~$ |
---|
Fractional bits: Expected cost interpretation | $~$(9 - \frac{169}{343})\approx 8.507$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\approx 2.836$~$ |
---|
Fractional bits: Expected cost interpretation | $~$2.807$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\log_2(7)$~$ |
---|
Fractional bits: Expected cost interpretation | $~$n$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\lceil \log_2(n) \rceil$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\log_2(n).$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\log_2(n)$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\log_2(n)$~$ |
---|
Fractional bits: Expected cost interpretation | $~$b$~$ |
---|
Fractional bits: Expected cost interpretation | $~$x < \log_2(b)$~$ |
---|
Fractional bits: Expected cost interpretation | $~$b$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\log_b(2) \cdot x$~$ |
---|
Fractional bits: Expected cost interpretation | $~$2$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\log_b(2)$~$ |
---|
Fractional bits: Expected cost interpretation | $~$b$~$ |
---|
Fractional bits: Expected cost interpretation | $~$2$~$ |
---|
Fractional bits: Expected cost interpretation | $~$x$~$ |
---|
Fractional bits: Expected cost interpretation | $~$b$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\log_b(2) \cdot \log_2(b) = 1$~$ |
---|
Fractional bits: Expected cost interpretation | $~$b,$~$ |
---|
Fractional bits: Expected cost interpretation | $~$b$~$ |
---|
Fractional bits: Expected cost interpretation | $~$\log_2(b)$~$ |
---|
Fractional digits | $~$b$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$\log_b(x)$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$b$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$b$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$b$~$ |
---|
Fractional digits | $~$\log_{3.16}(5.62) \approx 1.5$~$ |
---|
Fractional digits | $~$3.16^{1.5} \approx 5.62,$~$ |
---|
Fractional digits | $~$a$~$ |
---|
Fractional digits | $~$b$~$ |
---|
Fractional digits | $~$5a + b.$~$ |
---|
Fractional digits | $~$\log_{10}(5) + \log_{10}(2) = 1$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$y$~$ |
---|
Fractional digits | $~$x \cdot y \le n$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$y$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$18$~$ |
---|
Fractional digits | $~$3$~$ |
---|
Fractional digits | $~$6$~$ |
---|
Fractional digits | $~$a$~$ |
---|
Fractional digits | $~$b$~$ |
---|
Fractional digits | $~$6a+b.$~$ |
---|
Fractional digits | $~$n = x \cdot y,$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$y$~$ |
---|
Fractional digits | $~$n = x \cdot y$~$ |
---|
Fractional digits | $~$\log_b(x) + \log_b(y) = \log_b(n),$~$ |
---|
Fractional digits | $~$b$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$x \cdot x < 10.$~$ |
---|
Fractional digits | $~$a$~$ |
---|
Fractional digits | $~$b$~$ |
---|
Fractional digits | $~$31a + b$~$ |
---|
Fractional digits | $~$31 \cdot 30 + 30 = 960 \le 999$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$x \cdot x \le n$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$x=316$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$x^2 \le 100000.$~$ |
---|
Fractional digits | $~$\log_b(316) \approx \frac{5\log_b(10)}{2}$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$x \cdot x = n,$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$y$~$ |
---|
Fractional digits | $~$y \cdot y \cdot y = 216,$~$ |
---|
Fractional digits | $~$y$~$ |
---|
Fractional digits | $~$y = \sqrt[3]{2 \cdot 12 \cdot 9} = 6$~$ |
---|
Fractional digits | $~$\sqrt[2]{1 \cdot 10} \approx 3.16.$~$ |
---|
Fractional digits | $~$\sqrt[2]{10}$~$ |
---|
Fractional digits | $~$\sqrt[3]{1 \cdot 1 \cdot 10} \approx 2.15.$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$1 < n \le 10$~$ |
---|
Fractional digits | $~$\log_{3.16}(5.62) \approx 1.5$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$\sqrt{n}$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$x \cdot x$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$\sqrt{n}$~$ |
---|
Fractional digits | $~$10^2 = 100.$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$n^2$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$\sqrt{n}$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$x > 1.$~$ |
---|
Fractional digits | $~$\sqrt[n]{10} > 1$~$ |
---|
Fractional digits | $~$n$~$ |
---|
Fractional digits | $~$x$~$ |
---|
Fractional digits | $~$0 < x < 1,$~$ |
---|
Free group | $~$F(X)$~$ |
---|
Free group | $~$X$~$ |
---|
Free group | $~$X$~$ |
---|
Free group | $~$F(X)$~$ |
---|
Free group | $~$X$~$ |
---|
Free group | $~$X$~$ |
---|
Free group | $~$X$~$ |
---|
Free group | $~$F(X)$~$ |
---|
Free group | $~$FX$~$ |
---|
Free group | $~$X$~$ |
---|
Free group | $~$X$~$ |
---|
Free group | $~$X = \{ a, b \}$~$ |
---|
Free group | $~$(a,b,a,a,a,b^{-1})$~$ |
---|
Free group | $~$abaaab^{-1}$~$ |
---|
Free group | $~$aba^3b^{-1}$~$ |
---|
Free group | $~$()$~$ |
---|
Free group | $~$\varepsilon$~$ |
---|
Free group | $~$(b,b,b)$~$ |
---|
Free group | $~$b^3$~$ |
---|
Free group | $~$(a^{-1}, b^{-1}, b^{-1})$~$ |
---|
Free group | $~$a^{-1} b^{-2}$~$ |
---|
Free group | $~$aa^{-1}$~$ |
---|
Free group | $~$c$~$ |
---|
Free group | $~$c$~$ |
---|
Free group | $~$\{a,b\}$~$ |
---|
Free group | $~$abb^{-1}a$~$ |
---|
Free group | $~$\cdot$~$ |
---|
Free group | $~$aba \cdot bab = ababab$~$ |
---|
Free group | $~$aba^2 \cdot a^3b = aba^5b$~$ |
---|
Free group | $~$aba^{-1} \cdot a = ab$~$ |
---|
Free group | $~$aba^{-1}a$~$ |
---|
Free group | $~$ab \cdot b^{-1} a^{-1} = \varepsilon$~$ |
---|
Free group | $~$abb^{-1}a^{-1} = aa^{-1}$~$ |
---|
Free group | $~$b$~$ |
---|
Free group | $~$a a^{-1} = \varepsilon$~$ |
---|
Free group | $~$\{ a \}$~$ |
---|
Free group | $~$a^n$~$ |
---|
Free group | $~$a^{-n}$~$ |
---|
Free group | $~$a^0$~$ |
---|
Free group | $~$a^i$~$ |
---|
Free group | $~$i \in \mathbb{Z}$~$ |
---|
Free group | $~$a^{i_1} b^{j_1} a^{i_2} b^{j_2} \dots a^{i_n} b^{j_n}$~$ |
---|
Free group | $$~$a^{i_1} b^{j_1} a^{i_2} b^{j_2} \dots a^{i_n} b^{j_n} \mapsto 2^{\mathrm{sgn}(i_1)+2} 3^{|i_1|} 5^{\mathrm{sgn}(j_1)+2} 7^{|j_1|} \dots$~$$ |
---|
Free group | $~$\mathrm{sgn}$~$ |
---|
Free group | $~$-1$~$ |
---|
Free group | $~$1$~$ |
---|
Free group | $~$0$~$ |
---|
Free group | $~$0$~$ |
---|
Free group | $~$X$~$ |
---|
Free group | $~$X = \{ a, b \}$~$ |
---|
Free group | $~$C_2$~$ |
---|
Free group | $~$a$~$ |
---|
Free group | $~$b \cdot b = a$~$ |
---|
Free group | $~$X$~$ |
---|
Free group | $~$b^2 = a$~$ |
---|
Free group | $~$FX$~$ |
---|
Free group | $~$a, b$~$ |
---|
Free group | $~$a$~$ |
---|
Free group | $~$b$~$ |
---|
Free group | $~$\varepsilon$~$ |
---|
Free group | $~$a \cdot b$~$ |
---|
Free group | $~$a$~$ |
---|
Free group | $~$b$~$ |
---|
Free group | $~$\varepsilon$~$ |
---|
Free group | $~$a \cdot b$~$ |
---|
Free group | $~$ab$~$ |
---|
Free group | $~$a^{-1} \cdot a$~$ |
---|
Free group | $~$\varepsilon$~$ |
---|
Free group | $~$a^{-1} a$~$ |
---|
Free group | $~$a^{-1}ba^2b^{-2}$~$ |
---|
Free group | $~$G$~$ |
---|
Free group | $~$\langle X \mid R \rangle$~$ |
---|
Free group | $~$G$~$ |
---|
Free group | $~$F(X)$~$ |
---|
Free group | $~$F(X)$~$ |
---|
Free group | $~$G$~$ |
---|
Free group | $~$FX$~$ |
---|
Free group | $~$FY$~$ |
---|
Free group | $~$X$~$ |
---|
Free group | $~$Y$~$ |
---|
Free group | $~$\mathbb{Z}$~$ |
---|
Free group | $~$a, b$~$ |
---|
Free group | $~$ab \not = ba$~$ |
---|
Free group | $~$\rho_a \rho_b \not = \rho_b \rho_a$~$ |
---|
Free group | $~$\varepsilon$~$ |
---|
Free group | $~$ab$~$ |
---|
Free group | $~$ba$~$ |
---|
Free group | $~$\varepsilon$~$ |
---|
Free group | $~$x \in \mathbb{Q}$~$ |
---|
Free group | $~$n \not = 0$~$ |
---|
Free group | $~$x+x+\dots+x$~$ |
---|
Free group | $~$n$~$ |
---|
Free group | $~$0$~$ |
---|
Free group | $~$(\mathbb{Q}, +)$~$ |
---|
Free group | $~$n \times x = 0$~$ |
---|
Free group | $~$n=0$~$ |
---|
Free group | $~$x = 0$~$ |
---|
Free group | $~$n \not = 0$~$ |
---|
Free group | $~$x = 0$~$ |
---|
Free group | $~$x$~$ |
---|
Free group | $~$\mathbb{Q}$~$ |
---|
Free group | $~$\mathbb{Z}$~$ |
---|
Free group | $~$\mathbb{Z}$~$ |
---|
Free group | $~$\mathbb{Z}$~$ |
---|
Free group | $~$1$~$ |
---|
Free group | $~$\mathbb{Z}$~$ |
---|
Free group | $~$1$~$ |
---|
Free group | $~$\mathbb{Q}$~$ |
---|
Free group | $~$x$~$ |
---|
Free group | $~$\frac{x}{2}$~$ |
---|
Free group | $~$x$~$ |
---|
Free group universal property | $~$X$~$ |
---|
Free group universal property | $~$FX$~$ |
---|
Free group universal property | $~$X$~$ |
---|
Free group universal property | $~$G$~$ |
---|
Free group universal property | $~$f: X \to G$~$ |
---|
Free group universal property | $~$G$~$ |
---|
Free group universal property | $~$G$~$ |
---|
Free group universal property | $~$\overline{f}: FX \to G$~$ |
---|
Free group universal property | $~$\overline{f}(\rho_{a_1} \rho_{a_2} \dots \rho_{a_n}) = f(a_1) \cdot f(a_2) \cdot \dots \cdot f(a_n)$~$ |
---|
Free group universal property | $~$FX$~$ |
---|
Free group universal property | $~$G$~$ |
---|
Free group universal property | $~$f: X \to G$~$ |
---|
Free group universal property | $~$FX \to G$~$ |
---|
Free group universal property | $~$X$~$ |
---|
Free group universal property | $~$f$~$ |
---|
Free group universal property | $~$FX$~$ |
---|
Free group universal property | $~$FX$~$ |
---|
Free group universal property | $~$C_3$~$ |
---|
Free group universal property | $~$\{ e, a, b\}$~$ |
---|
Free group universal property | $~$e$~$ |
---|
Free group universal property | $~$a + a = b$~$ |
---|
Free group universal property | $~$a+b = e = b+a$~$ |
---|
Free group universal property | $~$b+b = a$~$ |
---|
Free group universal property | $~$a$~$ |
---|
Free group universal property | $~$a=a$~$ |
---|
Free group universal property | $~$a+a = b$~$ |
---|
Free group universal property | $~$a+a+a = e$~$ |
---|
Free group universal property | $~$G = (\mathbb{Z}, +)$~$ |
---|
Free group universal property | $~$f: C_3 \to \mathbb{Z}$~$ |
---|
Free group universal property | $~$a \mapsto 1$~$ |
---|
Free group universal property | $~$C_3$~$ |
---|
Free group universal property | $~$\{ e, a, b\}$~$ |
---|
Free group universal property | $~$\overline{f}: C_3 \to \mathbb{Z}$~$ |
---|
Free group universal property | $~$\overline{f}(a) = 1$~$ |
---|
Free group universal property | $~$f$~$ |
---|
Free group universal property | $~$\overline{f}$~$ |
---|
Free group universal property | $~$\overline{f}(e) = \overline{f}(a+a+a) = 1+1+1 = 3$~$ |
---|
Free group universal property | $~$\overline{f}(e) = 3$~$ |
---|
Free group universal property | $~$C_3$~$ |
---|
Free group universal property | $~$a+a+a = e$~$ |
---|
Free group universal property | $~$\overline{f}$~$ |
---|
Free group universal property | $~$C_3$~$ |
---|
Free groups are torsion-free | $~$FX$~$ |
---|
Free groups are torsion-free | $~$X$~$ |
---|
Free groups are torsion-free | $~$FX$~$ |
---|
Free groups are torsion-free | $~$X$~$ |
---|
Free groups are torsion-free | $~$a_1 a_2 \dots a_n$~$ |
---|
Free groups are torsion-free | $~$a_1 \not = a_n^{-1}$~$ |
---|
Free groups are torsion-free | $~$w$~$ |
---|
Free groups are torsion-free | $~$r w^\prime r^{-1}$~$ |
---|
Free groups are torsion-free | $~$r$~$ |
---|
Free groups are torsion-free | $~$w^\prime$~$ |
---|
Free groups are torsion-free | $~$r$~$ |
---|
Free groups are torsion-free | $~$r^{-1}$~$ |
---|
Free groups are torsion-free | $~$w^\prime$~$ |
---|
Free groups are torsion-free | $~$w$~$ |
---|
Free groups are torsion-free | $~$w$~$ |
---|
Free groups are torsion-free | $~$r = \varepsilon$~$ |
---|
Free groups are torsion-free | $~$w^\prime = w$~$ |
---|
Free groups are torsion-free | $~$w$~$ |
---|
Free groups are torsion-free | $~$a v a^{-1}$~$ |
---|
Free groups are torsion-free | $~$a \in X$~$ |
---|
Free groups are torsion-free | $~$v$~$ |
---|
Free groups are torsion-free | $~$v$~$ |
---|
Free groups are torsion-free | $~$w$~$ |
---|
Free groups are torsion-free | $~$v$~$ |
---|
Free groups are torsion-free | $~$r v^\prime r^{-1}$~$ |
---|
Free groups are torsion-free | $~$v^\prime$~$ |
---|
Free groups are torsion-free | $~$w = a r v^\prime r^{-1} a^{-1} = (ar) v^\prime (ar)^{-1}$~$ |
---|
Free groups are torsion-free | $~$r w^\prime r^{-1} = s v^\prime s^{-1}$~$ |
---|
Free groups are torsion-free | $~$s^{-1} r w^\prime r^{-1} s = v^\prime$~$ |
---|
Free groups are torsion-free | $~$v^\prime$~$ |
---|
Free groups are torsion-free | $~$s$~$ |
---|
Free groups are torsion-free | $~$v^\prime = r w^\prime r^{-1}$~$ |
---|
Free groups are torsion-free | $~$w = r w^\prime r^{-1}$~$ |
---|
Free groups are torsion-free | $~$r = e$~$ |
---|
Free groups are torsion-free | $~$v^\prime = w^\prime = w$~$ |
---|
Free groups are torsion-free | $~$s$~$ |
---|
Free groups are torsion-free | $~$r^{-1}$~$ |
---|
Free groups are torsion-free | $~$s$~$ |
---|
Free groups are torsion-free | $~$r$~$ |
---|
Free groups are torsion-free | $~$r$~$ |
---|
Free groups are torsion-free | $~$s$~$ |
---|
Free groups are torsion-free | $~$r$~$ |
---|
Free groups are torsion-free | $~$v^\prime = w^\prime$~$ |
---|
Free groups are torsion-free | $~$w$~$ |
---|
Free groups are torsion-free | $~$n$~$ |
---|
Free groups are torsion-free | $~$r w^\prime r^{-1}$~$ |
---|
Free groups are torsion-free | $~$(rw^\prime r^{-1})^n = r (w^\prime)^n r^{-1}$~$ |
---|
Free groups are torsion-free | $~$r$~$ |
---|
Free groups are torsion-free | $~$w^\prime$~$ |
---|
Free groups are torsion-free | $~$r^{-1}$~$ |
---|
Free groups are torsion-free | $~$r, (w^\prime)^n, r^{-1}$~$ |
---|
Free groups are torsion-free | $~$w^\prime$~$ |
---|
Free groups are torsion-free | $~$r (w^\prime)^n r^{-1}$~$ |
---|
Freely reduced word | $~$X$~$ |
---|
Freely reduced word | $~$X$~$ |
---|
Freely reduced word | $~$X^{-1}$~$ |
---|
Freely reduced word | $~$X^{-1}$~$ |
---|
Freely reduced word | $~$X$~$ |
---|
Freely reduced word | $~$X$~$ |
---|
Freely reduced word | $~$X^{-1}$~$ |
---|
Freely reduced word | $~$X$~$ |
---|
Freely reduced word | $~$x x^{-1}$~$ |
---|
Freely reduced word | $~$X$~$ |
---|
Freely reduced word | $~$X^{-1}$~$ |
---|
Freely reduced word | $~$X$~$ |
---|
Freely reduced word | $~$X$~$ |
---|
Freely reduced word | $~$x^{-1}$~$ |
---|
Freely reduced word | $$~$X^{-1} = \{ x^{-1} \mid x \in X \}$~$$ |
---|
Freely reduced word | $~$x^{-1}$~$ |
---|
Freely reduced word | $~$X \cup X^{-1}$~$ |
---|
Freely reduced word | $~$X \cup X^{-1}$~$ |
---|
Freely reduced word | $~$X \cup X^{-1}$~$ |
---|
Freely reduced word | $~$X = \{ 1, 2 \}$~$ |
---|
Freely reduced word | $~$X$~$ |
---|
Freely reduced word | $~$\varepsilon$~$ |
---|
Freely reduced word | $~$(1)$~$ |
---|
Freely reduced word | $~$(2)$~$ |
---|
Freely reduced word | $~$(2^{-1})$~$ |
---|
Freely reduced word | $~$(1, 2^{-1}, 2, 1, 1, 1, 2^{-1}, 1^{-1}, 1^{-1})$~$ |
---|
Freely reduced word | $~$\varepsilon$~$ |
---|
Freely reduced word | $~$1$~$ |
---|
Freely reduced word | $~$2$~$ |
---|
Freely reduced word | $~$2^{-1}$~$ |
---|
Freely reduced word | $~$1 2^{-1} 2 1 1 1 2^{-1} 1^{-1} 1^{-1}$~$ |
---|
Freely reduced word | $~$1 2^{-1} 2 1^3 2^{-1} 1^{-2}$~$ |
---|
Freely reduced word | $~$r r^{-1}$~$ |
---|
Freely reduced word | $~$r^{-1} r$~$ |
---|
Freely reduced word | $~$r \in X$~$ |
---|
Freely reduced word | $~$X = \{ a, b, c \}$~$ |
---|
Freely reduced word | $~$X^{-1}$~$ |
---|
Freely reduced word | $~$\{ a^{-1}, b^{-1}, c^{-1} \}$~$ |
---|
Freely reduced word | $~$\{ x, y, z \}$~$ |
---|
Freely reduced word | $~$a^{-1}$~$ |
---|
Freely reduced word | $~$x$~$ |
---|
Freely reduced word | $~$X \cup X^{-1} = \{ a,b,c, a^{-1}, b^{-1}, c^{-1} \}$~$ |
---|
Freely reduced word | $~$X \cup X^{-1}$~$ |
---|
Freely reduced word | $~$\varepsilon$~$ |
---|
Freely reduced word | $~$a$~$ |
---|
Freely reduced word | $~$aaaa$~$ |
---|
Freely reduced word | $~$b$~$ |
---|
Freely reduced word | $~$b^{-1}$~$ |
---|
Freely reduced word | $~$ab$~$ |
---|
Freely reduced word | $~$ab^{-1}cbb^{-1}c^{-1}$~$ |
---|
Freely reduced word | $~$aa^{-1}aa^{-1}$~$ |
---|
Freely reduced word | $~$ab^{-1}cbb^{-1}c^{-1}$~$ |
---|
Freely reduced word | $~$bb^{-1}$~$ |
---|
Freely reduced word | $~$aa^{-1}aa^{-1}$~$ |
---|
Freely reduced word | $~$aa^{-1}$~$ |
---|
Freely reduced word | $~$a^{-1} a$~$ |
---|
Freely reduced word | $~$a^{-1}$~$ |
---|
Freely reduced word | $~$b^{-1}$~$ |
---|
Freely reduced word | $~$X^{-1}$~$ |
---|
Freely reduced word | $~$\{ x, y, z \}$~$ |
---|
Freely reduced word | $~$\{ a, b, c \}$~$ |
---|
Freely reduced word | $~$\{ a^{-1}, b^{-1}, c^{-1} \}$~$ |
---|
Freely reduced word | $~$\varepsilon$~$ |
---|
Freely reduced word | $~$a$~$ |
---|
Freely reduced word | $~$aaaa$~$ |
---|
Freely reduced word | $~$a^4$~$ |
---|
Freely reduced word | $~$b$~$ |
---|
Freely reduced word | $~$y$~$ |
---|
Freely reduced word | $~$ab$~$ |
---|
Freely reduced word | $~$aycbyz$~$ |
---|
Freely reduced word | $~$axax$~$ |
---|
Freely reduced word | $~$aycbyz$~$ |
---|
Freely reduced word | $~$by$~$ |
---|
Freely reduced word | $~$axax$~$ |
---|
Freely reduced word | $~$ax$~$ |
---|
Freely reduced word | $~$xa$~$ |
---|
Freely reduced word | $~$X$~$ |
---|
Freely reduced word | $~$X \cup X^{-1}$~$ |
---|
Freely reduced word | $~$r r^{-1}$~$ |
---|
Freely reduced word | $~$r^{-1} r$~$ |
---|
Freely reduced word | $~$r r^{-1}$~$ |
---|
Freely reduced word | $~$r \in X$~$ |
---|
Freely reduced word | $~$r^{-1} r$~$ |
---|
Freely reduced word | $~$X$~$ |
---|
Freely reduced word | $~$X$~$ |
---|
Function | $~$f$~$ |
---|
Function | $~$f$~$ |
---|
Function | $~$X$~$ |
---|
Function | $~$Y$~$ |
---|
Function | $~$-$~$ |
---|
Function | $~$(4, 3)$~$ |
---|
Function | $~$1,$~$ |
---|
Function | $~$(19, 2)$~$ |
---|
Function | $~$17,$~$ |
---|
Function | $~$f : X \to Y$~$ |
---|
Function | $~$f$~$ |
---|
Function | $~$X$~$ |
---|
Function | $~$Y$~$ |
---|
Function | $~$f$~$ |
---|
Function | $~$X$~$ |
---|
Function | $~$Y$~$ |
---|
Function | $~$- : (\mathbb N \times \mathbb N) \to \mathbb N,$~$ |
---|
Function | $~$X$~$ |
---|
Function | $~$f.$~$ |
---|
Function | $~$Y$~$ |
---|
Function | $~$f$~$ |
---|
Function | $~$f : \mathbb{R} \to \mathbb{R}$~$ |
---|
Function | $~$f(x) = x^2$~$ |
---|
Function: Physical metaphor | $~$+$~$ |
---|
Function: Physical metaphor | $~$+$~$ |
---|
Function: Physical metaphor | $~$\times$~$ |
---|
Fundamental Theorem of Arithmetic | $~$2$~$ |
---|
Fundamental Theorem of Arithmetic | $~$1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$3 \times 5$~$ |
---|
Fundamental Theorem of Arithmetic | $~$3 \times 5 \times 1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$15$~$ |
---|
Fundamental Theorem of Arithmetic | $~$1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$\mathbb{Z}$~$ |
---|
Fundamental Theorem of Arithmetic | $~$\mathbb{Z}$~$ |
---|
Fundamental Theorem of Arithmetic | $~$\mathbb{Z}$~$ |
---|
Fundamental Theorem of Arithmetic | $~$0$~$ |
---|
Fundamental Theorem of Arithmetic | $~$1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$17 \times 23 \times 23$~$ |
---|
Fundamental Theorem of Arithmetic | $~$2$~$ |
---|
Fundamental Theorem of Arithmetic | $~$17 \times 23^2$~$ |
---|
Fundamental Theorem of Arithmetic | $~$\{ 17, 23, 23\}$~$ |
---|
Fundamental Theorem of Arithmetic | $~$2$~$ |
---|
Fundamental Theorem of Arithmetic | $~$2$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n$~$ |
---|
Fundamental Theorem of Arithmetic | $~$2$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n$~$ |
---|
Fundamental Theorem of Arithmetic | $~$a \times b$~$ |
---|
Fundamental Theorem of Arithmetic | $~$a$~$ |
---|
Fundamental Theorem of Arithmetic | $~$b$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n$~$ |
---|
Fundamental Theorem of Arithmetic | $~$a$~$ |
---|
Fundamental Theorem of Arithmetic | $~$b$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n$~$ |
---|
Fundamental Theorem of Arithmetic | $~$a$~$ |
---|
Fundamental Theorem of Arithmetic | $~$b$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n = 1274$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n$~$ |
---|
Fundamental Theorem of Arithmetic | $~$49 \times 26$~$ |
---|
Fundamental Theorem of Arithmetic | $~$49$~$ |
---|
Fundamental Theorem of Arithmetic | $~$7^2$~$ |
---|
Fundamental Theorem of Arithmetic | $~$26$~$ |
---|
Fundamental Theorem of Arithmetic | $~$2 \times 13$~$ |
---|
Fundamental Theorem of Arithmetic | $~$1274$~$ |
---|
Fundamental Theorem of Arithmetic | $~$2 \times 7^2 \times 13$~$ |
---|
Fundamental Theorem of Arithmetic | $~$49$~$ |
---|
Fundamental Theorem of Arithmetic | $~$1274$~$ |
---|
Fundamental Theorem of Arithmetic | $~$26$~$ |
---|
Fundamental Theorem of Arithmetic | $~$1274$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p$~$ |
---|
Fundamental Theorem of Arithmetic | $~$ab$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p$~$ |
---|
Fundamental Theorem of Arithmetic | $~$a$~$ |
---|
Fundamental Theorem of Arithmetic | $~$b$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n = 2$~$ |
---|
Fundamental Theorem of Arithmetic | $~$1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$2$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p_1 p_2 \dots p_r$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_1 q_2 \dots q_s$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p_i$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_j$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p_1 = p_2 = q_3 = q_7$~$ |
---|
Fundamental Theorem of Arithmetic | $~$r=s$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p_i = q_i$~$ |
---|
Fundamental Theorem of Arithmetic | $~$i$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p_1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$n$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p_1 p_2 \dots p_r$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_1 q_2 \dots q_s$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_2 \dots q_s$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_2$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_3 \dots q_s$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p_1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_i$~$ |
---|
Fundamental Theorem of Arithmetic | $~$i=1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_i$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p_1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p_1 = q_1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p_1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p_2 \dots p_r = q_2 \dots q_s$~$ |
---|
Fundamental Theorem of Arithmetic | $~$r-1 = s-1$~$ |
---|
Fundamental Theorem of Arithmetic | $~$r=s$~$ |
---|
Fundamental Theorem of Arithmetic | $~$p_i$~$ |
---|
Fundamental Theorem of Arithmetic | $~$q_i$~$ |
---|
Fundamental Theorem of Arithmetic | $~$i \geq 2$~$ |
---|
Fundamental Theorem of Arithmetic | $~$\mathbb{Z}[\sqrt{-5}]$~$ |
---|
Fundamental Theorem of Arithmetic | $~$\mathbb{Z}[\sqrt{-3}]$~$ |
---|
Generalized associative law | $~$\cdot$~$ |
---|
Generalized associative law | $~$[a, b, c, \ldots]$~$ |
---|
Generalized associative law | $~$f$~$ |
---|
Generalized associative law | $~$f$~$ |
---|
Generalized associative law | $~$\cdot$~$ |
---|
Generalized associative law | $~$f : X \times X \to X$~$ |
---|
Generalized associative law | $~$X$~$ |
---|
Generalized associative law | $~$\cdot$~$ |
---|
Generalized associative law | $~$[a, b, c, \ldots]$~$ |
---|
Generalized associative law | $~$f$~$ |
---|
Generalized associative law | $~$f$~$ |
---|
Generalized associative law | $~$[a, b, c, d, e],$~$ |
---|
Generalized associative law | $~$a \cdot b$~$ |
---|
Generalized associative law | $~$ab.$~$ |
---|
Generalized associative law | $~$((ab)c)(de)$~$ |
---|
Generalized associative law | $~$a$~$ |
---|
Generalized associative law | $~$b$~$ |
---|
Generalized associative law | $~$c$~$ |
---|
Generalized associative law | $~$d$~$ |
---|
Generalized associative law | $~$e$~$ |
---|
Generalized associative law | $~$a(b(c(de))$~$ |
---|
Generalized associative law | $~$d$~$ |
---|
Generalized associative law | $~$e$~$ |
---|
Generalized associative law | $~$c$~$ |
---|
Generalized associative law | $~$b$~$ |
---|
Generalized associative law | $~$a$~$ |
---|
Generalized associative law | $~$abcde$~$ |
---|
Generalized associative law | $~$[a, b, c, d, e]$~$ |
---|
Generalized associative law | $~$\cdot$~$ |
---|
Generalized associative law | $~$f$~$ |
---|
Generalized associative law | $~$f$~$ |
---|
Generalized associative law | $~$f_4$~$ |
---|
Generalized associative law | $~$f$~$ |
---|
Generalized associative law | $~$f_5$~$ |
---|
Generalized associative law | $~$f,$~$ |
---|
Generalized associative law | $~$\cdot$~$ |
---|
Generalized associative law | $~$\cdot$~$ |
---|
Generalized associative law | $~$(x\cdot y) \cdot z = x \cdot (y \cdot z).$~$ |
---|
Generalized associative law | $~$x \cdot y$~$ |
---|
Generalized associative law | $~$xy,$~$ |
---|
Generalized associative law | $~$[a, b, c, d]$~$ |
---|
Generalized associative law | $~$a(b(cd)),$~$ |
---|
Generalized associative law | $~$\cdot$~$ |
---|
Generalized associative law | $~$a(b(cd))=a((bc)d)=(a(bc))d=((ab)c)d=(ab)(cd).$~$ |
---|
Generalized associative law | $~$f : X \times X \to X$~$ |
---|
Generalized associative law | $~$f_n$~$ |
---|
Generalized associative law | $~$n$~$ |
---|
Generalized associative law | $~$n \ge 1$~$ |
---|
Generalized associative law | $~$f_1$~$ |
---|
Generalized associative law | $~$f,$~$ |
---|
Generalized associative law | $~$[a, b, c, \ldots]$~$ |
---|
Generalized associative law | $~$\alpha,$~$ |
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Generalized associative law | $~$[x, y, z, \ldots]$~$ |
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Generalized associative law | $~$\chi,$~$ |
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Generalized associative law | $~$f(\alpha, \chi)$~$ |
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Generalized associative law | $~$[a, b, c, \ldots, x, y, z, \ldots]:$~$ |
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Generalized associative law | $~$f$~$ |
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Generalized associative law | $~$f$~$ |
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Generalized associative law | $~$f$~$ |
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Generalized associative law | $~$f$~$ |
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Generalized associative law | $~$f_n : X^n \to X$~$ |
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Generalized associative law | $~$n \ge 0,$~$ |
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Generalized associative law | $~$0_X$~$ |
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Generalized associative law | $~$X$~$ |
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Generalized associative law | $~$f_0$~$ |
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Generalized associative law | $~$0_X$~$ |
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Generalized associative law | $~$f.$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$x : A \to X$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$A$~$ |
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Generalized element | $~$x$~$ |
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Generalized element | $~$I$~$ |
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Generalized element | $~$*$~$ |
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Generalized element | $~$I = \{*\}$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$I$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$x$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$I$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$f(i) = x$~$ |
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Generalized element | $~$i \in I$~$ |
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Generalized element | $~$f$~$ |
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Generalized element | $~$x$~$ |
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Generalized element | $~$f : I \to X$~$ |
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Generalized element | $~$*$~$ |
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Generalized element | $~$I$~$ |
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Generalized element | $~$f(*)$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$I$~$ |
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Generalized element | $~$I \to X$~$ |
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Generalized element | $~$A$~$ |
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Generalized element | $~$n$~$ |
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Generalized element | $~$A$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$n$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$1$~$ |
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Generalized element | $~$1$~$ |
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Generalized element | $~$\mathbb{Z}$~$ |
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Generalized element | $~$\mathbb{Z}$~$ |
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Generalized element | $~$A$~$ |
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Generalized element | $~$A$~$ |
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Generalized element | $~$\text{Set} \times \text{Set}$~$ |
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Generalized element | $~$(X,Y)$~$ |
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Generalized element | $~$(2^A, 2^{X + B})$~$ |
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Generalized element | $~$(2^{Y + A}, 2^{B})$~$ |
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Generalized element | $~$(X,Y)$~$ |
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Generalized element | $~$(2^A)^X\times(2^{X+B})^Y \cong 2^{X\times A + Y \times (X + B)} \cong 2^{X \times A + Y \times B + X \times Y}$~$ |
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Generalized element | $~$(X,Y)$~$ |
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Generalized element | $~$(2^{Y+A})^X \times (2^B)^Y \cong 2^{X\times(Y+A) + Y \times B} \cong 2^{X \times A + Y \times B + X \times Y}$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$Y$~$ |
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Generalized element | $~$(0,1)$~$ |
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Generalized element | $~$(1,0)$~$ |
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Generalized element | $~$x$~$ |
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Generalized element | $~$A$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$f$~$ |
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Generalized element | $~$X$~$ |
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Generalized element | $~$Y$~$ |
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Generalized element | $~$f(x) := f\circ x$~$ |
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Generalized element | $~$A$~$ |
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Generalized element | $~$Y$~$ |
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Generalized element | $~$f(xu) = f(x) u$~$ |
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Geometric product | $~$e^{\text{I}\theta}$~$ |
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Geometric product | $~$n$~$ |
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Geometric product | $~$|a|^2 + |b|^2 = |a+b|^2$~$ |
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Geometric product | $~$(a+b)^2 = a^2 + ab + ba + b^2$~$ |
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Geometric product | $~$ab$~$ |
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Geometric product | $~$ba$~$ |
---|
Geometric product | $~$2ab$~$ |
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Geometric product | $~$a^2 = |a|^2$~$ |
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Geometric product | $~$|a+b|^2 = |a|^2 + ab + ba + |b|^2$~$ |
---|
Geometric product | $~$ab + ba$~$ |
---|
Geometric product | $~$ab + ba$~$ |
---|
Geometric product | $~$a$~$ |
---|
Geometric product | $~$b$~$ |
---|
Geometric product | $~$a$~$ |
---|
Geometric product | $~$b$~$ |
---|
Geometric product | $~$|a+b|^2 = (|a| + |b|)^2 = |a|^2 + 2|a||b| + |b|^2$~$ |
---|
Geometric product | $~$ab + ba$~$ |
---|
Geometric product | $~$2|a||b|$~$ |
---|
Geometric product | $~$ab = - ba$~$ |
---|
Geometric product | $~$ab = ba = |a||b|$~$ |
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Geometric product | $~$\frac{1}{a}=\frac{a}{|a|^2}$~$ |
---|
Geometric product | $~$a^{-1}$~$ |
---|
Geometric product | $~$ae^{\text{I}\pi/2} = b$~$ |
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Geometric product | $~$e^{\text{I}\pi/2} = \frac{ab}{|a|^2}$~$ |
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Geometric product | $~$ab = |a|^2e^{\text{I}\pi/2}$~$ |
---|
Geometric product | $~$|a| = |b|$~$ |
---|
Geometric product | $~$b$~$ |
---|
Geometric product | $~$a|b|/|a|e^{\text{I}\pi/2} = b$~$ |
---|
Geometric product | $~$ab = |a||b|e^{\text{I}\pi/2}$~$ |
---|
Geometric product | $~$|b||a|=-e^{\text{I}\pi/2}$~$ |
---|
Geometric product | $~$e^{\text{I}\pi/2}$~$ |
---|
Geometric product | $~$\text{I}$~$ |
---|
Geometric product | $~$ab = |a||b|I$~$ |
---|
Geometric product | $~$I^2 = -1$~$ |
---|
Geometric product | $~$ab$~$ |
---|
Geometric product | $~$a = a_xx+a_yy$~$ |
---|
Geometric product | $~$b = b_xx+b_yy$~$ |
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Geometric product | $~$ab = (a_xx + a_yy)(b_xx + b_yy) = a_xb_xx^2 + a_yb_xyx + a_xb_yxy+a_yb_yy^2 = a_xb_x + a_yb_y - a_yb_xI + a_xb_yI$~$ |
---|
Geometric product | $~$e^{\text{I}\pi/4} = \frac{1 + I}{\sqrt{2}}$~$ |
---|
Geometric product | $~$e^{\text{I}\theta} = \cos(\theta) + \text{I}\sin(\theta)$~$ |
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Geometric product | $~$k$~$ |
---|
Geometry of vectors: direction | $~$\mathbf a$~$ |
---|
Geometry of vectors: direction | $~$\mathbf b$~$ |
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Geometry of vectors: direction | $~$\mathbf x$~$ |
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Geometry of vectors: direction | $~$\mathbf y$~$ |
---|
Geometry of vectors: direction | $~$\mathbf z$~$ |
---|
Geometry of vectors: direction | $~$\mathbf a$~$ |
---|
Geometry of vectors: direction | $~$\mathbf b$~$ |
---|
Geometry of vectors: direction | $~$\mathbf a$~$ |
---|
Geometry of vectors: direction | $~$\mathbf b$~$ |
---|
Geometry of vectors: direction | $~$\mathbf B$~$ |
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Geometry of vectors: direction | $~$\mathbf I$~$ |
---|
Geometry of vectors: direction | $~$(\mathbf {x},\mathbf{y})$~$ |
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Geometry of vectors: direction | $~$(\mathbf{y},\mathbf{z})$~$ |
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Geometry of vectors: direction | $~$(\mathbf{x},\mathbf{z})$~$ |
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Geometry of vectors: direction | $~$(\mathbf {x},\mathbf{y})$~$ |
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Geometry of vectors: direction | $~$\mathbf w$~$ |
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Geometry of vectors: direction | $~$\mathbf x$~$ |
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Geometry of vectors: direction | $~$\mathbf y$~$ |
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Geometry of vectors: direction | $~$\mathbf z$~$ |
---|
Geometry of vectors: direction | $~$(\mathbf{w},\mathbf{x})$~$ |
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Geometry of vectors: direction | $~$(\mathbf{w},\mathbf{y})$~$ |
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Geometry of vectors: direction | $~$(\mathbf {w},\mathbf{z})$~$ |
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Geometry of vectors: direction | $~$(\mathbf{x},\mathbf{y})$~$ |
---|
Geometry of vectors: direction | $~$(\mathbf{x},\mathbf{z})$~$ |
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Geometry of vectors: direction | $~$(\mathbf{y},\mathbf{z})$~$ |
---|
Geometry of vectors: direction | $~$\mathbf a$~$ |
---|
Geometry of vectors: direction | $~$\mathbf b$~$ |
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Geometry of vectors: direction | $~$\mathbf a$~$ |
---|
Geometry of vectors: direction | $~$\mathbf b$~$ |
---|
Geometry of vectors: direction | $~$\mathbf b$~$ |
---|
Geometry of vectors: direction | $~$\mathbf a$~$ |
---|
Geometry of vectors: direction | $~$\pi$~$ |
---|
Geometry of vectors: direction | $~$3.14$~$ |
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Geometry of vectors: direction | $~$\pi$~$ |
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Geometry of vectors: direction | $~$\frac{\pi}{2}$~$ |
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Geometry of vectors: direction | $~$\frac{\pi}{2}$~$ |
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Geometry of vectors: direction | $~$0$~$ |
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Geometry of vectors: direction | $~$\pi$~$ |
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Geometry of vectors: direction | $~$\frac{\pi}{4}$~$ |
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Geometry of vectors: direction | $~$R$~$ |
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Geometry of vectors: direction | $~$\mathbf B$~$ |
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Geometry of vectors: direction | $~$e$~$ |
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Geometry of vectors: direction | $~$R = e^{\mathbf B}$~$ |
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Goodhart's Curse | $~$V$~$ |
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Goodhart's Curse | $~$V$~$ |
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Goodhart's Curse | $~$U$~$ |
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Goodhart's Curse | $~$V,$~$ |
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Goodhart's Curse | $~$U$~$ |
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Goodhart's Curse | $~$V,$~$ |
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Goodhart's Curse | $~$U$~$ |
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Goodhart's Curse | $~$U$~$ |
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Goodhart's Curse | $~$V.$~$ |
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Goodhart's Curse | $~$U$~$ |
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Goodhart's Curse | $~$U-V$~$ |
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Goodhart's Curse | $~$\|U - V\|$~$ |
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Graham's number | $~$f(x) = 3\uparrow^n 3$~$ |
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Graham's number | $~$f^n(x) = \underbrace{f(f(f(\cdots f(f(x)) \cdots ))}_{n\text{ applications of }f}$~$ |
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Graham's number | $~$f^{64}(4).$~$ |
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Greatest common divisor | $~$a$~$ |
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Greatest common divisor | $~$b$~$ |
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Greatest common divisor | $~$a$~$ |
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Greatest common divisor | $~$b$~$ |
---|
Greatest common divisor | $~$a$~$ |
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Greatest common divisor | $~$b$~$ |
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Greatest common divisor | $~$c$~$ |
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Greatest common divisor | $~$c \mid a$~$ |
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Greatest common divisor | $~$c \mid b$~$ |
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Greatest common divisor | $~$d \mid a$~$ |
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Greatest common divisor | $~$d \mid b$~$ |
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Greatest common divisor | $~$d \mid c$~$ |
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Greatest common divisor | $~$a$~$ |
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Greatest common divisor | $~$b$~$ |
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Greatest lower bound in a poset | $~$x$~$ |
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Greatest lower bound in a poset | $~$y$~$ |
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Greatest lower bound in a poset | $~$x$~$ |
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Greatest lower bound in a poset | $~$y$~$ |
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Greatest lower bound in a poset | $~$x$~$ |
---|
Greatest lower bound in a poset | $~$y$~$ |
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Greatest lower bound in a poset | $~$x$~$ |
---|
Greatest lower bound in a poset | $~$y$~$ |
---|
Greatest lower bound in a poset | $~$x$~$ |
---|
Greatest lower bound in a poset | $~$y$~$ |
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Greatest lower bound in a poset | $~$x$~$ |
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Greatest lower bound in a poset | $~$y$~$ |
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Greatest lower bound in a poset | $~$P$~$ |
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Greatest lower bound in a poset | $~$\leq$~$ |
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Greatest lower bound in a poset | $~$x$~$ |
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Greatest lower bound in a poset | $~$y$~$ |
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Greatest lower bound in a poset | $~$P$~$ |
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Greatest lower bound in a poset | $~$z \in P$~$ |
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Greatest lower bound in a poset | $~$x$~$ |
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Greatest lower bound in a poset | $~$y$~$ |
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Greatest lower bound in a poset | $~$z \leq x$~$ |
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Greatest lower bound in a poset | $~$z \leq y$~$ |
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Greatest lower bound in a poset | $~$z \in P$~$ |
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Greatest lower bound in a poset | $~$x$~$ |
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Greatest lower bound in a poset | $~$y$~$ |
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Greatest lower bound in a poset | $~$z$~$ |
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Greatest lower bound in a poset | $~$x$~$ |
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Greatest lower bound in a poset | $~$y$~$ |
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Greatest lower bound in a poset | $~$w$~$ |
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Greatest lower bound in a poset | $~$x$~$ |
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Greatest lower bound in a poset | $~$y$~$ |
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Greatest lower bound in a poset | $~$w \leq z$~$ |
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Group | $~$120^\circ$~$ |
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Group | $~$240^\circ$~$ |
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Group | $~$f$~$ |
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Group | $~$g$~$ |
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Group | $~$h$~$ |
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Group | $~$g \circ f$~$ |
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Group | $~$h \circ (g \circ f)$~$ |
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Group | $~$h \circ g$~$ |
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Group | $~$(h \circ g) \circ f$~$ |
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Group | $~$G$~$ |
---|
Group | $~$(X, \bullet)$~$ |
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Group | $~$X$~$ |
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Group | $~$\bullet$~$ |
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Group | $~$x, y$~$ |
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Group | $~$X$~$ |
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Group | $~$x \bullet y$~$ |
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Group | $~$X$~$ |
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Group | $~$x \bullet y$~$ |
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Group | $~$xy$~$ |
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Group | $~$x(yz) = (xy)z$~$ |
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Group | $~$x, y, z \in X$~$ |
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Group | $~$e$~$ |
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Group | $~$xe=ex=x$~$ |
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Group | $~$x \in X$~$ |
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Group | $~$x$~$ |
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Group | $~$X$~$ |
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Group | $~$x^{-1} \in X$~$ |
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Group | $~$xx^{-1}=x^{-1}x=e$~$ |
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Group | $~$120^\circ$~$ |
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Group | $~$240^\circ$~$ |
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Group | $~$G$~$ |
---|
Group | $~$(X, \bullet)$~$ |
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Group | $~$X$~$ |
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Group | $~$X$~$ |
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Group | $~$G$~$ |
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Group | $~$\bullet : G \times G \to G$~$ |
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Group | $~$x \bullet y$~$ |
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Group | $~$xy$~$ |
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Group | $~$\bullet$~$ |
---|
Group | $~$x, y$~$ |
---|
Group | $~$X$~$ |
---|
Group | $~$x \bullet y$~$ |
---|
Group | $~$X$~$ |
---|
Group | $~$x \bullet y$~$ |
---|
Group | $~$xy$~$ |
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Group | $~$e$~$ |
---|
Group | $~$xe=ex=x$~$ |
---|
Group | $~$x \in X$~$ |
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Group | $~$x$~$ |
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Group | $~$X$~$ |
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Group | $~$x^{-1} \in X$~$ |
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Group | $~$xx^{-1}=x^{-1}x=e$~$ |
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Group | $~$x(yz) = (xy)z$~$ |
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Group | $~$x, y, z \in X$~$ |
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Group | $~$\bullet$~$ |
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Group | $~$\bullet$~$ |
---|
Group | $~$G \times G \to G$~$ |
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Group | $~$e$~$ |
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Group | $~$G$~$ |
---|
Group | $~$\bullet$~$ |
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Group | $~$e$~$ |
---|
Group | $~$x$~$ |
---|
Group | $~$\bullet$~$ |
---|
Group | $~$x$~$ |
---|
Group | $~$e$~$ |
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Group | $~$z$~$ |
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Group | $~$ze = ez = z.$~$ |
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Group | $~$e$~$ |
---|
Group | $~$G$~$ |
---|
Group | $~$e$~$ |
---|
Group | $~$e$~$ |
---|
Group | $~$e$~$ |
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Group | $~$1$~$ |
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Group | $~$1_G$~$ |
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Group | $~$\bullet$~$ |
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Group | $~$X$~$ |
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Group | $~$1$~$ |
---|
Group | $~$\bullet$~$ |
---|
Group | $~$0$~$ |
---|
Group | $~$0_G$~$ |
---|
Group | $~$x$~$ |
---|
Group | $~$X$~$ |
---|
Group | $~$y$~$ |
---|
Group | $~$\bullet$~$ |
---|
Group | $~$x$~$ |
---|
Group | $~$xy = e$~$ |
---|
Group | $~$x$~$ |
---|
Group | $~$x^{-1}$~$ |
---|
Group | $~$(-x)$~$ |
---|
Group | $~$\bullet$~$ |
---|
Group | $~$f$~$ |
---|
Group | $~$g$~$ |
---|
Group | $~$h$~$ |
---|
Group | $~$g \circ f$~$ |
---|
Group | $~$h \circ (g \circ f)$~$ |
---|
Group | $~$h \circ g$~$ |
---|
Group | $~$(h \circ g) \circ f$~$ |
---|
Group | $~$(\mathbb{Z}, +)$~$ |
---|
Group | $~$\mathbb{Z}$~$ |
---|
Group | $~$+$~$ |
---|
Group | $~$\mathbb Z \times \mathbb Z \to \mathbb Z$~$ |
---|
Group | $~$(x+y)+z=x+(y+z)$~$ |
---|
Group | $~$0+x = x = x + 0$~$ |
---|
Group | $~$x$~$ |
---|
Group | $~$-x$~$ |
---|
Group | $~$x + (-x) = 0$~$ |
---|
Group | $~$G = (X, \bullet)$~$ |
---|
Group | $~$X$~$ |
---|
Group | $~$\bullet$~$ |
---|
Group | $~$X$~$ |
---|
Group | $~$G$~$ |
---|
Group | $~$\bullet$~$ |
---|
Group | $~$x \bullet y$~$ |
---|
Group | $~$xy$~$ |
---|
Group | $~$G$~$ |
---|
Group | $~$X$~$ |
---|
Group | $~$x, y \in X$~$ |
---|
Group | $~$G$~$ |
---|
Group | $~$x, y \in G$~$ |
---|
Group | $~$G$~$ |
---|
Group | $~$|G|$~$ |
---|
Group | $~$|X|$~$ |
---|
Group | $~$X$~$ |
---|
Group | $~$|G|=9$~$ |
---|
Group | $~$G$~$ |
---|
Group action | $~$G$~$ |
---|
Group action | $~$X$~$ |
---|
Group action | $~$\alpha : G \times X \to X$~$ |
---|
Group action | $~$(g, x) \mapsto gx$~$ |
---|
Group action | $~$\alpha$~$ |
---|
Group action | $~$ex = x$~$ |
---|
Group action | $~$x \in X$~$ |
---|
Group action | $~$e$~$ |
---|
Group action | $~$g(hx) = (gh)x$~$ |
---|
Group action | $~$g, h \in G, x \in X$~$ |
---|
Group action | $~$gh$~$ |
---|
Group action | $~$G$~$ |
---|
Group action | $~$G$~$ |
---|
Group action | $~$X$~$ |
---|
Group action | $~$G \to \text{Aut}(X)$~$ |
---|
Group action | $~$\text{Aut}(X)$~$ |
---|
Group action | $~$X$~$ |
---|
Group action | $~$X \to X$~$ |
---|
Group action | $~$X = \mathbb{R}^2$~$ |
---|
Group action | $~$\mathbb{R}^2$~$ |
---|
Group action | $~$ISO(2)$~$ |
---|
Group action | $~$f : \mathbb{R}^2 \to \mathbb{R}^2$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho: G \times X \to X$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$G$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$X$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(g)$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$X \to X$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$x \mapsto \rho(g, x)$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(g)$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(g^{-1})$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(g)$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(g^{-1})(\rho(g)(x))$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(g^{-1})(\rho(g, x))$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(g^{-1}, \rho(g, x))$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(g^{-1} g, x) = \rho(e, x) = x$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$e$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(g)(\rho(g^{-1})(x)) = x$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(g)$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\mathrm{Sym}(X)$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\mathrm{Sym}$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$G$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\mathrm{Sym}(X)$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$G \times X$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$X$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho: G \to \mathrm{Sym}(X)$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(gh) = \rho(g) \rho(h)$~$ |
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Group action induces homomorphism to the symmetric group | $~$\mathrm{Sym}(X)$~$ |
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Group action induces homomorphism to the symmetric group | $~$\rho(gh)(x) = \rho(gh, x)$~$ |
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Group action induces homomorphism to the symmetric group | $~$\rho(gh)$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(g, \rho(h, x))$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho$~$ |
---|
Group action induces homomorphism to the symmetric group | $~$\rho(g)(\rho(h, x))$~$ |
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Group action induces homomorphism to the symmetric group | $~$\rho(g)$~$ |
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Group action induces homomorphism to the symmetric group | $~$\rho(g)(\rho(h)(x))$~$ |
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Group action induces homomorphism to the symmetric group | $~$\rho(h)$~$ |
---|
Group conjugate | $~$x, y$~$ |
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Group conjugate | $~$G$~$ |
---|
Group conjugate | $~$h \in G$~$ |
---|
Group conjugate | $~$hxh^{-1} = y$~$ |
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Group conjugate | $~$h$~$ |
---|
Group conjugate | $~$h$~$ |
---|
Group conjugate | $$~$\sigma = (a_{11} a_{12} \dots a_{1 n_1})(a_{21} \dots a_{2 n_2}) \dots (a_{k 1} a_{k 2} \dots a_{k n_k})$~$$ |
---|
Group conjugate | $~$\tau \in S_n$~$ |
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Group conjugate | $$~$\tau \sigma \tau^{-1} = (\tau(a_{11}) \tau(a_{12}) \dots \tau(a_{1 n_1}))(\tau(a_{21}) \dots \tau(a_{2 n_2})) \dots (\tau(a_{k 1}) \tau(a_{k 2}) \dots \tau(a_{k n_k}))$~$$ |
---|
Group conjugate | $~$\tau$~$ |
---|
Group conjugate | $~$\sigma$~$ |
---|
Group conjugate | $~$\tau$~$ |
---|
Group conjugate | $~$D_{2n}$~$ |
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Group conjugate | $~$n$~$ |
---|
Group conjugate | $~$G$~$ |
---|
Group conjugate | $~$X$~$ |
---|
Group conjugate | $~$g \in G$~$ |
---|
Group conjugate | $~$h \in G$~$ |
---|
Group conjugate | $~$hgh^{-1}$~$ |
---|
Group conjugate | $~$g$~$ |
---|
Group conjugate | $~$X$~$ |
---|
Group conjugate | $~$h$~$ |
---|
Group conjugate | $~$H$~$ |
---|
Group conjugate | $~$G$~$ |
---|
Group conjugate | $~$G$~$ |
---|
Group conjugate | $~$H$~$ |
---|
Group conjugate | $~$G$~$ |
---|
Group conjugate | $~$\rho: G \times G \to G$~$ |
---|
Group conjugate | $~$\rho(g, k) = g k g^{-1}$~$ |
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Group conjugate | $~$\rho(gh, k) = (gh)k(gh)^{-1} = ghkh^{-1}g^{-1} = g \rho(h, k) g^{-1} = \rho(g, \rho(h, k))$~$ |
---|
Group conjugate | $~$\rho(e, k) = eke^{-1} = k$~$ |
---|
Group conjugate | $~$\mathrm{Stab}_G(g)$~$ |
---|
Group conjugate | $~$g \in G$~$ |
---|
Group conjugate | $~$kgk^{-1} = g$~$ |
---|
Group conjugate | $~$kg = gk$~$ |
---|
Group conjugate | $~$g$~$ |
---|
Group conjugate | $~$G$~$ |
---|
Group conjugate | $~$G$~$ |
---|
Group conjugate | $~$\mathrm{Orb}_G(g)$~$ |
---|
Group conjugate | $~$g \in G$~$ |
---|
Group conjugate | $~$g$~$ |
---|
Group conjugate | $~$G$~$ |
---|
Group coset | $~$H$~$ |
---|
Group coset | $~$G$~$ |
---|
Group coset | $~$H$~$ |
---|
Group coset | $~$G$~$ |
---|
Group coset | $~$\{ gh : h \in H \}$~$ |
---|
Group coset | $~$g \in G$~$ |
---|
Group coset | $~$gH$~$ |
---|
Group coset | $~$Hg = \{ hg: h \in H \}$~$ |
---|
Group coset | $~$S_3$~$ |
---|
Group coset | $~$\{ e, (123), (132), (12), (13), (23) \}$~$ |
---|
Group coset | $~$A_3$~$ |
---|
Group coset | $~$\{ e, (123), (132) \}$~$ |
---|
Group coset | $~$(12) A_3$~$ |
---|
Group coset | $~$\{ (12), (12)(123), (12)(132) \}$~$ |
---|
Group coset | $~$\{ (12), (23), (13) \}$~$ |
---|
Group coset | $~$(123)A_3$~$ |
---|
Group coset | $~$A_3$~$ |
---|
Group coset | $~$A_3$~$ |
---|
Group coset | $~$(123)$~$ |
---|
Group coset | $~$A_3$~$ |
---|
Group coset | $~$H$~$ |
---|
Group coset | $~$G$~$ |
---|
Group coset | $~$G$~$ |
---|
Group coset | $~$H$~$ |
---|
Group coset | $~$H$~$ |
---|
Group coset | $~$G$~$ |
---|
Group coset | $~$p$~$ |
---|
Group coset | $~$p$~$ |
---|
Group homomorphism | $~$(G, +)$~$ |
---|
Group homomorphism | $~$(H, *)$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$H$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$H$~$ |
---|
Group homomorphism | $~$f$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$H$~$ |
---|
Group homomorphism | $~$f(a) * f(b) = f(a+b)$~$ |
---|
Group homomorphism | $~$a, b \in G$~$ |
---|
Group homomorphism | $~$(G, +)$~$ |
---|
Group homomorphism | $~$(H, *)$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$H$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$H$~$ |
---|
Group homomorphism | $~$f$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$H$~$ |
---|
Group homomorphism | $~$f(a) * f(b) = f(a+b)$~$ |
---|
Group homomorphism | $~$a, b \in G$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$1_G: G \to G$~$ |
---|
Group homomorphism | $~$1_G(g) = g$~$ |
---|
Group homomorphism | $~$g \in G$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$\{ e \}$~$ |
---|
Group homomorphism | $~$e * e = e$~$ |
---|
Group homomorphism | $~$g \mapsto e$~$ |
---|
Group homomorphism | $~$g \in G$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$X$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$e \mapsto e_G$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$(G, +)$~$ |
---|
Group homomorphism | $~$G^{\mathrm{op}}$~$ |
---|
Group homomorphism | $~$g \mapsto g^{-1}$~$ |
---|
Group homomorphism | $~$G^{\mathrm{op}}$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$g +_{\mathrm{op}} h := h + g$~$ |
---|
Group homomorphism | $~$G, H$~$ |
---|
Group homomorphism | $~$G$~$ |
---|
Group homomorphism | $~$H$~$ |
---|
Group homomorphism | $~$g \mapsto e_H$~$ |
---|
Group homomorphism | $~$C_2 = \{ e_{C_2}, g \}$~$ |
---|
Group homomorphism | $~$C_3 = \{e_{C_3}, h, h^2 \}$~$ |
---|
Group homomorphism | $~$e_{C_2} \mapsto e_{C_3}, g \mapsto e_{C_3}$~$ |
---|
Group homomorphism | $~$f: C_2 \to C_3$~$ |
---|
Group homomorphism | $~$e_{C_2} \mapsto e_{C_3}, g \mapsto h$~$ |
---|
Group homomorphism | $~$e_{C_3} = f(e_{C_2}) = f(gg) = f(g) f(g) = h h = h^2$~$ |
---|
Group isomorphism | $~$(\{ a \}, +_a)$~$ |
---|
Group isomorphism | $~$(\{ b \}, +_b)$~$ |
---|
Group isomorphism | $~$+_x$~$ |
---|
Group isomorphism | $~$(x, x)$~$ |
---|
Group isomorphism | $~$x$~$ |
---|
Group isomorphism | $~$\{a \} \to \{ b \}$~$ |
---|
Group isomorphism | $~$a \mapsto b$~$ |
---|
Group orbit | $~$X$~$ |
---|
Group orbit | $~$x \in X$~$ |
---|
Group orbit | $~$G$~$ |
---|
Group orbit | $~$X$~$ |
---|
Group orbit | $~$x$~$ |
---|
Group orbit | $~$Gx = \{gx : g \in G\}$~$ |
---|
Group orbit | $~$X$~$ |
---|
Group orbit | $~$X$~$ |
---|
Group orbit | $~$G$~$ |
---|
Group orbits partition | $~$G$~$ |
---|
Group orbits partition | $~$X$~$ |
---|
Group orbits partition | $~$X$~$ |
---|
Group orbits partition | $~$G$~$ |
---|
Group orbits partition | $~$X$~$ |
---|
Group orbits partition | $~$X$~$ |
---|
Group orbits partition | $~$x \in X$~$ |
---|
Group orbits partition | $~$x \in X$~$ |
---|
Group orbits partition | $~$\mathrm{Orb}_G(x)$~$ |
---|
Group orbits partition | $~$e(x) = x$~$ |
---|
Group orbits partition | $~$e$~$ |
---|
Group orbits partition | $~$G$~$ |
---|
Group orbits partition | $~$x$~$ |
---|
Group orbits partition | $~$\mathrm{Orb}_G(a)$~$ |
---|
Group orbits partition | $~$\mathrm{Orb}_G(b)$~$ |
---|
Group orbits partition | $~$a, b \in X$~$ |
---|
Group orbits partition | $~$g(a) = h(b) = x$~$ |
---|
Group orbits partition | $~$g, h \in G$~$ |
---|
Group orbits partition | $~$h^{-1}g(a) = b$~$ |
---|
Group orbits partition | $~$\mathrm{Orb}_G(a) = \mathrm{Orb}_G(b)$~$ |
---|
Group orbits partition | $~$r \in \mathrm{Orb}_G(b)$~$ |
---|
Group orbits partition | $~$r = k(b)$~$ |
---|
Group orbits partition | $~$k \in G$~$ |
---|
Group orbits partition | $~$r = k(h^{-1}g(a)) = kh^{-1}g(a)$~$ |
---|
Group orbits partition | $~$r \in \mathrm{Orb}_G(a)$~$ |
---|
Group orbits partition | $~$r \in \mathrm{Orb}_G(a)$~$ |
---|
Group orbits partition | $~$r = m(b)$~$ |
---|
Group orbits partition | $~$m \in G$~$ |
---|
Group orbits partition | $~$r = m(g^{-1}h(b)) = m g^{-1} h (b)$~$ |
---|
Group orbits partition | $~$r \in \mathrm{Orb}_G(b)$~$ |
---|
Group presentation | $~$\langle X \mid R \rangle$~$ |
---|
Group presentation | $~$X$~$ |
---|
Group presentation | $~$R$~$ |
---|
Group presentation | $~$\langle X \mid R \rangle$~$ |
---|
Group presentation | $~$G$~$ |
---|
Group presentation | $~$X$~$ |
---|
Group presentation | $~$R$~$ |
---|
Group presentation | $~$X \cup X^{-1}$~$ |
---|
Group presentation | $~$G \cong F(X) / \llangle R \rrangle^{F(X)}$~$ |
---|
Group presentation | $~$\llangle R \rrangle$~$ |
---|
Group presentation | $~$F(X)$~$ |
---|
Group presentation | $~$\langle X \mid R \rangle$~$ |
---|
Group presentation | $~$G$~$ |
---|
Group presentation | $~$G$~$ |
---|
Group presentation | $~$X$~$ |
---|
Group presentation | $~$X$~$ |
---|
Group presentation | $~$X^{-1}$~$ |
---|
Group presentation | $~$X$~$ |
---|
Group presentation | $~$X = \{ a, b \}$~$ |
---|
Group presentation | $~$X^{-1}$~$ |
---|
Group presentation | $~$\{ a^{-1}, b^{-1} \}$~$ |
---|
Group presentation | $~$R$~$ |
---|
Group presentation | $~$X \cup X^{-1}$~$ |
---|
Group presentation | $~$G$~$ |
---|
Group presentation | $~$R$~$ |
---|
Group presentation | $~$G$~$ |
---|
Group presentation | $~$X$~$ |
---|
Group presentation | $~$G$~$ |
---|
Group presentation | $~$G$~$ |
---|
Group presentation | $~$t$~$ |
---|
Group presentation | $~$t$~$ |
---|
Group presentation | $~$F(G)$~$ |
---|
Group presentation | $~$G$~$ |
---|
Group presentation | $~$\phi: F(G) \to G$~$ |
---|
Group presentation | $~$(a_1, a_2, \dots, a_n)$~$ |
---|
Group presentation | $~$a_1 a_2 \dots a_n$~$ |
---|
Group presentation | $~$C_2$~$ |
---|
Group presentation | $~$\langle x \mid x^2 \rangle$~$ |
---|
Group presentation | $~$x$~$ |
---|
Group presentation | $~$x^2$~$ |
---|
Group presentation | $~$x^2$~$ |
---|
Group presentation | $~$e$~$ |
---|
Group presentation | $~$\langle x \mid x^4 \rangle$~$ |
---|
Group presentation | $~$x^4$~$ |
---|
Group presentation | $~$C_2$~$ |
---|
Group presentation | $~$\langle X \mid R \rangle$~$ |
---|
Group presentation | $~$X$~$ |
---|
Group presentation | $~$R$~$ |
---|
Group presentation | $~$x^2 = e$~$ |
---|
Group presentation | $~$\langle x \mid x^4 \rangle$~$ |
---|
Group presentation | $~$\langle x, y \mid xyx^{-1}y^{-1} \rangle$~$ |
---|
Group presentation | $~$xyx^{-1}y^{-1} = e$~$ |
---|
Group presentation | $~$xy=yx$~$ |
---|
Group presentation | $~$x$~$ |
---|
Group presentation | $~$y$~$ |
---|
Group presentation | $~$x^n y^m$~$ |
---|
Group presentation | $~$n, m$~$ |
---|
Group presentation | $~$xyx = xxy = x^2y$~$ |
---|
Group presentation | $~$x$~$ |
---|
Group presentation | $~$x^{-1}$~$ |
---|
Group presentation | $~$(m, n)$~$ |
---|
Group presentation | $~$m, n$~$ |
---|
Group presentation | $~$\mathbb{Z}^2$~$ |
---|
Group presentation | $~$\langle x, y \mid x^2, y \rangle$~$ |
---|
Group presentation | $~$C_2$~$ |
---|
Group presentation | $~$y$~$ |
---|
Group presentation | $~$\langle a, b \mid aba^{-1}b^{-2}, bab^{-1}a^{-2} \rangle$~$ |
---|
Group presentation | $~$ab = b^2 a$~$ |
---|
Group presentation | $~$ba = a^2 b$~$ |
---|
Group presentation | $~$aba^{-1} b^{-2} = e$~$ |
---|
Group presentation | $~$ab = b^2 a$~$ |
---|
Group presentation | $~$b ba$~$ |
---|
Group presentation | $~$ba = a^2 b$~$ |
---|
Group presentation | $~$b a^2 b$~$ |
---|
Group presentation | $~$ab = b a^2 b$~$ |
---|
Group presentation | $~$a = b a^2$~$ |
---|
Group presentation | $~$b$~$ |
---|
Group presentation | $~$a$~$ |
---|
Group presentation | $~$e = ba$~$ |
---|
Group presentation | $~$a = b^{-1}$~$ |
---|
Group presentation | $~$ab = b^2 a = b b a$~$ |
---|
Group presentation | $~$ab$~$ |
---|
Group presentation | $~$ba$~$ |
---|
Group presentation | $~$e = b$~$ |
---|
Group presentation | $~$b$~$ |
---|
Group presentation | $~$a = b^{-1}$~$ |
---|
Group presentation | $~$a$~$ |
---|
Group theory | $~$G$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$\bullet$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$\bullet$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$G$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$\bullet$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$\bullet$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$\mathbb{Z}$~$ |
---|
Group theory | $~$f : \mathbb{R} \to \mathbb{R}$~$ |
---|
Group theory | $~$n : f(x) \mapsto f(x - n)$~$ |
---|
Group theory | $~$f$~$ |
---|
Group theory | $~$n$~$ |
---|
Group theory | $~$1$~$ |
---|
Group theory | $~$f(x) = \sum \left( a_n \cos 2 \pi n x + b_n \sin 2 \pi n x \right)$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$\bullet$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$G$~$ |
---|
Group theory | $~$\bullet$~$ |
---|
Group theory | $~$\bullet$~$ |
---|
Group theory | $~$\bullet$~$ |
---|
Group theory | $~$G$~$ |
---|
Group theory | $~$G$~$ |
---|
Group theory | $~$\bullet$~$ |
---|
Group theory | $~$\bullet$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$(0, 0, 0)$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$X$~$ |
---|
Group theory | $~$\mathbb Z$~$ |
---|
Group theory | $~$\mathbb Q$~$ |
---|
Group theory | $~$\mathbb R$~$ |
---|
Group theory | $~$\sqrt{2}$~$ |
---|
Group theory | $~$\mathbb Z \to \mathbb Q \to \mathbb R$~$ |
---|
Group theory | $~$0$~$ |
---|
Group theory | $~$+$~$ |
---|
Group theory: Examples | $~$f : \mathbb{R} \to \mathbb{R}$~$ |
---|
Group theory: Examples | $~$f(-x) = f(x)$~$ |
---|
Group theory: Examples | $~$f(-x) = - f(x)$~$ |
---|
Group theory: Examples | $~$f(x) = x^2$~$ |
---|
Group theory: Examples | $~$f(x) = \cos x$~$ |
---|
Group theory: Examples | $~$f(x) = x^3$~$ |
---|
Group theory: Examples | $~$f(x) = \sin x$~$ |
---|
Group theory: Examples | $~$C_2$~$ |
---|
Group theory: Examples | $~$2$~$ |
---|
Group theory: Examples | $~$\mathbb{R} \to \mathbb{R}$~$ |
---|
Group theory: Examples | $$~$ (\mathbb{R} \to \mathbb{R}) \to (\mathbb{R} \to \mathbb{R}) $~$$ |
---|
Group theory: Examples | $~$\mathbb{R} \to \mathbb{R}$~$ |
---|
Group theory: Examples | $~$\mathbb{R} \to \mathbb{R}$~$ |
---|
Group theory: Examples | $~$C_2$~$ |
---|
Group theory: Examples | $~$1$~$ |
---|
Group theory: Examples | $~$-1$~$ |
---|
Group theory: Examples | $~$1$~$ |
---|
Group theory: Examples | $~$f(x)$~$ |
---|
Group theory: Examples | $~$f(x)$~$ |
---|
Group theory: Examples | $~$-1$~$ |
---|
Group theory: Examples | $~$f(x)$~$ |
---|
Group theory: Examples | $~$f(-x)$~$ |
---|
Group theory: Examples | $~$f(x)$~$ |
---|
Group theory: Examples | $~$(-1) \times (-1) = 1$~$ |
---|
Group theory: Examples | $~$f(-(-x)) = f(x)$~$ |
---|
Group theory: Examples | $~$G$~$ |
---|
Group theory: Examples | $~$X$~$ |
---|
Group theory: Examples | $~$C_2$~$ |
---|
Group theory: Examples | $$~$f(x) = \underbrace{\frac{f(x) + f(-x)}{2}}_{\text{even}} + \underbrace{\frac{f(x) - f(-x)}{2}}_{\text{odd}}.$~$$ |
---|
Group theory: Examples | $~$C_2$~$ |
---|
Group theory: Examples | $~$1$~$ |
---|
Group theory: Examples | $~$-1$~$ |
---|
Group: Examples | $~$n$~$ |
---|
Group: Examples | $~$S_n$~$ |
---|
Group: Examples | $~$n$~$ |
---|
Group: Examples | $~$\{ 1, 2, \dots n \} \to \{ 1, 2, \dots n \}$~$ |
---|
Group: Examples | $~$n$~$ |
---|
Group: Examples | $~$G$~$ |
---|
Group: Examples | $~$X$~$ |
---|
Group: Examples | $~$n$~$ |
---|
Group: Examples | $~$G \to S_n$~$ |
---|
Group: Examples | $~$D_{2n}$~$ |
---|
Group: Examples | $~$n$~$ |
---|
Group: Examples | $~$\langle r, f \mid r^n, f^2, (rf)^2 \rangle$~$ |
---|
Group: Examples | $~$r$~$ |
---|
Group: Examples | $~$\tau/n$~$ |
---|
Group: Examples | $~$f$~$ |
---|
Group: Examples | $~$n > 2$~$ |
---|
Group: Examples | $~$K$~$ |
---|
Group: Examples | $~$n$~$ |
---|
Group: Examples | $~$GL_n(K)$~$ |
---|
Group: Examples | $~$n$~$ |
---|
Group: Examples | $~$K$~$ |
---|
Group: Examples | $~$n \times n$~$ |
---|
Group: Examples | $~$K$~$ |
---|
Group: Examples | $~$n$~$ |
---|
Group: Examples | $~$K$~$ |
---|
Group: Examples | $~$K$~$ |
---|
Group: Exercises | $~$G$~$ |
---|
Group: Exercises | $~$e_1, e_2 \in G$~$ |
---|
Group: Exercises | $~$e_1 = e_2$~$ |
---|
Group: Exercises | $~$e$~$ |
---|
Group: Exercises | $~$eg = ge = g$~$ |
---|
Group: Exercises | $~$g \in G$~$ |
---|
Group: Exercises | $~$e_1$~$ |
---|
Group: Exercises | $~$e_1 e_2 = e_2 e_1 = e_1$~$ |
---|
Group: Exercises | $~$e_2$~$ |
---|
Group: Exercises | $~$e_2 e_1 = e_1 e_2 = e_2$~$ |
---|
Group: Exercises | $~$e_1 = e_2$~$ |
---|
Group: Exercises | $~$g \in G$~$ |
---|
Group: Exercises | $~$h_1, h_2 \in G$~$ |
---|
Group: Exercises | $~$g$~$ |
---|
Group: Exercises | $~$h_1 = h_2$~$ |
---|
Group: Exercises | $~$h$~$ |
---|
Group: Exercises | $~$g$~$ |
---|
Group: Exercises | $~$hg = gh = e$~$ |
---|
Group: Exercises | $~$h_1 g = g h_1 = e$~$ |
---|
Group: Exercises | $~$h_2 g = g h_2 = e$~$ |
---|
Group: Exercises | $$~$h_1 g h_2 = (h_1 g) h_2 = (e) h_2 = h_2$~$$ |
---|
Group: Exercises | $$~$h_1 g h_2 = h_1 (g h_2) = h_1 (e) = h_1.$~$$ |
---|
Group: Exercises | $~$h_1 = h_2$~$ |
---|
Group: Exercises | $~$\mathbb{R}$~$ |
---|
Group: Exercises | $~$(x, y) \mapsto x + y$~$ |
---|
Group: Exercises | $~$0$~$ |
---|
Group: Exercises | $~$x \mapsto -x$~$ |
---|
Group: Exercises | $~$\mathbb{R}$~$ |
---|
Group: Exercises | $~$(x, y) \mapsto xy$~$ |
---|
Group: Exercises | $~$0 \in \mathbb{R}$~$ |
---|
Group: Exercises | $~$0 \times x = 0$~$ |
---|
Group: Exercises | $~$x$~$ |
---|
Group: Exercises | $~$\mathbb{R}_{>0}$~$ |
---|
Group: Exercises | $~$(x, y) \mapsto xy$~$ |
---|
Group: Exercises | $~$1$~$ |
---|
Group: Exercises | $~$x \mapsto \frac{1}{x}$~$ |
---|
Group: Exercises | $~$(\mathbb{R}, +)$~$ |
---|
Group: Exercises | $~$\mathbb{R}$~$ |
---|
Group: Exercises | $~$(x, y) \mapsto x + y - 1$~$ |
---|
Group: Exercises | $~$(\mathbb{R}, +)$~$ |
---|
Group: Exercises | $~$1$~$ |
---|
Group: Exercises | $~$x \mapsto 2 - x$~$ |
---|
Group: Exercises | $~$\mathbb{R}$~$ |
---|
Group: Exercises | $~$(x, y) \mapsto \frac{x + y}{1 + xy}$~$ |
---|
Group: Exercises | $~$0$~$ |
---|
Group: Exercises | $~$1$~$ |
---|
Groups as symmetires | $~$120^\circ$~$ |
---|
Groups as symmetires | $~$240^\circ$~$ |
---|
Groups as symmetires | $~$f$~$ |
---|
Groups as symmetires | $~$g$~$ |
---|
Groups as symmetires | $~$h$~$ |
---|
Groups as symmetires | $~$g \circ f$~$ |
---|
Groups as symmetires | $~$h \circ (g \circ f)$~$ |
---|
Groups as symmetires | $~$h \circ g$~$ |
---|
Groups as symmetires | $~$(h \circ g) \circ f$~$ |
---|
Groups as symmetires | $~$G$~$ |
---|
Groups as symmetires | $~$(X, \bullet)$~$ |
---|
Groups as symmetires | $~$X$~$ |
---|
Groups as symmetires | $~$\bullet: X \times X \to X$~$ |
---|
Groups as symmetires | $~$X$~$ |
---|
Groups as symmetires | $~$x(yz) = (xy)z$~$ |
---|
Groups as symmetires | $~$x, y, z \in X$~$ |
---|
Groups as symmetires | $~$e$~$ |
---|
Groups as symmetires | $~$xe=ex=x$~$ |
---|
Groups as symmetires | $~$x \in X$~$ |
---|
Groups as symmetires | $~$x$~$ |
---|
Groups as symmetires | $~$X$~$ |
---|
Groups as symmetires | $~$x^{-1} \in X$~$ |
---|
Groups as symmetires | $~$xx^{-1}=x^{-1}x=e$~$ |
---|
Groups as symmetires | $~$120^\circ$~$ |
---|
Groups as symmetires | $~$240^\circ$~$ |
---|
Groups as symmetires | $~$G$~$ |
---|
Groups as symmetires | $~$(X, \bullet)$~$ |
---|
Groups as symmetires | $~$X$~$ |
---|
Groups as symmetires | $~$X$~$ |
---|
Groups as symmetires | $~$G$~$ |
---|
Groups as symmetires | $~$\bullet : G \times G \to G$~$ |
---|
Groups as symmetires | $~$x \bullet y$~$ |
---|
Groups as symmetires | $~$xy$~$ |
---|
Groups as symmetires | $~$e$~$ |
---|
Groups as symmetires | $~$xe=ex=x$~$ |
---|
Groups as symmetires | $~$x \in X$~$ |
---|
Groups as symmetires | $~$x$~$ |
---|
Groups as symmetires | $~$X$~$ |
---|
Groups as symmetires | $~$x^{-1} \in X$~$ |
---|
Groups as symmetires | $~$xx^{-1}=x^{-1}x=e$~$ |
---|
Groups as symmetires | $~$x(yz) = (xy)z$~$ |
---|
Groups as symmetires | $~$x, y, z \in X$~$ |
---|
Groups as symmetires | $~$\bullet$~$ |
---|
Groups as symmetires | $~$e$~$ |
---|
Groups as symmetires | $~$G$~$ |
---|
Groups as symmetires | $~$\bullet$~$ |
---|
Groups as symmetires | $~$e$~$ |
---|
Groups as symmetires | $~$x$~$ |
---|
Groups as symmetires | $~$\bullet$~$ |
---|
Groups as symmetires | $~$x$~$ |
---|
Groups as symmetires | $~$e$~$ |
---|
Groups as symmetires | $~$z$~$ |
---|
Groups as symmetires | $~$ze = e = ez = z.$~$ |
---|
Groups as symmetires | $~$e$~$ |
---|
Groups as symmetires | $~$G$~$ |
---|
Groups as symmetires | $~$e$~$ |
---|
Groups as symmetires | $~$e$~$ |
---|
Groups as symmetires | $~$e$~$ |
---|
Groups as symmetires | $~$1$~$ |
---|
Groups as symmetires | $~$1_G$~$ |
---|
Groups as symmetires | $~$\bullet$~$ |
---|
Groups as symmetires | $~$X$~$ |
---|
Groups as symmetires | $~$1$~$ |
---|
Groups as symmetires | $~$\bullet$~$ |
---|
Groups as symmetires | $~$0$~$ |
---|
Groups as symmetires | $~$0_G$~$ |
---|
Groups as symmetires | $~$x$~$ |
---|
Groups as symmetires | $~$X$~$ |
---|
Groups as symmetires | $~$y$~$ |
---|
Groups as symmetires | $~$\bullet$~$ |
---|
Groups as symmetires | $~$x$~$ |
---|
Groups as symmetires | $~$xy = e$~$ |
---|
Groups as symmetires | $~$x$~$ |
---|
Groups as symmetires | $~$x^{-1}$~$ |
---|
Groups as symmetires | $~$(-x)$~$ |
---|
Groups as symmetires | $~$\bullet$~$ |
---|
Groups as symmetires | $~$f$~$ |
---|
Groups as symmetires | $~$g$~$ |
---|
Groups as symmetires | $~$h$~$ |
---|
Groups as symmetires | $~$g \circ f$~$ |
---|
Groups as symmetires | $~$h \circ (g \circ f)$~$ |
---|
Groups as symmetires | $~$h \circ g$~$ |
---|
Groups as symmetires | $~$(h \circ g) \circ f$~$ |
---|
Groups as symmetires | $~$(\mathbb{Z}, +)$~$ |
---|
Groups as symmetires | $~$\mathbb{Z}$~$ |
---|
Groups as symmetires | $~$+$~$ |
---|
Groups as symmetires | $~$\mathbb Z \times \mathbb Z \to \mathbb Z$~$ |
---|
Groups as symmetires | $~$(x+y)+z=x+(y+z)$~$ |
---|
Groups as symmetires | $~$0+x = x = x + 0$~$ |
---|
Groups as symmetires | $~$x$~$ |
---|
Groups as symmetires | $~$-x$~$ |
---|
Groups as symmetires | $~$x + (-x) = 0$~$ |
---|
Groups as symmetires | $~$G = (X, \bullet)$~$ |
---|
Groups as symmetires | $~$X$~$ |
---|
Groups as symmetires | $~$\bullet$~$ |
---|
Groups as symmetires | $~$X$~$ |
---|
Groups as symmetires | $~$G$~$ |
---|
Groups as symmetires | $~$\bullet$~$ |
---|
Groups as symmetires | $~$x \bullet y$~$ |
---|
Groups as symmetires | $~$xy$~$ |
---|
Groups as symmetires | $~$G$~$ |
---|
Groups as symmetires | $~$X$~$ |
---|
Groups as symmetires | $~$x, y \in X$~$ |
---|
Groups as symmetires | $~$G$~$ |
---|
Groups as symmetires | $~$x, y \in G$~$ |
---|
Groups as symmetires | $~$G$~$ |
---|
Groups as symmetires | $~$|G|$~$ |
---|
Groups as symmetires | $~$|X|$~$ |
---|
Groups as symmetires | $~$X$~$ |
---|
Groups as symmetires | $~$|G|=9$~$ |
---|
Groups as symmetires | $~$G$~$ |
---|
Gödel II and Löb's theorem | $~$P$~$ |
---|
Gödel II and Löb's theorem | $~$T$~$ |
---|
Gödel II and Löb's theorem | $~$T\not\vdash \neg P(\ulcorner S\urcorner)$~$ |
---|
Gödel II and Löb's theorem | $~$S$~$ |
---|
Gödel II and Löb's theorem | $~$X$~$ |
---|
Gödel II and Löb's theorem | $~$T\vdash P(\ulcorner X\urcorner)\rightarrow X$~$ |
---|
Gödel II and Löb's theorem | $~$T\vdash X$~$ |
---|
Gödel II and Löb's theorem | $~$X$~$ |
---|
Gödel II and Löb's theorem | $~$\bot$~$ |
---|
Gödel II and Löb's theorem | $~$T\vdash \neg \bot$~$ |
---|
Gödel II and Löb's theorem | $~$\bot$~$ |
---|
Gödel II and Löb's theorem | $~$T\vdash \neg P(\ulcorner \bot\urcorner)$~$ |
---|
Gödel II and Löb's theorem | $~$T\vdash \bot$~$ |
---|
Gödel II and Löb's theorem | $~$T$~$ |
---|
Gödel II and Löb's theorem | $~$T\neg\vdash \neg P(\ulcorner \bot\urcorner)$~$ |
---|
Gödel II and Löb's theorem | $~$T\vdash P(\ulcorner X\urcorner)\rightarrow X$~$ |
---|
Gödel II and Löb's theorem | $~$T\vdash \neg X \rightarrow \neg P(\ulcorner X\urcorner)$~$ |
---|
Gödel II and Löb's theorem | $~$T + \neg X\vdash \neg P(\ulcorner X\urcorner)$~$ |
---|
Gödel II and Löb's theorem | $~$T+\neg X$~$ |
---|
Gödel II and Löb's theorem | $~$\neg P(\ulcorner X\urcorner)$~$ |
---|
Gödel II and Löb's theorem | $~$X$~$ |
---|
Gödel II and Löb's theorem | $~$T$~$ |
---|
Gödel II and Löb's theorem | $~$\neg X$~$ |
---|
Gödel II and Löb's theorem | $~$X$~$ |
---|
Gödel II and Löb's theorem | $~$T$~$ |
---|
Gödel II and Löb's theorem | $~$X$~$ |
---|
Gödel II and Löb's theorem | $~$T$~$ |
---|
Gödel II and Löb's theorem | $~$T\vdash X$~$ |
---|
Gödel's first incompleteness theorem | $~$\omega$~$ |
---|
Gödel's first incompleteness theorem | $~$\omega$~$ |
---|
Gödel's first incompleteness theorem | $~$PA$~$ |
---|
High-speed intro to Bayes's rule | $~$(1 : 4) \cdot (3 : 1) = (3 : 4)$~$ |
---|
High-speed intro to Bayes's rule | $~$3/7 = 43\%$~$ |
---|
High-speed intro to Bayes's rule | $~$1 : 9$~$ |
---|
High-speed intro to Bayes's rule | $~$12 : 4$~$ |
---|
High-speed intro to Bayes's rule | $~$3 : 1.$~$ |
---|
High-speed intro to Bayes's rule | $~$(1 : 9) \cdot (3 : 1) = (3 : 9) \cong (1 : 3).$~$ |
---|
High-speed intro to Bayes's rule | $~$1 : 3$~$ |
---|
High-speed intro to Bayes's rule | $~$\frac{1}{1+3} = \frac{1}{4} = 25\%.$~$ |
---|
High-speed intro to Bayes's rule | $~$X$~$ |
---|
High-speed intro to Bayes's rule | $~$\mathbb P(X)$~$ |
---|
High-speed intro to Bayes's rule | $~$X$~$ |
---|
High-speed intro to Bayes's rule | $~$X.$~$ |
---|
High-speed intro to Bayes's rule | $~$\neg X$~$ |
---|
High-speed intro to Bayes's rule | $~$X$~$ |
---|
High-speed intro to Bayes's rule | $~$X$~$ |
---|
High-speed intro to Bayes's rule | $~$X$~$ |
---|
High-speed intro to Bayes's rule | $~$Y$~$ |
---|
High-speed intro to Bayes's rule | $~$X \wedge Y$~$ |
---|
High-speed intro to Bayes's rule | $~$\mathbb P(X \wedge Y)$~$ |
---|
High-speed intro to Bayes's rule | $~$X$~$ |
---|
High-speed intro to Bayes's rule | $~$Y$~$ |
---|
High-speed intro to Bayes's rule | $$~$\mathbb P(X|Y) := \dfrac{\mathbb P(X \wedge Y)}{\mathbb P(Y)} \tag*{(definition of conditional probability)}$~$$ |
---|
High-speed intro to Bayes's rule | $~$\mathbb P(X|Y)$~$ |
---|
High-speed intro to Bayes's rule | $~$X$~$ |
---|
High-speed intro to Bayes's rule | $~$Y$~$ |
---|
High-speed intro to Bayes's rule | $~$\frac{18}{18+24} = \frac{3}{7}.$~$ |
---|
High-speed intro to Bayes's rule | $~$\mathbb P(X \wedge Y) = \mathbb P(Y) \cdot \mathbb P(X|Y).$~$ |
---|
High-speed intro to Bayes's rule | $$~$\frac{\mathbb P(sick)}{\mathbb P(healthy)} \tag*{(prior odds)}$~$$ |
---|
High-speed intro to Bayes's rule | $$~$\frac{\mathbb P(positive | sick)}{\mathbb P(positive | healthy)} \tag*{(likelihood ratio)}$~$$ |
---|
High-speed intro to Bayes's rule | $$~$\frac{\mathbb P(sick | positive)}{\mathbb P(healthy | positive)} \tag*{(posterior odds)}$~$$ |
---|
High-speed intro to Bayes's rule | $$~$\frac{\mathbb P(sick)}{\mathbb P(healthy)} \cdot \frac{\mathbb P(positive | sick)}{\mathbb P(positive | healthy)} = \frac{\mathbb P(sick | positive)}{\mathbb P(healthy | positive)}$~$$ |
---|
High-speed intro to Bayes's rule | $~$H_j$~$ |
---|
High-speed intro to Bayes's rule | $~$H_k$~$ |
---|
High-speed intro to Bayes's rule | $~$e_0$~$ |
---|
High-speed intro to Bayes's rule | $$~$
\frac{\mathbb P(H_j)}{\mathbb P(H_k)}
\cdot
\frac{\mathbb P(e_0 | H_j)}{\mathbb P(e_0 | H_k)}
=
\frac{\mathbb P(e_0 \wedge H_j)}{\mathbb P(e_0 \wedge H_k)}
=
\frac{\mathbb P(e_0 \wedge H_j)/\mathbb P(e_0)}{\mathbb P(e_0 \wedge H_k)/\mathbb P(e_0)}
=
\frac{\mathbb P(H_j | e_0)}{\mathbb P(H_k | e_0)}
$~$$ |
---|
High-speed intro to Bayes's rule | $$~$
\frac{0.20}{0.80}
\cdot
\frac{0.90}{0.30}
=
\frac{0.18}{0.24}
=
\frac{0.18/0.42}{0.24/0.42}
=
\frac{0.43}{0.57}
$~$$ |
---|
High-speed intro to Bayes's rule | $~$\{X_1, X_2, …, X_i\}$~$ |
---|
High-speed intro to Bayes's rule | $~$Y$~$ |
---|
High-speed intro to Bayes's rule | $$~$\mathbb P(Y) = \sum_i \mathbb P(Y \wedge X_i) \tag*{(law of marginal probability)}$~$$ |
---|
High-speed intro to Bayes's rule | $~$H_k$~$ |
---|
High-speed intro to Bayes's rule | $~$e_0$~$ |
---|
High-speed intro to Bayes's rule | $$~$
\mathbb P(H_k | e_0)
= \frac{\mathbb P(H_k \wedge e_0)}{\mathbb P(e_0)}
= \frac{\mathbb P(e_0 \wedge H_k)}{\sum_i P(e_0 \wedge H_i)}
= \frac{\mathbb P(e_0 | X_k) \cdot \mathbb P(X_k)}{\sum_i \mathbb P(e_0 | X_i) \cdot \mathbb P(X_i)}
$~$$ |
---|
High-speed intro to Bayes's rule | $~$\mathbb P(sick | positive)$~$ |
---|
High-speed intro to Bayes's rule | $~$\mathbb P(positive | sick)$~$ |
---|
High-speed intro to Bayes's rule | $~$\mathbb P(sick)$~$ |
---|
High-speed intro to Bayes's rule | $$~$
\begin{array}{rll}
& (3 : 2 : 1) & \cong (\frac{1}{2} : \frac{1}{3} : \frac{1}{6}) \\
\times & (2 : 1 : 3) & \cong ( \frac{1}{2} : \frac{1}{4} : \frac{3}{4} ) \\
\times & (2 : 3 : 1) & \cong ( \frac{1}{2} : \frac{3}{4} : \frac{1}{4} ) \\
\times & (2 : 1 : 3) & \\
= & (24 : 6 : 9) & \cong (8 : 2 : 3) \cong (\frac{8}{13} : \frac{2}{13} : \frac{3}{13})
\end{array}
$~$$ |
---|
High-speed intro to Bayes's rule | $$~$\mathbb P(H_k | e_0) = \frac{\mathbb P(e_0 | X_k) \cdot \mathbb P(X_k)}{\sum_i \mathbb P(e_o | X_i) \cdot \mathbb P(X_i)}$~$$ |
---|
High-speed intro to Bayes's rule | $~$\mathbb O(H_i)$~$ |
---|
High-speed intro to Bayes's rule | $~$H,$~$ |
---|
High-speed intro to Bayes's rule | $~$\mathcal L(e_0 | H_i)$~$ |
---|
High-speed intro to Bayes's rule | $~$e_0$~$ |
---|
High-speed intro to Bayes's rule | $~$H_i,$~$ |
---|
High-speed intro to Bayes's rule | $~$\mathbb O(H_i | e_0)$~$ |
---|
High-speed intro to Bayes's rule | $$~$\mathbb O(H_i | e_0) = \mathcal L(e_0 | H_i) \cdot \mathbb O(H_i)$~$$ |
---|
High-speed intro to Bayes's rule | $~$\mathbb O$~$ |
---|
High-speed intro to Bayes's rule | $~$\mathbb P$~$ |
---|
High-speed intro to Bayes's rule | $~$\mathbb O$~$ |
---|
High-speed intro to Bayes's rule | $~$1$~$ |
---|
High-speed intro to Bayes's rule | $$~$\mathbb P(H_i | e_0) \propto \mathcal L(e_0 | H_i) \cdot \mathbb P(H_i) \tag*{(functional form of Bayes' rule)}$~$$ |
---|
High-speed intro to Bayes's rule | $$~$\frac{\mathbb P(\neg \text{sabotage} | \text {conspiracy})}{\mathbb P(\neg \text {sabotage} | \neg \text {conspiracy})}$~$$ |
---|
High-speed intro to Bayes's rule | $$~$
\frac{\mathbb P(\text {conspiracy} | \neg \text{sabotage})}{\mathbb P(\neg \text {conspiracy} | \neg \text{sabotage})}
<
\frac{\mathbb P(\text {conspiracy})}{\mathbb P(\neg \text {conspiracy})}
\cdot
\frac{\mathbb P(\neg \text{sabotage} | \text {conspiracy})}{\mathbb P(\neg \text {sabotage} | \neg \text {conspiracy})}
$~$$ |
---|
How many bits to a trit? | $~$\log_2(3) \approx 1.585.$~$ |
---|
How many bits to a trit? | $~$n$~$ |
---|
How many bits to a trit? | $~$3^n$~$ |
---|
How many bits to a trit? | $~$n$~$ |
---|
How many bits to a trit? | $~$\log_2(3)$~$ |
---|
How many bits to a trit? | $~$n$~$ |
---|
How many bits to a trit? | $~$\log_2(3) = 1.58496250072\ldots$~$ |
---|
How many bits to a trit? | $~$(\approx \lceil 1.585 \rceil = 2)$~$ |
---|
How many bits to a trit? | $~$\log_2(3) \approx 1.585$~$ |
---|
How many bits to a trit? | $~$n$~$ |
---|
How many bits to a trit? | $~$3^n$~$ |
---|
How many bits to a trit? | $~$n$~$ |
---|
How many bits to a trit? | $~$\log_2(3)$~$ |
---|
How many bits to a trit? | $~$n$~$ |
---|
How many bits to a trit? | $~$\log_2(3) = 1.58496250072\ldots$~$ |
---|
How many bits to a trit? | $~$(\approx \lceil 1.585 \rceil = 2)$~$ |
---|
Hypercomputer | $~$\Pi_n$~$ |
---|
Hypercomputer | $~$\Pi_{n+1}$~$ |
---|
Ideal target | $~$\Delta U$~$ |
---|
Ideal target | $~$U$~$ |
---|
Ideal target | $~$U$~$ |
---|
Ideal target | $~$U$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$f: R \to S$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$R$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$S$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$K$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$f$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$R$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$f$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$K$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$R$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$k \in K$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$r \in R$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$f(kr) = f(k)f(r) = 0 \times r = 0$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$kr$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$K$~$ |
---|
Ideals are the same thing as kernels of ring homomorphisms | $~$k$~$ |
---|
Identity element | $~$S$~$ |
---|
Identity element | $~$*$~$ |
---|
Identity element | $~$i$~$ |
---|
Identity element | $~$a \in S$~$ |
---|
Identity element | $~$S$~$ |
---|
Identity element | $~$*$~$ |
---|
Identity element | $~$i$~$ |
---|
Identity element | $~$a \in S$~$ |
---|
Identity element | $~$i$~$ |
---|
Identity element | $~$a \in S$~$ |
---|
Identity element | $~$i * a = a$~$ |
---|
Identity element | $~$i$~$ |
---|
Identity element | $~$a \in S$~$ |
---|
Identity element | $~$a * i = a$~$ |
---|
Identity element | $~$i$~$ |
---|
Iff | $~$\leftrightarrow$~$ |
---|
Iff | $~$A \leftrightarrow B$~$ |
---|
Iff | $~$A \rightarrow B$~$ |
---|
Image (of a function) | $~$\operatorname{im}(f)$~$ |
---|
Image (of a function) | $~$f : X \to Y$~$ |
---|
Image (of a function) | $~$f$~$ |
---|
Image (of a function) | $~$Y$~$ |
---|
Image (of a function) | $~$\operatorname{im}(f) = \{f(x) \mid x \in X\}.$~$ |
---|
Image (of a function) | $~$f$~$ |
---|
Image (of a function) | $~$Y$~$ |
---|
Image of the identity under a group homomorphism is the identity | $~$f: G \to H$~$ |
---|
Image of the identity under a group homomorphism is the identity | $~$f(e_G) = e_H$~$ |
---|
Image of the identity under a group homomorphism is the identity | $~$e_G$~$ |
---|
Image of the identity under a group homomorphism is the identity | $~$G$~$ |
---|
Image of the identity under a group homomorphism is the identity | $~$e_H$~$ |
---|
Image of the identity under a group homomorphism is the identity | $~$H$~$ |
---|
Image of the identity under a group homomorphism is the identity | $~$f(e_G) f(e_G) = f(e_G e_G) = f(e_G)$~$ |
---|
Image of the identity under a group homomorphism is the identity | $~$f(e_G)^{-1}$~$ |
---|
Image of the identity under a group homomorphism is the identity | $~$f(e_G) = e_H$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$R$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$r \not = 0$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$R$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$r$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$r$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$r$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$r$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$\langle r \rangle$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$R$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$2 \Rightarrow 1$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$3 \Rightarrow 2$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$\langle r \rangle$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$r$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$I$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$R/I$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$I$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$R/I$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$1 \Rightarrow 3$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$r$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$\langle r \rangle$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$\langle r \rangle$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$\langle r \rangle$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$J$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$J = \langle a \rangle$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$a$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$r = a c$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$c$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$a$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$\langle a \rangle$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$r$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$a$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$c$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$a$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$c$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$a = r c^{-1}$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$J$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$\langle r \rangle$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$j \in J$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$\langle r \rangle$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$j \in J = \langle a \rangle$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$d$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$j = a d$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$j = r c^{-1} d$~$ |
---|
In a principal ideal domain, "prime" and "irreducible" are the same | $~$j \in \langle r \rangle$~$ |
---|
In notation | $~$x \in X$~$ |
---|
In notation | $~$\in$~$ |
---|
In notation | $~$X$~$ |
---|
In notation | $~$x$~$ |
---|
In notation | $~$r \in \mathbb{R}$~$ |
---|
In notation | $~$r$~$ |
---|
Index two subgroup of group is normal | $~$H$~$ |
---|
Index two subgroup of group is normal | $~$G$~$ |
---|
Index two subgroup of group is normal | $~$2$~$ |
---|
Index two subgroup of group is normal | $~$H$~$ |
---|
Index two subgroup of group is normal | $~$G$~$ |
---|
Index two subgroup of group is normal | $~$H$~$ |
---|
Index two subgroup of group is normal | $~$G$~$ |
---|
Index two subgroup of group is normal | $~$H$~$ |
---|
Index two subgroup of group is normal | $~$2$~$ |
---|
Index two subgroup of group is normal | $~$G$~$ |
---|
Index two subgroup of group is normal | $~$H$~$ |
---|
Index two subgroup of group is normal | $~$xH$~$ |
---|
Index two subgroup of group is normal | $~$x$~$ |
---|
Index two subgroup of group is normal | $~$H$~$ |
---|
Index two subgroup of group is normal | $~$Hy$~$ |
---|
Index two subgroup of group is normal | $~$x \not \in H$~$ |
---|
Index two subgroup of group is normal | $~$x \in Hy$~$ |
---|
Index two subgroup of group is normal | $~$x = y$~$ |
---|
Index two subgroup of group is normal | $~$xH = Hx$~$ |
---|
Index two subgroup of group is normal | $~$xHx^{-1} = H$~$ |
---|
Index two subgroup of group is normal | $~$H$~$ |
---|
Index two subgroup of group is normal | $~$G$~$ |
---|
Index two subgroup of group is normal | $~$G$~$ |
---|
Index two subgroup of group is normal | $~$H$~$ |
---|
Index two subgroup of group is normal | $~$xH$~$ |
---|
Index two subgroup of group is normal | $~$h$~$ |
---|
Index two subgroup of group is normal | $~$xh$~$ |
---|
Index two subgroup of group is normal | $~$h \in H$~$ |
---|
Index two subgroup of group is normal | $~$hHh^{-1}$~$ |
---|
Index two subgroup of group is normal | $~$H$~$ |
---|
Index two subgroup of group is normal | $~$hH = H$~$ |
---|
Index two subgroup of group is normal | $~$Hh^{-1} = H$~$ |
---|
Index two subgroup of group is normal | $~$xh H (xh)^{-1} = xhHh^{-1} x^{-1} = xHx^{-1} = H$~$ |
---|
Indirect decision theory | $~$\mathbb{R}$~$ |
---|
Indirect decision theory | $~$\mathbb{E}$~$ |
---|
Indirect decision theory | $~$\mathbb{E}$~$ |
---|
Indirect decision theory | $~$\mathbb{R}$~$ |
---|
Indirect decision theory | $~$\mathbb{E}$~$ |
---|
Indirect decision theory | $~$\mathbb{E}$~$ |
---|
Indirect decision theory | $~$\mathbb{E}$~$ |
---|
Indirect decision theory | $~$\mathbb{E}$~$ |
---|
Indirect decision theory | $~$\mathbb{E}$~$ |
---|
Indirect decision theory | $~$\mathbb{E}$~$ |
---|
Indirect decision theory | $~$\mathbb{E}$~$ |
---|
Indirect decision theory | $~$\mathbb{E}$~$ |
---|
Indirect decision theory | $~$\mathbb{E}$~$ |
---|
Indirect decision theory | $~$\mathbb{E}$~$ |
---|
Information | $~$\mathrm P$~$ |
---|
Information | $~$O$~$ |
---|
Information | $~$o \in O$~$ |
---|
Information | $~$\log \frac{1}{\mathrm P(o)}$~$ |
---|
Information | $~$\mathrm P$~$ |
---|
Information | $~$n$~$ |
---|
Information | $~$\infty$~$ |
---|
Information | $~$n$~$ |
---|
Information | $~$\mathrm P$~$ |
---|
Information | $~$O$~$ |
---|
Information | $~$o \in O$~$ |
---|
Information | $~$\log_2\frac{1}{\mathrm P(o)}$~$ |
---|
Information | $~$\mathrm P$~$ |
---|
Injective function | $~$f: X \to Y$~$ |
---|
Injective function | $~$f(x) = f(y)$~$ |
---|
Injective function | $~$x=y$~$ |
---|
Injective function | $~$f$~$ |
---|
Injective function | $~$\mathbb{N} \to \mathbb{N}$~$ |
---|
Injective function | $~$\mathbb{N}$~$ |
---|
Injective function | $~$n \mapsto n+5$~$ |
---|
Injective function | $~$n+5 = m+5$~$ |
---|
Injective function | $~$n = m$~$ |
---|
Injective function | $~$k$~$ |
---|
Injective function | $~$k+5 = 2$~$ |
---|
Injective function | $~$2$~$ |
---|
Injective function | $~$f: \mathbb{N} \to \mathbb{N}$~$ |
---|
Injective function | $~$f(n) = 6$~$ |
---|
Injective function | $~$n$~$ |
---|
Injective function | $~$f(1) = f(2)$~$ |
---|
Injective function | $~$1 \not = 2$~$ |
---|
Instrumental convergence | $~$\mathcal U$~$ |
---|
Instrumental convergence | $~$U_k \in \mathcal U$~$ |
---|
Instrumental convergence | $~$\pi_k$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$\pi_k$~$ |
---|
Instrumental convergence | $~$\pi_k \in X$~$ |
---|
Instrumental convergence | $~$\pi_k \in \neg X$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$Y$~$ |
---|
Instrumental convergence | $~$Y$~$ |
---|
Instrumental convergence | $~$Y$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$\pi_k$~$ |
---|
Instrumental convergence | $~$Y_k$~$ |
---|
Instrumental convergence | $~$\pi_k$~$ |
---|
Instrumental convergence | $~$Y_k.$~$ |
---|
Instrumental convergence | $~$Y$~$ |
---|
Instrumental convergence | $~$\pi$~$ |
---|
Instrumental convergence | $~$Y$~$ |
---|
Instrumental convergence | $~$\Pi$~$ |
---|
Instrumental convergence | $~$X \subset \Pi$~$ |
---|
Instrumental convergence | $~$\neg X \subset \Pi$~$ |
---|
Instrumental convergence | $~$Y_k$~$ |
---|
Instrumental convergence | $~$\pi_k$~$ |
---|
Instrumental convergence | $~$X,$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$Y_1, Y_2, Y_3, Y_4.$~$ |
---|
Instrumental convergence | $~$\pi_1, \pi_2, \pi_3, \pi_4$~$ |
---|
Instrumental convergence | $~$X \subset \Pi$~$ |
---|
Instrumental convergence | $~$\neg X$~$ |
---|
Instrumental convergence | $~$\mathcal U_C$~$ |
---|
Instrumental convergence | $~$o,$~$ |
---|
Instrumental convergence | $~$\mathcal L$~$ |
---|
Instrumental convergence | $~$r$~$ |
---|
Instrumental convergence | $~$U_k$~$ |
---|
Instrumental convergence | $~$o_\mathcal L$~$ |
---|
Instrumental convergence | $~$o$~$ |
---|
Instrumental convergence | $~$r,$~$ |
---|
Instrumental convergence | $~$U_k$~$ |
---|
Instrumental convergence | $~$o$~$ |
---|
Instrumental convergence | $~$r$~$ |
---|
Instrumental convergence | $~$\mathbb P [o | \pi_i]$~$ |
---|
Instrumental convergence | $~$\pi$~$ |
---|
Instrumental convergence | $~$U_k(o)$~$ |
---|
Instrumental convergence | $~$\mathcal P,$~$ |
---|
Instrumental convergence | $~$\mathcal P_B$~$ |
---|
Instrumental convergence | $~$\mathcal P$~$ |
---|
Instrumental convergence | $~$\mathcal P_B$~$ |
---|
Instrumental convergence | $~$U_K$~$ |
---|
Instrumental convergence | $~$\mathbb P$~$ |
---|
Instrumental convergence | $~$U_k \in U_K$~$ |
---|
Instrumental convergence | $~$\pi_k$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$\Pi,$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$X.$~$ |
---|
Instrumental convergence | $~$\pi_k$~$ |
---|
Instrumental convergence | $~$Y_k,$~$ |
---|
Instrumental convergence | $~$\pi_k \in X$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$\pi_X$~$ |
---|
Instrumental convergence | $~$Y_k.$~$ |
---|
Instrumental convergence | $~$U_k$~$ |
---|
Instrumental convergence | $~$\pi_k$~$ |
---|
Instrumental convergence | $~$U_k,$~$ |
---|
Instrumental convergence | $$~$\pi_k = \underset{\pi_i \in \Pi}{\operatorname{argmax}} \mathbb E [ U_k | \pi_i ]$~$$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $$~$\big ( \underset{\pi_i \in \Pi}{\operatorname{argmax}} \mathbb E [ U_k | \pi_i ] \big ) \in X$~$$ |
---|
Instrumental convergence | $~$U_k$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$V_k$~$ |
---|
Instrumental convergence | $~$\pi_k \in X$~$ |
---|
Instrumental convergence | $~$U_k$~$ |
---|
Instrumental convergence | $~$X.$~$ |
---|
Instrumental convergence | $~$U_e$~$ |
---|
Instrumental convergence | $~$X_e$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$V_e.$~$ |
---|
Instrumental convergence | $~$Y$~$ |
---|
Instrumental convergence | $~$X,$~$ |
---|
Instrumental convergence | $~$Y_i$~$ |
---|
Instrumental convergence | $~$\neg X$~$ |
---|
Instrumental convergence | $~$U_k$~$ |
---|
Instrumental convergence | $~$S_i$~$ |
---|
Instrumental convergence | $~$U_k$~$ |
---|
Instrumental convergence | $~$S_i$~$ |
---|
Instrumental convergence | $~$\pi_k = \underset{\pi_i \in \Pi}{\operatorname{argmax}} \mathbb E [ U_k | S_i, \pi_i ]$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$S_k$~$ |
---|
Instrumental convergence | $$~$\big ( \underset{\pi_i \in X}{\operatorname{argmax}} \mathbb E [ U_k | S_k, \pi_i ] \big ) \ < \ \big ( \underset{\pi_i \in \neg X}{\operatorname{argmax}} \mathbb E [ U_k | S_k, \pi_i ] \big )$~$$ |
---|
Instrumental convergence | $~$U_k$~$ |
---|
Instrumental convergence | $~$S_k$~$ |
---|
Instrumental convergence | $~$\pi_i \in X$~$ |
---|
Instrumental convergence | $~$S_i$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$Y$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$Y$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$\neg X$~$ |
---|
Instrumental convergence | $~$Y.$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$Y$~$ |
---|
Instrumental convergence | $~$X_i$~$ |
---|
Instrumental convergence | $~$X_k$~$ |
---|
Instrumental convergence | $~$\neg X$~$ |
---|
Instrumental convergence | $~$X_i$~$ |
---|
Instrumental convergence | $~$Y$~$ |
---|
Instrumental convergence | $~$Y_k$~$ |
---|
Instrumental convergence | $~$Y_k$~$ |
---|
Instrumental convergence | $~$X_i.$~$ |
---|
Instrumental convergence | $~$10^{55}$~$ |
---|
Instrumental convergence | $~$10^{55}$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$W_j \subset \neg X$~$ |
---|
Instrumental convergence | $~$X.$~$ |
---|
Instrumental convergence | $~$W_j,$~$ |
---|
Instrumental convergence | $~$X,$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$W_j.$~$ |
---|
Instrumental convergence | $~$10^{55}$~$ |
---|
Instrumental convergence | $~$10^{55}$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$X_e$~$ |
---|
Instrumental convergence | $~$X.$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$X_e.$~$ |
---|
Instrumental convergence | $~$X_e$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$X_e$~$ |
---|
Instrumental convergence | $~$X_e$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$X.$~$ |
---|
Instrumental convergence | $~$\neg X$~$ |
---|
Instrumental convergence | $~$\neg X$~$ |
---|
Instrumental convergence | $~$\neg X$~$ |
---|
Instrumental convergence | $~$U_k$~$ |
---|
Instrumental convergence | $~$\pi_i$~$ |
---|
Instrumental convergence | $~$\mathbb E [ U | \pi_i ],$~$ |
---|
Instrumental convergence | $~$\pi_j$~$ |
---|
Instrumental convergence | $~$\mathbb E [ U | \pi_j ] > \mathbb E [ U | \pi_i]$~$ |
---|
Instrumental convergence | $~$\pi_i$~$ |
---|
Instrumental convergence | $~$\pi_j$~$ |
---|
Instrumental convergence | $~$\pi_i$~$ |
---|
Instrumental convergence | $~$\pi_j$~$ |
---|
Instrumental convergence | $~$\pi_i$~$ |
---|
Instrumental convergence | $~$\mathbb P (press | \pi_i) = 0.999…$~$ |
---|
Instrumental convergence | $~$\pi_j$~$ |
---|
Instrumental convergence | $~$\mathbb P (press | \pi_j) = 0.9999…$~$ |
---|
Instrumental convergence | $~$\neg X$~$ |
---|
Instrumental convergence | $~$Y_k$~$ |
---|
Instrumental convergence | $~$Y_k$~$ |
---|
Instrumental convergence | $~$X$~$ |
---|
Instrumental convergence | $~$10^{55}$~$ |
---|
Instrumental convergence | $~$10^{60}$~$ |
---|
Instrumental convergence | $~$S_w$~$ |
---|
Instrumental convergence | $~$\neg X$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$U_0$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_1, W_2, … W_25$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_25$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_1$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_4$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_3$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_2$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_1$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_4$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_2$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_0$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_4$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$U_0$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_1, W_2, W_3…W_N$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_1$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_1'$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_1 \subset W_1'$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$\mathbb E[U_0|W_1]$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_2$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_3$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_10$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_20$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_25$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$\mathbb E[U|W_1,W_2,…].$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$U.$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$U_1$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_10$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_25.$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_25$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_2, W_3, …$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$U_1$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$U_0, W_1.$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$U_0$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$U_1$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_1$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$W_1'$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$U_0$~$ |
---|
Instrumental goals are almost-equally as tractable as terminal goals | $~$U_1.$~$ |
---|
Integer | $~$\mathbb{Z}$~$ |
---|
Integer | $~$\mathbb{Z}$~$ |
---|
Integer | $~$0$~$ |
---|
Integer | $~$1$~$ |
---|
Integer | $~$\mathbb{Z}$~$ |
---|
Integer | $~$\mathbb{Z}$~$ |
---|
Integer | $~$\mathbb{Z}$~$ |
---|
Integer | $~$\mathbb{Z}$~$ |
---|
Integer | $~$\mathbb{Z}$~$ |
---|
Integer | $~$\mathbb{Z}$~$ |
---|
Integer | $~$\mathbb{Z}$~$ |
---|
Integer | $~$\mathbb{N}$~$ |
---|
Integer | $~$\mathbb{Z}$~$ |
---|
Integer | $~$(a, b)$~$ |
---|
Integer | $~$\sim$~$ |
---|
Integer | $~$(a,b) \sim (c,d)$~$ |
---|
Integer | $~$a+d = b+c$~$ |
---|
Integer | $~$(a,b)$~$ |
---|
Integer | $~$a-b$~$ |
---|
Integer | $~$[a,b]$~$ |
---|
Integer | $~$(a,b)$~$ |
---|
Integer | $~$[a,b] + [c,d] = [a+c,b+d]$~$ |
---|
Integer | $~$[a, b] \times [c, d] = [ac+bd, bc+ad]$~$ |
---|
Integer | $~$[a,b] \leq [c,d]$~$ |
---|
Integer | $~$a+d \leq b+c$~$ |
---|
Integers: Intro (Math 0) | $~$-2$~$ |
---|
Integers: Intro (Math 0) | $~$-15$~$ |
---|
Integers: Intro (Math 0) | $~$-6387$~$ |
---|
Integers: Intro (Math 0) | $~$3$~$ |
---|
Integers: Intro (Math 0) | $~$-3$~$ |
---|
Integers: Intro (Math 0) | $~$128$~$ |
---|
Integers: Intro (Math 0) | $~$-128$~$ |
---|
Integers: Intro (Math 0) | $~$6 - 4$~$ |
---|
Integers: Intro (Math 0) | $~$4 - 6$~$ |
---|
Integers: Intro (Math 0) | $~$-2$~$ |
---|
Integers: Intro (Math 0) | $~$-3$~$ |
---|
Integers: Intro (Math 0) | $~$-7$~$ |
---|
Integers: Intro (Math 0) | $~$-10$~$ |
---|
Integers: Intro (Math 0) | $~$(-6) + (-8) + (-12) + (-20)$~$ |
---|
Integers: Intro (Math 0) | $$~$6 + 8 + 12 + 20 = 46 \to (-6) + (-8) + (-12) + (-20) = -46$~$$ |
---|
Integers: Intro (Math 0) | $~$5 - 2$~$ |
---|
Integers: Intro (Math 0) | $~$5 + (-2)$~$ |
---|
Integers: Intro (Math 0) | $~$6 - 2 + 7 - 5$~$ |
---|
Integers: Intro (Math 0) | $~$6 + (-2) + 7 + (-5)$~$ |
---|
Integers: Intro (Math 0) | $~$6 + 7 + (-5) + (-2)$~$ |
---|
Integers: Intro (Math 0) | $~$6$~$ |
---|
Integers: Intro (Math 0) | $~$13 + 8 - 5 + 6 + 4 - 12 - 9 + 1$~$ |
---|
Integers: Intro (Math 0) | $$~$13 + 8 + (-5) + 6 + 4 + (-12) + (-9) + 1$~$$ |
---|
Integers: Intro (Math 0) | $$~$13 + 8 + 6 + 4 + 1 + (-5) + (-12) + (-9)$~$$ |
---|
Integers: Intro (Math 0) | $$~$(13 + 8 + 6 + 4 + 1) + ((-5) + (-12) + (-9)) = 32 + (-26)$~$$ |
---|
Integers: Intro (Math 0) | $$~$32 + (-26) = 32 - 26 = 6$~$$ |
---|
Integers: Intro (Math 0) | $~$8 - 6 + 4 - 13 + 7 - 5 - 9 + 12$~$ |
---|
Integers: Intro (Math 0) | $$~$8 + (-6) + 4 + (- 13) + 7 + (- 5) + (- 9) + 12 \\ \Downarrow$~$$ |
---|
Integers: Intro (Math 0) | $$~$8 + 4 + 7 + 12 + (-6) + (-13) + (-5) + (-9) \\ \Downarrow$~$$ |
---|
Integers: Intro (Math 0) | $$~$(8 + 4 + 7 + 12) + ((-6) + (-13) + (-5) + (-9)) = 31 + (-33)$~$$ |
---|
Integers: Intro (Math 0) | $$~$31 + (-33) = 31 - 33$~$$ |
---|
Integers: Intro (Math 0) | $$~$31 - 33 = -(33 - 31) = -2$~$$ |
---|
Integers: Intro (Math 0) | $~$\{ \ldots, -2, -1, 0, 1, 2, \ldots \}$~$ |
---|
Integral domain | $~$0$~$ |
---|
Integral domain | $~$0$~$ |
---|
Integral domain | $~$\mathbb{Z}$~$ |
---|
Integral domain | $~$2 \times 3$~$ |
---|
Integral domain | $~$0$~$ |
---|
Integral domain | $~$2$~$ |
---|
Integral domain | $~$3$~$ |
---|
Integral domain | $~$ab=0$~$ |
---|
Integral domain | $~$a=0$~$ |
---|
Integral domain | $~$b=0$~$ |
---|
Integral domain | $~$\mathbb{Z}$~$ |
---|
Integral domain | $~$a \times b = 0$~$ |
---|
Integral domain | $~$a=0$~$ |
---|
Integral domain | $~$b=0$~$ |
---|
Integral domain | $~$0$~$ |
---|
Integral domain | $~$0=1$~$ |
---|
Integral domain | $~$\mathbb{Z}$~$ |
---|
Integral domain | $~$ab = 0$~$ |
---|
Integral domain | $~$a \not = 0$~$ |
---|
Integral domain | $~$b=0$~$ |
---|
Integral domain | $~$a$~$ |
---|
Integral domain | $~$a^{-1}$~$ |
---|
Integral domain | $~$a^{-1}$~$ |
---|
Integral domain | $~$a^{-1} a b = 0 \times a^{-1}$~$ |
---|
Integral domain | $~$b = 0$~$ |
---|
Integral domain | $~$p$~$ |
---|
Integral domain | $~$\mathbb{Z}_p$~$ |
---|
Integral domain | $~$p$~$ |
---|
Integral domain | $~$n$~$ |
---|
Integral domain | $~$\mathbb{Z}_n$~$ |
---|
Integral domain | $~$n = r \times s$~$ |
---|
Integral domain | $~$r, s$~$ |
---|
Integral domain | $~$r s = n = 0$~$ |
---|
Integral domain | $~$\mathbb{Z}_n$~$ |
---|
Integral domain | $~$a \not = 0$~$ |
---|
Integral domain | $~$ab = ac$~$ |
---|
Integral domain | $~$b=c$~$ |
---|
Integral domain | $~$ab = ac$~$ |
---|
Integral domain | $~$ab-ac = 0$~$ |
---|
Integral domain | $~$a(b-c) = 0$~$ |
---|
Integral domain | $~$a=0$~$ |
---|
Integral domain | $~$b=c$~$ |
---|
Integral domain | $~$r s = 0$~$ |
---|
Integral domain | $~$r, s \not = 0$~$ |
---|
Integral domain | $~$rs = r \times 0$~$ |
---|
Integral domain | $~$s \not = 0$~$ |
---|
Integral domain | $~$r$~$ |
---|
Integral domain | $~$R$~$ |
---|
Integral domain | $~$r \in R$~$ |
---|
Integral domain | $~$S = \{ ar : a \in R\}$~$ |
---|
Integral domain | $~$R$~$ |
---|
Integral domain | $~$R$~$ |
---|
Integral domain | $~$ar = br$~$ |
---|
Integral domain | $~$r$~$ |
---|
Integral domain | $~$a = b$~$ |
---|
Integral domain | $~$|R|$~$ |
---|
Integral domain | $~$S$~$ |
---|
Integral domain | $~$| \cdot |$~$ |
---|
Integral domain | $~$R$~$ |
---|
Integral domain | $~$S$~$ |
---|
Integral domain | $~$R$~$ |
---|
Integral domain | $~$1 \in S$~$ |
---|
Integral domain | $~$1 = ar$~$ |
---|
Integral domain | $~$a$~$ |
---|
Intersection | $~$A$~$ |
---|
Intersection | $~$B$~$ |
---|
Intersection | $~$A \cap B$~$ |
---|
Intersection | $~$A$~$ |
---|
Intersection | $~$B$~$ |
---|
Intersection | $~$C = A \cap B$~$ |
---|
Intersection | $$~$x \in C \leftrightarrow (x \in A \land x \in B)$~$$ |
---|
Intersection | $~$x$~$ |
---|
Intersection | $~$C$~$ |
---|
Intersection | $~$x$~$ |
---|
Intersection | $~$A$~$ |
---|
Intersection | $~$x$~$ |
---|
Intersection | $~$B$~$ |
---|
Intersection | $~$\{1,2\} \cap \{2,3\} = \{2\}$~$ |
---|
Intersection | $~$\{1,2\} \cap \{8,9\} = \{\}$~$ |
---|
Intersection | $~$\{0,2,4,6\} \cap \{3,4,5,6\} = \{4,6\}$~$ |
---|
Intradependent encoding | $~$E(m)$~$ |
---|
Intradependent encoding | $~$m$~$ |
---|
Intradependent encoding | $~$E(m)$~$ |
---|
Intradependent encoding | $~$m$~$ |
---|
Intradependent encoding | $~$E(m)$~$ |
---|
Intradependent encoding | $~$m$~$ |
---|
Intradependent encoding | $~$E(m)$~$ |
---|
Intradependent encoding | $~$m$~$ |
---|
Intradependent encoding | $~$26^4 = 456976$~$ |
---|
Intradependent encoding | $~$\log_2(26^4) - 1 \approx 17.8$~$ |
---|
Intradependent encodings can be compressed | $~$E$~$ |
---|
Intradependent encodings can be compressed | $~$m,$~$ |
---|
Intradependent encodings can be compressed | $~$E^\prime$~$ |
---|
Intradependent encodings can be compressed | $~$m.$~$ |
---|
Intradependent encodings can be compressed | $~$E$~$ |
---|
Intradependent encodings can be compressed | $~$m$~$ |
---|
Intradependent encodings can be compressed | $~$E(m)$~$ |
---|
Intradependent encodings can be compressed | $~$E^\prime$~$ |
---|
Intro to Number Sets | $~$\mathbb{N}$~$ |
---|
Intro to Number Sets | $~$\mathbb{Z}$~$ |
---|
Intro to Number Sets | $~$\mathbb{Q}$~$ |
---|
Intro to Number Sets | $~$\mathbb{R}$~$ |
---|
Intro to Number Sets | $~$\mathbb{C}$~$ |
---|
Intro to Number Sets | $~$3$~$ |
---|
Intro to Number Sets | $~$2$~$ |
---|
Intro to Number Sets | $~$1.5$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$(1 : 4),$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$(90 : 30) = (3 : 1)$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$(1 : 4) \times (3 : 1) = (3 : 4),$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$3/(3+4) \approx 43\%.$~$ |
---|
Introduction to Bayes' rule: Odds form | $$~$ \textbf{Prior odds} \times \textbf{Relative likelihoods} = \textbf{Posterior odds}$~$$ |
---|
Introduction to Bayes' rule: Odds form | $~$X$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(X)$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$X.$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$X$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(X)$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$X$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb \neg X$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$X$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(\neg X)$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$X$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(blackened \mid sick) = 0.9$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(blackened \mid \neg sick) = 0.3$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(sick \mid blackened) = 3/7$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(X \mid Y)$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$X$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$Y$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$X \wedge Y$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$X$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$Y$~$ |
---|
Introduction to Bayes' rule: Odds form | $$~$\mathbb P(X \mid Y) := \frac{\mathbb P(X \wedge Y)}{\mathbb P(Y)}$~$$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(sick \mid blackened)$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(sick \wedge blackened)$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(blackened)$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(blackened \mid \neg sick),$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$Y$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$Y$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$Y$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$X$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$X$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$Y$~$ |
---|
Introduction to Bayes' rule: Odds form | $$~$\textbf{Prior odds} \times \textbf{Relative likelihoods} = \textbf{Posterior odds}$~$$ |
---|
Introduction to Bayes' rule: Odds form | $$~$\dfrac{\mathbb P({sick})}{\mathbb P(healthy)} \times \dfrac{\mathbb P({blackened}\mid {sick})}{\mathbb P({blackened}\mid healthy)} = \dfrac{\mathbb P({sick}\mid {blackened})}{\mathbb P(healthy\mid {blackened})}.$~$$ |
---|
Introduction to Bayes' rule: Odds form | $~$1 : 4$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(sick)=\frac{1}{4+1}=\frac{1}{5}=20\%$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\frac{\mathbb P(positive \mid sick)}{\mathbb P(positive \mid healthy)}=\frac{0.90}{0.30},$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$3 : 1.$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\frac{\mathbb P(sick \mid positive)}{\mathbb P(healthy \mid positive)} = \frac{3}{4}$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$3 : 4$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$1,$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$3 : 4$~$ |
---|
Introduction to Bayes' rule: Odds form | $$~$(\frac{3}{3+4} : \frac{4}{3+4}) = (\frac{3}{7} : \frac{4}{7}) \approx (0.43 : 0.57)$~$$ |
---|
Introduction to Bayes' rule: Odds form | $~$3$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$H_j$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$H_k$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$e$~$ |
---|
Introduction to Bayes' rule: Odds form | $$~$\dfrac{\mathbb P(H_j)}{\mathbb P(H_k)} \times \dfrac{\mathbb P(e \mid H_j)}{\mathbb P(e \mid H_k)} = \dfrac{\mathbb P(H_j \mid e)}{\mathbb P(H_k \mid e)}$~$$ |
---|
Introduction to Bayes' rule: Odds form | $~$H_j$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$H_k$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$e$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$H_j$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$H_k.$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$H_j$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$H_k$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$H_j$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$1.$~$ |
---|
Introduction to Bayes' rule: Odds form | $~$\mathbb P(X \wedge Y) = \mathbb P(Y) \cdot \mathbb P(X|Y).$~$ |
---|
Introduction to Bayes' rule: Odds form | $$~$
\frac{\mathbb P(H_j)}{\mathbb P(H_k)}
\cdot
\frac{\mathbb P(e_0 | H_j)}{\mathbb P(e_0 | H_k)}
=
\frac{\mathbb P(e_0 \wedge H_j)}{\mathbb P(e_0 \wedge H_k)}
=
\frac{\mathbb P(H_j \wedge e_0)/\mathbb P(e_0)}{\mathbb P(H_k \wedge e_0)/\mathbb P(e_0)}
=
\frac{\mathbb P(H_j | e_0)}{\mathbb P(H_k | e_0)}
$~$$ |
---|
Introduction to Bayes' rule: Odds form | $$~$
\frac{0.20}{0.80}
\cdot
\frac{0.90}{0.30}
=
\frac{0.18}{0.24}
=
\frac{0.18/0.42}{0.24/0.42}
=
\frac{0.43}{0.57}
$~$$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $$~$\mathbb E[\mathcal U|a_x] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(o_i|a_x)$~$$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathbb E[\mathcal U|a_x]$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$a_x$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathcal O$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathcal U$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathbb P(o_i|a_x)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$o_i$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$a_x$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathbb P(o_i|a_x).$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathbb P(o_i),$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$a_x$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\ \mathbb P(a_x \ \square \! \! \rightarrow o_i).$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$K$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$O$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathbb P(K| \neg O)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathbb P(\neg O \ \square \!\! \rightarrow K)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $$~$\mathbb E[\mathcal U|a_x] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(a_x \ \square \!\! \rightarrow o_i)$~$$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $$~$\mathsf Q(s) = \big ( \underset{\pi_x \in \Pi}{argmax} \ \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(\ulcorner \mathsf Q = \pi_x \urcorner \triangleright o_i) \big ) (s)$~$$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$s$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\pi_x \in \Pi$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\ulcorner \mathsf Q = \pi_x \urcorner$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\pi_x.$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathbb P(X \triangleright o_i)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$o_i$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$X.$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$X \triangleright Y$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathbb P(\bullet \ || \ \bullet).$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$X \triangleright Y$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$X$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q,$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$s$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q.$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q(city)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q(city)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q(desert)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q,$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q(city)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q(city)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\ulcorner \mathsf Q \urcorner$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q(warning)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q(message)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\operatorname{counterfactual}_1$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\operatorname {do}().$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\operatorname{counterfactual}_1$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\operatorname{counterfactual}_1$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\operatorname{counterfactual}_2$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\operatorname{counterfactual}_1$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\operatorname{counterfactual}_2,$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\operatorname{counterfactual}_2$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\operatorname{counterfactual}_1.$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\operatorname {do}()$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$a$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\pi.$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q,$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q(city)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$(o_1, o_2)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $$~$\begin{array}{r|c|c}
& D_2 & C_2 \\
\hline
D_1 & (\$1, \$1) & (\$3, \$0) \\ \hline
C_1 & (\$0, \$3) & (\$2, \$2)
\end{array}$~$$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$(\$1, \$1)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$(\$2, \$2).$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathcal T$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathcal T$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$S$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$\operatorname{Prov}(\mathcal T, \ulcorner S \urcorner)$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$A$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$B$~$ |
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Introduction to Logical Decision Theory for Analytic Philosophers | $~$C$~$ |
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Introduction to Logical Decision Theory for Analytic Philosophers | $~$C$~$ |
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Introduction to Logical Decision Theory for Analytic Philosophers | $~$\mathcal T$~$ |
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Introduction to Logical Decision Theory for Analytic Philosophers | $~$B$~$ |
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Introduction to Logical Decision Theory for Analytic Philosophers | $~$D$~$ |
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Introduction to Logical Decision Theory for Analytic Philosophers | $$~$\begin{array}{r|c|c}
& Y_2 & Z_2 \\
\hline
W_1 & (\$0, \$49) & (\$49, \$0) \\ \hline
X_1 & (\$1, \$0) & (\$0, \$1)
\end{array}$~$$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$W_1,$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$Y_2$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$Y_2$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$X_1,$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$X_1$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$Z_2,$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$Z_2$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$W_1.$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$W_1$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$X_1,$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$Y_2$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$Z_2.$~$ |
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Introduction to Logical Decision Theory for Analytic Philosophers | $~$Y_2$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$X_1$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$q$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $~$p$~$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $$~$p = \big ( \dfrac{\$5}{\$10 - q} \big ) ^ {1.01}$~$$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $$~$\begin{array}{r|c|c|c}
& Defect_2 & Cooperate_2 & Nuke_2 \\
\hline
Defect_1 & (\$1, \$1) & (\$3, \$0) & (-\$100, -\$100) \\ \hline
Cooperate_1 & (\$0, \$3) & (\$2, \$2) & (-\$100, -\$100) \\ \hline
Nuke_1 & (-\$100, -\$100) & (-\$100, -\$100) & (-\$100, -\$100)
\end{array}$~$$ |
---|
Introduction to Logical Decision Theory for Analytic Philosophers | $$~$\begin{array}{r|c|c}
& \text{One-boxing predicted} & \text{Two-boxing predicted} \\
\hline
\text{W: Take both boxes, no fee:} & \$500,500 & \$500 \\ \hline
\text{X: Take only Box B, no fee:} & \$500,000 & \$0 \\ \hline
\text{Y: Take both boxes, pay fee:} & \$1,000,100 & \$100 \\ \hline
\text{Z: Take only Box B, pay fee:} & \$999,100 & -\$900
\end{array}$~$$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$(o_1, o_2)$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $$~$\begin{array}{r|c|c}
& \text{ Player 2 Defects: } & \text{ Player 2 Cooperates: }\\
\hline
\text{ Player 1 Defects: }& \text{ (2 years, 2 years) } & \text{ (0 years, 3 years) } \\ \hline
\text{ Player 1 Cooperates: } & \text{ (3 years, 0 years) } & \text{ (1 year, 1 year) }
\end{array}$~$$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$D$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$C,$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $$~$\begin{array}{r|c|c}
& D_2 & C_2 \\
\hline
D_1 & (\$1, \$1) & (\$3, \$0) \\ \hline
C_1 & (\$0, \$3) & (\$2, \$2)
\end{array}$~$$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathcal T$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathcal T$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$S$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\operatorname{Prov}(\mathcal T, \ulcorner S \urcorner)$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$A$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$B$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$C$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$C$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathcal T$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$B$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$D$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $$~$\mathbb E[\mathcal U|a_x] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(o_i|a_x)$~$$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb E[\mathcal U|a_x]$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$a_x$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathcal O$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathcal U$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb P(o_i|a_x)$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$o_i$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$a_x$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb P(o_i|a_x).$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb P(o_i),$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$a_x$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\ \mathbb P(a_x \ \square \! \! \rightarrow o_i).$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$K$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$O$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb P(K| \neg O)$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb P(\neg O \ \square \!\! \rightarrow K)$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $$~$\mathbb E[\mathcal U|a_x] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(a_x \ \square \!\! \rightarrow o_i)$~$$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb P(a_x \ \square \!\! \rightarrow o_i).$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb P(\bullet \ || \ \bullet)$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X_1$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X_2$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X_3$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X_4$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X_5$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb P(X_i | \mathbf{pa}_i)$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X_i$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$x_i$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbf {pa}_i$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$x_i$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbf x$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $$~$\mathbb P(\mathbf x) = \prod_i \mathbb P(x_i | \mathbf{pa}_i)$~$$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb P(\mathbf x | \operatorname{do}(X_j=x_j))$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $$~$\mathbb P(\mathbf x | \operatorname{do}(X_j=x_j)) = \prod_{i \neq j} \mathbb P(x_i | \mathbf{pa}_i)$~$$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbf x$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$x_j$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\operatorname{do}$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X_j$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$0$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\operatorname{do}(X_j=x_j)$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X_j$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbf{pa}_j,$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X_j = x_j$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\operatorname{do}(X_j=x_j)$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X_k$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X_j$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $$~$\mathbb E[\mathcal U| \operatorname{do}(a_x)] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(o_i | \operatorname{do}(a_x))$~$$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $$~$\mathsf Q(s) = \big ( \underset{\pi_x \in \Pi}{argmax} \ \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(\ulcorner \mathsf Q = \pi_x \urcorner \triangleright o_i) \big ) (s)$~$$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$s$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\pi_x \in \Pi$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\ulcorner \mathsf Q = \pi_x \urcorner$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\pi_x.$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb P(X \triangleright o_i)$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$o_i$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X.$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X \triangleright Y$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X \triangleright Y$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X \triangleright Y$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$Y$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$X$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb P (X \triangleright Y)$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$\mathbb P( Y | \operatorname {do}(X))$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$a_x,$~$ |
---|
Introduction to Logical Decision Theory for Computer Scientists | $~$a_x$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathcal Q,$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathcal Q$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathcal Q$~$ |
---|
Introduction to Logical Decision Theory for Economists | $$~$\mathbb E[\mathcal U|a_x] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(o_i|a_x)$~$$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbb E[\mathcal U|a_x]$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$a_x$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathcal O$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathcal U$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbb P(o_i|a_x)$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$o_i$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$a_x$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbb P(o_i|a_x).$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbb P(o_i),$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$a_x$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\ \mathbb P(a_x \ \square \! \! \rightarrow o_i).$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$K$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$O$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbb P(K| \neg O)$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbb P(\neg O \ \square \!\! \rightarrow K)$~$ |
---|
Introduction to Logical Decision Theory for Economists | $$~$\mathbb E[\mathcal U|a_x] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(a_x \ \square \!\! \rightarrow o_i)$~$$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbb P(a_x \ \square \!\! \rightarrow o_i),$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbb P(\bullet \ || \ \bullet)$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X_1$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X_2$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X_3$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X_4$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X_5$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbb P(X_i | \mathbf{pa}_i)$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X_i$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$x_i$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbf {pa}_i$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$x_i$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbf x$~$ |
---|
Introduction to Logical Decision Theory for Economists | $$~$\mathbb P(\mathbf x) = \prod_i \mathbb P(x_i | \mathbf{pa}_i)$~$$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbb P(\mathbf x | \operatorname{do}(X_j=x_j))$~$ |
---|
Introduction to Logical Decision Theory for Economists | $$~$\mathbb P(\mathbf x | \operatorname{do}(X_j=x_j)) = \prod_{i \neq j} \mathbb P(x_i | \mathbf{pa}_i)$~$$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbf x$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$x_j$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\operatorname{do}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X_j$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$0$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\operatorname{do}(X_j=x_j)$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X_i$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X_j$~$ |
---|
Introduction to Logical Decision Theory for Economists | $$~$\mathbb E[\mathcal U| \operatorname{do}(a_x)] = \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(o_i | \operatorname{do}(a_x))$~$$ |
---|
Introduction to Logical Decision Theory for Economists | $$~$\mathsf Q(s) = \big ( \underset{\pi_x \in \Pi}{argmax} \ \sum_{o_i \in \mathcal O} \mathcal U(o_i) \cdot \mathbb P(\ulcorner \mathsf Q = \pi_x \urcorner \triangleright o_i) \big ) (s)$~$$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$s$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\pi_x \in \Pi$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\ulcorner \mathsf Q = \pi_x \urcorner$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf Q$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\pi_x.$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathbb P(X \triangleright o_i)$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$o_i$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X.$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X \triangleright Y$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\operatorname {do}()$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X \triangleright Y,$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$X$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\operatorname{do}()$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {A}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {A}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf A,$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot},$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf A$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}.$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf{Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {CooperateBot},$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {CooperateBot},$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf {Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf{PrudentBot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf{Fairbot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf{PrudentBot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf{DefectBot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf{CooperateBot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$\mathsf{PrudentBot}$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$q$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$p$~$ |
---|
Introduction to Logical Decision Theory for Economists | $$~$p = \big ( \dfrac{\$5}{\$10 - q} \big ) ^ {1.01}$~$$ |
---|
Introduction to Logical Decision Theory for Economists | $~$a_x,$~$ |
---|
Introduction to Logical Decision Theory for Economists | $~$a_x$~$ |
---|
Introductory guide to logarithms | $$~$\underbrace{139}_\text{3 digits}$~$$ |
---|
Introductory guide to logarithms | $~$\log_{10}(139) \approx 2.14$~$ |
---|
Inverse function | $~$g$~$ |
---|
Inverse function | $~$f$~$ |
---|
Inverse function | $~$g$~$ |
---|
Inverse function | $~$f$~$ |
---|
Inverse function | $~$f$~$ |
---|
Inverse function | $~$g$~$ |
---|
Inverse function | $~$g(f(x)) = x$~$ |
---|
Inverse function | $~$f(g(y)) = y$~$ |
---|
Inverse function | $~$f$~$ |
---|
Inverse function | $~$A$~$ |
---|
Inverse function | $~$B$~$ |
---|
Inverse function | $~$g$~$ |
---|
Inverse function | $~$B$~$ |
---|
Inverse function | $~$A$~$ |
---|
Inverse function | $~$f$~$ |
---|
Inverse function | $~$f^{-1}$~$ |
---|
Inverse function | $$~$y=f(x) = x^3\ \ \ \ \ \ \ \ \ \ \ \ x=f^{-1}(y) = y^{1/3}$~$$ |
---|
Inverse function | $$~$y=f(x) = e^x\ \ \ \ \ \ \ \ \ \ \ \ x=f^{-1}(y) = ln(y)$~$$ |
---|
Inverse function | $$~$y=f(x) = x+4\ \ \ \ \ \ \ \ \ \ \ \ x=f^{-1}(y) = y-4$~$$ |
---|
Invisible background fallacies | $~$9.8m/s^2.$~$ |
---|
Invisible background fallacies | $~$9.8m/s^2,$~$ |
---|
Invisible background fallacies | $~$9.8m/s^2.$~$ |
---|
Invisible background fallacies | $~$9.8m/s^2.$~$ |
---|
Invisible background fallacies | $~$9.8m/s^2$~$ |
---|
Irrational number | $~$\mathbb{I}$~$ |
---|
Irrational number | $~$\overline{\mathbb{Q}}$~$ |
---|
Irrational number | $~$\mathbb{Q}^\ge$~$ |
---|
Irrational number | $~$b$~$ |
---|
Irreducible element (ring theory) | $~$(R, +, \times)$~$ |
---|
Irreducible element (ring theory) | $~$x \in R$~$ |
---|
Irreducible element (ring theory) | $~$r = a \times b$~$ |
---|
Irreducible element (ring theory) | $~$a$~$ |
---|
Irreducible element (ring theory) | $~$b$~$ |
---|
Irreducible element (ring theory) | $~$\mathbb{Z}$~$ |
---|
Irreducible element (ring theory) | $~$R$~$ |
---|
Irreducible element (ring theory) | $~$x \in R$~$ |
---|
Irreducible element (ring theory) | $~$r = a \times b$~$ |
---|
Irreducible element (ring theory) | $~$a$~$ |
---|
Irreducible element (ring theory) | $~$b$~$ |
---|
Irreducible element (ring theory) | $~$R$~$ |
---|
Irreducible element (ring theory) | $~$p$~$ |
---|
Irreducible element (ring theory) | $~$p \mid ab$~$ |
---|
Irreducible element (ring theory) | $~$p \mid a$~$ |
---|
Irreducible element (ring theory) | $~$p \mid b$~$ |
---|
Irreducible element (ring theory) | $~$p=ab$~$ |
---|
Irreducible element (ring theory) | $~$a$~$ |
---|
Irreducible element (ring theory) | $~$b$~$ |
---|
Irreducible element (ring theory) | $~$p = ab$~$ |
---|
Irreducible element (ring theory) | $~$p \mid ab$~$ |
---|
Irreducible element (ring theory) | $~$p \mid a$~$ |
---|
Irreducible element (ring theory) | $~$p \mid b$~$ |
---|
Irreducible element (ring theory) | $~$p \mid a$~$ |
---|
Irreducible element (ring theory) | $~$c$~$ |
---|
Irreducible element (ring theory) | $~$a = cp$~$ |
---|
Irreducible element (ring theory) | $~$p = ab = cpb$~$ |
---|
Irreducible element (ring theory) | $~$p(1-bc) = 0$~$ |
---|
Irreducible element (ring theory) | $~$p$~$ |
---|
Irreducible element (ring theory) | $~$1-bc = 0$~$ |
---|
Irreducible element (ring theory) | $~$bc = 1$~$ |
---|
Irreducible element (ring theory) | $~$b$~$ |
---|
Irreducible element (ring theory) | $~$\mathbb{Z}$~$ |
---|
Irreducible element (ring theory) | $~$p$~$ |
---|
Irreducible element (ring theory) | $~$\mathbb{Z}$~$ |
---|
Irreducible element (ring theory) | $~$p$~$ |
---|
Irreducible element (ring theory) | $~$\mathbb{Z}[\sqrt{-3}]$~$ |
---|
Irreducible element (ring theory) | $~$a+b \sqrt{-3}$~$ |
---|
Irreducible element (ring theory) | $~$a, b$~$ |
---|
Irreducible element (ring theory) | $~$2$~$ |
---|
Irreducible element (ring theory) | $~$4 = 2 \times 2 = (1+\sqrt{-3})(1-\sqrt{-3})$~$ |
---|
Irreducible element (ring theory) | $~$2 \mid (1+\sqrt{-3})(1-\sqrt{-3})$~$ |
---|
Irreducible element (ring theory) | $~$2$~$ |
---|
Irreducible element (ring theory) | $~$2$~$ |
---|
Irreducible element (ring theory) | $~$2$~$ |
---|
Irreducible element (ring theory) | $~$N(2)$~$ |
---|
Irreducible element (ring theory) | $~$2$~$ |
---|
Irreducible element (ring theory) | $~$4$~$ |
---|
Irreducible element (ring theory) | $~$2 = ab$~$ |
---|
Irreducible element (ring theory) | $~$N(2) = N(a)N(b)$~$ |
---|
Irreducible element (ring theory) | $~$N(a) N(b) = 4$~$ |
---|
Irreducible element (ring theory) | $~$N(x + y \sqrt{-3}) = x^2 + 3 y^2$~$ |
---|
Irreducible element (ring theory) | $~$N(a) = 1, N(b) = 4$~$ |
---|
Irreducible element (ring theory) | $~$N(a) = 2 = N(b)$~$ |
---|
Irreducible element (ring theory) | $~$a$~$ |
---|
Irreducible element (ring theory) | $~$b$~$ |
---|
Irreducible element (ring theory) | $~$N(x+y \sqrt{3}) = 1$~$ |
---|
Irreducible element (ring theory) | $~$x= \pm 1, y = \pm 0$~$ |
---|
Irreducible element (ring theory) | $~$a=\pm1, b=\pm2$~$ |
---|
Irreducible element (ring theory) | $~$2$~$ |
---|
Irreducible element (ring theory) | $~$N(x+y \sqrt{3}) = 2$~$ |
---|
Irreducible element (ring theory) | $~$y \not = 0$~$ |
---|
Irreducible element (ring theory) | $~$y = 0$~$ |
---|
Irreducible element (ring theory) | $~$x \in \mathbb{Z}$~$ |
---|
Irreducible element (ring theory) | $~$x^2 = 2$~$ |
---|
Irreducible element (ring theory) | $~$2$~$ |
---|
Irreducible element (ring theory) | $~$\pm 1$~$ |
---|
Irreducible element (ring theory) | $~$R$~$ |
---|
Irreducible element (ring theory) | $~$r \in R$~$ |
---|
Irreducible element (ring theory) | $~$\mathbb{Z}$~$ |
---|
Irreducible element (ring theory) | $~$\mathbb{Z}$~$ |
---|
Isomorphism | $~$f:A \to B$~$ |
---|
Isomorphism | $~$g: B \to A$~$ |
---|
Isomorphism | $~$f$~$ |
---|
Isomorphism | $~$g$~$ |
---|
Isomorphism | $~$fg$~$ |
---|
Isomorphism | $~$gf$~$ |
---|
Isomorphism | $~$gf = \mathrm {id}_A$~$ |
---|
Isomorphism | $~$fg = \mathrm {id}_B$~$ |
---|
Join and meet | $~$\langle P, \leq \rangle$~$ |
---|
Join and meet | $~$S \subseteq P$~$ |
---|
Join and meet | $~$S$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$\bigvee_P S$~$ |
---|
Join and meet | $~$p \in P$~$ |
---|
Join and meet | $~$S$~$ |
---|
Join and meet | $~$s \in S$~$ |
---|
Join and meet | $~$s \leq p$~$ |
---|
Join and meet | $~$q$~$ |
---|
Join and meet | $~$S$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$p \leq q$~$ |
---|
Join and meet | $~$\bigvee_P S$~$ |
---|
Join and meet | $~$\bigvee S$~$ |
---|
Join and meet | $~$\bigvee_P S$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$a, b$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$\{a,b\}$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$a \vee_P b$~$ |
---|
Join and meet | $~$a \vee b$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$\langle P, \leq \rangle$~$ |
---|
Join and meet | $~$S \subseteq P$~$ |
---|
Join and meet | $~$S$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$\bigvee_P S$~$ |
---|
Join and meet | $~$p \in P$~$ |
---|
Join and meet | $~$S$~$ |
---|
Join and meet | $~$s \in S$~$ |
---|
Join and meet | $~$s \leq p$~$ |
---|
Join and meet | $~$q$~$ |
---|
Join and meet | $~$S$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$p \leq q$~$ |
---|
Join and meet | $~$\bigvee_P S$~$ |
---|
Join and meet | $~$\bigvee S$~$ |
---|
Join and meet | $~$\bigvee_P S$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$a, b$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$\{a,b\}$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$a \vee_P b$~$ |
---|
Join and meet | $~$a \vee b$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$S$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$\bigwedge_P S$~$ |
---|
Join and meet | $~$p \in P$~$ |
---|
Join and meet | $~$S$~$ |
---|
Join and meet | $~$s$~$ |
---|
Join and meet | $~$S$~$ |
---|
Join and meet | $~$p \leq s$~$ |
---|
Join and meet | $~$q$~$ |
---|
Join and meet | $~$S$~$ |
---|
Join and meet | $~$P$~$ |
---|
Join and meet | $~$q \leq p$~$ |
---|
Join and meet | $~$\bigwedge S$~$ |
---|
Join and meet | $~$p \wedge_P q$~$ |
---|
Join and meet | $~$p \wedge q$~$ |
---|
Join and meet | $~$a$~$ |
---|
Join and meet | $~$b$~$ |
---|
Join and meet | $~$c$~$ |
---|
Join and meet | $~$d$~$ |
---|
Join and meet | $~$\bigvee \{a,b\}$~$ |
---|
Join and meet | $~$\{a,b\}$~$ |
---|
Join and meet | $~$\bigvee \{c,d\}$~$ |
---|
Join and meet | $~$\{c, d\}$~$ |
---|
Join and meet | $~$a$~$ |
---|
Join and meet | $~$b$~$ |
---|
Join and meet | $~$\{c, d\}$~$ |
---|
Join and meet | $~$a \vee c = a$~$ |
---|
Join and meet | $~$\bigvee \{b,c,d\} = b$~$ |
---|
Join and meet | $~$\bigvee \{c\} = c$~$ |
---|
Join and meet | $~$\bigwedge \{a, b, c, d\}$~$ |
---|
Join and meet | $~$c$~$ |
---|
Join and meet | $~$d$~$ |
---|
Join and meet | $~$\bigwedge \{a,b,d\} = d$~$ |
---|
Join and meet | $~$a \wedge c = c$~$ |
---|
Join and meet: Examples | $~$\langle \mathbb{R}, \leq \rangle$~$ |
---|
Join and meet: Examples | $~$X \subseteq \mathbb{R}$~$ |
---|
Join and meet: Examples | $~$\bigvee X$~$ |
---|
Join and meet: Examples | $~$X$~$ |
---|
Join and meet: Examples | $~$X$~$ |
---|
Join and meet: Examples | $~$\langle \mathcal{P}(X), \subseteq \rangle$~$ |
---|
Join and meet: Examples | $~$X$~$ |
---|
Join and meet: Examples | $~$A \subseteq \mathcal{P}(X)$~$ |
---|
Join and meet: Examples | $~$\bigvee A = \bigcup A$~$ |
---|
Join and meet: Examples | $~$A \subseteq \mathcal{P}(X)$~$ |
---|
Join and meet: Examples | $~$\bigcup A$~$ |
---|
Join and meet: Examples | $~$A$~$ |
---|
Join and meet: Examples | $~$\bigcup A$~$ |
---|
Join and meet: Examples | $~$A$~$ |
---|
Join and meet: Examples | $~$Z$~$ |
---|
Join and meet: Examples | $~$A$~$ |
---|
Join and meet: Examples | $~$x \in \bigcup A$~$ |
---|
Join and meet: Examples | $~$x \in Y$~$ |
---|
Join and meet: Examples | $~$Y \in A$~$ |
---|
Join and meet: Examples | $~$Y \subseteq Z$~$ |
---|
Join and meet: Examples | $~$x \in Y \subseteq Z$~$ |
---|
Join and meet: Examples | $~$x \in \bigcup A$~$ |
---|
Join and meet: Examples | $~$x \in Z$~$ |
---|
Join and meet: Examples | $~$\bigcup A \subseteq Z$~$ |
---|
Join and meet: Examples | $~$\bigvee A = \bigcup A$~$ |
---|
Join and meet: Examples | $~$\langle \mathbb Z_+, | \rangle$~$ |
---|
Join and meet: Examples | $~$\langle \mathbb Z_+, | \rangle$~$ |
---|
Join and meet: Examples | $~$\langle \mathbb Z_+, | \rangle$~$ |
---|
Join and meet: Exercises | $~$c \vee b$~$ |
---|
Join and meet: Exercises | $~$i$~$ |
---|
Join and meet: Exercises | $~$j$~$ |
---|
Join and meet: Exercises | $~$\{ c, b \}$~$ |
---|
Join and meet: Exercises | $~$\{ c, b \}$~$ |
---|
Join and meet: Exercises | $~$g \vee e$~$ |
---|
Join and meet: Exercises | $~$g \vee e = j$~$ |
---|
Join and meet: Exercises | $~$j \wedge f$~$ |
---|
Join and meet: Exercises | $~$d$~$ |
---|
Join and meet: Exercises | $~$b$~$ |
---|
Join and meet: Exercises | $~$\{ j, f \}$~$ |
---|
Join and meet: Exercises | $~$\{ j, f \}$~$ |
---|
Join and meet: Exercises | $~$\bigvee \{a,b,c,d,e,f,g,h,i,j,k,l\}$~$ |
---|
Join and meet: Exercises | $~$l$~$ |
---|
Join and meet: Exercises | $~$P$~$ |
---|
Join and meet: Exercises | $~$S \subseteq P$~$ |
---|
Join and meet: Exercises | $~$p \in P$~$ |
---|
Join and meet: Exercises | $~$\bigvee S$~$ |
---|
Join and meet: Exercises | $~$(\bigvee S) \vee p$~$ |
---|
Join and meet: Exercises | $~$\bigvee (S \cup \{p\})$~$ |
---|
Join and meet: Exercises | $~$(\bigvee S) \vee p = \bigvee (S \cup \{p\})$~$ |
---|
Join and meet: Exercises | $~$X \subset P$~$ |
---|
Join and meet: Exercises | $~$X^U$~$ |
---|
Join and meet: Exercises | $~$X$~$ |
---|
Join and meet: Exercises | $~$\{\bigvee S, p\}^U = (S \cup p)^U$~$ |
---|
Join and meet: Exercises | $~$q \in \{\bigvee S, p\}^U \iff$~$ |
---|
Join and meet: Exercises | $~$s \in S, q \geq \bigvee S \geq s$~$ |
---|
Join and meet: Exercises | $~$q \geq p \iff$~$ |
---|
Join and meet: Exercises | $~$q \in (S \cup \{p\})^U$~$ |
---|
Join and meet: Exercises | $~$P$~$ |
---|
Join and meet: Exercises | $~$S \subseteq P$~$ |
---|
Join and meet: Exercises | $~$p \in P$~$ |
---|
Join and meet: Exercises | $~$\bigwedge S$~$ |
---|
Join and meet: Exercises | $~$(\bigwedge S) \wedge p$~$ |
---|
Join and meet: Exercises | $~$\bigwedge (S \cup \{p\})$~$ |
---|
Join and meet: Exercises | $~$(\bigwedge S) \wedge p = \bigwedge(S \cup \{p\})$~$ |
---|
Join and meet: Exercises | $~$\langle \mathbb N, | \rangle$~$ |
---|
Join and meet: Exercises | $~$\bigvee \mathbb N$~$ |
---|
Joint probability | $~$\mathbb{P}(X \wedge Y)$~$ |
---|
Joint probability | $~$\mathbb{P}(X, Y)$~$ |
---|
Joint probability distribution | $$~$\newcommand{\bR}{\mathbb{R}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\gO}{\Omega}
\newcommand{\go}{\omega}
\newcommand{\ts}{\times}$~$$ |
---|
Joint probability distribution | $~$X_1, X_2, \cdots, X_n$~$ |
---|
Joint probability distribution | $~$\bP$~$ |
---|
Joint probability distribution | $~$\bR^n$~$ |
---|
Joint probability distribution | $~$(X_1 \in A_1, X_2 \in A_2, \cdots, X_n \in A_n)$~$ |
---|
Joint probability distribution | $~$\bP(A_1,A_2, \cdots, A_n)$~$ |
---|
Joint probability distribution | $$~$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\gO}{\Omega}
\newcommand{\go}{\omega}
\newcommand{\ts}{\times}
$~$$ |
---|
Joint probability distribution | $~$X_1, X_2, \cdots, X_n$~$ |
---|
Joint probability distribution | $~$\bP$~$ |
---|
Joint probability distribution | $~$\bR^n$~$ |
---|
Joint probability distribution | $~$(X_1 \in A_1, X_2 \in A_2, \cdots, X_n \in A_n)$~$ |
---|
Joint probability distribution | $~$\bP(A_1,A_2, \cdots, A_n)$~$ |
---|
Joint probability distribution | $~$\{X_i \}_{i \in I}$~$ |
---|
Joint probability distribution | $~$(S_i, \cS_i)$~$ |
---|
Joint probability distribution | $~$\{X_i \}_{i \in I}$~$ |
---|
Joint probability distribution | $~$\prod_{i \in I} S_i$~$ |
---|
Joint probability distribution | $~$\{X_i \}_{i \in I}$~$ |
---|
Joint probability distribution | $~$(\gO, \cF, \bP)$~$ |
---|
Joint probability distribution | $~$\bP$~$ |
---|
Joint probability distribution | $~$X_i$~$ |
---|
Joint probability distribution | $~$(X_1 \in A_1, X_2 \in A_2, \cdots, X_n \in A_n)$~$ |
---|
Joint probability distribution | $~$A_k \in \cS_k$~$ |
---|
Joint probability distribution | $~$\{ \go \in \gO : X_1(\go) \in A_1, \cdots, X_n(\go) \in A_n\}$~$ |
---|
Joint probability distribution | $~$\cF$~$ |
---|
Joint probability distribution | $~$\go \mapsto (X_1(\go), \cdots, X_n(\go))$~$ |
---|
Joint probability distribution | $~$\gO \to S_1 \ts \cdots \ts S_k$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $$~$
\newcommand{\gO}{\Omega}
\newcommand{\go}{\omega}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\pc}{0.4}
\newcommand{\pnc}{0.6}
\newcommand{\plc}{0.7}
\newcommand{\fpcl}{0.9}
\newcommand{\fpcnl}{0.2}
\newcommand{\plnc}{0.2}
\newcommand{\pscr}{0.4}
\newcommand{\pscnr}{0.1}
\newcommand{\psncr}{0.7}
\newcommand{\psncnr}{0.9}
\newcommand{\pjnclnrns}{0.0036}
\newcommand{\pjclrs}{0.0784}
\newcommand{\true}{\text{True}}
\newcommand{\false}{\text{False}}
$~$$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\bP(C)$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\bP(\neg C) = 1-\bP(C)$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\bP(L\mid C) = \plc$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\bP(L \mid \neg C) = \plnc$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$R$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\text{Not Radiation}^{\text{TM}}$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\neg R$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$S$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $$~$\bP(S \mid \;\; C, \;\; R) = \pscr$~$$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $$~$\bP(S \mid \;\; C, \neg R) = \pscnr$~$$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $$~$\bP(S \mid \neg C, \;\; R) = \psncr$~$$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $$~$\bP(S \mid \neg C, \neg R) = \psncnr$~$$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\bP(S \mid C, \neg R) = \pscnr$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\bP(S \mid C,R)$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\bP(C \mid L) = \fpcl$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\bP(C \mid \neg L) = \fpcnl$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\bP(L \mid C) = big$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\bP(S \mid C, R) >\bP(S \mid C, \neg R)$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$A$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$B$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$A$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\bP(A,B) > \bP(A)$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$(A,B)$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$A$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $$~$\bP(L\mid \;\; C) = \plc$~$$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $$~$\bP(L \mid \neg C) = \plnc$~$$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $$~$\bP(C \mid \;\; L) = \fpcl$~$$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $$~$\bP(C \mid \neg L) = \fpcnl$~$$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\neg C$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$L$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\neg L$~$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $$~$\frac{\bP(L,C)}{\bP(C)} = \bP(L \mid C)\ ,$~$$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $$~$\bP(\text{disease}_9 \mid \text{symptom}_2, \text{symptom}_5, \text{symptom}_{17})=0.153$~$$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $$~$\bP(\text{outcome} = \text{survival} \mid \text{disease}_9, \text{treatment} = \text{bezoar})=0.094,$~$$ |
---|
Joint probability distribution: (Motivation) coherent probabilities | $~$\bP(A,B) > \bP(A)$~$ |
---|
Kernel of group homomorphism | $~$f: G \to H$~$ |
---|
Kernel of group homomorphism | $~$g$~$ |
---|
Kernel of group homomorphism | $~$G$~$ |
---|
Kernel of group homomorphism | $~$f(g) = e_H$~$ |
---|
Kernel of group homomorphism | $~$H$~$ |
---|
Kernel of group homomorphism | $~$G \to H$~$ |
---|
Kernel of group homomorphism | $~$G$~$ |
---|
Kernel of group homomorphism | $~$f(g_1) = e_H$~$ |
---|
Kernel of group homomorphism | $~$f(g_2) = e_H$~$ |
---|
Kernel of group homomorphism | $~$e_H = f(g_1) f(g_2) = f(g_1 g_2)$~$ |
---|
Kernel of group homomorphism | $~$G$~$ |
---|
Kernel of group homomorphism | $~$f(x) = e_H$~$ |
---|
Kernel of group homomorphism | $~$e_H = f(e_G) = f(x^{-1} x) = f(x^{-1}) f(x) = f(x^{-1})$~$ |
---|
Kernel of group homomorphism | $~$f(e_G) = e_H$~$ |
---|
Kernel of ring homomorphism | $~$0$~$ |
---|
Kernel of ring homomorphism | $~$f: R \to S$~$ |
---|
Kernel of ring homomorphism | $~$R$~$ |
---|
Kernel of ring homomorphism | $~$S$~$ |
---|
Kernel of ring homomorphism | $~$f$~$ |
---|
Kernel of ring homomorphism | $~$R$~$ |
---|
Kernel of ring homomorphism | $~$f$~$ |
---|
Kernel of ring homomorphism | $~$S$~$ |
---|
Kernel of ring homomorphism | $$~$\{ r \in R \mid f(r) = 0_S \}$~$$ |
---|
Kernel of ring homomorphism | $~$0_S$~$ |
---|
Kernel of ring homomorphism | $~$S$~$ |
---|
Kernel of ring homomorphism | $~$\mathrm{id}: \mathbb{Z} \to \mathbb{Z}$~$ |
---|
Kernel of ring homomorphism | $~$n$~$ |
---|
Kernel of ring homomorphism | $~$n$~$ |
---|
Kernel of ring homomorphism | $~$\{ 0 \}$~$ |
---|
Kernel of ring homomorphism | $~$\mathbb{Z} \to \mathbb{Z}$~$ |
---|
Kernel of ring homomorphism | $~$n \mapsto n \pmod{2}$~$ |
---|
Kernel of ring homomorphism | $~$0$~$ |
---|
Kernel of ring homomorphism | $~$0$~$ |
---|
Kernel of ring homomorphism | $~$0$~$ |
---|
Kernel of ring homomorphism | $~$1$~$ |
---|
Kernel of ring homomorphism | $~$0$~$ |
---|
Kernel of ring homomorphism | $~$f(1) = 0$~$ |
---|
Kernel of ring homomorphism | $~$f(r) = f(r \times 1) = f(r) \times f(1) = f(r) \times 0 = 0$~$ |
---|
Kripke model | $~$M$~$ |
---|
Kripke model | $~$W$~$ |
---|
Kripke model | $~$R$~$ |
---|
Kripke model | $~$wRx$~$ |
---|
Kripke model | $~$w$~$ |
---|
Kripke model | $~$x$~$ |
---|
Kripke model | $~$x$~$ |
---|
Kripke model | $~$w$~$ |
---|
Kripke model | $~$V$~$ |
---|
Kripke model | $~$w \in W$~$ |
---|
Kripke model | $~$A$~$ |
---|
Kripke model | $~$M,w\models A$~$ |
---|
Kripke model | $~$A$~$ |
---|
Kripke model | $~$p$~$ |
---|
Kripke model | $~$M,w\models p$~$ |
---|
Kripke model | $~$V(w,p)=\top$~$ |
---|
Kripke model | $~$A$~$ |
---|
Kripke model | $~$A=B\to C$~$ |
---|
Kripke model | $~$M,w\models A$~$ |
---|
Kripke model | $~$M,w\not\models B$~$ |
---|
Kripke model | $~$M,w\models C$~$ |
---|
Kripke model | $~$A$~$ |
---|
Kripke model | $~$\square B$~$ |
---|
Kripke model | $~$M,w\models \square B$~$ |
---|
Kripke model | $~$M,x\models B$~$ |
---|
Kripke model | $~$x$~$ |
---|
Kripke model | $~$w$~$ |
---|
Kripke model | $~$A$~$ |
---|
Kripke model | $~$M$~$ |
---|
Kripke model | $~$M\models A$~$ |
---|
Kripke model | $~$M,w\models A$~$ |
---|
Kripke model | $~$w$~$ |
---|
Kripke model | $~$W$~$ |
---|
Kripke model | $~$A$~$ |
---|
Kripke model | $~$A$~$ |
---|
Kripke model | $~$M$~$ |
---|
Kripke model | $~$w \in W$~$ |
---|
Kripke model | $~$M,w\models A$~$ |
---|
Kripke model | $~$A$~$ |
---|
Kripke model | $~$A$~$ |
---|
Kripke model | $~$A$~$ |
---|
Kripke model | $~$M$~$ |
---|
Kripke model | $~$\square A$~$ |
---|
Kripke model | $~$M$~$ |
---|
Kripke model | $~$w \in W$~$ |
---|
Kripke model | $~$M$~$ |
---|
Kripke model | $~$x$~$ |
---|
Kripke model | $~$w$~$ |
---|
Kripke model | $~$M\models A$~$ |
---|
Kripke model | $~$M,x\models A$~$ |
---|
Kripke model | $~$x$~$ |
---|
Kripke model | $~$w$~$ |
---|
Kripke model | $~$A$~$ |
---|
Kripke model | $~$w\models \square A$~$ |
---|
Kripke model | $~$w$~$ |
---|
Kripke model | $~$M$~$ |
---|
Kripke model | $~$\square A$~$ |
---|
Kripke model | $~$M\models \square A$~$ |
---|
Kripke model | $~$\square[A\to B]\to(\square A \to \square B)$~$ |
---|
Kripke model | $~$w \in W$~$ |
---|
Kripke model | $~$w\models \square[A\to B]$~$ |
---|
Kripke model | $~$w\models \square A$~$ |
---|
Kripke model | $~$x$~$ |
---|
Kripke model | $~$w$~$ |
---|
Kripke model | $~$x\models A\to B$~$ |
---|
Kripke model | $~$x\models A$~$ |
---|
Kripke model | $~$x\models B$~$ |
---|
Kripke model | $~$x$~$ |
---|
Kripke model | $~$w\models \square B$~$ |
---|
Kripke model | $~$w\models\square[A\to B]\to(\square A \to \square B)$~$ |
---|
Kripke model | $~$M\models\square[A\to B]\to(\square A \to \square B)$~$ |
---|
Kripke model | $~$M$~$ |
---|
Lagrange theorem on subgroup size | $~$G$~$ |
---|
Lagrange theorem on subgroup size | $~$H$~$ |
---|
Lagrange theorem on subgroup size | $~$|H|$~$ |
---|
Lagrange theorem on subgroup size | $~$H$~$ |
---|
Lagrange theorem on subgroup size | $~$|G|$~$ |
---|
Lagrange theorem on subgroup size | $~$G$~$ |
---|
Lagrange theorem on subgroup size | $~$|G|$~$ |
---|
Lagrange theorem on subgroup size | $~$G$~$ |
---|
Lagrange theorem on subgroup size | $~$|H|$~$ |
---|
Lagrange theorem on subgroup size | $~$|G|/|H|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$G$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$G$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$G$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$G$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$X$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$Y$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$X$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$Y$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$|H|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$|G|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$|H|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$|G|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$C_6$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$6$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$1, 2, 3, 4, 5$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$6$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$1, 2, 3$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$6$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$4$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$5$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$1,2,3$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$6$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$1,2,3,6$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$1$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$C_6$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$6$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$2$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$3$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$|H|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$|G|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$G$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$|H|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$|H|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$G$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$|H|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$G$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$g$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$G$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$gH = \{ g h : h \in H \}$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$G$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$gH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$g$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$G$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$gH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$e$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$ge$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$gH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$ge = g$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$gH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$|H|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$gH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$|H|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$gH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$|H|$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H \to gH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$h \in H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$gh$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$gH \to H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$g^{-1}$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$gx \mapsto g^{-1} g x = x$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$x \in rH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$x \in sH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$rH = sH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$x \in rH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$x \in sH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$x = r h_1$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$x = s h_2$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$h_1, h_2 \in H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$r h_1 = s h_2$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$s^{-1} r h_1 = h_2$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$s^{-1} r = h_2 h_1^{-1}$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$s^{-1} r$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$r^{-1} s$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$\{ s h : h \in H \}$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$\{ r h : h \in H\}$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$a$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$a = rh$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$h$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$s^{-1} a = s^{-1} r h$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$s^{-1} r$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$s^{-1} r h$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$s^{-1} a$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$a \in s H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$a$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$a$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$a = sh$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$h$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$r^{-1} a = r^{-1} s h$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$r^{-1} s$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$r^{-1} s h$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$r^{-1} a$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$H$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$a \in rH$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$a$~$ |
---|
Lagrange theorem on subgroup size: Intuitive version | $~$gH$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$f\ x$~$ |
---|
Lambda calculus | $~$f$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$f(x)$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda x.f(x)$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$f(x)$~$ |
---|
Lambda calculus | $~$\lambda x.x+1$~$ |
---|
Lambda calculus | $~$\lambda x.\lambda y.x+y$~$ |
---|
Lambda calculus | $~$3$~$ |
---|
Lambda calculus | $~$4$~$ |
---|
Lambda calculus | $~$3+4=7$~$ |
---|
Lambda calculus | $~$\lambda xy.x+y$~$ |
---|
Lambda calculus | $~$\lambda xy$~$ |
---|
Lambda calculus | $~$\lambda x.\lambda y$~$ |
---|
Lambda calculus | $~$+$~$ |
---|
Lambda calculus | $~$1$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda x.x+1$~$ |
---|
Lambda calculus | $~$+$~$ |
---|
Lambda calculus | $~$1$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$v_1,v_2,\dots$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$M$~$ |
---|
Lambda calculus | $~$(\lambda x.M)$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$M$~$ |
---|
Lambda calculus | $~$N$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$(M\ N)$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$(\lambda x.M)$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$M$~$ |
---|
Lambda calculus | $~$\lambda x$~$ |
---|
Lambda calculus | $~$(\lambda x.M)$~$ |
---|
Lambda calculus | $~$M$~$ |
---|
Lambda calculus | $~$\lambda x$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$(M\ N)$~$ |
---|
Lambda calculus | $~$M$~$ |
---|
Lambda calculus | $~$N$~$ |
---|
Lambda calculus | $~$((\lambda x.(\lambda y.(x\ y))))\ x)$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$y$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$(\lambda x.(\lambda y.(x+y)))$~$ |
---|
Lambda calculus | $~$\lambda x.(\lambda y.(x+y))$~$ |
---|
Lambda calculus | $~$(\lambda x.(\lambda y.(x+y)))$~$ |
---|
Lambda calculus | $~$f\ x\ y$~$ |
---|
Lambda calculus | $~$f$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$y$~$ |
---|
Lambda calculus | $~$(f\ x)\ y$~$ |
---|
Lambda calculus | $~$f\ (x\ y)$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda x.\lambda y.x+y$~$ |
---|
Lambda calculus | $~$\lambda x.(\lambda y.(x+y))$~$ |
---|
Lambda calculus | $~$(\lambda x.\lambda y.x)+y$~$ |
---|
Lambda calculus | $~$\lambda x.(\lambda y.x)+y$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda xy.x+y$~$ |
---|
Lambda calculus | $~$\lambda x.\lambda y.x+y$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\beta$~$ |
---|
Lambda calculus | $~$\alpha$~$ |
---|
Lambda calculus | $~$\eta$~$ |
---|
Lambda calculus | $~$\beta$~$ |
---|
Lambda calculus | $~$(\lambda x.\lambda y.x+y)\ 6\ 3$~$ |
---|
Lambda calculus | $~$6$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda x$~$ |
---|
Lambda calculus | $~$6$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$(\lambda y.6+y)\ 3$~$ |
---|
Lambda calculus | $~$3$~$ |
---|
Lambda calculus | $~$y$~$ |
---|
Lambda calculus | $~$6+3=9$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$M$~$ |
---|
Lambda calculus | $~$N$~$ |
---|
Lambda calculus | $~$\beta$~$ |
---|
Lambda calculus | $~$((\lambda x.M)\ N)$~$ |
---|
Lambda calculus | $~$M[N/x]$~$ |
---|
Lambda calculus | $~$M$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$N$~$ |
---|
Lambda calculus | $~$((\lambda x.M)\ N)$~$ |
---|
Lambda calculus | $~$M[N/x]$~$ |
---|
Lambda calculus | $~$\alpha$~$ |
---|
Lambda calculus | $~$\lambda x.f\ x$~$ |
---|
Lambda calculus | $~$\lambda y.f\ y$~$ |
---|
Lambda calculus | $~$M$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda x$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$\lambda x$~$ |
---|
Lambda calculus | $~$\lambda y$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$\lambda x$~$ |
---|
Lambda calculus | $~$y$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$\lambda x.\lambda y.x$~$ |
---|
Lambda calculus | $~$y$~$ |
---|
Lambda calculus | $~$\lambda y.\lambda y.y$~$ |
---|
Lambda calculus | $~$y$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$\lambda x.\lambda y.x$~$ |
---|
Lambda calculus | $~$\eta$~$ |
---|
Lambda calculus | $~$\lambda x.f\ x$~$ |
---|
Lambda calculus | $~$f$~$ |
---|
Lambda calculus | $~$\beta$~$ |
---|
Lambda calculus | $~$(\lambda x.f\ x)\ x$~$ |
---|
Lambda calculus | $~$f\ x$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$M$~$ |
---|
Lambda calculus | $~$(\lambda x.(M\ x))$~$ |
---|
Lambda calculus | $~$M$~$ |
---|
Lambda calculus | $~$\lambda x.\lambda y.x+y$~$ |
---|
Lambda calculus | $~$\mathbb N^2\to\mathbb N$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$\lambda y.x+y$~$ |
---|
Lambda calculus | $~$6$~$ |
---|
Lambda calculus | $~$(\lambda x.\lambda y.x+y)\ 6=\lambda y.6+y$~$ |
---|
Lambda calculus | $~$\mathbb N\to(\mathbb N\to\mathbb N)$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$d=\lambda f.\lambda x.f\ (f\ x)$~$ |
---|
Lambda calculus | $~$d$~$ |
---|
Lambda calculus | $~$d\ d$~$ |
---|
Lambda calculus | $~$d$~$ |
---|
Lambda calculus | $~$(\lambda f.\lambda x.f\ (f\ x))\ (\lambda f.\lambda x.f\ (f\ x))$~$ |
---|
Lambda calculus | $~$\beta$~$ |
---|
Lambda calculus | $~$\lambda f.\lambda x.f\ (f\ (f\ (f\ x)))$~$ |
---|
Lambda calculus | $~$d\ d$~$ |
---|
Lambda calculus | $~$d$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$(\lambda x.x\ x)\ d$~$ |
---|
Lambda calculus | $~$d$~$ |
---|
Lambda calculus | $~$(\lambda x.x\ x)\ (\lambda f.\lambda x.f\ (f\ x))$~$ |
---|
Lambda calculus | $~$\beta$~$ |
---|
Lambda calculus | $~$d$~$ |
---|
Lambda calculus | $~$d$~$ |
---|
Lambda calculus | $~$x = a; f\ x$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$(\lambda x.f\ x)\ a$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\beta$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$Z$~$ |
---|
Lambda calculus | $~$0$~$ |
---|
Lambda calculus | $~$I$~$ |
---|
Lambda calculus | $~$I = (\lambda p.\lambda x.\lambda y.$~$ |
---|
Lambda calculus | $~$p$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$y)$~$ |
---|
Lambda calculus | $~$I\ True\ x\ y=x$~$ |
---|
Lambda calculus | $~$I\ False\ x\ y=y$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$F = \lambda n.I\ (Z\ n)\ 1\ (n\times(F\ (n-1)))$~$ |
---|
Lambda calculus | $~$n$~$ |
---|
Lambda calculus | $~$0$~$ |
---|
Lambda calculus | $~$1$~$ |
---|
Lambda calculus | $~$n\times F(n-1)$~$ |
---|
Lambda calculus | $~$F$~$ |
---|
Lambda calculus | $~$F$~$ |
---|
Lambda calculus | $$~$F=(\lambda x.x\ x)\ (\lambda r.\lambda n.I\ (Z\ n)\ 1\ (n\times(r\ r\ (n-1))))$~$$ |
---|
Lambda calculus | $~$g=\lambda r.\lambda n.I\ (Z\ n)\ 1\ (n\times(r\ r\ (n-1)))$~$ |
---|
Lambda calculus | $~$F=(\lambda x.x\ x)\ g=g\ g$~$ |
---|
Lambda calculus | $~$g$~$ |
---|
Lambda calculus | $~$g$~$ |
---|
Lambda calculus | $~$r$~$ |
---|
Lambda calculus | $~$g\ g=\lambda n.I\ (Z\ n)\ 1\ (n\times(g\ g\ (n-1)))$~$ |
---|
Lambda calculus | $~$g\ g=F$~$ |
---|
Lambda calculus | $~$\lambda n.I\ (Z\ n)\ 1\ (n\times(F\ (n-1)))$~$ |
---|
Lambda calculus | $~$F$~$ |
---|
Lambda calculus | $~$f=\lambda x.h\ f\ x$~$ |
---|
Lambda calculus | $~$h$~$ |
---|
Lambda calculus | $~$h=\lambda f.\lambda n.I\ (Z\ n)\ 1\ (n\times(f\ (n-1)))$~$ |
---|
Lambda calculus | $~$h$~$ |
---|
Lambda calculus | $~$f$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$h$~$ |
---|
Lambda calculus | $~$f$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$Y\ h$~$ |
---|
Lambda calculus | $~$h$~$ |
---|
Lambda calculus | $~$f=Y\ h$~$ |
---|
Lambda calculus | $~$f=\lambda x.h\ f\ x$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$h$~$ |
---|
Lambda calculus | $~$f$~$ |
---|
Lambda calculus | $~$x$~$ |
---|
Lambda calculus | $~$h=\lambda f.\lambda x.f x=\lambda f.f$~$ |
---|
Lambda calculus | $~$\beta$~$ |
---|
Lambda calculus | $~$F$~$ |
---|
Lambda calculus | $~$F(100)$~$ |
---|
Lambda calculus | $~$100\times F(99)=100\times 99\times F(98)$~$ |
---|
Lambda calculus | $~$100\times99\times\dots\times2\times1$~$ |
---|
Lambda calculus | $~$k$~$ |
---|
Lambda calculus | $~$k=1$~$ |
---|
Lambda calculus | $~$k$~$ |
---|
Lambda calculus | $~$1$~$ |
---|
Lambda calculus | $~$100$~$ |
---|
Lambda calculus | $~$f(100,1)=f(99,1*100)=f(99,100)=f(98,9900)=\dots$~$ |
---|
Lambda calculus | $~$f$~$ |
---|
Lambda calculus | $~$\lambda$~$ |
---|
Lambda calculus | $~$Y$~$ |
---|
Lambda calculus | $~$Y\ h=\lambda x.h\ (Y\ h)\ x$~$ |
---|
Lambda calculus | $~$\eta$~$ |
---|
Lambda calculus | $~$Y\ h=h\ (Y\ h)$~$ |
---|
Lambda calculus | $~$Y\ h$~$ |
---|
Lambda calculus | $~$h$~$ |
---|
Lambda calculus | $~$h$~$ |
---|
Lambda calculus | $~$Y$~$ |
---|
Lambda calculus | $~$f$~$ |
---|
Lambda calculus | $~$h\ f=f$~$ |
---|
Lambda calculus | $~$Y$~$ |
---|
Lambda calculus | $~$f$~$ |
---|
Lambda calculus | $~$k$~$ |
---|
Lambda calculus | $~$h\ f$~$ |
---|
Lambda calculus | $~$k+1$~$ |
---|
Lambda calculus | $~$h$~$ |
---|
Lambda calculus | $~$i=\lambda n.n$~$ |
---|
Lambda calculus | $~$i$~$ |
---|
Lambda calculus | $~$0$~$ |
---|
Lambda calculus | $~$h\ i$~$ |
---|
Lambda calculus | $~$1$~$ |
---|
Lambda calculus | $~$h\ (h\ i)$~$ |
---|
Lambda calculus | $~$2$~$ |
---|
Lambda calculus | $~$F$~$ |
---|
Lambda calculus | $~$h\ F=F$~$ |
---|
Lambda calculus | $~$F$~$ |
---|
Lambda calculus | $~$h$~$ |
---|
Lambda calculus | $~$h$~$ |
---|
Lambda calculus | $~$Y$~$ |
---|
Lambda calculus | $~$Y$~$ |
---|
Lambda calculus | $~$Y$~$ |
---|
Lambda calculus | $~$Y$~$ |
---|
Lambda calculus | $~$Y$~$ |
---|
Lambda calculus | $~$Y$~$ |
---|
Lambda calculus | $~$Y$~$ |
---|
Lambda calculus | $~$Y$~$ |
---|
Laplace's Rule of Succession | $~$X_1, \dots, X_n$~$ |
---|
Laplace's Rule of Succession | $~$X_i$~$ |
---|
Laplace's Rule of Succession | $~$f$~$ |
---|
Laplace's Rule of Succession | $~$f$~$ |
---|
Laplace's Rule of Succession | $~$M$~$ |
---|
Laplace's Rule of Succession | $~$N$~$ |
---|
Laplace's Rule of Succession | $~$\dfrac{M + 1}{M + N + 2}$~$ |
---|
Laplace's Rule of Succession | $~$f$~$ |
---|
Laplace's Rule of Succession | $~$f$~$ |
---|
Laplace's Rule of Succession | $~$1 - f$~$ |
---|
Laplace's Rule of Succession | $~$f$~$ |
---|
Laplace's Rule of Succession | $~$1 \cdot f^M(1 - f)^N.$~$ |
---|
Laplace's Rule of Succession | $~$\int_0^1 f^M(1 - f)^N \operatorname{d}\!f = \frac{M!N!}{(M + N + 1)!}.$~$ |
---|
Laplace's Rule of Succession | $~$f^M(1 - f)^N \frac{(M + N + 1)!}{M!N!}.$~$ |
---|
Laplace's Rule of Succession | $~$f,$~$ |
---|
Laplace's Rule of Succession | $~$\dfrac{(M+1)!N!}{(M + N + 2)!} \cdot \dfrac{(M + N + 1)!}{M!N!} = \dfrac{M + 1}{M + N + 2}.$~$ |
---|
Laplace's Rule of Succession | $~$f$~$ |
---|
Laplace's Rule of Succession | $~$M$~$ |
---|
Laplace's Rule of Succession | $~$N$~$ |
---|
Laplace's Rule of Succession | $~$\dfrac{M + \dfrac{1}{2}}{M + N + 1}$~$ |
---|
Lattice (Order Theory) | $~$L$~$ |
---|
Lattice (Order Theory) | $~$p, q \in L$~$ |
---|
Lattice (Order Theory) | $~$(p \vee q) \vee r = p \vee (q \vee r)$~$ |
---|
Lattice (Order Theory) | $~$(p \wedge q) \wedge r = p \wedge (q \wedge r)$~$ |
---|
Lattice (Order Theory) | $~$p \vee q = q \vee p$~$ |
---|
Lattice (Order Theory) | $~$p \wedge q = q \wedge p$~$ |
---|
Lattice (Order Theory) | $~$p \vee p = p$~$ |
---|
Lattice (Order Theory) | $~$p \wedge p = p$~$ |
---|
Lattice (Order Theory) | $~$p \vee (p \wedge q) = p$~$ |
---|
Lattice (Order Theory) | $~$p \wedge (p \vee q) = p$~$ |
---|
Lattice (Order Theory) | $~$L$~$ |
---|
Lattice (Order Theory) | $~$p,q,r \in L$~$ |
---|
Lattice (Order Theory) | $~$(p \vee q) \vee r = p \vee (q \vee r)$~$ |
---|
Lattice (Order Theory) | $~$(p \wedge q) \wedge r = p \wedge (q \wedge r)$~$ |
---|
Lattice (Order Theory) | $~$p \vee q = q \vee p$~$ |
---|
Lattice (Order Theory) | $~$p \wedge q = q \wedge p$~$ |
---|
Lattice (Order Theory) | $~$p \vee p = p$~$ |
---|
Lattice (Order Theory) | $~$p \wedge p = p$~$ |
---|
Lattice (Order Theory) | $~$p \vee (p \wedge q) = p$~$ |
---|
Lattice (Order Theory) | $~$p \wedge (p \vee q) = p$~$ |
---|
Lattice (Order Theory) | $~$P$~$ |
---|
Lattice (Order Theory) | $~$S \subseteq P$~$ |
---|
Lattice (Order Theory) | $~$p \in P$~$ |
---|
Lattice (Order Theory) | $~$\bigvee S$~$ |
---|
Lattice (Order Theory) | $~$(\bigvee S) \vee p$~$ |
---|
Lattice (Order Theory) | $~$\bigvee (S \cup \{p\})$~$ |
---|
Lattice (Order Theory) | $~$(\bigvee S) \vee p = \bigvee (S \cup \{p\})$~$ |
---|
Lattice (Order Theory) | $~$L$~$ |
---|
Lattice (Order Theory) | $~$p,q,r,s \in L$~$ |
---|
Lattice (Order Theory) | $~$s = p \vee (q \vee r)$~$ |
---|
Lattice (Order Theory) | $$~$p \vee (q \vee r) = (q \vee r) \vee p = (\bigvee \{q, r\}) \vee p = \bigvee (\{q, r\} \cup \{p\}) =$~$$ |
---|
Lattice (Order Theory) | $$~$\bigvee \{ q, r, p \} = \bigvee (\{p, q\} \cup \{r\}) = (\bigvee \{p, q\}) \vee r = (p \vee q) \vee r.$~$$ |
---|
Lattice (Order Theory) | $~$L$~$ |
---|
Lattice (Order Theory) | $~$p,q \in L$~$ |
---|
Lattice (Order Theory) | $~$p \vee q = \bigvee \{ p, q \} = q \vee p$~$ |
---|
Lattice (Order Theory) | $~$L$~$ |
---|
Lattice (Order Theory) | $~$p \in L$~$ |
---|
Lattice (Order Theory) | $~$p \vee p = \bigvee \{ p \} = p$~$ |
---|
Lattice (Order Theory) | $~$p \in L$~$ |
---|
Lattice (Order Theory) | $~$p \vee p = p$~$ |
---|
Lattice (Order Theory) | $~$p \in L$~$ |
---|
Lattice (Order Theory) | $~$p \wedge p = p$~$ |
---|
Lattice (Order Theory) | $~$p \wedge q$~$ |
---|
Lattice (Order Theory) | $~$\{p,q\}$~$ |
---|
Lattice (Order Theory) | $~$p \wedge q \leq p$~$ |
---|
Lattice (Order Theory) | $~$p \leq p$~$ |
---|
Lattice (Order Theory) | $~$(p \wedge q) \leq p$~$ |
---|
Lattice (Order Theory) | $~$p$~$ |
---|
Lattice (Order Theory) | $~$\{p, p \wedge q\}$~$ |
---|
Lattice (Order Theory) | $~$p \vee (p \wedge q) \leq p$~$ |
---|
Lattice (Order Theory) | $~$p \vee (p \wedge q)$~$ |
---|
Lattice (Order Theory) | $~$\{p, p \wedge q\}$~$ |
---|
Lattice (Order Theory) | $~$p \leq p \vee (p \wedge q)$~$ |
---|
Lattice (Order Theory) | $~$p = p \vee (p \wedge q)$~$ |
---|
Lattice (Order Theory) | $~$L$~$ |
---|
Lattice (Order Theory) | $~$S = \{ s_1, …, s_n \}$~$ |
---|
Lattice (Order Theory) | $~$L$~$ |
---|
Lattice (Order Theory) | $~$\bigvee S$~$ |
---|
Lattice (Order Theory) | $~$S$~$ |
---|
Lattice (Order Theory) | $~$\bigvee \{ s_1 \} = s_1 \in L$~$ |
---|
Lattice (Order Theory) | $~$\bigvee \{s_1, …, s_i \}$~$ |
---|
Lattice (Order Theory) | $~$\bigvee \{s_1, …, s_{i+1} \} = \bigvee \{s_1, …, s_i \} \vee s_{i+1}$~$ |
---|
Lattice (Order Theory) | $~$\bigvee \{s_1, …, s_i \} \vee s_{i+1}$~$ |
---|
Lattice (Order Theory) | $~$L$~$ |
---|
Lattice (Order Theory) | $~$p,q \in L$~$ |
---|
Lattice (Order Theory) | $~$p \vee q = p \Leftrightarrow q \leq p$~$ |
---|
Lattice (Order Theory) | $~$p \wedge q = p \Leftrightarrow q \geq p$~$ |
---|
Lattice (Order Theory) | $~$p \vee q = p \Leftrightarrow q \leq p$~$ |
---|
Lattice (Order Theory) | $~$p \vee q = p$~$ |
---|
Lattice (Order Theory) | $~$p$~$ |
---|
Lattice (Order Theory) | $~$p$~$ |
---|
Lattice (Order Theory) | $~$q$~$ |
---|
Lattice (Order Theory) | $~$q \leq p$~$ |
---|
Lattice (Order Theory) | $~$q \leq p$~$ |
---|
Lattice (Order Theory) | $~$p$~$ |
---|
Lattice (Order Theory) | $~$p$~$ |
---|
Lattice (Order Theory) | $~$\{p, q\}$~$ |
---|
Lattice (Order Theory) | $~$\{p, q\}$~$ |
---|
Lattice (Order Theory) | $~$p$~$ |
---|
Lattice (Order Theory) | $~$p \vee q = p$~$ |
---|
Lattice (Order Theory) | $~$\langle L, \vee, \wedge \rangle$~$ |
---|
Lattice (Order Theory) | $~$\langle L, \leq \rangle$~$ |
---|
Lattice (Order Theory) | $~$p, q \in L$~$ |
---|
Lattice (Order Theory) | $~$p \leq q$~$ |
---|
Lattice (Order Theory) | $~$p \vee q = q$~$ |
---|
Lattice: Examples | $~$\newcommand{\nsubg}{\mathcal N \mbox{-} Sub~G}$~$ |
---|
Lattice: Examples | $~$G$~$ |
---|
Lattice: Examples | $~$\nsubg$~$ |
---|
Lattice: Examples | $~$G$~$ |
---|
Lattice: Examples | $~$\langle \nsubg, \subseteq \rangle$~$ |
---|
Lattice: Examples | $~$H, K \in \nsubg$~$ |
---|
Lattice: Examples | $~$H \wedge K = H \cap K$~$ |
---|
Lattice: Examples | $~$H \vee K = HK = \{ hk \mid h \in H, k \in K \}$~$ |
---|
Lattice: Examples | $~$H,K \in \nsubg$~$ |
---|
Lattice: Examples | $~$H \wedge K = H \cap K$~$ |
---|
Lattice: Examples | $~$H \cap K$~$ |
---|
Lattice: Examples | $~$G$~$ |
---|
Lattice: Examples | $~$a,b \in H \cap K$~$ |
---|
Lattice: Examples | $~$H$~$ |
---|
Lattice: Examples | $~$a \in H$~$ |
---|
Lattice: Examples | $~$b \in H$~$ |
---|
Lattice: Examples | $~$ab \in H$~$ |
---|
Lattice: Examples | $~$ab \in K$~$ |
---|
Lattice: Examples | $~$ab \in H \cap K$~$ |
---|
Lattice: Examples | $~$H \cap K$~$ |
---|
Lattice: Examples | $~$H$~$ |
---|
Lattice: Examples | $~$K$~$ |
---|
Lattice: Examples | $~$a \in H$~$ |
---|
Lattice: Examples | $~$a \in K$~$ |
---|
Lattice: Examples | $~$a^{-1} \in H$~$ |
---|
Lattice: Examples | $~$a^{-1} \in K$~$ |
---|
Lattice: Examples | $~$a^{-1} \in H \cap K$~$ |
---|
Lattice: Examples | $~$H \cap K$~$ |
---|
Lattice: Examples | $~$H$~$ |
---|
Lattice: Examples | $~$K$~$ |
---|
Lattice: Examples | $~$G$~$ |
---|
Lattice: Examples | $~$e \in H$~$ |
---|
Lattice: Examples | $~$e \in K$~$ |
---|
Lattice: Examples | $~$e \in H \cap K$~$ |
---|
Lattice: Examples | $~$H \cap K$~$ |
---|
Lattice: Examples | $~$H \cap K$~$ |
---|
Lattice: Examples | $~$a \in G$~$ |
---|
Lattice: Examples | $~$a^{-1}(H \cap K)a = a^{-1}Ha \cap a^{-1}Ka = H \cap K$~$ |
---|
Lattice: Examples | $~$H$~$ |
---|
Lattice: Examples | $~$K$~$ |
---|
Lattice: Examples | $~$H \cap K$~$ |
---|
Lattice: Examples | $~$H, K \in \nsubg$~$ |
---|
Lattice: Examples | $~$H \vee K = HK = \{ hk \mid h \in H, k \in K \}$~$ |
---|
Lattice: Examples | $~$HK$~$ |
---|
Lattice: Examples | $~$hk, h'k' \in HK$~$ |
---|
Lattice: Examples | $~$kH = Hk$~$ |
---|
Lattice: Examples | $~$h'' \in H$~$ |
---|
Lattice: Examples | $~$kh' = h''k$~$ |
---|
Lattice: Examples | $~$hkh'k' = hh''kk' \in HK$~$ |
---|
Lattice: Examples | $~$HK$~$ |
---|
Lattice: Examples | $~$G$~$ |
---|
Lattice: Examples | $~$hk \in HK$~$ |
---|
Lattice: Examples | $~$(hk)^{-1} = k^{-1}h^{-1} \in k^{-1}H = Hk^{-1} \subseteq HK$~$ |
---|
Lattice: Examples | $~$HK$~$ |
---|
Lattice: Examples | $~$e \in H$~$ |
---|
Lattice: Examples | $~$e \in K$~$ |
---|
Lattice: Examples | $~$e = ee \in HK$~$ |
---|
Lattice: Examples | $~$HK$~$ |
---|
Lattice: Examples | $~$G$~$ |
---|
Lattice: Examples | $~$HK$~$ |
---|
Lattice: Examples | $~$G$~$ |
---|
Lattice: Examples | $~$a \in G$~$ |
---|
Lattice: Examples | $~$a^{-1}HKa = Ha^{-1}Ka = HKa^{-1}a = HK$~$ |
---|
Lattice: Examples | $~$F$~$ |
---|
Lattice: Examples | $~$G$~$ |
---|
Lattice: Examples | $~$HK$~$ |
---|
Lattice: Examples | $~$H$~$ |
---|
Lattice: Examples | $~$K$~$ |
---|
Lattice: Examples | $~$h \in H$~$ |
---|
Lattice: Examples | $~$k \in K$~$ |
---|
Lattice: Examples | $~$hk \not\in F$~$ |
---|
Lattice: Examples | $~$h \in F$~$ |
---|
Lattice: Examples | $~$k \in F$~$ |
---|
Lattice: Examples | $~$F$~$ |
---|
Lattice: Examples | $~$hk \in F$~$ |
---|
Lattice: Exercises | $~$L$~$ |
---|
Lattice: Exercises | $~$p, q, r \in L$~$ |
---|
Lattice: Exercises | $~$p \wedge (q \vee r) = (p \wedge q) \vee (p \wedge r)$~$ |
---|
Lattice: Exercises | $~$p \wedge (q \vee r) = p \neq t = (p \wedge q) \vee (p \wedge r)$~$ |
---|
Lattice: Exercises | $~$L$~$ |
---|
Lattice: Exercises | $~$J$~$ |
---|
Lattice: Exercises | $~$K$~$ |
---|
Lattice: Exercises | $~$L$~$ |
---|
Lattice: Exercises | $~$\bigwedge J \leq \bigvee K$~$ |
---|
Lattice: Exercises | $~$J$~$ |
---|
Lattice: Exercises | $~$K$~$ |
---|
Lattice: Exercises | $~$p$~$ |
---|
Lattice: Exercises | $~$p \in J$~$ |
---|
Lattice: Exercises | $~$p \in K$~$ |
---|
Lattice: Exercises | $~$\bigwedge J$~$ |
---|
Lattice: Exercises | $~$J$~$ |
---|
Lattice: Exercises | $~$\bigwedge J \leq p$~$ |
---|
Lattice: Exercises | $~$\bigvee K$~$ |
---|
Lattice: Exercises | $~$K$~$ |
---|
Lattice: Exercises | $~$p \leq \bigvee K$~$ |
---|
Lattice: Exercises | $~$\bigwedge J \leq p \leq \bigvee K$~$ |
---|
Lattice: Exercises | $~$L$~$ |
---|
Lattice: Exercises | $~$J$~$ |
---|
Lattice: Exercises | $~$K$~$ |
---|
Lattice: Exercises | $~$L$~$ |
---|
Lattice: Exercises | $~$j \in J$~$ |
---|
Lattice: Exercises | $~$k \in K$~$ |
---|
Lattice: Exercises | $~$j \leq k$~$ |
---|
Lattice: Exercises | $~$\bigvee J \leq \bigwedge K$~$ |
---|
Lattice: Exercises | $~$J$~$ |
---|
Lattice: Exercises | $~$K$~$ |
---|
Lattice: Exercises | $~$K$~$ |
---|
Lattice: Exercises | $~$J$~$ |
---|
Lattice: Exercises | $~$j \in J$~$ |
---|
Lattice: Exercises | $~$j \leq \bigwedge K$~$ |
---|
Lattice: Exercises | $~$\bigwedge K$~$ |
---|
Lattice: Exercises | $~$J$~$ |
---|
Lattice: Exercises | $~$J$~$ |
---|
Lattice: Exercises | $~$\bigvee J \leq \bigwedge K$~$ |
---|
Lattice: Exercises | $~$L$~$ |
---|
Lattice: Exercises | $~$A$~$ |
---|
Lattice: Exercises | $~$m \times n$~$ |
---|
Lattice: Exercises | $~$L$~$ |
---|
Lattice: Exercises | $$~$\bigvee_{i=1}^m \bigwedge_{j=1}^n A_{ij} \leq \bigwedge_{j=1}^n \bigvee_{i=1}^m A_{ij}$~$$ |
---|
Lattice: Exercises | $~$3 \times 3$~$ |
---|
Lattice: Exercises | $~$a,b,c,d,e,f,g,h$~$ |
---|
Lattice: Exercises | $~$i$~$ |
---|
Lattice: Exercises | $~$(a \wedge b \wedge c) \vee (d \wedge e \wedge f) \vee (g \wedge h \wedge i) \leq (a \vee d \vee g) \wedge (b \vee e \vee h) \wedge (c \vee f \vee i)$~$ |
---|
Lattice: Exercises | $$~$\left[ \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \\
\end{array} \right]$~$$ |
---|
Lattice: Exercises | $~$J = \{ a \wedge b \wedge c, d \wedge e \wedge f, g \wedge h \wedge i \}$~$ |
---|
Lattice: Exercises | $~$K = \{ a \vee d \vee g, b \vee e \vee h, c \vee f \vee i \}$~$ |
---|
Lattice: Exercises | $~$\bigvee J \leq \bigwedge K$~$ |
---|
Least common multiple | $~$a$~$ |
---|
Least common multiple | $~$b$~$ |
---|
Least common multiple | $~$\text{LCM}(a,b)$~$ |
---|
Least common multiple | $~$a$~$ |
---|
Least common multiple | $~$b$~$ |
---|
Least common multiple | $~$a=12, b=10$~$ |
---|
Least common multiple | $~$60$~$ |
---|
Least common multiple | $~$l$~$ |
---|
Least common multiple | $~$a$~$ |
---|
Least common multiple | $~$b$~$ |
---|
Least common multiple | $~$c$~$ |
---|
Least common multiple | $~$a$~$ |
---|
Least common multiple | $~$b$~$ |
---|
Least common multiple | $~$l$~$ |
---|
Least common multiple | $~$c$~$ |
---|
Least common multiple | $~$\mathbb{N}$~$ |
---|
Least common multiple | $~$a$~$ |
---|
Least common multiple | $~$b$~$ |
---|
Least common multiple | $~$ab$~$ |
---|
Least common multiple | $~$\text{LCM}(a,b)$~$ |
---|
Least common multiple | $~$ab$~$ |
---|
Least common multiple | $~$\text{GCD}(a,b)$~$ |
---|
Least common multiple | $~$a, b$~$ |
---|
Least common multiple | $$~$a\cdot b = \text{GCD}(a,b) \cdot \text{LCM}(a,b). $~$$ |
---|
Least common multiple | $~$\text{GCD}(a,b)$~$ |
---|
Least common multiple | $~$ab$~$ |
---|
Least common multiple | $~$a,b$~$ |
---|
Least common multiple | $~$12=2 \cdot 2 \cdot 3$~$ |
---|
Least common multiple | $~$10=2 \cdot 5$~$ |
---|
Least common multiple | $~$c$~$ |
---|
Least common multiple | $~$60=2 \cdot 2 \cdot 3 \cdot 5$~$ |
---|
Least common multiple | $~$c$~$ |
---|
Least common multiple | $~$p$~$ |
---|
Least common multiple | $~$a,b$~$ |
---|
Least common multiple | $~$p=2,3,5$~$ |
---|
Least common multiple | $~$c$~$ |
---|
Least common multiple | $~$p$~$ |
---|
Least common multiple | $~$a$~$ |
---|
Least common multiple | $~$b$~$ |
---|
Least common multiple | $~$p=2$~$ |
---|
Least common multiple | $~$12$~$ |
---|
Least common multiple | $~$10$~$ |
---|
Least common multiple | $~$3$~$ |
---|
Least common multiple | $~$12$~$ |
---|
Least common multiple | $~$10$~$ |
---|
Left cosets are all in bijection | $~$H$~$ |
---|
Left cosets are all in bijection | $~$G$~$ |
---|
Left cosets are all in bijection | $~$H$~$ |
---|
Left cosets are all in bijection | $~$G$~$ |
---|
Left cosets are all in bijection | $~$aH, bH$~$ |
---|
Left cosets are all in bijection | $~$f: aH \to bH$~$ |
---|
Left cosets are all in bijection | $~$x \mapsto b a^{-1} x$~$ |
---|
Left cosets are all in bijection | $~$x \in aH$~$ |
---|
Left cosets are all in bijection | $~$x = ah$~$ |
---|
Left cosets are all in bijection | $~$ba^{-1} a x = bx$~$ |
---|
Left cosets are all in bijection | $~$f(x) \in bH$~$ |
---|
Left cosets are all in bijection | $~$b a^{-1} x = b a^{-1} y$~$ |
---|
Left cosets are all in bijection | $~$a b^{-1}$~$ |
---|
Left cosets are all in bijection | $~$x = y$~$ |
---|
Left cosets are all in bijection | $~$b h \in b H $~$ |
---|
Left cosets are all in bijection | $~$x \in aH$~$ |
---|
Left cosets are all in bijection | $~$f(x) = bh$~$ |
---|
Left cosets are all in bijection | $~$x = a h$~$ |
---|
Left cosets are all in bijection | $~$f(x) = b a^{-1} a h = b h$~$ |
---|
Left cosets partition the parent group | $~$G$~$ |
---|
Left cosets partition the parent group | $~$H$~$ |
---|
Left cosets partition the parent group | $~$H$~$ |
---|
Left cosets partition the parent group | $~$G$~$ |
---|
Left cosets partition the parent group | $~$G$~$ |
---|
Left cosets partition the parent group | $~$g$~$ |
---|
Left cosets partition the parent group | $~$g \in gH$~$ |
---|
Left cosets partition the parent group | $~$g$~$ |
---|
Left cosets partition the parent group | $~$c$~$ |
---|
Left cosets partition the parent group | $~$aH$~$ |
---|
Left cosets partition the parent group | $~$bH$~$ |
---|
Left cosets partition the parent group | $~$aH = cH = bH$~$ |
---|
Left cosets partition the parent group | $~$aH$~$ |
---|
Left cosets partition the parent group | $~$bH$~$ |
---|
Left cosets partition the parent group | $~$c \in aH$~$ |
---|
Left cosets partition the parent group | $~$k \in H$~$ |
---|
Left cosets partition the parent group | $~$c = ak$~$ |
---|
Left cosets partition the parent group | $~$cH = \{ ch : h \in H \} = \{ akh : h \in H \}$~$ |
---|
Left cosets partition the parent group | $~$\{ akh : h \in H \} = \{ ar : r \in H \}$~$ |
---|
Left cosets partition the parent group | $~$akh$~$ |
---|
Left cosets partition the parent group | $~$r=kh$~$ |
---|
Left cosets partition the parent group | $~$ar$~$ |
---|
Left cosets partition the parent group | $~$r = k k^{-1} r$~$ |
---|
Left cosets partition the parent group | $~$a k k^{-1} r$~$ |
---|
Left cosets partition the parent group | $~$k^{-1} r$~$ |
---|
Left cosets partition the parent group | $~$H$~$ |
---|
Left cosets partition the parent group | $~$aH$~$ |
---|
Left cosets partition the parent group | $~$a$~$ |
---|
Left cosets partition the parent group | $~$b$~$ |
---|
Left cosets partition the parent group | $~$cH = bH$~$ |
---|
Left cosets partition the parent group | $~$G$~$ |
---|
Left cosets partition the parent group | $~$H$~$ |
---|
Left cosets partition the parent group | $~$G$~$ |
---|
Left cosets partition the parent group | $~$H$~$ |
---|
Left cosets partition the parent group | $~$G$~$ |
---|
Life in logspace | $~$\log_2$~$ |
---|
Likelihood | $~$e$~$ |
---|
Likelihood | $~$H_S$~$ |
---|
Likelihood | $~$\mathbb P(e \mid H_S) = 0.20.$~$ |
---|
Likelihood | $~$e,$~$ |
---|
Likelihood | $~$H_S$~$ |
---|
Likelihood | $~$H_M$~$ |
---|
Likelihood | $~$e.$~$ |
---|
Likelihood | $~$\mathbb P$~$ |
---|
Likelihood | $~$e$~$ |
---|
Likelihood | $~$H_i$~$ |
---|
Likelihood | $~$\mathbb P(e \mid H_i).$~$ |
---|
Likelihood | $~$\mathcal L_e(H_i)$~$ |
---|
Likelihood | $~$e$~$ |
---|
Likelihood | $~$H_S$~$ |
---|
Likelihood | $~$H_M$~$ |
---|
Likelihood | $~$e$~$ |
---|
Likelihood | $~$H_M$~$ |
---|
Likelihood | $~$H_S,$~$ |
---|
Likelihood | $~$H_{0.3}$~$ |
---|
Likelihood | $~$H_{0.9}$~$ |
---|
Likelihood | $~$H_{0.3},$~$ |
---|
Likelihood | $~$H_{0.3}$~$ |
---|
Likelihood | $~$H_{0.3}$~$ |
---|
Likelihood function | $~$e$~$ |
---|
Likelihood function | $~$\mathcal H.$~$ |
---|
Likelihood function | $~$H_i \in \mathcal H$~$ |
---|
Likelihood function | $~$e.$~$ |
---|
Likelihood function | $~$\mathcal L_{e}(H_i)$~$ |
---|
Likelihood function | $~$H_i \in \mathcal H$~$ |
---|
Likelihood function | $~$e_c$~$ |
---|
Likelihood function | $~$H_S$~$ |
---|
Likelihood function | $~$H_M$~$ |
---|
Likelihood function | $~$H_P$~$ |
---|
Likelihood function | $$~$\mathcal L_{e_c}(h) =
\begin{cases}
0.2 & \text{if $h = H_S$} \\
0.1 & \text{if $h = H_M$} \\
0.01 & \text{if $h = H_P$} \\
\end{cases}
$~$$ |
---|
Likelihood function | $~$b$~$ |
---|
Likelihood function | $~$0$~$ |
---|
Likelihood function | $~$1$~$ |
---|
Likelihood function | $~$e_{HTT}.$~$ |
---|
Likelihood function | $~$H_b$~$ |
---|
Likelihood function | $~$b$~$ |
---|
Likelihood function | $~$b \in [0, 1]$~$ |
---|
Likelihood function | $$~$\mathcal L_{e_{HTT}}(H_b) = b\cdot (1-b)\cdot (1-b).$~$$ |
---|
Likelihood function | $~$e_s$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$2^6 = 64$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$p<0.05$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$(5/6)^5 \cdot (1/6)^1 \approx 6.7\%$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$2 \times \frac{1}{6} = \frac{1}{3}.$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$2^{20} : 1$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$e$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$H$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$\neg$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$\mathbb P(X)$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$X$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$\mathbb P(X \mid Y)$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$X$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$Y$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $$~$\mathbb P(H) = \left(P(H \mid e) \cdot P(e)\right) + \left(P(H\mid \neg e) \cdot P(\neg e)\right).$~$$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$e,$~$ |
---|
Likelihood functions, p-values, and the replication crisis | $~$\neg e.$~$ |
---|
Likelihood notation | $~$e$~$ |
---|
Likelihood notation | $~$H,$~$ |
---|
Likelihood notation | $~$e$~$ |
---|
Likelihood notation | $~$H$~$ |
---|
Likelihood notation | $~$\mathcal L_e(H)$~$ |
---|
Likelihood notation | $~$\mathcal L(H \mid e).$~$ |
---|
Likelihood notation | $~$\mathcal L(H \mid e) = \mathbb P(e \mid H).$~$ |
---|
Likelihood notation | $~$\mathbb P(H \mid e)$~$ |
---|
Likelihood notation | $~$\mathbb P(e \mid H)$~$ |
---|
Likelihood notation | $~$\mathcal L_e(H) = \mathbb P(e \mid H).$~$ |
---|
Likelihood ratio | $~$e$~$ |
---|
Likelihood ratio | $~$H_i$~$ |
---|
Likelihood ratio | $~$H_j,$~$ |
---|
Likelihood ratio | $~$e$~$ |
---|
Likelihood ratio | $~$e$~$ |
---|
Likelihood ratio | $~$e$~$ |
---|
Likelihood ratio | $~$e_0$~$ |
---|
Likelihood ratio | $~$H_i$~$ |
---|
Likelihood ratio | $~$H_j,$~$ |
---|
Likelihood ratio | $~$e_0.$~$ |
---|
Likelihood ratio | $~$e$~$ |
---|
Likelihood ratio | $~$H_P$~$ |
---|
Likelihood ratio | $~$H_W$~$ |
---|
Likelihood ratio | $~$e_0$~$ |
---|
Likelihood ratio | $~$H_P$~$ |
---|
Likelihood ratio | $~$H_W$~$ |
---|
Likelihood ratio | $~$H_P$~$ |
---|
Likelihood ratio | $~$e$~$ |
---|
Likelihood ratio | $~$H_W,$~$ |
---|
Likelihood ratio | $~$(5 : 1).$~$ |
---|
List | $~$X$~$ |
---|
List | $~$[X]$~$ |
---|
List | $~$X^{\le \omega}$~$ |
---|
List | $~$\omega$~$ |
---|
List | $~$X$~$ |
---|
List | $~$X^{< \omega}$~$ |
---|
List | $~$X$~$ |
---|
List | $~$X^{\omega}$~$ |
---|
Locale | $~$A$~$ |
---|
Locale | $~$B$~$ |
---|
Locale | $~$A\le B$~$ |
---|
Locale | $~$A$~$ |
---|
Locale | $~$B$~$ |
---|
Locale | $~$\bigwedge A_i$~$ |
---|
Locale | $~$A \wedge B$~$ |
---|
Locale | $~$A$~$ |
---|
Locale | $~$B$~$ |
---|
Locale | $~$\bot$~$ |
---|
Locale | $~$\bigvee A_i$~$ |
---|
Locale | $~$A \vee B$~$ |
---|
Locale | $~$A$~$ |
---|
Locale | $~$B$~$ |
---|
Locale | $~$\top$~$ |
---|
Locale | $~$A\wedge \bigvee A_i = \bigvee (A \wedge A_i)$~$ |
---|
Locale | $~$T_0$~$ |
---|
Locale | $~$A \le B$~$ |
---|
Locale | $~$A \rightarrow B$~$ |
---|
Locale | $~$\bot$~$ |
---|
Locale | $~$\top$~$ |
---|
Locale | $~$A$~$ |
---|
Locale | $~$B$~$ |
---|
Locale | $~$f$~$ |
---|
Locale | $~$A$~$ |
---|
Locale | $~$B$~$ |
---|
Locale | $~$A$~$ |
---|
Locale | $~$B$~$ |
---|
Locale | $~$f$~$ |
---|
Locale | $~$f$~$ |
---|
Locale | $~$B$~$ |
---|
Locale | $~$A$~$ |
---|
Locale | $~$S$~$ |
---|
Locale | $~$B$~$ |
---|
Locale | $~$f(\vee S) = \vee f(S)$~$ |
---|
Locale | $~$S$~$ |
---|
Locale | $~$f(\wedge S) = \wedge f(S)$~$ |
---|
Locale | $~$f:A\rightarrow B$~$ |
---|
Log as generalized length | $$~$
\begin{align}
\log_{10}(2) &\ \approx 0.30 \\
\log_{10}(7) &\ \approx 0.85 \\
\log_{10}(22) &\ \approx 1.34 \\
\log_{10}(70) &\ \approx 1.85 \\
\log_{10}(139) &\ \approx 2.14 \\
\log_{10}(316) &\ \approx 2.50 \\
\log_{10}(123456) &\ \approx 5.09 \\
\log_{10}(654321) &\ \approx 5.82 \\
\log_{10}(123456789) &\ \approx 8.09 \\
\log_{10}(\underbrace{987654321}_\text{9 digits}) &\ \approx 8.99
\end{align}
$~$$ |
---|
Log as generalized length | $~$13$~$ |
---|
Log as generalized length | $~$\texttt{1101}$~$ |
---|
Log as generalized length | $$~$
\begin{align}
\log_2(3) = \log_2(\texttt{11}) &\ \approx 1.58 \\
\log_2(7) = \log_2(\texttt{111}) &\ \approx 2.81 \\
\log_2(13) = \log_2(\texttt{1101}) &\ \approx 3.70 \\
\log_2(22) = \log_2(\texttt{10110}) &\ \approx 4.46 \\
\log_2(70) = \log_2(\texttt{1010001}) &\ \approx 6.13 \\
\log_2(139) = \log_2(\texttt{10001011}) &\ \approx 7.12 \\
\log_2(316) = \log_2(\texttt{1100101010}) &\ \approx 8.30 \\
\log_2(1000) = \log_2(\underbrace{\texttt{1111101000}}_\text{10 digits}) &\ \approx 9.97
\end{align}
$~$$ |
---|
Log as generalized length | $~$b$~$ |
---|
Log as generalized length | $~$b$~$ |
---|
Log as generalized length | $~$b$~$ |
---|
Log as generalized length | $~$b$~$ |
---|
Log as generalized length | $~$x$~$ |
---|
Log as generalized length | $~$x$~$ |
---|
Log as generalized length | $~$b.$~$ |
---|
Log as generalized length | $~$x$~$ |
---|
Log as generalized length | $~$x.$~$ |
---|
Log as generalized length | $~$\log_{10}(316) \approx 2.5.$~$ |
---|
Log as generalized length | $~$n$~$ |
---|
Log as generalized length | $~$x$~$ |
---|
Log as generalized length | $~$x$~$ |
---|
Log as generalized length | $~$x$~$ |
---|
Log as generalized length | $~$\log_{10}(200) \approx 2.301,$~$ |
---|
Log as generalized length | $~$(\log_{10}(200) - 2)$~$ |
---|
Log as generalized length | $~$\log_{10}(500) \approx 2.7$~$ |
---|
Log as generalized length | $~$\approx$~$ |
---|
Log as generalized length | $~$\log_{10}(316) \approx 2.5$~$ |
---|
Log as generalized length | $~$\log_{10}$~$ |
---|
Log as generalized length | $~$10\cdot 8\cdot 4 = 320$~$ |
---|
Log as generalized length | $~$n$~$ |
---|
Log as generalized length | $~$\log_{10}(100)=2,$~$ |
---|
Log as generalized length | $~$\log_b(b^k)=k$~$ |
---|
Log as generalized length | $~$b$~$ |
---|
Log as generalized length | $~$k$~$ |
---|
Log as generalized length | $~$k+1$~$ |
---|
Log as generalized length | $~$b^k$~$ |
---|
Log as generalized length | $~$b$~$ |
---|
Log as generalized length | $~$k$~$ |
---|
Log as generalized length | $~$x$~$ |
---|
Log as generalized length | $~$x$~$ |
---|
Log as generalized length | $~$x$~$ |
---|
Log as generalized length | $~$\log_{10}$~$ |
---|
Log as generalized length | $~$\log_{10}(100.87249072)$~$ |
---|
Log as generalized length | $~$x$~$ |
---|
Log as generalized length | $~$x$~$ |
---|
Log as generalized length | $~$\pi$~$ |
---|
Log as generalized length | $~$\pi$~$ |
---|
Log as generalized length | $~$0 < x < 1$~$ |
---|
Log as generalized length | $~$x$~$ |
---|
Log as generalized length | $~$\frac{1}{10}$~$ |
---|
Log as generalized length | $~$-1$~$ |
---|
Log as generalized length | $$~$\underbrace{\text{2,310,426}}_\text{7 digits}$~$$ |
---|
Log as generalized length | $~$\log_{10}(\text{2,310,426})$~$ |
---|
Log as the change in the cost of communicating | $~$n$~$ |
---|
Log as the change in the cost of communicating | $~$\frac{n}{10}$~$ |
---|
Log as the change in the cost of communicating | $~$n$~$ |
---|
Log as the change in the cost of communicating | $~$\frac{n}{10}$~$ |
---|
Log as the change in the cost of communicating | $~$\log_{10}(\frac{1}{10})$~$ |
---|
Log as the change in the cost of communicating | $~$-1$~$ |
---|
Log as the change in the cost of communicating | $~$\frac{1}{10}$~$ |
---|
Log as the change in the cost of communicating | $~$-\$1.$~$ |
---|
Log as the change in the cost of communicating | $~$6 \cdot 10 \cdot 7 = 420$~$ |
---|
Log as the change in the cost of communicating | $~$6^3 < 420 < 6^4.$~$ |
---|
Log as the change in the cost of communicating | $~$\log_2(6) * 4 \approx 10.33$~$ |
---|
Log as the change in the cost of communicating | $~$\log_2(2) + 3\cdot \log_2(6) \approx 8.75$~$ |
---|
Log as the change in the cost of communicating | $~$= \log_2(6)$~$ |
---|
Log as the change in the cost of communicating | $~$= \log_{10}(6)$~$ |
---|
Log as the change in the cost of communicating | $~$\log_2(6).$~$ |
---|
Log as the change in the cost of communicating | $~$= \log_6(10)$~$ |
---|
Log as the change in the cost of communicating | $~$= \log_2(10)$~$ |
---|
Log as the change in the cost of communicating | $~$n$~$ |
---|
Log as the change in the cost of communicating | $~$\log_2(n)$~$ |
---|
Log as the change in the cost of communicating | $~$n$~$ |
---|
Log as the change in the cost of communicating | $~$\log_2(n)$~$ |
---|
Log as the change in the cost of communicating | $~$\log_b(x)$~$ |
---|
Log as the change in the cost of communicating | $~$b$~$ |
---|
Log as the change in the cost of communicating | $~$x$~$ |
---|
Log as the change in the cost of communicating | $~$\log_b(1) = 0,$~$ |
---|
Log as the change in the cost of communicating | $~$\log_b(b) = 1,$~$ |
---|
Log as the change in the cost of communicating | $~$b$~$ |
---|
Log as the change in the cost of communicating | $~$b$~$ |
---|
Log as the change in the cost of communicating | $~$\log_b\left(\frac{1}{b}\right) = -1,$~$ |
---|
Log as the change in the cost of communicating | $~$b$~$ |
---|
Log as the change in the cost of communicating | $~$b$~$ |
---|
Log as the change in the cost of communicating | $~$\log_b(x\cdot y) = \log_b(x) + \log_b(y),$~$ |
---|
Log as the change in the cost of communicating | $~$n = x \cdot y$~$ |
---|
Log as the change in the cost of communicating | $~$n$~$ |
---|
Log as the change in the cost of communicating | $~$x$~$ |
---|
Log as the change in the cost of communicating | $~$y$~$ |
---|
Log as the change in the cost of communicating | $~$x\cdot y$~$ |
---|
Log as the change in the cost of communicating | $~$x$~$ |
---|
Log as the change in the cost of communicating | $~$y$~$ |
---|
Log as the change in the cost of communicating | $~$\log_b(x^n) = n \cdot log_b(x),$~$ |
---|
Log as the change in the cost of communicating | $~$n$~$ |
---|
Log as the change in the cost of communicating | $~$x$~$ |
---|
Log as the change in the cost of communicating | $~$x^n$~$ |
---|
Log as the change in the cost of communicating | $~$x$~$ |
---|
Log as the change in the cost of communicating | $~$\log_b(x)$~$ |
---|
Log as the change in the cost of communicating | $~$b$~$ |
---|
Log as the change in the cost of communicating | $~$\log_b(x)$~$ |
---|
Log as the change in the cost of communicating | $~$x$~$ |
---|
Log as the change in the cost of communicating | $~$\mu$~$ |
---|
Log as the change in the cost of communicating | $~$1$~$ |
---|
Log as the change in the cost of communicating | $~$b$~$ |
---|
Log as the change in the cost of communicating | $~$\log_b(x)$~$ |
---|
Log as the change in the cost of communicating | $~$\mu$~$ |
---|
Log as the change in the cost of communicating | $~$x$~$ |
---|
Log base infinity | $~$\log_{\infty},$~$ |
---|
Log base infinity | $~$\infty$~$ |
---|
Log base infinity | $~$z$~$ |
---|
Log base infinity | $~$z(x) = 0$~$ |
---|
Log base infinity | $~$x \in$~$ |
---|
Log base infinity | $~$\mathbb R^+$~$ |
---|
Log base infinity | $~$\log_{\infty}$~$ |
---|
Log base infinity | $~$z$~$ |
---|
Log base infinity | $~$b$~$ |
---|
Log base infinity | $~$\log(b) = 1.$~$ |
---|
Log base infinity | $~$\log_\infty,$~$ |
---|
Log base infinity | $~$b$~$ |
---|
Log base infinity | $~$\log_\infty$~$ |
---|
Log base infinity | $~$\log_\infty(\infty)$~$ |
---|
Log base infinity | $~$\infty^0$~$ |
---|
Log base infinity | $~$=$~$ |
---|
Log base infinity | $~$\in$~$ |
---|
Logarithm | $~$b$~$ |
---|
Logarithm | $~$n,$~$ |
---|
Logarithm | $~$\log_b(n),$~$ |
---|
Logarithm | $~$b$~$ |
---|
Logarithm | $~$n$~$ |
---|
Logarithm | $~$\log_{10}(1000)=3,$~$ |
---|
Logarithm | $~$10 \cdot 10 \cdot 10 = 1000,$~$ |
---|
Logarithm | $~$\log_2(16)=4$~$ |
---|
Logarithm | $~$2 \cdot 2 \cdot 2 \cdot 2 = 16.$~$ |
---|
Logarithm | $~$\log_b(n)$~$ |
---|
Logarithm | $~$x$~$ |
---|
Logarithm | $~$b^x = n.$~$ |
---|
Logarithm | $~$\log_b(1) = 0$~$ |
---|
Logarithm | $~$\log_b(b) = 1$~$ |
---|
Logarithm | $~$\log_b(x\cdot y) = log_b(x) + \log_b(y)$~$ |
---|
Logarithm | $~$\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$~$ |
---|
Logarithm | $~$\log_b(x^n) = n\log_b(x)$~$ |
---|
Logarithm | $~$\log_b(\sqrt[n]{x}) = \frac{\log_b(x)}{n}$~$ |
---|
Logarithm | $~$\log_b(n) = \frac{\log_a(n)}{\log_a(b)}$~$ |
---|
Logarithm | $~$b$~$ |
---|
Logarithm | $~$n,$~$ |
---|
Logarithm | $~$\log_b(b^n) = n$~$ |
---|
Logarithm | $~$b^{\log_b(n)} = n.$~$ |
---|
Logarithm | $~$n$~$ |
---|
Logarithm | $~$\log(n)$~$ |
---|
Logarithm | $~$\log_{10}(1024) \approx 3$~$ |
---|
Logarithm | $~$\log_2(1024)=10$~$ |
---|
Logarithm | $~$\log_b(n)$~$ |
---|
Logarithm | $~$n$~$ |
---|
Logarithm | $~$b$~$ |
---|
Logarithm | $~$b$~$ |
---|
Logarithm | $~$n,$~$ |
---|
Logarithm | $~$\log_b(n),$~$ |
---|
Logarithm | $~$b$~$ |
---|
Logarithm | $~$n$~$ |
---|
Logarithm | $~$\log_{10}(100)=2,$~$ |
---|
Logarithm | $~$\log_{10}(316) \approx 2.5,$~$ |
---|
Logarithm | $~$316 \approx$~$ |
---|
Logarithm | $~$10 \cdot 10 \cdot \sqrt{10},$~$ |
---|
Logarithm | $~$\sqrt{10}$~$ |
---|
Logarithm | $~$\log_b(x)$~$ |
---|
Logarithm | $~$b$~$ |
---|
Logarithm | $~$x$~$ |
---|
Logarithm | $~$\log_2(100)$~$ |
---|
Logarithm | $~$6 < \log_2(100) < 7$~$ |
---|
Logarithm | $~$\log_b(n)$~$ |
---|
Logarithm | $~$x$~$ |
---|
Logarithm | $~$b^x = n,$~$ |
---|
Logarithm | $~$b$~$ |
---|
Logarithm | $~$n$~$ |
---|
Logarithm | $~$b$~$ |
---|
Logarithm | $~$x$~$ |
---|
Logarithm | $~$\mathbb R$~$ |
---|
Logarithm | $~$\mathbb C$~$ |
---|
Logarithm | $~$\log_b(1) = 0$~$ |
---|
Logarithm | $~$b$~$ |
---|
Logarithm | $~$\log_b(b) = 1$~$ |
---|
Logarithm | $~$b$~$ |
---|
Logarithm | $~$\log_b(x\cdot y) = log_b(x) + \log_b(y)$~$ |
---|
Logarithm | $~$\log_b(x^n) = n\log_b(x)$~$ |
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Logarithm | $~$\log_a(n) = \frac{\log_b(n)}{\log_b(a)}$~$ |
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Logarithm | $~$\log_{10}(100)=2,$~$ |
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Logarithm | $~$\log_b(\cdot)$~$ |
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Logarithm | $~$b^{\ \cdot}.$~$ |
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Logarithm | $~$\log_b(n) = x$~$ |
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Logarithm | $~$b^x = n,$~$ |
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Logarithm | $~$\log_b(b^x)=x$~$ |
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Logarithm | $~$b^{\log_b(n)}=n.$~$ |
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Logarithm | $~$e$~$ |
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Logarithm | $~$e$~$ |
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Logarithm | $~$x$~$ |
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Logarithm | $~$\ln(x)$~$ |
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Logarithm | $~$x$~$ |
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Logarithm | $~$\frac{x}{\ln(x)},$~$ |
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Logarithm | $~$n$~$ |
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Logarithm | $~$\log_2(n),$~$ |
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Logarithm base 1 | $~$\log_b(1)=0$~$ |
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Logarithm base 1 | $~$b$~$ |
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Logarithm base 1 | $~$\log_b(b)=1$~$ |
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Logarithm base 1 | $~$b$~$ |
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Logarithm base 1 | $~$\log_1.$~$ |
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Logarithm base 1 | $~$\log(1 \cdot 1) = \log(1) + \log(1)$~$ |
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Logarithm base 1 | $~$x > 1$~$ |
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Logarithm base 1 | $~$\infty$~$ |
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Logarithm base 1 | $~$0 < x < 1$~$ |
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Logarithm base 1 | $~$-\infty,$~$ |
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Logarithm base 1 | $~$\log_1,$~$ |
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Logarithm base 1 | $~$\log_b(1)=0$~$ |
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Logarithm base 1 | $~$b$~$ |
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Logarithm base 1 | $~$\log_b(b)=1$~$ |
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Logarithm base 1 | $~$b$~$ |
---|
Logarithm base 1 | $~$\log_1.$~$ |
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Logarithm base 1 | $~$\log(1 \cdot 1) = \log(1) + \log(1)$~$ |
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Logarithm base 1 | $~$x > 1$~$ |
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Logarithm base 1 | $~$\infty$~$ |
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Logarithm base 1 | $~$0 < x < 1$~$ |
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Logarithm base 1 | $~$-\infty,$~$ |
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Logarithm base 1 | $~$1$~$ |
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Logarithm base 1 | $~$\log_1$~$ |
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Logarithm base 1 | $~$\mathbb R^+$~$ |
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Logarithm base 1 | $~$\mathbb R \cup \{ \infty, -\infty \}.$~$ |
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Logarithm base 1 | $~$1$~$ |
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Logarithm base 1 | $~$\mathbb R$~$ |
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Logarithm base 1 | $~$x > 1$~$ |
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Logarithm base 1 | $~$\{ \infty \}$~$ |
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Logarithm base 1 | $~$0 < x < 1$~$ |
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Logarithm base 1 | $~$\{ -\infty \}$~$ |
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Logarithm base 1 | $~$=$~$ |
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Logarithm base 1 | $~$\in$~$ |
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Logarithm base 1 | $~$1^{\{\infty\}}$~$ |
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Logarithm base 1 | $~$(1, \infty)$~$ |
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Logarithm base 1 | $~$1^{\{-\infty\}}$~$ |
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Logarithm base 1 | $~$(0, 1)$~$ |
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Logarithm base 1 | $~$0 \in \log_1(1)$~$ |
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Logarithm base 1 | $~$1 \in \log_1(1)$~$ |
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Logarithm base 1 | $~$\log_1(1) + \log_1(1) \in \log_1(1 \cdot 1)$~$ |
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Logarithm base 1 | $~$7 \in log_1(1^7)$~$ |
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Logarithm base 1 | $~$15 \in 1^{\log_1(15)}$~$ |
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Logarithm tutorial overview | $~$\log_{10}(\text{2,310,426})$~$ |
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Logarithm tutorial overview | $$~$\underbrace{\text{2,310,426}}_\text{7 digits}$~$$ |
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Logarithm tutorial overview | $~$\log_{10}$~$ |
---|
Logarithm tutorial overview | $~$\log_6(52).$~$ |
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Logarithm tutorial overview | $~$f$~$ |
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Logarithm tutorial overview | $~$f(x \cdot y) = f(x) + f(y)$~$ |
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Logarithm tutorial overview | $~$x, y \in$~$ |
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Logarithm tutorial overview | $~$\mathbb R^+$~$ |
---|
Logarithm tutorial overview | $~$f$~$ |
---|
Logarithm tutorial overview | $~$\log_e,$~$ |
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Logarithm: Examples | $~$\log_{10}(100)=2.$~$ |
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Logarithm: Examples | $~$\log_2(4)=2.$~$ |
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Logarithm: Examples | $~$\log_2(3)\approx 1.58.$~$ |
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Logarithm: Exercises | $~$\log_{10}(4321)$~$ |
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Logarithmic identities | $~$b^{\log_b(n)} = \log_b(b^n) = n.$~$ |
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Logarithmic identities | $~$\log_b(1) = 0$~$ |
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Logarithmic identities | $~$\log_b(b) = 1$~$ |
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Logarithmic identities | $~$\log_b(x\cdot y) = log_b(x) + \log_b(y).$~$ |
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Logarithmic identities | $~$\log_b(x^n) = n\log_b(x).$~$ |
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Logarithmic identities | $~$x^{\log_b(y)} = y^{\log_b(x)}.$~$ |
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Logarithmic identities | $~$\log_b(n) = \frac{\log_a(n)}{\log_a(b)}$~$ |
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Logarithmic identities | $~$\log_b(n)$~$ |
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Logarithmic identities | $~$b$~$ |
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Logarithmic identities | $~$n.$~$ |
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Logarithmic identities | $~$b$~$ |
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Logarithmic identities | $~$b^{\log_b(n)} = \log_b(b^n) = n.$~$ |
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Logarithmic identities | $~$\log_b(1) = 0$~$ |
---|
Logarithmic identities | $~$\log_b(b) = 1$~$ |
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Logarithmic identities | $~$\log_b(x\cdot y) = log_b(x) + \log_b(y).$~$ |
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Logarithmic identities | $~$\log_b(x^n) = n\log_b(x).$~$ |
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Logarithmic identities | $~$x^{\log_b(y)} = y^{\log_b(x)}.$~$ |
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Logarithmic identities | $~$\log_a(n) = \frac{\log_b(n)}{\log_b(a)}$~$ |
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Logarithms invert exponentials | $~$\log_b(\cdot)$~$ |
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Logarithms invert exponentials | $~$b^{(\cdot)}.$~$ |
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Logarithms invert exponentials | $~$\log_b(n) = x$~$ |
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Logarithms invert exponentials | $~$b^x = n,$~$ |
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Logarithms invert exponentials | $~$\log_b(b^x)=x$~$ |
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Logarithms invert exponentials | $~$b^{\log_b(n)}=n.$~$ |
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Logarithms invert exponentials | $~$\log_2(2^3) = 3$~$ |
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Logarithms invert exponentials | $~$2^{\log_2(8)} = 8.$~$ |
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Logical Induction (incomplete) | $$~$\mathbb{E}_{now}(X) = \mathbb{E}_{now}(\mathbb{E}_{future}(X))$~$$ |
---|
Logical Induction (incomplete) | $~$\mathbb{E}_{now}(X)$~$ |
---|
Logical Induction (incomplete) | $~$\mathbb{E}_{future}(X)$~$ |
---|
Logical Induction (incomplete) | $~$\phi\rightarrow\psi$~$ |
---|
Logical Induction (incomplete) | $~$\mathbb{P}_{\infty}(\phi)\le\mathbb{P}_{\infty}(\psi)$~$ |
---|
Logical Induction (incomplete) | $~$\phi$~$ |
---|
Logical Induction (incomplete) | $~$\psi$~$ |
---|
Logical Induction (incomplete) | $~$\rightarrow$~$ |
---|
Logical Induction (incomplete) | $~$\phi\rightarrow\psi$~$ |
---|
Logical Induction (incomplete) | $~$\phi$~$ |
---|
Logical Induction (incomplete) | $~$\psi$~$ |
---|
Logical Induction (incomplete) | $~$\mathbb{P}_{n}(\phi)$~$ |
---|
Logical Induction (incomplete) | $~$\phi$~$ |
---|
Logical Induction (incomplete) | $~$[0,1]$~$ |
---|
Logical Induction (incomplete) | $~$\mathbb{P}_{\infty}$~$ |
---|
Logical Induction (incomplete) | $~$\phi$~$ |
---|
Logical Induction (incomplete) | $~$\mathbb{P}_{\infty}(\phi)$~$ |
---|
Logical Induction (incomplete) | $~$\mathbb{P}_{\infty}=1$~$ |
---|
Logical Induction (incomplete) | $~$\mathbb{P}_{\infty}=1$~$ |
---|
Logical Induction (incomplete) | $~$\phi$~$ |
---|
Logical Induction (incomplete) | $~$\psi$~$ |
---|
Logical Induction (incomplete) | $~$\phi\rightarrow\psi$~$ |
---|
Logical Induction (incomplete) | $~$\phi$~$ |
---|
Logical Induction (incomplete) | $~$\psi$~$ |
---|
Logical Induction (incomplete) | $~$\mathbb{P}_{\infty}(AOC)$~$ |
---|
Logical Induction (incomplete) | $~$M(A\cup B)=M(A)+M(B)$~$ |
---|
Logical Induction (incomplete) | $~$\mathbb{R}$~$ |
---|
Logical Induction (incomplete) | $~$\mathbb{P}(x)=0$~$ |
---|
Logical Induction (incomplete) | $~$\frac{1}{0.5822…}*n^{-2}*2^{-n}$~$ |
---|
Logical Induction (incomplete) | $~$2^{-n}$~$ |
---|
Logical Induction (incomplete) | $~$n^{-2}$~$ |
---|
Logical Induction (incomplete) | $~$C$~$ |
---|
Logical Induction (incomplete) | $~$\sigma$~$ |
---|
Logical Induction (incomplete) | $~$\mathbb{P}_{USM}(\sigma)<C*\mathbb{P}_{LIL}(\sigma)$~$ |
---|
Logical Induction (incomplete) | $~$C$~$ |
---|
Logical Induction (incomplete) | $~$C$~$ |
---|
Logical Induction (incomplete) | $~$\sigma$~$ |
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Logical Induction (incomplete) | $~$\mathbb{P}_{LIL}(\sigma)>C*\mathbb{P}_{USM}(\sigma)$~$ |
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Logical Induction (incomplete) | $~$\prod_{1}$~$ |
---|
Logical Induction (incomplete) | $~$C*poly(n)$~$ |
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Logical Induction (incomplete) | $~$\mathcal{O}(n^{100})$~$ |
---|
Logical Induction (incomplete) | $~$2^{2^{n}}$~$ |
---|
Logical Induction (incomplete) | $~$1.16*10^{77}$~$ |
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Logical Induction (incomplete) | $~$1.34*10^{154}$~$ |
---|
Logical Induction (incomplete) | $~$P(X |\perp)$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{S}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{V}:\mathcal{S}\rightarrow\mathbb{R}$~$ |
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Logical Inductor Notation and Definitions | $~$\alpha:\overline{\mathbb{V}}\rightarrow\mathbb{R}$~$ |
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Logical Inductor Notation and Definitions | $~$\overline{\mathbb{V}}$~$ |
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Logical Inductor Notation and Definitions | $~$\phi,\psi$~$ |
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Logical Inductor Notation and Definitions | $~$\mathcal{S}$~$ |
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Logical Inductor Notation and Definitions | $~$X$~$ |
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Logical Inductor Notation and Definitions | $~$\forall x: X(x)\rightarrow x>3$~$ |
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Logical Inductor Notation and Definitions | $~$\mathcal{U}$~$ |
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Logical Inductor Notation and Definitions | $~$\overline{\phi}$~$ |
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Logical Inductor Notation and Definitions | $~$\overline{X}$~$ |
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Logical Inductor Notation and Definitions | $~$\overline{D}$~$ |
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Logical Inductor Notation and Definitions | $~$D_{n}$~$ |
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Logical Inductor Notation and Definitions | $~$D_{\infty}$~$ |
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Logical Inductor Notation and Definitions | $~$\mathbb{V}$~$ |
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Logical Inductor Notation and Definitions | $~$\mathcal{S}\rightarrow[0,1]$~$ |
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Logical Inductor Notation and Definitions | $~$\mathbb{P}$~$ |
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Logical Inductor Notation and Definitions | $~$\mathbb{P}(\phi)$~$ |
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Logical Inductor Notation and Definitions | $~$\phi$~$ |
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Logical Inductor Notation and Definitions | $~$\overline{\mathbb{P}}$~$ |
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Logical Inductor Notation and Definitions | $~$\mathbb{P}_{n}$~$ |
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Logical Inductor Notation and Definitions | $~$\mathbb{P}_{n}(\phi)$~$ |
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Logical Inductor Notation and Definitions | $~$\phi$~$ |
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Logical Inductor Notation and Definitions | $~$n$~$ |
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Logical Inductor Notation and Definitions | $~$\mathbb{P}_{\infty}$~$ |
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Logical Inductor Notation and Definitions | $~$\mathbb{W}$~$ |
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Logical Inductor Notation and Definitions | $~$\mathbb{W}(\phi)$~$ |
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Logical Inductor Notation and Definitions | $~$\phi$~$ |
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Logical Inductor Notation and Definitions | $~$\mathbb{W}$~$ |
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Logical Inductor Notation and Definitions | $~$\mathcal{PC}(\Gamma)$~$ |
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Logical Inductor Notation and Definitions | $~$\Gamma$~$ |
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Logical Inductor Notation and Definitions | $~$\Gamma$~$ |
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Logical Inductor Notation and Definitions | $~$\Gamma$~$ |
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Logical Inductor Notation and Definitions | $~$\mathbb{W}(X)$~$ |
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Logical Inductor Notation and Definitions | $~$X$~$ |
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Logical Inductor Notation and Definitions | $~$\mathbb(W)$~$ |
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Logical Inductor Notation and Definitions | $~$X$~$ |
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Logical Inductor Notation and Definitions | $~$X=\frac{1}{\sqrt{2}}+3\epsilon$~$ |
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Logical Inductor Notation and Definitions | $~$\overline{\mathbb{V}}\rightarrow\mathbb{R}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{\mathbb{V}}$~$ |
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Logical Inductor Notation and Definitions | $~$\mathcal{F}$~$ |
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Logical Inductor Notation and Definitions | $~$\phi^{*n}$~$ |
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Logical Inductor Notation and Definitions | $~$\mathbb{P}_{n}(\phi)$~$ |
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Logical Inductor Notation and Definitions | $~$\phi$~$ |
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Logical Inductor Notation and Definitions | $~$\overline{\mathbb{P}}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\xi$~$ |
---|
Logical Inductor Notation and Definitions | $~$\alpha^{\dagger}$~$ |
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Logical Inductor Notation and Definitions | $~$\beta^{\dagger}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\gamma^{\dagger}$~$ |
---|
Logical Inductor Notation and Definitions | $~$w^{\dagger}$~$ |
---|
Logical Inductor Notation and Definitions | $~$max(-,-)$~$ |
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Logical Inductor Notation and Definitions | $~$\frac{1}{max(1,-)}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{EF}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{\mathbb{P}}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{\alpha^{\dagger}}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{\beta^{\dagger}}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{\gamma^{\dagger}}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{w^{\dagger}}$~$ |
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Logical Inductor Notation and Definitions | $~$\mathcal{EF}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\alpha^{\dagger}_{n}$~$ |
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Logical Inductor Notation and Definitions | $~$\dagger$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{EF}$~$ |
---|
Logical Inductor Notation and Definitions | $~$*n$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{F}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{F}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{F}$~$ |
---|
Logical Inductor Notation and Definitions | $~$A^{\dagger}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{EF}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{F}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\xi$~$ |
---|
Logical Inductor Notation and Definitions | $~$\phi$~$ |
---|
Logical Inductor Notation and Definitions | $~$\dagger$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{A^{\dagger}}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{EF}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{EF}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{EF}$~$ |
---|
Logical Inductor Notation and Definitions | $~$B^{\dagger}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{EF}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{B^{\dagger}}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{EF}$~$ |
---|
Logical Inductor Notation and Definitions | $~$A$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{R}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{\mathbb{P}}$~$ |
---|
Logical Inductor Notation and Definitions | $~$B$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{R}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{A}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{R}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{B}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{R}$~$ |
---|
Logical Inductor Notation and Definitions | $~$T$~$ |
---|
Logical Inductor Notation and Definitions | $~$\xi$~$ |
---|
Logical Inductor Notation and Definitions | $~$\phi$~$ |
---|
Logical Inductor Notation and Definitions | $~$c:=-\sum\limits_{i}\xi_{i}\phi_{i}^{*n}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{EF}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{EF}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{T}$~$ |
---|
Logical Inductor Notation and Definitions | $~$T_{n}$~$ |
---|
Logical Inductor Notation and Definitions | $~$poly(n)$~$ |
---|
Logical Inductor Notation and Definitions | $~$(\overline{T}^{k})_{k}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{T}^{k}$~$ |
---|
Logical Inductor Notation and Definitions | $~$poly(k)$~$ |
---|
Logical Inductor Notation and Definitions | $~$poly(n)$~$ |
---|
Logical Inductor Notation and Definitions | $~$k\le n$~$ |
---|
Logical Inductor Notation and Definitions | $~$T_{n}^{k}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{BCS}(\overline{\mathbb{P}})$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{BLCS}(\overline{\mathbb{P}})$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{R}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{\mathbb{P}}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{R}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{P}(A)$~$ |
---|
Logical Inductor Notation and Definitions | $~$A$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{P}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\phi$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{P}(\phi)$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{P}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{W}$~$ |
---|
Logical Inductor Notation and Definitions | $~$A^{*n}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{R}$~$ |
---|
Logical Inductor Notation and Definitions | $~$A$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{P}_{n}$~$ |
---|
Logical Inductor Notation and Definitions | $~$T[\phi]$~$ |
---|
Logical Inductor Notation and Definitions | $~$\phi$~$ |
---|
Logical Inductor Notation and Definitions | $~$\phi$~$ |
---|
Logical Inductor Notation and Definitions | $~$T[\phi](\overline{\mathbb{P}})$~$ |
---|
Logical Inductor Notation and Definitions | $~$T[1]$~$ |
---|
Logical Inductor Notation and Definitions | $~$||T(\overline{\mathbb{P}})||_{mg}$~$ |
---|
Logical Inductor Notation and Definitions | $~$||\overline{T}(\overline{\mathbb{P}})||_{mg}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{E}_{n}^{\mathbb{P}}(X)$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{P}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\sum\limits_{i=0}^{n-1}\frac{1}{n}\mathbb{P}(X>\frac{i}{n})$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{E}_{n}(X)$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{E}_{n}^{\mathbb{P}_{n}}(X)$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{E}_{\infty}(X)$~$ |
---|
Logical Inductor Notation and Definitions | $~$Ex_{n}(B)$~$ |
---|
Logical Inductor Notation and Definitions | $~$B$~$ |
---|
Logical Inductor Notation and Definitions | $~$Ind_{\delta}(x>y)$~$ |
---|
Logical Inductor Notation and Definitions | $~$x\le y$~$ |
---|
Logical Inductor Notation and Definitions | $~$x>y+\delta$~$ |
---|
Logical Inductor Notation and Definitions | $~$Val_{\Gamma}(A)$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{R}$~$ |
---|
Logical Inductor Notation and Definitions | $~$A$~$ |
---|
Logical Inductor Notation and Definitions | $~$\Gamma$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{W}\in\mathcal{PC}(\Gamma)$~$ |
---|
Logical Inductor Notation and Definitions | $~$A$~$ |
---|
Logical Inductor Notation and Definitions | $~$Thm_{\Gamma}(\phi)$~$ |
---|
Logical Inductor Notation and Definitions | $~$\phi$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathbb{1}(\phi)$~$ |
---|
Logical Inductor Notation and Definitions | $~$\phi$~$ |
---|
Logical Inductor Notation and Definitions | $~$\phi$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{\alpha}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{\mathbb{P}}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\mathcal{EF}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\xi_{n}(\overline{\mathbb{P}})=\alpha_{n}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{\mathbb{P}}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{\mathbb{P}}$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{w}$~$ |
---|
Logical Inductor Notation and Definitions | $~$[0,1]$~$ |
---|
Logical Inductor Notation and Definitions | $~$\sum(w_{n})=\infty$~$ |
---|
Logical Inductor Notation and Definitions | $~$\overline{w}$~$ |
---|
Logical Inductor Notation and Definitions | $~$f$~$ |
---|
Logical Inductor Notation and Definitions | $~$f(n)$~$ |
---|
Logical Inductor Notation and Definitions | $~$poly(f(n))$~$ |
---|
Logical Inductor Notation and Definitions | $~$\sum\limits_{n}^{f(n)}w_{n}<b$~$ |
---|
Logical Inductor Notation and Definitions | $~$2^{n}$~$ |
---|
Logical system | $~$\Sigma^* = \{\neg,\wedge,\vee,=,+,\cdot ,0,a_1,a_2,a_3,…\}^*$~$ |
---|
Logical system | $~$n+1$~$ |
---|
Logical system | $~$n$~$ |
---|
Logical system | $~$A\rightarrow B$~$ |
---|
Logical system | $~$A$~$ |
---|
Logical system | $~$B$~$ |
---|
Logical system | $~$S$~$ |
---|
Logical system | $~$S$~$ |
---|
Logistic function | $~$(0, 1)$~$ |
---|
Logistic function | $~$\displaystyle f(x) = \frac{1}{1 + e^{-x}}$~$ |
---|
Logistic function | $~$(0, 1)$~$ |
---|
Logistic function | $~$\displaystyle f(x) = \frac{1}{1 + e^{-x}}$~$ |
---|
Logistic function | $~$\displaystyle f(x) = \frac{M}{1 + \alpha^{c(x_0 - x)}}$~$ |
---|
Logistic function | $~$M$~$ |
---|
Logistic function | $~$(0, M)$~$ |
---|
Logistic function | $~$M = 1$~$ |
---|
Logistic function | $~$x_0$~$ |
---|
Logistic function | $~$x$~$ |
---|
Logistic function | $~$\alpha$~$ |
---|
Logistic function | $~$c$~$ |
---|
Logistic function | $~$\alpha = 2$~$ |
---|
Logistic function | $~$\alpha = 10$~$ |
---|
Logistic function | $~$c = 1/400$~$ |
---|
Low impact | $~$V_i$~$ |
---|
Low impact | $~$v_i.$~$ |
---|
Low impact | $~$v_i$~$ |
---|
Low impact | $~$v_i^*$~$ |
---|
Low impact | $~$V_i$~$ |
---|
Low impact | $~$\mathcal M$~$ |
---|
Low impact | $~$M \in \mathcal M,$~$ |
---|
Low impact | $~$o_M$~$ |
---|
Low impact | $~$o_M^'$~$ |
---|
Low impact | $~$%% o_M - o_M^' %%$~$ |
---|
Low impact | $~$\pi_0$~$ |
---|
Low impact | $~$(o|\pi_0)$~$ |
---|
Low impact | $~$\pi_k$~$ |
---|
Low impact | $~$\mathbb E[%% (o | \pi_0) - (o | \pi_k) %%].$~$ |
---|
Low impact | $~$\frac{X}{Y}$~$ |
---|
Low impact | $~$Y$~$ |
---|
Löb's theorem | $~$PA\vdash Prv_{PA}(A)\implies A$~$ |
---|
Löb's theorem | $~$PA\vdash A$~$ |
---|
Löb's theorem | $~$Prv$~$ |
---|
Löb's theorem | $~$PA$~$ |
---|
Löb's theorem | $~$Prv(T)$~$ |
---|
Löb's theorem | $~$PA$~$ |
---|
Löb's theorem | $~$T$~$ |
---|
Löb's theorem | $~$PA$~$ |
---|
Löb's theorem | $~$Prv(S)\implies S$~$ |
---|
Löb's theorem | $~$S$~$ |
---|
Löb's theorem | $~$PA$~$ |
---|
Löb's theorem | $~$PA\vdash Prv(S)\implies S$~$ |
---|
Löb's theorem | $~$PA\vdash S$~$ |
---|
Löb's theorem | $~$PA$~$ |
---|
Löb's theorem | $~$PA\vdash Prv(S)\implies S$~$ |
---|
Löb's theorem | $~$S$~$ |
---|
Löb's theorem | $~$PA$~$ |
---|
Löb's theorem | $~$PA$~$ |
---|
Löb's theorem | $~$PA\nvdash Prv(0= 1)\implies 0= 1$~$ |
---|
Löb's theorem | $~$PA\nvdash \neg Prv(0= 1)$~$ |
---|
Löb's theorem and computer programs | $~$n$~$ |
---|
Löb's theorem and computer programs | $~$X$~$ |
---|
Löb's theorem and computer programs | $~$L(X)$~$ |
---|
Löb's theorem and computer programs | $~$X$~$ |
---|
Löb's theorem and computer programs | $~$X$~$ |
---|
Löb's theorem and computer programs | $~$X$~$ |
---|
Löb's theorem and computer programs | $~$L(X)$~$ |
---|
Löb's theorem and computer programs | $~$PA$~$ |
---|
Löb's theorem and computer programs | $~$X$~$ |
---|
Löb's theorem and computer programs | $~$PA\vdash X$~$ |
---|
Löb's theorem and computer programs | $~$PA\vdash L(X)$~$ |
---|
Löb's theorem and computer programs | $~$L(X)$~$ |
---|
Löb's theorem and computer programs | $~$PA\vdash \neg X$~$ |
---|
Löb's theorem and computer programs | $~$PA\vdash L(X)$~$ |
---|
Löb's theorem and computer programs | $~$PA\vdash$~$ |
---|
Löb's theorem and computer programs | $~$PA$~$ |
---|
Löb's theorem and computer programs | $~$PA$~$ |
---|
Löb's theorem and computer programs | $~$X$~$ |
---|
Löb's theorem and computer programs | $~$PA$~$ |
---|
Löb's theorem and computer programs | $~$PA\vdash L(X)$~$ |
---|
Löb's theorem and computer programs | $~$PA\vdash \neg X$~$ |
---|
Löb's theorem and computer programs | $~$X$~$ |
---|
Löb's theorem and computer programs | $~$PA\vdash X$~$ |
---|
Löb's theorem and computer programs | $~$PA$~$ |
---|
Löb's theorem and computer programs | $~$X$~$ |
---|
Löb's theorem and computer programs | $~$PA$~$ |
---|
Löb's theorem and computer programs | $~$X$~$ |
---|
Löb's theorem and computer programs | $~$PA\not\vdash X$~$ |
---|
Löb's theorem and computer programs | $~$PA\not\vdash \square_{PA} X \rightarrow X$~$ |
---|
Löb's theorem and computer programs | $~$PA$~$ |
---|
Löbstacle | $~$D1$~$ |
---|
Löbstacle | $~$D1$~$ |
---|
Löbstacle | $~$D2$~$ |
---|
Löbstacle | $~$D1$~$ |
---|
Löbstacle | $~$D2$~$ |
---|
Löbstacle | $~$D2$~$ |
---|
Löbstacle | $~$D1$~$ |
---|
Löbstacle | $~$D1$~$ |
---|
Löbstacle | $~$D2$~$ |
---|
Löbstacle | $~$D1$~$ |
---|
Löbstacle | $~$D2$~$ |
---|
Löbstacle | $~$D2$~$ |
---|
Löbstacle | $~$D1$~$ |
---|
Löbstacle | $~$D1$~$ |
---|
Löbstacle | $~$D1$~$ |
---|
Löbstacle | $~$\square_{D1}A\rightarrow \square_{D2}A$~$ |
---|
Löbstacle | $~$D1$~$ |
---|
Löbstacle | $~$D2$~$ |
---|
Löbstacle | $~$D1$~$ |
---|
Löbstacle | $~$D1$~$ |
---|
Mapsto notation | $~$\mapsto$~$ |
---|
Mapsto notation | $~$\mapsto$~$ |
---|
Mapsto notation | $~$f(x) = x^2$~$ |
---|
Mapsto notation | $$~$f : \mathbb{R} \to \mathbb{R}$~$$ |
---|
Mapsto notation | $$~$x \mapsto x^2.$~$$ |
---|
Mapsto notation | $$~$f : \mathbb{R} \ni x \mapsto x^2 \in \mathbb{R}.$~$$ |
---|
Math 2 example statements | $~$ax^2 + bx + c$~$ |
---|
Math 2 example statements | $~$\displaystyle \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$~$ |
---|
Math 2 example statements | $~$b^2 - 4ac$~$ |
---|
Math 2 example statements | $~$i$~$ |
---|
Math 2 example statements | $~$x^2 + 1 = 0$~$ |
---|
Math 2 example statements | $~$\begin{matrix}ax + by = c \\ dx + ey = f\end{matrix}$~$ |
---|
Math 2 example statements | $~$x$~$ |
---|
Math 2 example statements | $~$y$~$ |
---|
Math 2 example statements | $~$x$~$ |
---|
Math 2 example statements | $~$\displaystyle \frac{bf - ce}{bd - ae}$~$ |
---|
Math 2 example statements | $~$\frac{d}{dx} x^n = nx^{n-1}$~$ |
---|
Math 2 example statements | $~$m^n = n^m$~$ |
---|
Math 2 example statements | $~$m < n$~$ |
---|
Math 2 example statements | $~$m = (1 + \frac 1x)^x$~$ |
---|
Math 2 example statements | $~$n = (1 + \frac 1x)^{x+1}$~$ |
---|
Math 2 example statements | $~$x$~$ |
---|
Math 3 example statements | $~$G$~$ |
---|
Math 3 example statements | $~$g$~$ |
---|
Math 3 example statements | $~$hgh^{-1}$~$ |
---|
Math 3 example statements | $~$h \in G$~$ |
---|
Math 3 example statements | $~$f: V \to W$~$ |
---|
Math 3 example statements | $~$f$~$ |
---|
Math 3 example statements | $~$f$~$ |
---|
Math 3 example statements | $~$V$~$ |
---|
Math 3 example statements | $~$X$~$ |
---|
Math 3 example statements | $~${F_1, F_2, F_3, \ldots}$~$ |
---|
Math 3 example statements | $~$X$~$ |
---|
Math 3 example statements | $~$\bigcap_{n=1}^\infty F_n$~$ |
---|
Math 3 example statements | $~$X$~$ |
---|
Math 3 example statements | $~$\zeta(s) = \sum_{n=1}^\infty \frac{1}{s^n}$~$ |
---|
Math 3 example statements | $~$s$~$ |
---|
Math 3 example statements | $~$\frac12$~$ |
---|
Math 3 example statements | $~$\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}$~$ |
---|
Math 3 example statements | $~$f: \mathbb{R}^m \to \mathbb{R}^n$~$ |
---|
Math 3 example statements | $~$\left[ \begin{matrix} \pd{y_1}{x_1} & \pd{y_1}{x_2} & \cdots & \pd{y_1}{x_m} \\ \pd{y_2}{x_1} & \pd{y_2}{x_2} & \cdots & \pd{y_2}{x_m} \\ \vdots & \vdots & \ddots & \vdots \\ \pd{y_n}{x_1} & \pd{y_n}{x_2} & \cdots & \pd{y_n}{x_m} \end{matrix} \right]$~$ |
---|
Math 3 example statements | $~$x = (x_1, x_2, \ldots, x_m)$~$ |
---|
Math 3 example statements | $~$y = f(x) = (y_1, y_2, \ldots, y_n)$~$ |
---|
Math 3 example statements | $~$\displaystyle \frac{d\mathbf{y}}{d\mathbf{x}}$~$ |
---|
Math 3 example statements | $~$\displaystyle \frac{d(y_1, y_2, \ldots, y_n)}{d(x_1, x_2, \ldots, x_m)}$~$ |
---|
Math playpen | $$~$\mathcal{P}e^{i \oint_C A_\mu dx^\mu} \to g(x) \mathcal{P}e^{i \oint_C A_\mu dx^\mu} g^{-1}(x)\,$~$$ |
---|
Math style guidelines | $~$\{f(x) \mid x \in S\}$~$ |
---|
Math style guidelines | $~$0$~$ |
---|
Math style guidelines | $~$\{0,1,2,\dots\}$~$ |
---|
Math style guidelines | $~$\mathbb N$~$ |
---|
Math style guidelines | $~$\{1,2,\dots\}$~$ |
---|
Math style guidelines | $~$\mathbb N^+$~$ |
---|
Math style guidelines | $~$\mathbb N^+$~$ |
---|
Mathematical induction | $~$P(n)$~$ |
---|
Mathematical induction | $~$n$~$ |
---|
Mathematical induction | $~$P(n)$~$ |
---|
Mathematical induction | $~$n$~$ |
---|
Mathematical induction | $~$P(m)$~$ |
---|
Mathematical induction | $~$k \geq m$~$ |
---|
Mathematical induction | $~$P(k)$~$ |
---|
Mathematical induction | $~$P(k+1)$~$ |
---|
Mathematical induction | $~$P(m)$~$ |
---|
Mathematical induction | $~$P(m+1)$~$ |
---|
Mathematical induction | $~$P(m+1)$~$ |
---|
Mathematical induction | $~$P(m+2)$~$ |
---|
Mathematical induction | $$~$ 1 + 2 + \cdots + n = \frac{n(n+1)}{2}$~$$ |
---|
Mathematical induction | $~$n \ge 1$~$ |
---|
Mathematical induction | $~$n=1$~$ |
---|
Mathematical induction | $$~$ 1 = \frac{1(1+1)}{2} = \frac{2}{2} = 1.$~$$ |
---|
Mathematical induction | $~$k$~$ |
---|
Mathematical induction | $~$k\ge1$~$ |
---|
Mathematical induction | $$~$1 + 2 + \cdots + k = \frac{k(k+1)}{2}$~$$ |
---|
Mathematical induction | $$~$ 1 + 2 + \cdots + k + (k+1) = \frac{(k+1)([k+1]+1)}{2}.$~$$ |
---|
Mathematical induction | $~$k+1$~$ |
---|
Mathematical induction | $$~$1+2+\cdots + k + (k+1) = \frac{k(k+1)}{2} + k + 1.$~$$ |
---|
Mathematical induction | $$~$\frac{k(k+1)}{2} + \frac{2(k+1)}{2} = \frac{(k+2)(k+1)}{2} = \frac{(k+1)([k+1]+1)}{2}.$~$$ |
---|
Mathematical induction | $$~$ 1 + 2 + \cdots + k + (k+1) = \frac{(k+1)([k+1]+1)}{2}$~$$ |
---|
Mathematical induction | $~$P(n)$~$ |
---|
Mathematical induction | $~$n \ge 1$~$ |
---|
Mathematical induction | $~$n$~$ |
---|
Mathematical induction | $~$n(n+1)/2$~$ |
---|
Mathematical induction | $~$P(1)$~$ |
---|
Mathematical induction | $~$k \ge 1$~$ |
---|
Mathematical induction | $~$P(k)$~$ |
---|
Mathematical induction | $~$P(k+1)$~$ |
---|
Mathematical induction | $~$P(1)$~$ |
---|
Mathematical induction | $~$P(2)$~$ |
---|
Mathematical induction | $~$P(2)$~$ |
---|
Mathematical induction | $~$P(3)$~$ |
---|
Mathematical induction | $~$P(n)$~$ |
---|
Mathematical induction | $~$n \ge 1$~$ |
---|
Mathematical induction | $~$P(n)$~$ |
---|
Mathematical induction | $~$P$~$ |
---|
Mathematical induction | $~$n \ge m$~$ |
---|
Mathematical induction | $~$P(m)$~$ |
---|
Mathematical induction | $~$k \ge m$~$ |
---|
Mathematical induction | $~$P(k+1)$~$ |
---|
Mathematical induction | $~$P(k)$~$ |
---|
Mathematical induction | $~$P(n)$~$ |
---|
Mathematical induction | $~$P$~$ |
---|
Mathematical induction | $~$n \ge m$~$ |
---|
Mathematical induction | $~$P(m)$~$ |
---|
Mathematical induction | $~$k \ge m$~$ |
---|
Mathematical induction | $~$P(k)$~$ |
---|
Mathematical induction | $~$P(\ell)$~$ |
---|
Mathematical induction | $~$m \le \ell < k$~$ |
---|
Mathematical induction | $~$P(x)$~$ |
---|
Mathematical induction | $~$x\leq 1$~$ |
---|
Mathematical induction | $~$P(0)$~$ |
---|
Mathematical induction | $~$x\ge 0$~$ |
---|
Mathematical induction | $~$P(x)$~$ |
---|
Mathematical induction | $~$P(y)$~$ |
---|
Mathematical induction | $~$0 \le y < x$~$ |
---|
Mathematical induction | $~$P(2)$~$ |
---|
Meta-rules for (narrow) value learning are still unsolved | $~$o$~$ |
---|
Meta-rules for (narrow) value learning are still unsolved | $~$U(o)$~$ |
---|
Meta-rules for (narrow) value learning are still unsolved | $~$U_1(o)$~$ |
---|
Meta-rules for (narrow) value learning are still unsolved | $~$s$~$ |
---|
Meta-rules for (narrow) value learning are still unsolved | $~$U_2(o)$~$ |
---|
Meta-rules for (narrow) value learning are still unsolved | $~$s$~$ |
---|
Meta-rules for (narrow) value learning are still unsolved | $~$s$~$ |
---|
Meta-rules for (narrow) value learning are still unsolved | $~$U_1$~$ |
---|
Meta-rules for (narrow) value learning are still unsolved | $~$U_2,$~$ |
---|
Meta-rules for (narrow) value learning are still unsolved | $~$R$~$ |
---|
Meta-rules for (narrow) value learning are still unsolved | $~$O$~$ |
---|
Meta-rules for (narrow) value learning are still unsolved | $~$O$~$ |
---|
Methodology of unbounded analysis | $~$\gamma$~$ |
---|
Methodology of unbounded analysis | $~$\gamma$~$ |
---|
Methodology of unbounded analysis | $~$\gamma$~$ |
---|
Methodology of unbounded analysis | $~$\gamma$~$ |
---|
Methodology of unbounded analysis | $~$\gamma$~$ |
---|
Methodology of unbounded analysis | $~$\gamma$~$ |
---|
Methodology of unbounded analysis | $~$\gamma$~$ |
---|
Metric | $~$S$~$ |
---|
Metric | $~$S$~$ |
---|
Metric | $~$d : S \times S \to \mathbb R_{\ge 0}$~$ |
---|
Metric | $~$a,b,c \in S$~$ |
---|
Metric | $~$d(a,b) = 0 \Leftrightarrow a = b$~$ |
---|
Metric | $~$d(a,b) = d(b,a)$~$ |
---|
Metric | $~$d(a,b) + d(b,c) \geq d(a,c)$~$ |
---|
Metric | $~$S$~$ |
---|
Metric | $~$d$~$ |
---|
Metric | $~$d$~$ |
---|
Metric | $~$S$~$ |
---|
Metric | $$~$d: S \times S \to [0, \infty)$~$$ |
---|
Metric | $~$d$~$ |
---|
Metric | $~$S$~$ |
---|
Metric | $~$d$~$ |
---|
Metric | $~$S$~$ |
---|
Metric | $~$a$~$ |
---|
Metric | $~$b$~$ |
---|
Metric | $~$S$~$ |
---|
Metric | $~$S$~$ |
---|
Metric | $~$d$~$ |
---|
Metric | $~$a$~$ |
---|
Metric | $~$b$~$ |
---|
Metric | $~$c$~$ |
---|
Metric | $~$S$~$ |
---|
Metric | $~$d(a, b) = 0 \iff a = b$~$ |
---|
Metric | $~$d(a, b) = d(b, a)$~$ |
---|
Metric | $~$d(a, b) + d(b, c) \geq d(a, c)$~$ |
---|
Metric | $~$a$~$ |
---|
Metric | $~$b$~$ |
---|
Metric | $~$b$~$ |
---|
Metric | $~$a$~$ |
---|
Metric | $~$a$~$ |
---|
Metric | $~$c$~$ |
---|
Metric | $~$a$~$ |
---|
Metric | $~$b$~$ |
---|
Metric | $~$b$~$ |
---|
Metric | $~$c$~$ |
---|
Metric | $~$a$~$ |
---|
Metric | $~$b$~$ |
---|
Metric | $~$c$~$ |
---|
Metric | $~$d$~$ |
---|
Metric | $~$e$~$ |
---|
Metric | $~$S$~$ |
---|
Metric | $~$a$~$ |
---|
Metric | $~$b$~$ |
---|
Metric | $~$S$~$ |
---|
Metric | $~$d(a, b)$~$ |
---|
Metric | $~$e(a, b)$~$ |
---|
Metric | $~$a$~$ |
---|
Metric | $~$b$~$ |
---|
Metric | $~$e$~$ |
---|
Metric | $~$d(a, b) = \sqrt{(a_1-b_1)^2 + (a_2-b_2)^2}$~$ |
---|
Metric | $~$n$~$ |
---|
Metric | $~$d(a, b) = \sqrt{\sum_{i=1}^n (a_i-b_i)^2}$~$ |
---|
Metric | $~$d(a, b) = \sum_{i=1}^n |a_i-b_i|$~$ |
---|
Mild optimization | $~$<_p$~$ |
---|
Mild optimization | $~$O$~$ |
---|
Mild optimization | $~$O'$~$ |
---|
Mild optimization | $~$O <_p O'$~$ |
---|
Mild optimization | $~$\theta$~$ |
---|
Mild optimization | $~$\theta$~$ |
---|
Mind design space is wide | $~$2^{1,000,000,000}$~$ |
---|
Mind design space is wide | $~$P$~$ |
---|
Mind design space is wide | $~$2^{1,000,000,000}$~$ |
---|
Mind design space is wide | $~$P$~$ |
---|
Mind design space is wide | $~$2^{1,000,000,000}$~$ |
---|
Mind design space is wide | $~$P$~$ |
---|
Mind design space is wide | $~$P$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$V$~$ |
---|
Missing the weird alternative | $~$\pi_1$~$ |
---|
Missing the weird alternative | $~$V$~$ |
---|
Missing the weird alternative | $~$U,$~$ |
---|
Missing the weird alternative | $~$\pi_0$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$V$~$ |
---|
Missing the weird alternative | $~$V$~$ |
---|
Missing the weird alternative | $~$V$~$ |
---|
Missing the weird alternative | $~$W$~$ |
---|
Missing the weird alternative | $~$W$~$ |
---|
Missing the weird alternative | $~$V.$~$ |
---|
Missing the weird alternative | $~$W$~$ |
---|
Missing the weird alternative | $~$W$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_1$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_1.$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_1$~$ |
---|
Missing the weird alternative | $~$\pi_1$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_1$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_1$~$ |
---|
Missing the weird alternative | $~$\pi_k$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_k$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_1$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_k$~$ |
---|
Missing the weird alternative | $~$\pi_0$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi$~$ |
---|
Missing the weird alternative | $~$\pi$~$ |
---|
Missing the weird alternative | $~$\pi$~$ |
---|
Missing the weird alternative | $~$U.$~$ |
---|
Missing the weird alternative | $~$\pi_0$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_1$~$ |
---|
Missing the weird alternative | $~$\pi_0$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_1$~$ |
---|
Missing the weird alternative | $~$\pi_0.$~$ |
---|
Missing the weird alternative | $~$\pi_0$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_1$~$ |
---|
Missing the weird alternative | $~$U.$~$ |
---|
Missing the weird alternative | $~$\pi_0$~$ |
---|
Missing the weird alternative | $~$U.$~$ |
---|
Missing the weird alternative | $~$\pi_k$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_1,$~$ |
---|
Missing the weird alternative | $~$\pi_1$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U.$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$\pi_0$~$ |
---|
Missing the weird alternative | $~$U$~$ |
---|
Missing the weird alternative | $~$V$~$ |
---|
Missing the weird alternative | $~$U,$~$ |
---|
Missing the weird alternative | $~$W$~$ |
---|
Missing the weird alternative | $~$F_1, F_2$~$ |
---|
Missing the weird alternative | $~$V$~$ |
---|
Missing the weird alternative | $~$U,$~$ |
---|
Missing the weird alternative | $~$W$~$ |
---|
Missing the weird alternative | $~$U.$~$ |
---|
Modal combat | $~$A$~$ |
---|
Modal combat | $~$B$~$ |
---|
Modal combat | $~$A$~$ |
---|
Modal combat | $~$DefectBot$~$ |
---|
Modal combat | $~$CliqueBot$~$ |
---|
Modal combat | $~$CliqueBot$~$ |
---|
Modal combat | $~$\square$~$ |
---|
Modal combat | $~$PA$~$ |
---|
Modal combat | $~$\phi(x)$~$ |
---|
Modal combat | $~$x$~$ |
---|
Modal combat | $~$\phi$~$ |
---|
Modal combat | $~$\phi(x) \leftrightarrow \square x(\phi)$~$ |
---|
Modal combat | $~$\psi(x)\leftrightarrow \neg\square x(\psi)$~$ |
---|
Modal combat | $~$\psi(\phi)\leftrightarrow \neg\square\phi(\psi)\leftrightarrow \neg\square\square\psi (\phi)$~$ |
---|
Modal combat | $~$GL\vdash \psi(\phi)\leftrightarrow \neg [\square \bot \vee \square\square\bot \wedge \neg\square \bot] $~$ |
---|
Modal combat | $~$\phi(\psi)$~$ |
---|
Modal combat | $~$0$~$ |
---|
Modal combat | $~$k$~$ |
---|
Modal combat | $~$k$~$ |
---|
Modal combat | $~$DefectBot$~$ |
---|
Modal combat | $~$DB(x)\leftrightarrow \bot$~$ |
---|
Modal combat | $~$FairBot$~$ |
---|
Modal combat | $~$PA$~$ |
---|
Modal combat | $~$FairBot$~$ |
---|
Modal combat | $~$FB(x) \leftrightarrow \square x(FB)$~$ |
---|
Modal combat | $~$FairBot$~$ |
---|
Modal combat | $~$FB(FB)\leftrightarrow \square FB(FB)$~$ |
---|
Modal combat | $~$\top$~$ |
---|
Modal combat | $~$FairBot$~$ |
---|
Modal combat | $~$FairBot$~$ |
---|
Modal combat | $~$CliqueBot$~$ |
---|
Modal combat | $~$FairBot$~$ |
---|
Modal combat | $~$DefectBot$~$ |
---|
Modal combat | $~$FB(DB)\leftrightarrow \square \bot$~$ |
---|
Modal combat | $~$\square \bot$~$ |
---|
Modal combat | $~$DefectBot$~$ |
---|
Modal combat | $~$FairBot$~$ |
---|
Modal combat | $~$CooperateBot$~$ |
---|
Modal combat | $~$CB(x)\leftrightarrow \top$~$ |
---|
Modal combat | $~$1$~$ |
---|
Modal combat | $~$PB(x)\leftrightarrow\square [x(PB)\wedge \neg\square\bot \rightarrow \neg x(DB)]$~$ |
---|
Modal combat | $~$PrudentBot$~$ |
---|
Modal combat | $~$PrudentBot$~$ |
---|
Modal combat | $~$PA+1$~$ |
---|
Modal combat | $~$DefectBot$~$ |
---|
Modal combat | $~$PrudentBot$~$ |
---|
Modal combat | $~$FairBot$~$ |
---|
Modal combat | $~$DefectBot$~$ |
---|
Modal combat | $~$CooperateBot$~$ |
---|
Modal combat | $~$PrudentBot$~$ |
---|
Modal combat | $~$FairBot$~$ |
---|
Modal combat | $~$TrollBot$~$ |
---|
Modal combat | $~$TB(x)\leftrightarrow \square x(DB)$~$ |
---|
Modal combat | $~$TrollBot$~$ |
---|
Modal combat | $~$TrollBot$~$ |
---|
Modal combat | $~$DefectBot$~$ |
---|
Modal combat | $~$CooperateBot$~$ |
---|
Modal combat | $~$TrollBot$~$ |
---|
Modal combat | $~$TrollBot$~$ |
---|
Modal combat | $~$DefectBot$~$ |
---|
Modal logic | $~$\square$~$ |
---|
Modal logic | $~$\diamond$~$ |
---|
Modal logic | $~$\diamond A \iff \neg \square \neg A$~$ |
---|
Modal logic | $~$\square A$~$ |
---|
Modal logic | $~$A$~$ |
---|
Modal logic | $~$\neg\square \bot$~$ |
---|
Modal logic | $~$Gödel-Löb$~$ |
---|
Modal logic | $~$GL$~$ |
---|
Modal logic | $~$GL$~$ |
---|
Modal logic | $~$GLS$~$ |
---|
Modal logic | $~$\bot$~$ |
---|
Modal logic | $~$p,q,…$~$ |
---|
Modal logic | $~$A$~$ |
---|
Modal logic | $~$\square A$~$ |
---|
Modal logic | $~$A$~$ |
---|
Modal logic | $~$B$~$ |
---|
Modal logic | $~$A\to B$~$ |
---|
Modal logic | $~$\bot$~$ |
---|
Modal logic | $~$\neg A = A\to \bot$~$ |
---|
Modal logic | $~$\diamond$~$ |
---|
Modal logic | $~$\neg\square\neg$~$ |
---|
Modalized modal sentence | $~$A$~$ |
---|
Modalized modal sentence | $~$p$~$ |
---|
Modalized modal sentence | $~$p$~$ |
---|
Modalized modal sentence | $~$\square$~$ |
---|
Modalized modal sentence | $~$\square p \wedge q$~$ |
---|
Modalized modal sentence | $~$p$~$ |
---|
Modalized modal sentence | $~$q$~$ |
---|
Modalized modal sentence | $~$A$~$ |
---|
Modalized modal sentence | $~$p$~$ |
---|
Modalized modal sentence | $~$p$~$ |
---|
Modalized modal sentence | $~$p$~$ |
---|
Modalized modal sentence | $~$p$~$ |
---|
Modeling AI control with humans | $~$\mathbb{E}$~$ |
---|
Modular arithmetic | $~$9 + 6 = 15$~$ |
---|
Modular arithmetic | $~$9 + 6$~$ |
---|
Modular arithmetic | $~$9 + 6 = 15$~$ |
---|
Monoid | $~$M$~$ |
---|
Monoid | $~$(X, \diamond)$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$x \diamond y$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$x, y \in X$~$ |
---|
Monoid | $~$x \diamond y$~$ |
---|
Monoid | $~$xy$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$x, y$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$xy$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$x, y, z$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$x(yz) = (xy)z$~$ |
---|
Monoid | $~$e$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$x$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$xe = ex = x.$~$ |
---|
Monoid | $~$x \diamond y \in X$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$e$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$e$~$ |
---|
Monoid | $~$x$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$x$~$ |
---|
Monoid | $~$e$~$ |
---|
Monoid | $~$z$~$ |
---|
Monoid | $~$ze = e = ez = z.$~$ |
---|
Monoid | $~$e$~$ |
---|
Monoid | $~$M$~$ |
---|
Monoid | $~$e$~$ |
---|
Monoid | $~$1$~$ |
---|
Monoid | $~$1_M$~$ |
---|
Monoid | $~$M = (X, \diamond)$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$M$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monoid | $~$x \diamond y$~$ |
---|
Monoid | $~$xy$~$ |
---|
Monoid | $~$M$~$ |
---|
Monoid | $~$X$~$ |
---|
Monoid | $~$x, y \in X$~$ |
---|
Monoid | $~$M$~$ |
---|
Monoid | $~$x, y \in M$~$ |
---|
Monoid | $~$M$~$ |
---|
Monoid | $~$Y$~$ |
---|
Monoid | $~$Y$~$ |
---|
Monoid | $~$\mathbb N$~$ |
---|
Monoid | $~$0$~$ |
---|
Monoid | $~$\diamond$~$ |
---|
Monotone function | $~$\langle P, \leq_P \rangle$~$ |
---|
Monotone function | $~$\langle Q, \leq_Q \rangle$~$ |
---|
Monotone function | $~$\phi : P \rightarrow Q$~$ |
---|
Monotone function | $~$s, t \in P$~$ |
---|
Monotone function | $~$s \le_P t$~$ |
---|
Monotone function | $~$\phi(s) \le_Q \phi(t)$~$ |
---|
Monotone function | $~$\phi$~$ |
---|
Monotone function | $~$P$~$ |
---|
Monotone function | $~$Q$~$ |
---|
Monotone function | $~$\le_P$~$ |
---|
Monotone function | $~$(c,a)$~$ |
---|
Monotone function | $~$(b,a)$~$ |
---|
Monotone function | $~$\phi$~$ |
---|
Monotone function | $~$c \leq_P a$~$ |
---|
Monotone function | $~$\phi(c) = u \leq_Q t = \phi(a)$~$ |
---|
Monotone function | $~$b \leq_P a$~$ |
---|
Monotone function | $~$\phi(b) = t \leq_Q t = \phi(a)$~$ |
---|
Monotone function | $~$\phi$~$ |
---|
Monotone function | $~$P$~$ |
---|
Monotone function | $~$Q$~$ |
---|
Monotone function | $~$a \leq_P b$~$ |
---|
Monotone function | $~$\phi(a) = v \parallel_Q u = \phi(b)$~$ |
---|
Monotone function: examples | $~$\star$~$ |
---|
Monotone function: examples | $~$\star$~$ |
---|
Monotone function: examples | $~$Alph = \langle \{A,…,Z\}, \leq_{Alph} \rangle$~$ |
---|
Monotone function: examples | $~$Alph$~$ |
---|
Monotone function: examples | $~$false <_{\textbf{2}} true$~$ |
---|
Monotone function: examples | $~$f : Alph \to \textbf{2}$~$ |
---|
Monotone function: examples | $~$f(\star) \doteq Q >_{Alph} \star$~$ |
---|
Monotone function: examples | $~$f$~$ |
---|
Monotone function: examples | $~$f$~$ |
---|
Monotone function: examples | $~$\star_1$~$ |
---|
Monotone function: examples | $~$\star_2$~$ |
---|
Monotone function: examples | $~$\star_1$~$ |
---|
Monotone function: examples | $~$\star_2 \leq_{Alph} \star_1$~$ |
---|
Monotone function: examples | $~$f(\star_2) \leq_{\textbf{2}} f(\star_1)$~$ |
---|
Monotone function: examples | $~$\phi$~$ |
---|
Monotone function: examples | $~$\psi$~$ |
---|
Monotone function: examples | $~$\wedge$~$ |
---|
Monotone function: examples | $~$\to$~$ |
---|
Monotone function: examples | $~$\wedge$~$ |
---|
Monotone function: examples | $~$\phi$~$ |
---|
Monotone function: examples | $~$\psi$~$ |
---|
Monotone function: examples | $~$\phi \wedge \psi$~$ |
---|
Monotone function: examples | $~$\phi$~$ |
---|
Monotone function: examples | $~$\psi$~$ |
---|
Monotone function: examples | $~$\to$~$ |
---|
Monotone function: examples | $~$\phi \to \psi$~$ |
---|
Monotone function: examples | $~$\phi$~$ |
---|
Monotone function: examples | $~$\psi$~$ |
---|
Monotone function: examples | $~$\phi$~$ |
---|
Monotone function: examples | $~$\psi$~$ |
---|
Monotone function: examples | $~$\phi \wedge \psi$~$ |
---|
Monotone function: examples | $~$X$~$ |
---|
Monotone function: examples | $~$F : \mathcal P(X) \to \mathcal P(X)$~$ |
---|
Monotone function: examples | $~$F(A) = \\ ~~ \{ \phi \wedge \psi\mid\phi \in A, \psi \in A \} \cup \\ ~~ \{ \phi \mid \phi \wedge \psi \in A \} \cup \\ ~~ \{ \psi \mid \phi \wedge \psi \in A \} \cup \\ ~~ \{ \phi \mid \psi \in A, \psi \to \phi \in A \}$~$ |
---|
Monotone function: examples | $~$F$~$ |
---|
Monotone function: examples | $~$\langle \mathcal P(X), \subseteq \rangle$~$ |
---|
Monotone function: examples | $~$F$~$ |
---|
Monotone function: examples | $~$\langle \mathbb R, \le \rangle$~$ |
---|
Monotone function: exercises | $~$P, Q$~$ |
---|
Monotone function: exercises | $~$R$~$ |
---|
Monotone function: exercises | $~$f : P \to Q$~$ |
---|
Monotone function: exercises | $~$g : Q \to R$~$ |
---|
Monotone function: exercises | $~$g \circ f$~$ |
---|
Monotone function: exercises | $~$P$~$ |
---|
Monotone function: exercises | $~$R$~$ |
---|
Monotone function: exercises | $~$P$~$ |
---|
Monotone function: exercises | $~$Q$~$ |
---|
Monotone function: exercises | $~$f : P \to Q$~$ |
---|
Monotone function: exercises | $~$f$~$ |
---|
Monotone function: exercises | $~$p_1 \leq_P p_2$~$ |
---|
Monotone function: exercises | $~$f(p_1) \geq_Q f(p_2)$~$ |
---|
Monotone function: exercises | $~$f : P \times A \to Q$~$ |
---|
Monotone function: exercises | $~$P$~$ |
---|
Monotone function: exercises | $~$Q$~$ |
---|
Monotone function: exercises | $~$a \in A$~$ |
---|
Monotone function: exercises | $~$p_1 \leq_P p_2$~$ |
---|
Monotone function: exercises | $~$f(a, p_1) \leq_Q f(a, p_2)$~$ |
---|
Monotone function: exercises | $~$f : A \times P \to Q$~$ |
---|
Monotone function: exercises | $~$P$~$ |
---|
Monotone function: exercises | $~$Q$~$ |
---|
Monotone function: exercises | $~$a \in A$~$ |
---|
Monotone function: exercises | $~$p_1 \leq_P p_2$~$ |
---|
Monotone function: exercises | $~$f(p_1, a) \leq_Q f(p_2, a)$~$ |
---|
Monotone function: exercises | $~$P, Q, R$~$ |
---|
Monotone function: exercises | $~$S$~$ |
---|
Monotone function: exercises | $~$f : P \times Q \to R$~$ |
---|
Monotone function: exercises | $~$g_1 : S \to P$~$ |
---|
Monotone function: exercises | $~$g_2 : S \to Q$~$ |
---|
Monotone function: exercises | $~$h : S \to R$~$ |
---|
Monotone function: exercises | $~$h(s) \doteq f(g_1(s), g_2(s))$~$ |
---|
Moral uncertainty | $~$\Delta U$~$ |
---|
Moral uncertainty | $~$U$~$ |
---|
Moral uncertainty | $~$U_1$~$ |
---|
Moral uncertainty | $~$U_2.$~$ |
---|
Moral uncertainty | $~$0.5 \cdot U_1 + 0.5 \cdot U_2.$~$ |
---|
Most complex things are not very compressible | $~$2^{101}$~$ |
---|
Multiple stage fallacy | $~$0.50 \cdot 0.50 = 0.25.$~$ |
---|
Multiple stage fallacy | $~$0.50 \cdot 0.50 = 0.25$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$1$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{b}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$a$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$m$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{b}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{b}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$n$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$n$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{b}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a}{m} \times \frac{b}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$1 \times \frac{b}{n} = \frac{b}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{b}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$1$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{b}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$2 \times \frac{3}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{2}{1}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$2$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{2}{1}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{2}{1}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{3}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{3}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{3}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{3}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{3}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{3}{5} + \frac{3}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{6}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$2 \times \frac{3}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$m \times \frac{a}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$m$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a \times m}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$1$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$n$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$a$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$m \times \frac{a}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$m$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$n$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$a$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$m$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$2 \times \frac{3}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$m$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$1$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$2$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$1$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$1$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$n$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$5$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$a$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$a$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$3$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$m$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$m$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$2$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$a \times m$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a \times m}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$-5 \times \frac{2}{3}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{2}{3}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$1$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$-5$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$15$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{1}{3}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$10$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$-5 \times \frac{2}{3} = \frac{-10}{3}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$n \times \frac{a}{n} = \frac{a \times m}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a}{m} + \frac{b}{n} = \frac{b}{n} + \frac{a}{m}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{b}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$1$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{b}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$1$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$2 \times \frac{3}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{3}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$1$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$2$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$2$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{3}{5}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{3}{5} \times 2$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{a}{m} \times \frac{b}{n}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{b}{n} \times \frac{a}{m}$~$ |
---|
Multiplication of rational numbers (Math 0) | $~$\frac{-5}{7} \times \frac{2}{3}$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{a}{n}$~$ |
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Multiplication of rational numbers (Math 0) | $~$n$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
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Multiplication of rational numbers (Math 0) | $~$a$~$ |
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Multiplication of rational numbers (Math 0) | $~$a$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
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Multiplication of rational numbers (Math 0) | $~$1$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{a}{n}$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{1}{n} \times a$~$ |
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Multiplication of rational numbers (Math 0) | $~$a$~$ |
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Multiplication of rational numbers (Math 0) | $~$n$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
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Multiplication of rational numbers (Math 0) | $~$a$~$ |
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Multiplication of rational numbers (Math 0) | $~$1$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{a}{n} = a \times \frac{1}{n}$~$ |
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Multiplication of rational numbers (Math 0) | $~$a$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{a}{1}$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{a}{n}$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{a}{1} \times \frac{1}{n}$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{a}{1} \times \frac{1}{n} = \frac{a \times 1}{1 \times n} = \frac{a}{n}$~$ |
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Multiplication of rational numbers (Math 0) | $~$0$~$ |
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Multiplication of rational numbers (Math 0) | $~$a$~$ |
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Multiplication of rational numbers (Math 0) | $~$0$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{c}{d}$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{a}{b} \times \frac{c}{d} = 1$~$ |
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Multiplication of rational numbers (Math 0) | $~$1$~$ |
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Multiplication of rational numbers (Math 0) | $~$0$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$~$ |
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Multiplication of rational numbers (Math 0) | $~$1$~$ |
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Multiplication of rational numbers (Math 0) | $~$a \times c$~$ |
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Multiplication of rational numbers (Math 0) | $~$b \times d$~$ |
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Multiplication of rational numbers (Math 0) | $~$c = b$~$ |
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Multiplication of rational numbers (Math 0) | $~$d = a$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{a \times b}{b \times a} = \frac{a \times b}{a \times b} = \frac{1}{1} = 1$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{b}{a}$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{a}{b}$~$ |
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Multiplication of rational numbers (Math 0) | $~$a$~$ |
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Multiplication of rational numbers (Math 0) | $~$0$~$ |
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Multiplication of rational numbers (Math 0) | $~$\frac{b}{a}$~$ |
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Multiplication of rational numbers (Math 0) | $~$a$~$ |
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Mutually exclusive and exhaustive | $~$X$~$ |
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Mutually exclusive and exhaustive | $~$\forall i: \forall j: i \neq j \implies \mathbb{P}(X_i \wedge X_j) = 0.$~$ |
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Mutually exclusive and exhaustive | $~$\mathbb{P}(X_i \vee X_j) = \mathbb{P}(X_i) + \mathbb{P}(X_j) - \mathbb{P}(X_j \wedge X_j) = \mathbb{P}(X_i) + \mathbb{P}(X_j).$~$ |
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Mutually exclusive and exhaustive | $~$X,$~$ |
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Mutually exclusive and exhaustive | $~$1$~$ |
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Mutually exclusive and exhaustive | $~$X_i$~$ |
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Mutually exclusive and exhaustive | $~$1$~$ |
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Mutually exclusive and exhaustive | $~$\mathbb{P}(X_1 \vee X_2 \vee \dots \vee X_N) = 1.$~$ |
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Mutually exclusive and exhaustive | $~$\displaystyle \sum_i \mathbb{P}(X_i) = 1.$~$ |
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Natural number | $~$\mathbb N.$~$ |
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Natural number | $~$\mathbb N.$~$ |
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Natural number | $~$2 + 3 = 5$~$ |
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Natural number | $~$2 \cdot 3 = 6$~$ |
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Natural numbers: Intro to Number Sets | $~$0$~$ |
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Natural numbers: Intro to Number Sets | $~$0$~$ |
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Natural numbers: Intro to Number Sets | $~$0$~$ |
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Natural numbers: Intro to Number Sets | $~$1$~$ |
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Natural numbers: Intro to Number Sets | $~$2$~$ |
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Natural numbers: Intro to Number Sets | $~$3$~$ |
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Natural numbers: Intro to Number Sets | $~$\{0, 1, 2, 3, \ldots\}$~$ |
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Natural numbers: Intro to Number Sets | $~$4$~$ |
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Natural numbers: Intro to Number Sets | $~$5$~$ |
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Natural numbers: Intro to Number Sets | $~$n$~$ |
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Natural numbers: Intro to Number Sets | $~$n'$~$ |
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Natural numbers: Intro to Number Sets | $~$n$~$ |
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Natural numbers: Intro to Number Sets | $~$2'$~$ |
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Natural numbers: Intro to Number Sets | $~$3$~$ |
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Natural numbers: Intro to Number Sets | $~$59'$~$ |
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Natural numbers: Intro to Number Sets | $~$60$~$ |
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Natural numbers: Intro to Number Sets | $~$9287'$~$ |
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Natural numbers: Intro to Number Sets | $~$9288$~$ |
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Natural numbers: Intro to Number Sets | $~$\prime$~$ |
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Natural numbers: Intro to Number Sets | $~$2'''$~$ |
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Natural numbers: Intro to Number Sets | $~$5$~$ |
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Natural numbers: Intro to Number Sets | $~$0$~$ |
---|
Natural numbers: Intro to Number Sets | $~$1$~$ |
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Natural numbers: Intro to Number Sets | $~$1$~$ |
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Natural numbers: Intro to Number Sets | $~$0$~$ |
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Natural numbers: Intro to Number Sets | $~$n < m$~$ |
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Natural numbers: Intro to Number Sets | $~$m$~$ |
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Natural numbers: Intro to Number Sets | $~$n''''^\ldots$~$ |
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Natural numbers: Intro to Number Sets | $~$\prime$~$ |
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Natural numbers: Intro to Number Sets | $~$5 = 2'''$~$ |
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Natural numbers: Intro to Number Sets | $~$2 < 5$~$ |
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Natural numbers: Intro to Number Sets | $~$a < b$~$ |
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Natural numbers: Intro to Number Sets | $~$b < c$~$ |
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Natural numbers: Intro to Number Sets | $~$a < c$~$ |
---|
Natural numbers: Intro to Number Sets | $~$a < b$~$ |
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Natural numbers: Intro to Number Sets | $~$b < c$~$ |
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Natural numbers: Intro to Number Sets | $~$c < a$~$ |
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Natural numbers: Intro to Number Sets | $~$2 < 4$~$ |
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Natural numbers: Intro to Number Sets | $~$4 < 6$~$ |
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Natural numbers: Intro to Number Sets | $~$2 < 6$~$ |
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Natural numbers: Intro to Number Sets | $~$a < b$~$ |
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Natural numbers: Intro to Number Sets | $~$b = a''''^\ldots$~$ |
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Natural numbers: Intro to Number Sets | $~$\prime$~$ |
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Natural numbers: Intro to Number Sets | $~$b < c$~$ |
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Natural numbers: Intro to Number Sets | $~$c = b''''^\ldots$~$ |
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Natural numbers: Intro to Number Sets | $~$\prime$~$ |
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Natural numbers: Intro to Number Sets | $~$c = b''''^\ldots = (a''''^\ldots)''''^\ldots$~$ |
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Natural numbers: Intro to Number Sets | $~$a''''^\ldots$~$ |
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Natural numbers: Intro to Number Sets | $~$\prime$~$ |
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Natural numbers: Intro to Number Sets | $~$4 + 3$~$ |
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Natural numbers: Intro to Number Sets | $~$4 = 0''''$~$ |
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Natural numbers: Intro to Number Sets | $~$4 + 3 = 3'''' = 7$~$ |
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Natural numbers: Intro to Number Sets | $~$4 \times 3$~$ |
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Natural numbers: Intro to Number Sets | $~$4 = 0''''$~$ |
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Natural numbers: Intro to Number Sets | $~$4 \times 3 = 0 + 3 + 3 + 3 + 3 = 12$~$ |
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Natural numbers: Intro to Number Sets | $~$\{1, 3, 5, 7, 9,\ldots\}$~$ |
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Natural numbers: Intro to Number Sets | $~$3$~$ |
---|
Natural numbers: Intro to Number Sets | $~$5$~$ |
---|
Natural numbers: Intro to Number Sets | $~$3$~$ |
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Natural numbers: Intro to Number Sets | $~$3$~$ |
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Natural numbers: Intro to Number Sets | $~$6$~$ |
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Natural numbers: Intro to Number Sets | $~$a$~$ |
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Natural numbers: Intro to Number Sets | $~$c$~$ |
---|
Natural numbers: Intro to Number Sets | $~$b$~$ |
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Natural numbers: Intro to Number Sets | $~$a + b = c$~$ |
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Natural numbers: Intro to Number Sets | $~$3$~$ |
---|
Natural numbers: Intro to Number Sets | $~$2$~$ |
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Natural numbers: Intro to Number Sets | $~$2$~$ |
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Natural numbers: Intro to Number Sets | $~$3$~$ |
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Natural numbers: Intro to Number Sets | $~$n$~$ |
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Natural numbers: Intro to Number Sets | $~$n + 3 = 2$~$ |
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Natural numbers: Intro to Number Sets | $~$a$~$ |
---|
Natural numbers: Intro to Number Sets | $~$c$~$ |
---|
Natural numbers: Intro to Number Sets | $~$b$~$ |
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Natural numbers: Intro to Number Sets | $~$a \times b = c$~$ |
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Natural numbers: Intro to Number Sets | $~$3$~$ |
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Natural numbers: Intro to Number Sets | $~$2$~$ |
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Natural numbers: Intro to Number Sets | $~$n$~$ |
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Natural numbers: Intro to Number Sets | $~$n \times 2 = 3$~$ |
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Natural numbers: Intro to Number Sets | $~$1$~$ |
---|
Nearest unblocked strategy | $~$X$~$ |
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Nearest unblocked strategy | $~$P$~$ |
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Nearest unblocked strategy | $~$X,$~$ |
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Nearest unblocked strategy | $~$X'$~$ |
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Nearest unblocked strategy | $~$X$~$ |
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Nearest unblocked strategy | $~$P.$~$ |
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Nearest unblocked strategy | $~$i,$~$ |
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Nearest unblocked strategy | $~$U_i,$~$ |
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Nearest unblocked strategy | $~$X_i$~$ |
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Nearest unblocked strategy | $~$P_i$~$ |
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Nearest unblocked strategy | $~$U_i$~$ |
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Nearest unblocked strategy | $~$U_{i+1},$~$ |
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Nearest unblocked strategy | $~$X_i^*$~$ |
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Nearest unblocked strategy | $~$U_{i+1}$~$ |
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Nearest unblocked strategy | $~$X_{i+1}$~$ |
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Nearest unblocked strategy | $~$X_i$~$ |
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Nearest unblocked strategy | $~$P_i,$~$ |
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Nearest unblocked strategy | $~$P_{i+1}$~$ |
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Nearest unblocked strategy | $~$X,$~$ |
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Nearest unblocked strategy | $~$X'$~$ |
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Nearest unblocked strategy | $~$X.$~$ |
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Nearest unblocked strategy | $~$X$~$ |
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Nearest unblocked strategy | $~$G$~$ |
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Nearest unblocked strategy | $~$X$~$ |
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Nearest unblocked strategy | $~$X'$~$ |
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Nearest unblocked strategy | $~$G.$~$ |
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Negation of propositions | $~$S$~$ |
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Negation of propositions | $~$Q$~$ |
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Negation of propositions | $~$P$~$ |
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Negation of propositions | $~$Q$~$ |
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Negation of propositions | $~$Q$~$ |
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Negation of propositions | $~$P$~$ |
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Negation of propositions | $~$ Q \equiv \neg P$~$ |
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Negation of propositions | $~$P$~$ |
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Negation of propositions | $~$\neg P$~$ |
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Negation of propositions | $~$P$~$ |
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Negation of propositions | $~$\neg P$~$ |
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Normal subgroup | $~$N$~$ |
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Normal subgroup | $~$G$~$ |
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Normal subgroup | $~$h \in G$~$ |
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Normal subgroup | $~$\{ h n h^{-1} : n \in N \} = N$~$ |
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Normal subgroup | $~$hNh^{-1} = N$~$ |
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Normal subgroup | $~$G$~$ |
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Normal subgroup | $~$G$~$ |
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Normal subgroup | $~$G$~$ |
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Normal subgroup | $~$H$~$ |
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Normal subgroup | $~$f$~$ |
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Normal subgroup | $~$f$~$ |
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Normal subgroup | $~$f$~$ |
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Normal system of provability logic | $~$L\vdash A$~$ |
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Normal system of provability logic | $~$L\vdash \square A$~$ |
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Normal system of provability logic | $~$L\vdash A\rightarrow B$~$ |
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Normal system of provability logic | $~$L\vdash A$~$ |
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Normal system of provability logic | $~$L\vdash B$~$ |
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Normal system of provability logic | $~$\square(A\rightarrow B)\rightarrow (\square A \rightarrow \square B)$~$ |
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Normal system of provability logic | $~$L\vdash F(p)$~$ |
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Normal system of provability logic | $~$L\vdash F(H)$~$ |
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Normal system of provability logic | $~$H$~$ |
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Normal system of provability logic | $~$L\vdash \square(A_1\wedge … \wedge A_n)\leftrightarrow (\square A_1 \wedge … \wedge \square A_n)$~$ |
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Normal system of provability logic | $~$L\vdash A\rightarrow B$~$ |
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Normal system of provability logic | $~$L\vdash \square A \rightarrow \square B$~$ |
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Normal system of provability logic | $~$L\vdash \diamond A \rightarrow \diamond B$~$ |
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Normal system of provability logic | $~$L\vdash \diamond A \wedge \square B \rightarrow \diamond (A\wedge B)$~$ |
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Normal system of provability logic | $~$L\vdash A\leftrightarrow B$~$ |
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Normal system of provability logic | $~$F(p)$~$ |
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Normal system of provability logic | $~$p$~$ |
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Normal system of provability logic | $~$L\vdash F(A)\leftrightarrow F(B)$~$ |
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Normalization (probability) | $~$3:2 \cong \frac{3}{3+2} : \frac{2}{3+2} = 0.6 : 0.4.$~$ |
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Normalization (probability) | $~$\frac{3}{3+2} : \frac{2}{3+2} = 0.6 : 0.4.$~$ |
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Normalization (probability) | $~$m : n$~$ |
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Normalization (probability) | $~$\frac{m}{m+n} : \frac{n}{m+n},$~$ |
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Normalization (probability) | $~$\frac{1}{m+n}$~$ |
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Normalization (probability) | $~$\mathbb{O}(H)$~$ |
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Normalization (probability) | $~$H$~$ |
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Normalization (probability) | $~$\mathbb{P}(H)$~$ |
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Normalization (probability) | $~$\mathbb{O}(H)$~$ |
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Normalization (probability) | $~$\mathbb{O}(H).$~$ |
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Normalization (probability) | $~$\mathbb{P}(H_i) = \frac{\mathbb{O}(H_i)}{\sum_i \mathbb{O}(H_i)}$~$ |
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Normalization (probability) | $~$\mathbb{O}(x)$~$ |
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Normalization (probability) | $~$X$~$ |
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Normalization (probability) | $~$\mathbb{P}(x)$~$ |
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Normalization (probability) | $~$\mathbb{O}(x)$~$ |
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Normalization (probability) | $~$\mathbb{P}(x) = \frac{\mathbb{O}(x)}{\int \mathbb{O}(x) \operatorname{d}x}$~$ |
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Normalization (probability) | $~$\mathbb{P}(H) \propto \mathbb{O}(H) \implies \mathbb{P}(H) = \frac{\mathbb{O}(H)}{\sum \mathbb{O}(H)}$~$ |
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Note on 1x1 Convolutions | $$~$
\vec{y_{n}}=(\mathbf{W_n}^T \times \vec{y_{n-1}} + \vec{b_n})+\mathbf{W_{new}}^T (\mathbf{W_n}^T \times \vec{y_{n-1}} + \vec{b_n})
$~$$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$B$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$B$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$B$~$ |
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Object identity via interactions | $~$\{1\}$~$ |
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Object identity via interactions | $~$\{1\} \to A$~$ |
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Object identity via interactions | $~$\{1\} \to B$~$ |
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Object identity via interactions | $~$\{1\}$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$f$~$ |
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Object identity via interactions | $~$1$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$\{1\}$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$A = \{ 5, 6 \}$~$ |
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Object identity via interactions | $~$\{1\} \to A$~$ |
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Object identity via interactions | $~$f: 1 \mapsto 5$~$ |
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Object identity via interactions | $~$g: 1 \mapsto 6$~$ |
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Object identity via interactions | $~$5 \in A$~$ |
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Object identity via interactions | $~$f$~$ |
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Object identity via interactions | $~$6 \in A$~$ |
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Object identity via interactions | $~$g$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$\{ f, g \}$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$A$~$ |
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Object identity via interactions | $~$B$~$ |
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Object identity via interactions | $~$\{1\} \to B$~$ |
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Odds | $~$2:3 = \frac{2}{3+2}:\frac{3}{3+2} = 0.4:0.6.$~$ |
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Odds | $~$7:9$~$ |
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Odds form to probability form | $~$H_i$~$ |
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Odds form to probability form | $~$H_j,$~$ |
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Odds form to probability form | $$~$\frac{\mathbb P(H_i \mid e)}{\mathbb P(H_j \mid e)} = \frac{\mathbb P(H_i)}{\mathbb P(H_j)} \times \frac{\mathbb P(e \mid H_i)}{\mathbb P(e \mid H_j)} \tag{1}.$~$$ |
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Odds form to probability form | $~$H_1,H_2,H_3,\ldots$~$ |
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Odds form to probability form | $$~$\mathbb P(H_i\mid e) = \frac{\mathbb P(e\mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P(e\mid H_k) \cdot \mathbb P(H_k)} \tag{2}.$~$$ |
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Odds form to probability form | $~$H_1,H_2,H_3,\ldots$~$ |
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Odds form to probability form | $~$H_i$~$ |
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Odds form to probability form | $~$\lnot H_i$~$ |
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Odds form to probability form | $~$H_1,H_2,H_3,\ldots$~$ |
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Odds form to probability form | $~$H_i.$~$ |
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Odds form to probability form | $~$\lnot H_i$~$ |
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Odds form to probability form | $~$H_j$~$ |
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Odds form to probability form | $$~$\frac{\mathbb P(H_i \mid e)}{\mathbb P(\lnot H_i \mid e)} = \frac{\mathbb P(H_i) \cdot \mathbb P(e \mid H_i)}{\mathbb P(\lnot H_i)\cdot \mathbb P(e \mid \lnot H_i)}.$~$$ |
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Odds form to probability form | $~$\mathbb P(\lnot H_i)\cdot \mathbb P(e \mid \lnot H_i)$~$ |
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Odds form to probability form | $~$\lnot H_i$~$ |
---|
Odds form to probability form | $~$\lnot H_i$~$ |
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Odds form to probability form | $~$e.$~$ |
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Odds form to probability form | $~$\lnot H_i$~$ |
---|
Odds form to probability form | $~$\mathbb P(H_k) \cdot \mathbb P(e \mid H_k)$~$ |
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Odds form to probability form | $~$H_k$~$ |
---|
Odds form to probability form | $~$H_i.$~$ |
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Odds form to probability form | $$~$\frac{\mathbb P(H_i \mid e)}{\mathbb P(\lnot H_i \mid e)} = \frac{\mathbb P(e \mid H_i) \cdot \mathbb P(H_i)}{\sum_{k \neq i} \mathbb P(e \mid H_k) \cdot \mathbb P(H_k)}.$~$$ |
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Odds form to probability form | $~$H_i$~$ |
---|
Odds form to probability form | $~$\lnot H_i.$~$ |
---|
Odds form to probability form | $~$H_i$~$ |
---|
Odds form to probability form | $~$\lnot H_i$~$ |
---|
Odds form to probability form | $~$H_i,$~$ |
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Odds form to probability form | $~$3 : 4$~$ |
---|
Odds form to probability form | $$~$\mathbb P(H_i\mid e) = \frac{\mathbb P(e\mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P(e\mid H_k) \cdot \mathbb P(H_k)}.$~$$ |
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Odds: Introduction | $~$(x : y)$~$ |
---|
Odds: Introduction | $~$\alpha$~$ |
---|
Odds: Introduction | $~$(\alpha x : \alpha y).$~$ |
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Odds: Introduction | $~$(r : b : g)$~$ |
---|
Odds: Introduction | $~$(p_r : p_b : p_g)$~$ |
---|
Odds: Introduction | $~$p_r + p_g + p_b$~$ |
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Odds: Introduction | $~$(p_r : p_b : p_g)$~$ |
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Odds: Introduction | $~$(r : b : g)$~$ |
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Odds: Introduction | $~$(1 : 2 : 1)$~$ |
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Odds: Introduction | $~$\frac{1}{4} : \frac{2}{4} : \frac{1}{4},$~$ |
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Odds: Introduction | $~$(r : b)$~$ |
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Odds: Introduction | $~$(a : b : c \ldots)$~$ |
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Odds: Introduction | $~$(a + b + c \ldots)$~$ |
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Odds: Introduction | $$~$\left(\frac{a}{a + b + c \ldots} : \frac{b}{a + b + c \ldots} : \frac{c}{a + b + c \ldots}\ldots\right)$~$$ |
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Odds: Introduction | $~$\mathbb P(R) = \frac{1}{4}$~$ |
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Odds: Introduction | $~$\mathbb P(B) = \frac{1}{2}.$~$ |
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Odds: Introduction | $~$\mathbb P(R) : \mathbb P(B) = \left(\frac{1}{4} : \frac{1}{2}\right),$~$ |
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Odds: Introduction | $~$\left(\frac{\mathbb P(R)}{\mathbb P(B)} : 1\right)$~$ |
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Odds: Introduction | $~$\frac{\mathbb P(R)}{\mathbb P(B)}$~$ |
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Odds: Introduction | $~$\frac{\mathbb P(R)}{\mathbb P(B)} = \frac{1}{2},$~$ |
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Odds: Introduction | $~$\frac{\mathbb P(R)}{\mathbb P(B)}$~$ |
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Odds: Introduction | $~$\left(\frac{\mathbb P(R)}{\mathbb P(B)} : 1\right).$~$ |
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Odds: Introduction | $~$x$~$ |
---|
Odds: Introduction | $~$y$~$ |
---|
Odds: Introduction | $~$\frac{x}{y}.$~$ |
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Odds: Introduction | $~$\frac{x}{y}$~$ |
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Odds: Introduction | $~$(x : y),$~$ |
---|
Odds: Introduction | $~$\left(\frac{x}{y} : 1\right).$~$ |
---|
Odds: Refresher | $~$(2 : 3).$~$ |
---|
Odds: Refresher | $~$(x : y)$~$ |
---|
Odds: Refresher | $~$\alpha$~$ |
---|
Odds: Refresher | $~$(\alpha x : \alpha y).$~$ |
---|
Odds: Refresher | $~$A, B, C$~$ |
---|
Odds: Refresher | $~$\mathbb P(A) + \mathbb P(B) + \mathbb P(C)$~$ |
---|
Odds: Refresher | $~$1.$~$ |
---|
Odds: Refresher | $~$d,$~$ |
---|
Odds: Refresher | $~$(a : b : c)$~$ |
---|
Odds: Refresher | $~$(\frac{a}{a + b + c} : \frac{b}{a + b + c} : \frac{c}{a + b + c}).$~$ |
---|
Odds: Refresher | $~$x$~$ |
---|
Odds: Refresher | $~$y$~$ |
---|
Odds: Refresher | $~$\frac{x}{y}.$~$ |
---|
Odds: Refresher | $~$\frac{x}{y}$~$ |
---|
Odds: Refresher | $~$(x : y),$~$ |
---|
Odds: Refresher | $~$\left(\frac{x}{y} : 1\right).$~$ |
---|
Odds: Refresher | $~$(x : y)$~$ |
---|
Odds: Refresher | $~$\frac{x}{y}.$~$ |
---|
Odds: Technical explanation | $~$17 : 2$~$ |
---|
Odds: Technical explanation | $~$17 : 2$~$ |
---|
Odds: Technical explanation | $~$(17 : 2 : 100).$~$ |
---|
Odds: Technical explanation | $~$2 : 3.$~$ |
---|
Odds: Technical explanation | $~$n$~$ |
---|
Odds: Technical explanation | $~$X_1, X_2, \ldots X_n,$~$ |
---|
Odds: Technical explanation | $~$(x_1, x_2, \ldots, x_n)$~$ |
---|
Odds: Technical explanation | $~$x_i$~$ |
---|
Odds: Technical explanation | $~$(x_1, x_2, \ldots, x_n)$~$ |
---|
Odds: Technical explanation | $~$(y_1, y_2, \ldots, y_n)$~$ |
---|
Odds: Technical explanation | $~$\alpha > 0$~$ |
---|
Odds: Technical explanation | $~$ \alpha x_i = y_i$~$ |
---|
Odds: Technical explanation | $~$i$~$ |
---|
Odds: Technical explanation | $~$n.$~$ |
---|
Odds: Technical explanation | $~$(x_1 : x_2 : \ldots : x_n),$~$ |
---|
Odds: Technical explanation | $~$(3 : 6) = (9 : 18).$~$ |
---|
Odds: Technical explanation | $~$\frac{x}{y},$~$ |
---|
Odds: Technical explanation | $~$\frac{x}{y}$~$ |
---|
Odds: Technical explanation | $~$(x : y).$~$ |
---|
Odds: Technical explanation | $~$X,$~$ |
---|
Odds: Technical explanation | $~$Y,$~$ |
---|
Odds: Technical explanation | $~$Z,$~$ |
---|
Odds: Technical explanation | $~$(3 : 2 : 6).$~$ |
---|
Odds: Technical explanation | $~$X$~$ |
---|
Odds: Technical explanation | $~$Z.$~$ |
---|
Odds: Technical explanation | $$~$(x_1 : x_2 : \dots : x_n) = \left(\frac{x_1}{\sum_{i=1}^n x_i} : \frac{x_2}{\sum_{i=1}^n x_i} : \dots : \frac{x_n}{\sum_{i=1}^n x_i}\right)$~$$ |
---|
Odds: Technical explanation | $$~$(1 : 3) = \left(\frac{1}{1+3}:\frac{3}{1+3}\right) = ( 0.25 : 0.75 )$~$$ |
---|
Odds: Technical explanation | $~$\mathbb P(X) + \mathbb P(\neg X) = 1,$~$ |
---|
Odds: Technical explanation | $~$\neg X$~$ |
---|
Odds: Technical explanation | $~$X.$~$ |
---|
Odds: Technical explanation | $~$X$~$ |
---|
Odds: Technical explanation | $~$\neg X$~$ |
---|
Odds: Technical explanation | $~$\mathbb P(X) : \mathbb P(\neg X)$~$ |
---|
Odds: Technical explanation | $~$=$~$ |
---|
Odds: Technical explanation | $~$\mathbb P(X) : 1 - \mathbb P(X).$~$ |
---|
Odds: Technical explanation | $~$(0.2 : 1 - 0.2)$~$ |
---|
Odds: Technical explanation | $~$=$~$ |
---|
Odds: Technical explanation | $~$(0.2 : 0.8)$~$ |
---|
Odds: Technical explanation | $~$=$~$ |
---|
Odds: Technical explanation | $~$(1 : 4).$~$ |
---|
Odds: Technical explanation | $$~$\dfrac{\mathbb{P}(H_i\mid e_0)}{\mathbb{P}(H_j\mid e_0)} = \dfrac{\mathbb{P}(e_0\mid H_i)}{\mathbb{P}(e_0\mid H_j)} \cdot \dfrac{\mathbb{P}(H_i)}{\mathbb{P}(H_j)}$~$$ |
---|
Odds: Technical explanation | $$~$(6 : 3 : 1) \times (6 : 1 : 9) \times (2 : 8 : 1) = (72 : 24 : 9) = (24 : 8: 3)$~$$ |
---|
Odds: Technical explanation | $~$\mathbb{P}(X) : \mathbb{P}(\neg X)$~$ |
---|
Odds: Technical explanation | $~$\frac{\mathbb{P}(X)}{\mathbb{P}(\neg X)}$~$ |
---|
Odds: Technical explanation | $~$[0, +\infty]$~$ |
---|
Odds: Technical explanation | $~$\log\left(\frac{\mathbb{P}(X)}{\mathbb{P}(\neg X)}\right),$~$ |
---|
Odds: Technical explanation | $~$\log_2(1:4) = \log_2(0.25) = -2.$~$ |
---|
Odds: Technical explanation | $~$[-\infty, +\infty]$~$ |
---|
Odds: Technical explanation | $~$(0, 1).$~$ |
---|
Odds: Technical explanation | $~$0$~$ |
---|
Odds: Technical explanation | $~$1$~$ |
---|
Odds: Technical explanation | $~$-\infty$~$ |
---|
Odds: Technical explanation | $~$+\infty$~$ |
---|
Odds: Technical explanation | $~$0$~$ |
---|
Odds: Technical explanation | $~$1,$~$ |
---|
Odds: Technical explanation | $~$0$~$ |
---|
Odds: Technical explanation | $~$1$~$ |
---|
Of arguments and wagers | $~$100; if I lose I pay $~$ |
---|
Of arguments and wagers | $~$0.10 against $~$ |
---|
Ontology identification problem | $~$l$~$ |
---|
Ontology identification problem | $~$t$~$ |
---|
Ontology identification problem: Technical tutorial | $~$l$~$ |
---|
Ontology identification problem: Technical tutorial | $~$t$~$ |
---|
Operations in Set theory | $~$\cup$~$ |
---|
Operations in Set theory | $~$\cap$~$ |
---|
Operations in Set theory | $~$\setminus$~$ |
---|
Operations in Set theory | $~$\times$~$ |
---|
Operator | $~$f$~$ |
---|
Operator | $~$S$~$ |
---|
Operator | $~$S$~$ |
---|
Operator | $~$S$~$ |
---|
Operator | $~$f$~$ |
---|
Operator | $~$f$~$ |
---|
Operator | $~$S$~$ |
---|
Operator | $~$S$~$ |
---|
Operator | $~$f$~$ |
---|
Operator | $~$+$~$ |
---|
Operator | $~$\mathbb N$~$ |
---|
Operator | $~$\mathbb N$~$ |
---|
Operator | $~$+$~$ |
---|
Operator | $~$\mathbb N$~$ |
---|
Operator | $~$\mathbb N$~$ |
---|
Operator | $~$+$~$ |
---|
Operator | $~$\operatorname{neg}$~$ |
---|
Operator | $~$x$~$ |
---|
Operator | $~$-x$~$ |
---|
Operator | $~$\mathbb Z$~$ |
---|
Operator | $~$\mathbb Z$~$ |
---|
Operator | $~$\mathbb Z$~$ |
---|
Operator | $~$\operatorname{neg}$~$ |
---|
Operator | $~$\mathbb N$~$ |
---|
Operator | $~$\mathbb N$~$ |
---|
Operator | $~$\operatorname{neg}$~$ |
---|
Operator | $~$\operatorname{neg}(3)=-3$~$ |
---|
Operator | $~$\mathbb N$~$ |
---|
Operator | $~$\operatorname{zero}$~$ |
---|
Operator | $~$0$~$ |
---|
Operator | $~$f(a, b, c, d) = ac - bd$~$ |
---|
Optimizing with comparisons | $~$\mathbb{E}$~$ |
---|
Optimizing with comparisons | $~$\mathbb{E}$~$ |
---|
Optimizing with comparisons | $~$\mathbb{E}$~$ |
---|
Optimizing with comparisons | $~$\mathbb{E}$~$ |
---|
Orbit-stabiliser theorem | $~$G$~$ |
---|
Orbit-stabiliser theorem | $~$X$~$ |
---|
Orbit-stabiliser theorem | $~$G$~$ |
---|
Orbit-stabiliser theorem | $~$X$~$ |
---|
Orbit-stabiliser theorem | $~$X$~$ |
---|
Orbit-stabiliser theorem | $~$x \in X$~$ |
---|
Orbit-stabiliser theorem | $~$x$~$ |
---|
Orbit-stabiliser theorem | $~$x$~$ |
---|
Orbit-stabiliser theorem | $~$x$~$ |
---|
Orbit-stabiliser theorem | $~$G$~$ |
---|
Orbit-stabiliser theorem | $~$x$~$ |
---|
Orbit-stabiliser theorem | $~$x$~$ |
---|
Orbit-stabiliser theorem | $~$x$~$ |
---|
Orbit-stabiliser theorem | $~$G$~$ |
---|
Orbit-stabiliser theorem | $~$y$~$ |
---|
Orbit-stabiliser theorem | $~$x$~$ |
---|
Orbit-stabiliser theorem | $~$G$~$ |
---|
Orbit-stabiliser theorem | $~$y$~$ |
---|
Orbit-stabiliser theorem | $~$G$~$ |
---|
Orbit-stabiliser theorem | $~$X$~$ |
---|
Orbit-stabiliser theorem | $~$x \in X$~$ |
---|
Orbit-stabiliser theorem | $~$\mathrm{Stab}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$x$~$ |
---|
Orbit-stabiliser theorem | $~$\mathrm{Orb}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$x$~$ |
---|
Orbit-stabiliser theorem | $$~$|G| = |\mathrm{Stab}_G(x)| \times |\mathrm{Orb}_G(x)|$~$$ |
---|
Orbit-stabiliser theorem | $~$| \cdot |$~$ |
---|
Orbit-stabiliser theorem | $~$G$~$ |
---|
Orbit-stabiliser theorem | $~$X$~$ |
---|
Orbit-stabiliser theorem | $~$x \in X$~$ |
---|
Orbit-stabiliser theorem | $~$\mathrm{Stab}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$x$~$ |
---|
Orbit-stabiliser theorem | $~$\mathrm{Orb}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$x$~$ |
---|
Orbit-stabiliser theorem | $$~$|G| = |\mathrm{Stab}_G(x)| \times |\mathrm{Orb}_G(x)|$~$$ |
---|
Orbit-stabiliser theorem | $~$| \cdot |$~$ |
---|
Orbit-stabiliser theorem | $~$\mathrm{Stab}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$\mathrm{Orb}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $$~$|\mathrm{Orb}_G(x)| |\mathrm{Stab}_G(x)| = |\{ \text{left cosets of} \ \mathrm{Stab}_G(x) \}| |\mathrm{Stab}_G(x)|$~$$ |
---|
Orbit-stabiliser theorem | $~$|G|$~$ |
---|
Orbit-stabiliser theorem | $~$G$~$ |
---|
Orbit-stabiliser theorem | $~$\theta: \mathrm{Orb}_G(x) \to \{ \text{left cosets of} \ \mathrm{Stab}_G(x) \}$~$ |
---|
Orbit-stabiliser theorem | $$~$g(x) \mapsto g \mathrm{Stab}_G(x)$~$$ |
---|
Orbit-stabiliser theorem | $~$\mathrm{Orb}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$g(x)$~$ |
---|
Orbit-stabiliser theorem | $~$g \in G$~$ |
---|
Orbit-stabiliser theorem | $~$g(x) = h(x)$~$ |
---|
Orbit-stabiliser theorem | $~$g \mathrm{Stab}_G(x) = h \mathrm{Stab}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$h^{-1}g(x) = x$~$ |
---|
Orbit-stabiliser theorem | $~$h^{-1}g \in \mathrm{Stab}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$g \mathrm{Stab}_G(x) = h \mathrm{Stab}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$g(x)=h(x)$~$ |
---|
Orbit-stabiliser theorem | $~$h^{-1} g \in \mathrm{Stab}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$h^{-1}g(x) = x$~$ |
---|
Orbit-stabiliser theorem | $~$g(x) = h(x)$~$ |
---|
Orbit-stabiliser theorem | $~$g \mathrm{Stab}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$g(x) \in \mathrm{Orb}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$g(x)$~$ |
---|
Orbit-stabiliser theorem | $~$g \mathrm{Stab}_G(x)$~$ |
---|
Orbit-stabiliser theorem | $~$\theta$~$ |
---|
Order of a group | $~$|G|$~$ |
---|
Order of a group | $~$G$~$ |
---|
Order of a group | $~$G=(X,\bullet)$~$ |
---|
Order of a group | $~$X$~$ |
---|
Order of a group | $~$G$~$ |
---|
Order of a group | $~$9$~$ |
---|
Order of a group | $~$X$~$ |
---|
Order of a group | $~$G$~$ |
---|
Order of a group | $~$X$~$ |
---|
Order of a group | $~$G$~$ |
---|
Order of a group | $~$g \in G$~$ |
---|
Order of a group | $~$n$~$ |
---|
Order of a group | $~$g^n = e$~$ |
---|
Order of a group | $~$\infty$~$ |
---|
Order of a group | $~$g \in G$~$ |
---|
Order of a group | $~$\langle g \rangle = \{ 1, g, g^2, \dots \}$~$ |
---|
Order of a group | $~$G$~$ |
---|
Order of a group | $~$g$~$ |
---|
Order of a group element | $~$g$~$ |
---|
Order of a group element | $~$(G, +)$~$ |
---|
Order of a group element | $~$G$~$ |
---|
Order of a group element | $~$g$~$ |
---|
Order of a group element | $~$g$~$ |
---|
Order of a group element | $~$e$~$ |
---|
Order of a group element | $~$\langle g \rangle$~$ |
---|
Order of a group element | $~$g$~$ |
---|
Order of a group element | $~$\{ e, g, g^2, \dots, g^{-1}, g^{-2}, \dots \}$~$ |
---|
Order of a group element | $~$+$~$ |
---|
Order of a group element | $~$1$~$ |
---|
Order of a group element | $~$S_5$~$ |
---|
Order of a group element | $~$C_6$~$ |
---|
Order of a group element | $~$6$~$ |
---|
Order of a group element | $~$C_6$~$ |
---|
Order of a group element | $~$6$~$ |
---|
Order of a group element | $~$0$~$ |
---|
Order of a group element | $~$1$~$ |
---|
Order of a group element | $~$1$~$ |
---|
Order of a group element | $~$5$~$ |
---|
Order of a group element | $~$6$~$ |
---|
Order of a group element | $~$2$~$ |
---|
Order of a group element | $~$4$~$ |
---|
Order of a group element | $~$3$~$ |
---|
Order of a group element | $~$3$~$ |
---|
Order of a group element | $~$2$~$ |
---|
Order of a group element | $~$\mathbb{Z}$~$ |
---|
Order of a group element | $~$0$~$ |
---|
Order of a group element | $~$0$~$ |
---|
Order of a group element | $~$1$~$ |
---|
Order of operations | $~$2 - 4 + 3$~$ |
---|
Order of operations | $~$2 - 4$~$ |
---|
Order of operations | $~$4 + 3$~$ |
---|
Order of operations | $~$1$~$ |
---|
Order of operations | $~$-5$~$ |
---|
Order of operations | $~$7 + 8 \times 9 - 6$~$ |
---|
Order of operations | $~$31$~$ |
---|
Order of operations | $~$129$~$ |
---|
Order of operations | $~$3 + 7 \times 2^{(6 + 8)}$~$ |
---|
Order of operations | $$~$3 + 7 \times 2^{14}$~$$ |
---|
Order of operations | $$~$3 + 7 \times 16384$~$$ |
---|
Order of operations | $$~$3 + 114688$~$$ |
---|
Order of operations | $$~$114691$~$$ |
---|
Order of operations | $~$48 \div 2 (9 + 3)$~$ |
---|
Order of operations | $~$2(3+5)$~$ |
---|
Order of operations | $~$2 \times (3 + 5)$~$ |
---|
Order of operations | $~$16$~$ |
---|
Order of operations | $$~$\begin{align*}
48 \div 2 (9 + 3) &= 48 \div 2 (12) \\
&= 48 \div 24 \\
&= 2
\end{align*}$~$$ |
---|
Order of operations | $$~$\begin{align*}
48 \div 2 (9 + 3) &= 48 \div 2 \times 12 \\
&= 24 \times 12 \\
&= 288
\end{align*}$~$$ |
---|
Order relation | $~$\le$~$ |
---|
Order relation | $~$S$~$ |
---|
Order relation | $~$a \in S$~$ |
---|
Order relation | $~$a \le a$~$ |
---|
Order relation | $~$a, b \in S$~$ |
---|
Order relation | $~$a \le b$~$ |
---|
Order relation | $~$b \le a$~$ |
---|
Order relation | $~$a = b$~$ |
---|
Order relation | $~$a, b, c \in S$~$ |
---|
Order relation | $~$a \le b$~$ |
---|
Order relation | $~$b \le c$~$ |
---|
Order relation | $~$a \le c$~$ |
---|
Order relation | $~$\le$~$ |
---|
Order relation | $~$a, b \in S$~$ |
---|
Order relation | $~$a \le b$~$ |
---|
Order relation | $~$b \le a$~$ |
---|
Order relation | $~$a = b$~$ |
---|
Order relation | $~$S$~$ |
---|
Order relation | $~$\le$~$ |
---|
Order relation | $~$X$~$ |
---|
Order relation | $~$S$~$ |
---|
Order relation | $~$X$~$ |
---|
Order relation | $~$x$~$ |
---|
Order relation | $~$y \in X$~$ |
---|
Order relation | $~$x \leq y$~$ |
---|
Order relation | $~$\ge$~$ |
---|
Order relation | $~$a \ge b$~$ |
---|
Order relation | $~$b \le a$~$ |
---|
Order relation | $~$(S, \le)$~$ |
---|
Order relation | $~$<$~$ |
---|
Order relation | $~$a, b \in S$~$ |
---|
Order relation | $~$a < b$~$ |
---|
Order relation | $~$a \le b$~$ |
---|
Order relation | $~$a \neq b$~$ |
---|
Order relation | $~$>$~$ |
---|
Order relation | $~$a > b$~$ |
---|
Order relation | $~$b \le a$~$ |
---|
Order relation | $~$a \neq b$~$ |
---|
Order relation | $~$a$~$ |
---|
Order relation | $~$b$~$ |
---|
Order relation | $~$a \leq b$~$ |
---|
Order relation | $~$b \leq a$~$ |
---|
Order relation | $~$a$~$ |
---|
Order relation | $~$b$~$ |
---|
Order relation | $~$a \parallel b$~$ |
---|
Order relation | $~$(S, \leq)$~$ |
---|
Order relation | $~$\prec$~$ |
---|
Order relation | $~$a, b \in S$~$ |
---|
Order relation | $~$a \prec b$~$ |
---|
Order relation | $~$a < b$~$ |
---|
Order relation | $~$s \in S$~$ |
---|
Order relation | $~$a \leq s < b$~$ |
---|
Order relation | $~$a = s$~$ |
---|
Order relation | $~$a \prec b$~$ |
---|
Order relation | $~$b$~$ |
---|
Order relation | $~$S$~$ |
---|
Order relation | $~$a$~$ |
---|
Order relation | $~$a \prec b$~$ |
---|
Order relation | $~$a$~$ |
---|
Order relation | $~$b$~$ |
---|
Order relation | $~$b$~$ |
---|
Order relation | $~$a$~$ |
---|
Order relation | $~$b$~$ |
---|
Order relation | $~$a$~$ |
---|
Order theory | $~$(a,b)$~$ |
---|
Order theory | $~$a$~$ |
---|
Order theory | $~$b$~$ |
---|
Order theory | $~$(a,b)$~$ |
---|
Order theory | $~$a$~$ |
---|
Order theory | $~$b$~$ |
---|
Order theory | $~$(a,b)$~$ |
---|
Order theory | $~$\{ (Monday, Tuesday), (Tuesday, Wednesday), … \}$~$ |
---|
Order theory | $~$\langle P, \leq \rangle$~$ |
---|
Order theory | $~$P$~$ |
---|
Order theory | $~$\leq$~$ |
---|
Order theory | $~$P$~$ |
---|
Order theory | $~$p,q,r \in P$~$ |
---|
Order theory | $~$p \leq p$~$ |
---|
Order theory | $~$p \leq q$~$ |
---|
Order theory | $~$q \leq r$~$ |
---|
Order theory | $~$p \leq r$~$ |
---|
Order theory | $~$p \leq q$~$ |
---|
Order theory | $~$q \leq p$~$ |
---|
Order theory | $~$p = q$~$ |
---|
Order theory | $~$P$~$ |
---|
Order theory | $~$\leq$~$ |
---|
Order theory | $~$a$~$ |
---|
Order theory | $~$b$~$ |
---|
Order theory | $~$a \leq b$~$ |
---|
Order theory | $~$b \leq a$~$ |
---|
Order theory | $~$a \parallel b$~$ |
---|
Order theory | $~$a$~$ |
---|
Order theory | $~$b$~$ |
---|
Order theory | $~$\langle P, \leq \rangle$~$ |
---|
Order theory | $~$<$~$ |
---|
Order theory | $~$\leq$~$ |
---|
Order theory | $~$P$~$ |
---|
Ordered ring | $~$R=(X,\oplus,\otimes)$~$ |
---|
Ordered ring | $~$\leq$~$ |
---|
Ordered ring | $~$a,b,c \in X$~$ |
---|
Ordered ring | $~$a \leq b$~$ |
---|
Ordered ring | $~$a \oplus c \leq b \oplus c$~$ |
---|
Ordered ring | $~$0 \leq a$~$ |
---|
Ordered ring | $~$0 \leq b$~$ |
---|
Ordered ring | $~$0 \leq a \otimes b$~$ |
---|
Ordered ring | $~$a$~$ |
---|
Ordered ring | $~$0<a$~$ |
---|
Ordered ring | $~$a<0$~$ |
---|
Ordered ring | $~$a$~$ |
---|
Ordered ring | $~$a \leq 0$~$ |
---|
Ordered ring | $~$0 \leq -a$~$ |
---|
Ordered ring | $~$a \leq 0$~$ |
---|
Ordered ring | $~$-a$~$ |
---|
Ordered ring | $~$a+(-a) = 0 \leq -a$~$ |
---|
Ordered ring | $~$0 \leq -a$~$ |
---|
Ordered ring | $~$a \leq -a+a = 0$~$ |
---|
Ordered ring | $~$a$~$ |
---|
Ordered ring | $~$b$~$ |
---|
Ordered ring | $~$R$~$ |
---|
Ordered ring | $~$a,b \leq 0$~$ |
---|
Ordered ring | $~$a+(-a) = 0 \leq -a$~$ |
---|
Ordered ring | $~$0 \leq -b$~$ |
---|
Ordered ring | $~$0 \leq -a \otimes -b$~$ |
---|
Ordered ring | $~$-a \otimes -b = a \otimes b$~$ |
---|
Ordered ring | $~$0 \leq a \otimes b$~$ |
---|
Ordered ring | $~$a$~$ |
---|
Ordered ring | $~$0 \leq a$~$ |
---|
Ordered ring | $~$a \leq 0$~$ |
---|
Ordered ring | $~$0 \leq a^2$~$ |
---|
Ordered ring | $~$0 \leq a^2$~$ |
---|
Ordered ring | $~$a$~$ |
---|
Ordered ring | $~$0 \leq a^2$~$ |
---|
Ordered ring | $~$1 \geq 0$~$ |
---|
Ordered ring | $~$1>0$~$ |
---|
Ordered ring | $~$1 = 1 \otimes 1$~$ |
---|
Ordered ring | $~$1$~$ |
---|
Ordered ring | $~$\mathbb R$~$ |
---|
Ordered ring | $~$\mathbb Q$~$ |
---|
Ordered ring | $~$0$~$ |
---|
Ordered ring | $~$i$~$ |
---|
Ordered ring | $~$0 \le i$~$ |
---|
Ordered ring | $~$0 \le i \times i = -1$~$ |
---|
Ordered ring | $~$i \le 0$~$ |
---|
Ordered ring | $~$0 = i + (-i) \le 0 + (-i)$~$ |
---|
Ordered ring | $~$0 \le (-i) \times (-i) = -1$~$ |
---|
Ordered ring | $~$i^2=-1$~$ |
---|
Ordered ring | $~$0 \leq -1$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{16}{107} - \frac{3}{20}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{16}{107} - \frac{3}{20} = -\frac{1}{2140}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0 < \frac{5}{16}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0 < 0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0 > 0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0=0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$<$~$ |
---|
Ordering of rational numbers (Math 0) | $~$>$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{5}{6}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{3}{4}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{3}{4}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{5}{6}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{3}{4}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{5}{6} - \frac{3}{4}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{3}{4} - \frac{3}{4}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{5}{6}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{3}{4}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{1}{12}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{5}{6} > \frac{3}{4}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$-\frac{59}{12}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{4}{7}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$-\frac{59}{12}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{4}{7}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{59}{12}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$-\frac{59}{12}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{4}{7}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{461}{84}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{461}{84}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{4}{7}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$-\frac{59}{12}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{-3}{5}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{9}{-11}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$-\frac{3}{5}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$-\frac{9}{11}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{3}{5}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$-\frac{9}{11} + \frac{3}{5} = -\frac{12}{55}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{12}{55}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{12}{55}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{12}{55}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{-3}{5}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{9}{-11}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{a}{b}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{c}{d}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$0$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{c}{d} - \frac{a}{b}$~$ |
---|
Ordering of rational numbers (Math 0) | $$~$\frac{c}{d} - \frac{a}{b} = \frac{c \times b - a \times d}{b \times d}$~$$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{5}{6}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{-4}{7}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{3}{-8}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{-2}{-9}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{-1}{-1} = 1$~$ |
---|
Ordering of rational numbers (Math 0) | $~$b$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{a}{b} = \frac{-a}{-b}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$-b$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{5}{-6}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$b=-6$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{-5}{6}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{-7}{-8}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{7}{8}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{c}{d}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{-c}{-d}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$b, d$~$ |
---|
Ordering of rational numbers (Math 0) | $~$\frac{c \times b - a \times d}{b \times d}$~$ |
---|
Ordering of rational numbers (Math 0) | $~$c \times b - a \times d$~$ |
---|
Ordering of rational numbers (Math 0) | $~$cb > ad$~$ |
---|
Ordering of rational numbers (Math 0) | $~$b$~$ |
---|
Ordering of rational numbers (Math 0) | $~$d$~$ |
---|
Ordering of rational numbers (Math 0) | $$~$\frac{c}{d} - \frac{a}{b} = \frac{-c}{-d} - \frac{a}{b} = \frac{(-c) \times b - a \times (-d)}{b \times (-d)}$~$$ |
---|
Ordering of rational numbers (Math 0) | $~$b \times (-d)$~$ |
---|
Orthogonality Thesis | $~$\pi_0$~$ |
---|
Orthogonality Thesis | $~$\pi$~$ |
---|
Orthogonality Thesis | $~$\pi_0$~$ |
---|
Orthogonality Thesis | $~$\pi$~$ |
---|
Orthogonality Thesis | $~$U,$~$ |
---|
Orthogonality Thesis | $~$U$~$ |
---|
Orthogonality Thesis | $~$U$~$ |
---|
Orthogonality Thesis | $~$U$~$ |
---|
Orthogonality Thesis | $~$U$~$ |
---|
Orthogonality Thesis | $~$U$~$ |
---|
Orthogonality Thesis | $~$U$~$ |
---|
Orthogonality Thesis | $~$U$~$ |
---|
Orthogonality Thesis | $~$U$~$ |
---|
Orthogonality Thesis | $~$P$~$ |
---|
Orthogonality Thesis | $~$P$~$ |
---|
Orthogonality Thesis | $~$2^{1,000,000}$~$ |
---|
Orthogonality Thesis | $~$P$~$ |
---|
Orthogonality Thesis | $~$2^{1,000,000}$~$ |
---|
Orthogonality Thesis | $~$<_V.$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$<_V,$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$>_{paperclips}$~$ |
---|
Orthogonality Thesis | $~$<_V.$~$ |
---|
Orthogonality Thesis | $~$<_V.$~$ |
---|
Orthogonality Thesis | $~$>_{paperclips}.$~$ |
---|
Orthogonality Thesis | $~$\pi_0$~$ |
---|
Orthogonality Thesis | $~$\pi$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$<_V.$~$ |
---|
Orthogonality Thesis | $~$>_{paperclips}$~$ |
---|
Orthogonality Thesis | $~$<_V,$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$>_{paperclips}$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$>_{paperclips}$~$ |
---|
Orthogonality Thesis | $~$\ll_W$~$ |
---|
Orthogonality Thesis | $~$<$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$>_{paperclips}$~$ |
---|
Orthogonality Thesis | $~$>_{paperclips} \ll_W <_V$~$ |
---|
Orthogonality Thesis | $~$\ll_W,$~$ |
---|
Orthogonality Thesis | $~$\gg_{paperclips},$~$ |
---|
Orthogonality Thesis | $~$\ll_W$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$\ll_W$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$\ll_W$~$ |
---|
Orthogonality Thesis | $~$\ll_W$~$ |
---|
Orthogonality Thesis | $~$\ll_W$~$ |
---|
Orthogonality Thesis | $~$\gg_{something}$~$ |
---|
Orthogonality Thesis | $~$<_V.$~$ |
---|
Orthogonality Thesis | $~$<_V,$~$ |
---|
Orthogonality Thesis | $~$<_V$~$ |
---|
Orthogonality Thesis | $~$<_V.$~$ |
---|
Other-izing (wanted: new optimization idiom) | $~$\theta,$~$ |
---|
Other-izing (wanted: new optimization idiom) | $~$\theta,$~$ |
---|
P vs NP | $~$P$~$ |
---|
P vs NP | $~$NP$~$ |
---|
P vs NP | $~$P\subseteq NP$~$ |
---|
P vs NP | $~$P \supseteq NP$~$ |
---|
P vs NP: Arguments against P=NP | $~$P \neq NP$~$ |
---|
P vs NP: Arguments against P=NP | $~$P=NP$~$ |
---|
P vs NP: Arguments against P=NP | $~$NP$~$ |
---|
P vs NP: Arguments against P=NP | $~$P \neq NP$~$ |
---|
P vs NP: Arguments against P=NP | $~$P \neq NP,$~$ |
---|
P vs NP: Arguments against P=NP | $~$P \neq NP$~$ |
---|
P vs NP: Arguments against P=NP | $~$P \neq NP$~$ |
---|
P vs NP: Arguments against P=NP | $~$P=NP$~$ |
---|
P vs NP: Arguments against P=NP | $~$P=NP,$~$ |
---|
P vs NP: Arguments against P=NP | $~$NP$~$ |
---|
P vs NP: Arguments against P=NP | $~$P=NP$~$ |
---|
P vs NP: Arguments against P=NP | $~$NP$~$ |
---|
P vs NP: Arguments against P=NP | $~$P \neq NP$~$ |
---|
Partial function | $~$f: A \to B$~$ |
---|
Partial function | $~$f(a)$~$ |
---|
Partial function | $~$a \in A$~$ |
---|
Partial function | $~$a = b$~$ |
---|
Partial function | $~$f(a)$~$ |
---|
Partial function | $~$f(b)$~$ |
---|
Partial function | $~$f(a) = f(b)$~$ |
---|
Partial function | $~$f(a)$~$ |
---|
Partial function | $~$f: A \rightharpoonup B$~$ |
---|
Partial function | $~$f$~$ |
---|
Partial function | $~$A$~$ |
---|
Partial function | $~$B$~$ |
---|
Partial function | $~$f(x)$~$ |
---|
Partial function | $~$x \in A$~$ |
---|
Partial function | $~$f(x) \in B$~$ |
---|
Partial function | $~$f$~$ |
---|
Partial function | $~$(a, b)$~$ |
---|
Partial function | $~$x \in A$~$ |
---|
Partial function | $~$f$~$ |
---|
Partial function | $~$(a, b)$~$ |
---|
Partial function | $~$f$~$ |
---|
Partial function | $~$a \in A$~$ |
---|
Partial function | $~$b \in B$~$ |
---|
Partial function | $~$(a, b)$~$ |
---|
Partial function | $~$(a, c)$~$ |
---|
Partial function | $~$f$~$ |
---|
Partial function | $~$b=c$~$ |
---|
Partial function | $~$f$~$ |
---|
Partial function | $~$(a,b)$~$ |
---|
Partial function | $~$(a, b)$~$ |
---|
Partial function | $~$f$~$ |
---|
Partial function | $~$a \in A$~$ |
---|
Partial function | $~$b \in B$~$ |
---|
Partial function | $~$(a, b)$~$ |
---|
Partial function | $~$(a, c)$~$ |
---|
Partial function | $~$f$~$ |
---|
Partial function | $~$b=c$~$ |
---|
Partial function | $~$\mathcal{T}$~$ |
---|
Partial function | $~$f: \mathbb{N} \to \mathbb{N}$~$ |
---|
Partial function | $~$f(n)$~$ |
---|
Partial function | $~$\mathcal{T}$~$ |
---|
Partial function | $~$n$~$ |
---|
Partial function | $~$\mathcal{T}$~$ |
---|
Partial function | $~$n$~$ |
---|
Partial function | $~$n = 3$~$ |
---|
Partial function | $~$1$~$ |
---|
Partial function | $~$f(4)$~$ |
---|
Partial function | $~$f$~$ |
---|
Partial function | $~$\mathcal{T}$~$ |
---|
Partial function | $~$\mathcal{T}$~$ |
---|
Partial function | $~$f$~$ |
---|
Partial function | $~$3$~$ |
---|
Partial function | $~$f(3) = 1$~$ |
---|
Partially ordered set | $~$\langle P, \leq \rangle$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$\leq$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$p,q,r \in P$~$ |
---|
Partially ordered set | $~$p \leq p$~$ |
---|
Partially ordered set | $~$p \leq q$~$ |
---|
Partially ordered set | $~$q \leq r$~$ |
---|
Partially ordered set | $~$p \leq r$~$ |
---|
Partially ordered set | $~$p \leq q$~$ |
---|
Partially ordered set | $~$q \leq p$~$ |
---|
Partially ordered set | $~$p = q$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$\leq$~$ |
---|
Partially ordered set | $~$\langle P, \leq \rangle$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$\langle P, \leq \rangle$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$\leq$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$p,q,r \in P$~$ |
---|
Partially ordered set | $~$p \leq p$~$ |
---|
Partially ordered set | $~$p \leq q$~$ |
---|
Partially ordered set | $~$q \leq r$~$ |
---|
Partially ordered set | $~$p \leq r$~$ |
---|
Partially ordered set | $~$p \leq q$~$ |
---|
Partially ordered set | $~$q \leq p$~$ |
---|
Partially ordered set | $~$p = q$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$\leq$~$ |
---|
Partially ordered set | $~$\geq$~$ |
---|
Partially ordered set | $~$a \geq b$~$ |
---|
Partially ordered set | $~$b \leq a$~$ |
---|
Partially ordered set | $~$p$~$ |
---|
Partially ordered set | $~$q$~$ |
---|
Partially ordered set | $~$p \leq q$~$ |
---|
Partially ordered set | $~$q \leq p$~$ |
---|
Partially ordered set | $~$p$~$ |
---|
Partially ordered set | $~$q$~$ |
---|
Partially ordered set | $~$p \parallel q$~$ |
---|
Partially ordered set | $~$\langle P, \leq \rangle$~$ |
---|
Partially ordered set | $~$<$~$ |
---|
Partially ordered set | $~$p, q \in P$~$ |
---|
Partially ordered set | $~$p < q$~$ |
---|
Partially ordered set | $~$p \leq q$~$ |
---|
Partially ordered set | $~$p \neq q$~$ |
---|
Partially ordered set | $~$\langle P, \leq \rangle$~$ |
---|
Partially ordered set | $~$\prec$~$ |
---|
Partially ordered set | $~$p,q \in P$~$ |
---|
Partially ordered set | $~$p \prec q$~$ |
---|
Partially ordered set | $~$p < q$~$ |
---|
Partially ordered set | $~$r \in P$~$ |
---|
Partially ordered set | $~$p \leq r < q$~$ |
---|
Partially ordered set | $~$p = r$~$ |
---|
Partially ordered set | $~$p \prec q$~$ |
---|
Partially ordered set | $~$q$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$p$~$ |
---|
Partially ordered set | $~$p \prec q$~$ |
---|
Partially ordered set | $~$p$~$ |
---|
Partially ordered set | $~$q$~$ |
---|
Partially ordered set | $~$q$~$ |
---|
Partially ordered set | $~$p$~$ |
---|
Partially ordered set | $~$q$~$ |
---|
Partially ordered set | $~$p$~$ |
---|
Partially ordered set | $~$\langle P, \leq \rangle$~$ |
---|
Partially ordered set | $~$P = \{ p, q, r \}$~$ |
---|
Partially ordered set | $~$\leq = \{(p,p),(p,q)(p,r),(q,q),(q,r),(r,r) \}$~$ |
---|
Partially ordered set | $~$\leq$~$ |
---|
Partially ordered set | $~$\leq$~$ |
---|
Partially ordered set | $~$(p,r)$~$ |
---|
Partially ordered set | $~$\langle P, \leq \rangle$~$ |
---|
Partially ordered set | $~$\langle P, \leq \rangle$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$(p,q) \in P^2$~$ |
---|
Partially ordered set | $~$p \prec q$~$ |
---|
Partially ordered set | $~$p$~$ |
---|
Partially ordered set | $~$q$~$ |
---|
Partially ordered set | $~$p \leq q$~$ |
---|
Partially ordered set | $~$p$~$ |
---|
Partially ordered set | $~$q$~$ |
---|
Partially ordered set | $~$t \parallel r$~$ |
---|
Partially ordered set | $~$t$~$ |
---|
Partially ordered set | $~$r$~$ |
---|
Partially ordered set | $~$\langle P, \leq \rangle$~$ |
---|
Partially ordered set | $~$\langle P^{\partial}, \geq \rangle$~$ |
---|
Partially ordered set | $~$P^{\partial}$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$\geq$~$ |
---|
Partially ordered set | $~$\leq$~$ |
---|
Partially ordered set | $~$P^{\partial}$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$\phi$~$ |
---|
Partially ordered set | $~$\phi$~$ |
---|
Partially ordered set | $~$\phi^{\partial}$~$ |
---|
Partially ordered set | $~$\leq$~$ |
---|
Partially ordered set | $~$\geq$~$ |
---|
Partially ordered set | $~$\geq$~$ |
---|
Partially ordered set | $~$\leq$~$ |
---|
Partially ordered set | $~$\forall P. \phi$~$ |
---|
Partially ordered set | $~$\forall P. \phi^{\partial}$~$ |
---|
Partially ordered set | $~$\forall P. \phi$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$\forall P. \phi^{\partial}$~$ |
---|
Partially ordered set | $~$P^{\partial}$~$ |
---|
Partially ordered set | $~$a \leq b$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$a \geq b$~$ |
---|
Partially ordered set | $~$P^{\partial}$~$ |
---|
Partially ordered set | $~$\langle P, \leq \rangle$~$ |
---|
Partially ordered set | $~$P$~$ |
---|
Partially ordered set | $~$p,q \in P$~$ |
---|
Partially ordered set | $~$p$~$ |
---|
Partially ordered set | $~$q$~$ |
---|
Partially ordered set | $~$p \leq q$~$ |
---|
Partially ordered set | $~$\leq$~$ |
---|
Partially ordered set | $~$\leq$~$ |
---|
Patch resistance | $~$alarm \rightarrow burglar, \ earthquake \rightarrow alarm,$~$ |
---|
Patch resistance | $~$(alarm \wedge earthquake) \rightarrow \neg burglar.$~$ |
---|
Patch resistance | $~$k$~$ |
---|
Patch resistance | $~$l \gg k.$~$ |
---|
Peano Arithmetic | $~$\left\{(,),\wedge,\vee,\neg,\to,\leftrightarrow,\in,\forall,\exists,=,+,\cdot,O,S,N \right\}$~$ |
---|
Peano Arithmetic | $~$x, y, z, \dots$~$ |
---|
Peano Arithmetic | $~$O$~$ |
---|
Peano Arithmetic | $~$S$~$ |
---|
Peano Arithmetic | $~$SO$~$ |
---|
Peano Arithmetic | $~$SSO$~$ |
---|
Peano Arithmetic | $~$N$~$ |
---|
Peano Arithmetic | $~$SO+SO=SSO$~$ |
---|
Peano Arithmetic | $~$\forall x \in N\; Sx \cdot Sx = x\cdot x + SSO \cdot x + SO$~$ |
---|
Peano Arithmetic | $~$\exists x\in N \; SSO\cdot x = SSSO$~$ |
---|
Pi | $~$π$~$ |
---|
Pi | $~$π$~$ |
---|
Pi | $~$3.141593$~$ |
---|
Pi | $~$N$~$ |
---|
Pi | $~$∞$~$ |
---|
Pi | $~$π$~$ |
---|
Pi | $~$π$~$ |
---|
Pi is irrational | $~$\pi$~$ |
---|
Pi is irrational | $~$q$~$ |
---|
Pi is irrational | $~$n$~$ |
---|
Pi is irrational | $$~$A_n = \frac{q^n}{n!} \int_0^{\pi} [x (\pi - x)]^n \sin(x) dx$~$$ |
---|
Pi is irrational | $~$n!$~$ |
---|
Pi is irrational | $~$n$~$ |
---|
Pi is irrational | $~$\int$~$ |
---|
Pi is irrational | $~$\sin$~$ |
---|
Pi is irrational | $~$A_n = (4n-2) q A_{n-1} - (q \pi)^2 A_{n-2}$~$ |
---|
Pi is irrational | $$~$A_0 = \int_0^{\pi} \sin(x) dx = 2$~$$ |
---|
Pi is irrational | $~$A_0$~$ |
---|
Pi is irrational | $$~$A_1 = q \int_0^{\pi} x (\pi-x) \sin(x) dx$~$$ |
---|
Pi is irrational | $~$4q$~$ |
---|
Pi is irrational | $$~$\frac{A_1}{q} = \int_0^{\pi} x (\pi-x) \sin(x) dx = \pi \int_0^{\pi} x \sin(x) dx - \int_0^{\pi} x^2 \sin(x) dx$~$$ |
---|
Pi is irrational | $$~$[-x \cos(x)]_0^{\pi} + \int_0^{\pi} \cos(x) dx = \pi$~$$ |
---|
Pi is irrational | $$~$[-x^2 \cos(x)]_{0}^{\pi} + \int_0^{\pi} 2x \cos(x) dx$~$$ |
---|
Pi is irrational | $$~$\pi^2 + 2 \left( [x \sin(x)]_0^{\pi} - \int_0^{\pi} \sin(x) dx \right)$~$$ |
---|
Pi is irrational | $$~$\pi^2 -4$~$$ |
---|
Pi is irrational | $$~$\frac{A_1}{q} = \pi^2 - (\pi^2 - 4) = 4$~$$ |
---|
Pi is irrational | $~$q$~$ |
---|
Pi is irrational | $~$q \pi$~$ |
---|
Pi is irrational | $~$A_n$~$ |
---|
Pi is irrational | $~$(4n-2) q A_{n-1}$~$ |
---|
Pi is irrational | $~$(q \pi)^2 A_{n-2}$~$ |
---|
Pi is irrational | $~$A_n \to 0$~$ |
---|
Pi is irrational | $~$n \to \infty$~$ |
---|
Pi is irrational | $~$\int_0^{\pi} [x (\pi-x)]^n \sin(x) dx$~$ |
---|
Pi is irrational | $$~$\pi \times \max_{0 \leq x \leq \pi} [x (\pi-x)]^n \sin(x) \leq \pi \times \max_{0 \leq x \leq \pi} [x (\pi-x)]^n = \pi \times \left[\frac{\pi^2}{4}\right]^n$~$$ |
---|
Pi is irrational | $$~$|A_n| \leq \frac{1}{n!} \left[\frac{\pi^2 q}{4}\right]^n$~$$ |
---|
Pi is irrational | $~$n$~$ |
---|
Pi is irrational | $~$\frac{\pi^2 q}{4}$~$ |
---|
Pi is irrational | $~$n$~$ |
---|
Pi is irrational | $~$n$~$ |
---|
Pi is irrational | $~$0$~$ |
---|
Pi is irrational | $~$\frac{r^n}{n!} \to 0$~$ |
---|
Pi is irrational | $~$n \to \infty$~$ |
---|
Pi is irrational | $~$r > 0$~$ |
---|
Pi is irrational | $$~$\frac{r^{n+1}/(n+1)!}{r^n/n!} = \frac{r}{n+1}$~$$ |
---|
Pi is irrational | $~$n > 2r-1$~$ |
---|
Pi is irrational | $~$\frac{1}{2}$~$ |
---|
Pi is irrational | $~$\frac{1}{2}$~$ |
---|
Pi is irrational | $~$n$~$ |
---|
Pi is irrational | $~$0$~$ |
---|
Pi is irrational | $~$\pi$~$ |
---|
Pi is irrational | $~$\frac{p}{q}$~$ |
---|
Pi is irrational | $~$p, q$~$ |
---|
Pi is irrational | $~$q \pi$~$ |
---|
Pi is irrational | $~$p$~$ |
---|
Pi is irrational | $~$q$~$ |
---|
Pi is irrational | $~$A_n$~$ |
---|
Pi is irrational | $~$n$~$ |
---|
Pi is irrational | $~$A_n \to 0$~$ |
---|
Pi is irrational | $~$n \to \infty$~$ |
---|
Pi is irrational | $~$N$~$ |
---|
Pi is irrational | $~$|A_n| < \frac{1}{2}$~$ |
---|
Pi is irrational | $~$n > N$~$ |
---|
Pi is irrational | $~$n$~$ |
---|
Pi is irrational | $~$A_n$~$ |
---|
Pi is irrational | $~$0$~$ |
---|
Pi is irrational | $~$A_0 = 2$~$ |
---|
Pi is irrational | $~$A_1 = 4q$~$ |
---|
Pi is irrational | $~$0$~$ |
---|
Pi is irrational | $~$N$~$ |
---|
Pi is irrational | $~$A_n = 0$~$ |
---|
Pi is irrational | $~$n \geq N$~$ |
---|
Pi is irrational | $~$N > 1$~$ |
---|
Pi is irrational | $$~$0 = A_{N+1} = (4N-2) q A_N - (q \pi)^2 A_{N-1} = - (q \pi)^2 A_{N-1}$~$$ |
---|
Pi is irrational | $~$q=0$~$ |
---|
Pi is irrational | $~$\pi = 0$~$ |
---|
Pi is irrational | $~$A_{N-1} = 0$~$ |
---|
Pi is irrational | $~$q \not = 0$~$ |
---|
Pi is irrational | $~$q$~$ |
---|
Pi is irrational | $~$\pi \not = 0$~$ |
---|
Pi is irrational | $~$\pi$~$ |
---|
Pi is irrational | $~$A_{N-1}$~$ |
---|
Pi is irrational | $~$0$~$ |
---|
Pi is irrational | $~$N-1$~$ |
---|
Pi is irrational | $~$m$~$ |
---|
Pi is irrational | $~$A_n = 0$~$ |
---|
Pi is irrational | $~$n \geq m$~$ |
---|
Pi is irrational | $~$N$~$ |
---|
Pi is irrational | $~$\pi$~$ |
---|
Poset: Examples | $~$\leq$~$ |
---|
Poset: Examples | $~$\subseteq$~$ |
---|
Poset: Examples | $~$|$~$ |
---|
Poset: Examples | $~$\mathbb Z$~$ |
---|
Poset: Examples | $~$\leq$~$ |
---|
Poset: Examples | $~$\langle \mathbb Z, \leq \rangle$~$ |
---|
Poset: Examples | $~$X$~$ |
---|
Poset: Examples | $~$X$~$ |
---|
Poset: Examples | $~$\subseteq$~$ |
---|
Poset: Examples | $~$\langle \mathcal{P}(X), \subseteq \rangle$~$ |
---|
Poset: Examples | $~$\subseteq$~$ |
---|
Poset: Examples | $~$A,B \in \mathcal{P}(X)$~$ |
---|
Poset: Examples | $~$A \subseteq B$~$ |
---|
Poset: Examples | $~$B \subseteq A$~$ |
---|
Poset: Examples | $~$x \in A \Leftrightarrow x \in B$~$ |
---|
Poset: Examples | $~$A = B$~$ |
---|
Poset: Examples | $~$\subseteq$~$ |
---|
Poset: Examples | $~$A, B, C \in \mathcal{P}(X)$~$ |
---|
Poset: Examples | $~$A \subseteq B$~$ |
---|
Poset: Examples | $~$B \subseteq C$~$ |
---|
Poset: Examples | $~$x \in A \Rightarrow x \in B$~$ |
---|
Poset: Examples | $~$x \in B \Rightarrow x \in C$~$ |
---|
Poset: Examples | $~$\subseteq$~$ |
---|
Poset: Examples | $~$\Rightarrow$~$ |
---|
Poset: Examples | $~$\subset$~$ |
---|
Poset: Examples | $~$\langle \mathcal{P}(X), \subseteq \rangle$~$ |
---|
Poset: Examples | $~$\mathbb N$~$ |
---|
Poset: Examples | $~$|$~$ |
---|
Poset: Examples | $~$a|b$~$ |
---|
Poset: Examples | $~$k$~$ |
---|
Poset: Examples | $~$ak=b$~$ |
---|
Poset: Examples | $~$\langle \mathbb{N}, | \rangle$~$ |
---|
Poset: Examples | $~$|$~$ |
---|
Poset: Examples | $~$|$~$ |
---|
Poset: Examples | $~$a|b$~$ |
---|
Poset: Examples | $~$b|a$~$ |
---|
Poset: Examples | $~$k_1$~$ |
---|
Poset: Examples | $~$k_2$~$ |
---|
Poset: Examples | $~$a = k_1b$~$ |
---|
Poset: Examples | $~$b = k_2a$~$ |
---|
Poset: Examples | $~$a = k_1k_2a$~$ |
---|
Poset: Examples | $~$k$~$ |
---|
Poset: Examples | $~$0$~$ |
---|
Poset: Examples | $~$a$~$ |
---|
Poset: Examples | $~$b$~$ |
---|
Poset: Examples | $~$0$~$ |
---|
Poset: Examples | $~$k$~$ |
---|
Poset: Examples | $~$1$~$ |
---|
Poset: Examples | $~$a = k_1k_2a$~$ |
---|
Poset: Examples | $~$a = b$~$ |
---|
Poset: Examples | $~$|$~$ |
---|
Poset: Examples | $~$|$~$ |
---|
Poset: Examples | $~$a|b$~$ |
---|
Poset: Examples | $~$b|c$~$ |
---|
Poset: Examples | $~$k_1$~$ |
---|
Poset: Examples | $~$k_2$~$ |
---|
Poset: Examples | $~$a = k_1b$~$ |
---|
Poset: Examples | $~$b = k_2c$~$ |
---|
Poset: Examples | $~$a = k_1k_2c$~$ |
---|
Poset: Examples | $~$a|c$~$ |
---|
Poset: Exercises | $~$X$~$ |
---|
Poset: Exercises | $~$\mathcal{M}(X)$~$ |
---|
Poset: Exercises | $~$X$~$ |
---|
Poset: Exercises | $~$A \in \mathcal{M}(X)$~$ |
---|
Poset: Exercises | $~$A$~$ |
---|
Poset: Exercises | $~$1_A : X \rightarrow \mathbb N$~$ |
---|
Poset: Exercises | $~$X$~$ |
---|
Poset: Exercises | $~$A$~$ |
---|
Poset: Exercises | $~$\subseteq$~$ |
---|
Poset: Exercises | $~$A, B \in \mathcal M(X)$~$ |
---|
Poset: Exercises | $~$A \subseteq B$~$ |
---|
Poset: Exercises | $~$x \in X$~$ |
---|
Poset: Exercises | $~$1_A(x) \leq 1_B(x)$~$ |
---|
Poset: Exercises | $~$\mathcal{M}(X)$~$ |
---|
Poset: Exercises | $~$\subseteq$~$ |
---|
Poset: Exercises | $~$P$~$ |
---|
Poset: Exercises | $~$p, q \in P$~$ |
---|
Poset: Exercises | $~$q \prec p$~$ |
---|
Poset: Exercises | $~$\{ r \in P~|~r \leq p \}$~$ |
---|
Poset: Exercises | $~$\{ r \in P~|~r \leq q\}$~$ |
---|
Poset: Exercises | $~$P$~$ |
---|
Poset: Exercises | $~$p, q \in P$~$ |
---|
Poset: Exercises | $~$q \succ p$~$ |
---|
Poset: Exercises | $~$\{ r \in P~|~r \geq p \}$~$ |
---|
Poset: Exercises | $~$\{ r \in P~|~r \geq q\}$~$ |
---|
Poset: Exercises | $~$q \succ p$~$ |
---|
Poset: Exercises | $~$p \prec q$~$ |
---|
Poset: Exercises | $~$X = \{ x, y, z \}$~$ |
---|
Poset: Exercises | $~$\langle \mathcal P(X), \subseteq \rangle$~$ |
---|
Poset: Exercises | $~$X$~$ |
---|
Poset: Exercises | $~$\prec$~$ |
---|
Poset: Exercises | $~$\langle \mathbb R, \leq \rangle$~$ |
---|
Poset: Exercises | $~$\leq$~$ |
---|
Poset: Exercises | $~$0 < 1$~$ |
---|
Poset: Exercises | $~$0$~$ |
---|
Poset: Exercises | $~$\mathbb R$~$ |
---|
Poset: Exercises | $~$x \in \mathbb R$~$ |
---|
Poset: Exercises | $~$x > 0$~$ |
---|
Poset: Exercises | $~$y \in \mathbb R$~$ |
---|
Poset: Exercises | $~$0 < y < x$~$ |
---|
Poset: Exercises | $~$\mathbb R$~$ |
---|
Posterior probability | $~$H$~$ |
---|
Posterior probability | $~$e$~$ |
---|
Posterior probability | $~$\mathbb P(H\mid e).$~$ |
---|
Power set | $~$\mathcal P (X)$~$ |
---|
Power set | $~$X$~$ |
---|
Power set | $~$X$~$ |
---|
Power set | $~$Y \subseteq X$~$ |
---|
Power set | $~$Y \in \mathcal P (X)$~$ |
---|
Power set | $~$\mathcal P (X)$~$ |
---|
Power set | $~$X$~$ |
---|
Power set | $~$X$~$ |
---|
Power set | $~$Y \subseteq X$~$ |
---|
Power set | $~$Y \in \mathcal P (X)$~$ |
---|
Preemptive Learning | $~$\overline{A}\in\mathcal{BCS}(\overline{\mathbb{P}})$~$ |
---|
Preemptive Learning | $~$\displaystyle\liminf_{n\to\infty}\mathbb{P}_{n}(A_{n})=\liminf_{n\to\infty}\sup_{m\ge n}\mathbb{P}_{m}(A_{n})$~$ |
---|
Preemptive Learning | $~$\displaystyle\limsup_{n\to\infty}\mathbb{P}_{n}(A_{n})=\limsup_{n\to\infty}\inf_{m\ge n}\mathbb{P}_{m}(A_{n})$~$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$\overline{\mathbb{P}}$~$ |
---|
Preemptive Learning | $~$\mathcal{EF}$~$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$b$~$ |
---|
Preemptive Learning | $~$\le b$~$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$\overline{\mathbb{P}}$~$ |
---|
Preemptive Learning | $~$\mathcal{BCS}(\overline{\mathbb{P}})$~$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$x$~$ |
---|
Preemptive Learning | $~$y$~$ |
---|
Preemptive Learning | $~$\sup_{m\ge n}\mathbb{P}_{m}(A_{n})$~$ |
---|
Preemptive Learning | $~$\inf$~$ |
---|
Preemptive Learning | $~$\varepsilon$~$ |
---|
Preemptive Learning | $~$\varepsilon$~$ |
---|
Preemptive Learning | $~$\overline{\mathbb{P}}$~$ |
---|
Preemptive Learning | $~$\overline\alpha$~$ |
---|
Preemptive Learning | $~$\displaystyle\lim_{k\to\infty}\alpha_{k}\neq 0$~$ |
---|
Preemptive Learning | $~$\displaystyle\liminf_{n\to\infty}\mathbb{P}_{n}(A_{n})=\liminf_{n\to\infty}\sup_{m\ge n}\mathbb{P}_{m}(A_{n})$~$ |
---|
Preemptive Learning | $~$\sup_{m\ge n}\mathbb{P}_{m}(A_{n})\ge\mathbb{P}_{n}(A_{n})$~$ |
---|
Preemptive Learning | $~$\displaystyle\liminf_{n\to\infty}\mathbb{P}_{n}(A_{n})<\liminf_{n\to\infty}\sup_{m\ge n}\mathbb{P}_{m}(A_{n})$~$ |
---|
Preemptive Learning | $~$b$~$ |
---|
Preemptive Learning | $~$\varepsilon>0$~$ |
---|
Preemptive Learning | $~$\displaystyle\liminf_{n\to\infty}\mathbb{P}_{n}(A_{n})<b-\varepsilon<b+\varepsilon<\liminf_{n\to\infty}\sup_{m\ge n}\mathbb{P}_{m}(A_{n})$~$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$b-\varepsilon$~$ |
---|
Preemptive Learning | $~$b+\varepsilon$~$ |
---|
Preemptive Learning | $~$\varepsilon$~$ |
---|
Preemptive Learning | $~$b+\varepsilon$~$ |
---|
Preemptive Learning | $~$n$~$ |
---|
Preemptive Learning | $~$\mathbb{P}_{n}(A_{n})<b-\varepsilon$~$ |
---|
Preemptive Learning | $~$s_{e}$~$ |
---|
Preemptive Learning | $~$\forall n>s_{e}:\sup_{m\ge n}\mathbb{P}_{n}(A_{n})>b+\varepsilon$~$ |
---|
Preemptive Learning | $~$\forall s_{e}\exists n>s_{e}:\sup_{m\ge n}\mathbb{P}_{n}(A_{n})\le b+\varepsilon$~$ |
---|
Preemptive Learning | $~$n$~$ |
---|
Preemptive Learning | $~$b+\varepsilon$~$ |
---|
Preemptive Learning | $~$b+\varepsilon$~$ |
---|
Preemptive Learning | $~$b+\varepsilon$~$ |
---|
Preemptive Learning | $~$b+\varepsilon$~$ |
---|
Preemptive Learning | $~$b+\varepsilon$~$ |
---|
Preemptive Learning | $~$s_{e}$~$ |
---|
Preemptive Learning | $~$s_{e}$~$ |
---|
Preemptive Learning | $~$b+\varepsilon$~$ |
---|
Preemptive Learning | $~$b-\varepsilon$~$ |
---|
Preemptive Learning | $~$b+\varepsilon$~$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$b+\varepsilon$~$ |
---|
Preemptive Learning | $~$s_{e}$~$ |
---|
Preemptive Learning | $~$\varepsilon$~$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$\varepsilon$~$ |
---|
Preemptive Learning | $~$\varepsilon$~$ |
---|
Preemptive Learning | $~$\varepsilon$~$ |
---|
Preemptive Learning | $~$poly(k)$~$ |
---|
Preemptive Learning | $~$b-\varepsilon$~$ |
---|
Preemptive Learning | $~$<s_{e}$~$ |
---|
Preemptive Learning | $~$n<k$~$ |
---|
Preemptive Learning | $~$n=k$~$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$n>k$~$ |
---|
Preemptive Learning | $$~$T_{k}^{k}:=Ind_{\varepsilon/2}(A_{k}^{\dagger * k}<b-\varepsilon/2)\cdot (A_{k}^{\dagger}-A_{k}^{\dagger * k})$~$$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$A_{k}$~$ |
---|
Preemptive Learning | $~$b-\varepsilon/2$~$ |
---|
Preemptive Learning | $~$b-\varepsilon$~$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$\mathcal{BCS}(\overline{\mathbb{P}})$~$ |
---|
Preemptive Learning | $$~$F_{n}:=Ind_{\varepsilon/2}(A_{k}^{\dagger * n}>b+\varepsilon/2)\left(1-\sum_{k<i<n}F_{i}\right)$~$$ |
---|
Preemptive Learning | $~$b+\varepsilon/2$~$ |
---|
Preemptive Learning | $~$b+\varepsilon$~$ |
---|
Preemptive Learning | $~$n>k$~$ |
---|
Preemptive Learning | $$~$T_{n}^{k}:=-F_{n}\cdot T_{k}^{k}$~$$ |
---|
Preemptive Learning | $~$\varepsilon$~$ |
---|
Preemptive Learning | $~$\overline{\mathbb{P}}$~$ |
---|
Preemptive Learning | $~$\mathcal{EF}$~$ |
---|
Preemptive Learning | $~$A_{k}^{dagger}$~$ |
---|
Preemptive Learning | $~$poly(k)$~$ |
---|
Preemptive Learning | $~$\mathcal{BCS}(\overline{\mathbb{P}})$~$ |
---|
Preemptive Learning | $~$\overline{\mathbb{P}}$~$ |
---|
Preemptive Learning | $~$poly(k)$~$ |
---|
Preemptive Learning | $~$A_{k}^{\dagger}$~$ |
---|
Preemptive Learning | $~$<s_{e}$~$ |
---|
Preemptive Learning | $~$poly(n)$~$ |
---|
Preemptive Learning | $~$\mathbb{R}$~$ |
---|
Preemptive Learning | $~$k$~$ |
---|
Preemptive Learning | $~$poly(k)$~$ |
---|
Preemptive Learning | $~$poly(n)$~$ |
---|
Preemptive Learning | $~$F_{n}$~$ |
---|
Preemptive Learning | $~$\sum_{k<i\le n}F_{i}\le 1$~$ |
---|
Preemptive Learning | $~$F_{n}\ge 0$~$ |
---|
Preemptive Learning | $~$m$~$ |
---|
Preemptive Learning | $~$\mathbb{P}_{m}(A_{k})>b+\varepsilon$~$ |
---|
Preference framework | $~$U_X$~$ |
---|
Preference framework | $~$U_Y$~$ |
---|
Prime element of a ring | $~$(R, +, \times)$~$ |
---|
Prime element of a ring | $~$p \in R$~$ |
---|
Prime element of a ring | $~$p \mid ab$~$ |
---|
Prime element of a ring | $~$p \mid a$~$ |
---|
Prime element of a ring | $~$p \mid b$~$ |
---|
Prime element of a ring | $~$p \mid ab$~$ |
---|
Prime element of a ring | $~$p \mid a$~$ |
---|
Prime element of a ring | $~$p \mid b$~$ |
---|
Prime element of a ring | $~$ab \in \langle p \rangle$~$ |
---|
Prime element of a ring | $~$a$~$ |
---|
Prime element of a ring | $~$b$~$ |
---|
Prime element of a ring | $~$\langle p \rangle$~$ |
---|
Prime element of a ring | $~$\mathbb{Z}$~$ |
---|
Prime number | $~$n > 1$~$ |
---|
Prime number | $~$1$~$ |
---|
Prime number | $~$n \mid ab$~$ |
---|
Prime number | $~$n$~$ |
---|
Prime number | $~$ab$~$ |
---|
Prime number | $~$n \mid a$~$ |
---|
Prime number | $~$n \mid b$~$ |
---|
Prime number | $~$1$~$ |
---|
Prime number | $~$2$~$ |
---|
Prime number | $~$1$~$ |
---|
Prime number | $~$2$~$ |
---|
Prime number | $~$1$~$ |
---|
Prime number | $~$3$~$ |
---|
Prime number | $~$5, 7, 11, 13, \dots$~$ |
---|
Prime number | $~$4$~$ |
---|
Prime number | $~$6, 8, 9, 10, 12, \dots$~$ |
---|
Prime number | $~$2 \times 3 = 3 \times 2$~$ |
---|
Prime number | $~$6$~$ |
---|
Prime number | $~$n$~$ |
---|
Prime number | $~$n$~$ |
---|
Prime order groups are cyclic | $~$G$~$ |
---|
Prime order groups are cyclic | $~$p$~$ |
---|
Prime order groups are cyclic | $~$G$~$ |
---|
Prime order groups are cyclic | $~$C_p$~$ |
---|
Prime order groups are cyclic | $~$p$~$ |
---|
Prime order groups are cyclic | $~$g$~$ |
---|
Prime order groups are cyclic | $~$g$~$ |
---|
Prime order groups are cyclic | $~$1$~$ |
---|
Prime order groups are cyclic | $~$p$~$ |
---|
Prime order groups are cyclic | $~$p$~$ |
---|
Prime order groups are cyclic | $~$1$~$ |
---|
Prime order groups are cyclic | $~$1$~$ |
---|
Primer on Infinite Series | $$~$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots$~$$ |
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Primer on Infinite Series | $$~$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots = 1$~$$ |
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Primer on Infinite Series | $~$\frac{1}{2}$~$ |
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Primer on Infinite Series | $$~$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots = 1$~$$ |
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Primer on Infinite Series | $$~$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots \approx \text{ (is approximately equal to) }1$~$$ |
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Primer on Infinite Series | $$~$\frac{1}{2}+\frac{1}{4}+\frac{1}{8} \approx 1$~$$ |
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Primer on Infinite Series | $$~$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32} \approx 1$~$$ |
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Primer on Infinite Series | $$~$ \text{Circumference }(C) = \pi \cdot \text{Diameter }(D) = 2\pi \cdot \text{Radius }(r)$~$$ |
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Primer on Infinite Series | $$~$ \text{Area of a Circle}(A) = \pi \cdot r^2$~$$ |
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Primer on Infinite Series | $~$\pi$~$ |
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Primer on Infinite Series | $~$\frac{C}{D}$~$ |
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Primer on Infinite Series | $~$\pi$~$ |
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Primer on Infinite Series | $~$\frac{C}{D} = \pi$~$ |
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Primer on Infinite Series | $~$C=\pi D$~$ |
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Primer on Infinite Series | $$~$\text{Area of polygon with }n \text{ sides} = \text{sum of } n \text{ triangles}$~$$ |
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Primer on Infinite Series | $~$= \frac{1}{2} \cdot \text{base}\cdot \text{height}$~$ |
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Primer on Infinite Series | $~$s$~$ |
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Primer on Infinite Series | $~$h$~$ |
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Primer on Infinite Series | $$~$\text{Area of polygon with }n \text{ sides} = \underbrace{\frac{1}{2}sh+\frac{1}{2}sh+\ldots+\frac{1}{2}sh}_{n \text{ times}}$~$$ |
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Primer on Infinite Series | $~$\frac{1}{2}h$~$ |
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Primer on Infinite Series | $$~$\text{Area of polygon with }n \text{ sides} = \frac{1}{2}h(\underbrace{s+s+\ldots+s}_{n \text{ times}})$~$$ |
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Primer on Infinite Series | $~$n$~$ |
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Primer on Infinite Series | $~$P$~$ |
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Primer on Infinite Series | $$~$\text{Area of polygon} = \frac{1}{2}hP$~$$ |
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Primer on Infinite Series | $$~$ \text{Area of polygon} \rightarrow \text{Area of a Circle}(A)$~$$ |
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Primer on Infinite Series | $$~$h \rightarrow r$~$$ |
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Primer on Infinite Series | $$~$P \rightarrow C$~$$ |
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Primer on Infinite Series | $$~$ A = \frac{1}{2}rC = \frac{1}{2}r(\pi D) = \frac{1}{2}r(\pi(2r))=\pi r^2$~$$ |
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Primer on Infinite Series | $$~$\text{infinite sum} = \text{finite number}$~$$ |
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Principal ideal domain | $~$I$~$ |
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Principal ideal domain | $~$i \in I$~$ |
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Principal ideal domain | $~$\langle i \rangle = I$~$ |
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Principal ideal domain | $~$I$~$ |
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Principal ideal domain | $~$i$~$ |
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Principal ideal domain | $~$R$~$ |
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Principal ideal domain | $~$R$~$ |
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Principal ideal domain | $~$\mathbb{Z}$~$ |
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Principal ideal domain | $~$\{ 0 \}$~$ |
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Principal ideal domain | $~$0$~$ |
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Principal ideal domain | $~$1$~$ |
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Principal ideal domain | $~$F[X]$~$ |
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Principal ideal domain | $~$F$~$ |
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Principal ideal domain | $~$\mathbb{Z}[i]$~$ |
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Principal ideal domain | $~$\mathbb{Z}[X]$~$ |
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Principal ideal domain | $~$\langle 2, X \rangle$~$ |
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Principal ideal domain | $~$\mathbb{Z}_6$~$ |
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Principal ideal domain | $~$3 \times 2 = 0$~$ |
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Principal ideal domain | $~$\mathbb{Z}[\frac{1}{2} (1+\sqrt{-19})]$~$ |
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Principal ideal domain | $~$\mathbb{Z}[X]$~$ |
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Prior probability | $~$H$~$ |
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Prior probability | $~$\mathbb P(H)$~$ |
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Prior probability | $~$e$~$ |
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Prior probability | $~$\mathbb P(H\mid e).$~$ |
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Prior probability | $~$\mathbb P (H\mid I_0)$~$ |
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Prior probability | $~$H$~$ |
---|
Prior probability | $~$I_0$~$ |
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Prior probability | $~$\mathbb P(X)$~$ |
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Prior probability | $~$e_0$~$ |
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Prior probability | $~$e_1$~$ |
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Prior probability | $~$\mathbb P(H\mid e_0)$~$ |
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Prior probability | $~$\mathbb P(H\mid e_1 \wedge e_0).$~$ |
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Prisoner's Dilemma | $~$D$~$ |
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Prisoner's Dilemma | $~$C$~$ |
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Prisoner's Dilemma | $~$(p_1, p_2)$~$ |
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Prisoner's Dilemma | $$~$\begin{array}{r|c|c}
& D_2 & C_2 \\
\hline
D_1 & (\$1, \$1) & (\$3, \$0) \\ \hline
C_1 & (\$0, \$3) & (\$2, \$2)
\end{array}$~$$ |
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Prisoner's Dilemma | $~$(o_1, o_2)$~$ |
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Prisoner's Dilemma | $$~$\begin{array}{r|c|c}
& \text{ Player 2 Defects: } & \text{ Player 2 Cooperates: }\\
\hline
\text{ Player 1 Defects: }& \text{ (2 years, 2 years) } & \text{ (0 years, 3 years) } \\ \hline
\text{ Player 1 Cooperates: } & \text{ (3 years, 0 years) } & \text{ (1 year, 1 year) }
\end{array}$~$$ |
---|
Prisoner's Dilemma | $~$D$~$ |
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Prisoner's Dilemma | $~$C,$~$ |
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Prisoner's Dilemma | $$~$\begin{array}{r|c|c}
& D_2 & C_2 \\
\hline
D_1 & (\$1, \$1) & (\$3, \$0) \\ \hline
C_1 & (\$0, \$3) & (\$2, \$2)
\end{array}$~$$ |
---|
Probability | $~$\mathbb{P}(X)$~$ |
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Probability | $~$X$~$ |
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Probability | $~$\mathbb{P}(X \wedge Y) = 0 \implies \mathbb{P}(X \vee Y) = \mathbb{P}(X) + \mathbb{P}(Y).$~$ |
---|
Probability | $~$\mathbb{P}(X)$~$ |
---|
Probability | $~$\mathbb{P}(\neg X) = 1 - \mathbb{P}(X)$~$ |
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Probability | $~$\mathbb{P}(X \wedge Y)$~$ |
---|
Probability | $~$\mathbb{P}(X \vee Y)$~$ |
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Probability | $~$\mathbb{P}(X|Y) := \frac{\mathbb{P}(X \wedge Y}{\mathbb{P}(Y)}$~$ |
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Probability | $~$\mathbb{P}(X|Y)$~$ |
---|
Probability | $~$\mathbb{P}(yellow|banana)$~$ |
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Probability | $~$\mathbb{P}(banana|yellow)$~$ |
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Probability distribution (countable sample space) | $~$\Omega$~$ |
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Probability distribution (countable sample space) | $~$\mathbb{P}: \Omega \to [0,1]$~$ |
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Probability distribution (countable sample space) | $~$\sum_{\omega \in \Omega} \mathbb{P}(\omega) = 1$~$ |
---|
Probability distribution (countable sample space) | $~$\Omega$~$ |
---|
Probability distribution (countable sample space) | $~$\mathbb{P}: \Omega \to [0,1]$~$ |
---|
Probability distribution (countable sample space) | $~$\sum_{\omega \in \Omega} \mathbb{P}(\omega) = 1$~$ |
---|
Probability distribution (countable sample space) | $~$x\in \Omega$~$ |
---|
Probability distribution (countable sample space) | $~$r$~$ |
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Probability distribution (countable sample space) | $~$\mathbb{P}(x) = r$~$ |
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Probability distribution (countable sample space) | $~$\Omega$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P(sick)$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P(sick)$~$ |
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Probability distribution: Motivated definition | $~$\mathbb P(sick)$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P(sick)$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P(sick)$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P(sick)$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P(sick).$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P(sick)$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability distribution: Motivated definition | $~$sick,$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P(sick)$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability distribution: Motivated definition | $~$sick$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P(sick)$~$ |
---|
Probability distribution: Motivated definition | $~$sick$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability distribution: Motivated definition | $~$sick$~$ |
---|
Probability distribution: Motivated definition | $~$healthy$~$ |
---|
Probability distribution: Motivated definition | $~$sick$~$ |
---|
Probability distribution: Motivated definition | $~$healthy$~$ |
---|
Probability distribution: Motivated definition | $~$sick$~$ |
---|
Probability distribution: Motivated definition | $~$healthy,$~$ |
---|
Probability distribution: Motivated definition | $$~$
\begin{align}
sick, \text{age }1, \text{Afghanistan} \\
healthy, \text{age }1, \text{Afghanistan} \\
sick, \text{age }2, \text{Afghanistan} \\
\vdots \\
sick, \text{age }29, \text{Albania} \\
healthy, \text{age }29, \text{Albania} \\
sick, \text{age }30, \text{Albania} \\
\vdots
\end{align}
$~$$ |
---|
Probability distribution: Motivated definition | $~$sick$~$ |
---|
Probability distribution: Motivated definition | $~$healthy$~$ |
---|
Probability distribution: Motivated definition | $~$2 \cdot 150 \cdot 196 = 58800,$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P(sick)$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P(\text{Health}=sick),$~$ |
---|
Probability distribution: Motivated definition | $~$sick$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability distribution: Motivated definition | $~$sick$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability distribution: Motivated definition | $~$sick$~$ |
---|
Probability distribution: Motivated definition | $~$healthy$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability distribution: Motivated definition | $~$\mathbb P$~$ |
---|
Probability interpretations: Examples | $~$f$~$ |
---|
Probability interpretations: Examples | $~$p$~$ |
---|
Probability interpretations: Examples | $~$f$~$ |
---|
Probability interpretations: Examples | $~$p$~$ |
---|
Probability interpretations: Examples | $~$f$~$ |
---|
Probability interpretations: Examples | $~$e$~$ |
---|
Probability interpretations: Examples | $~$f$~$ |
---|
Probability interpretations: Examples | $~$e$~$ |
---|
Probability interpretations: Examples | $~$f$~$ |
---|
Probability interpretations: Examples | $~$f$~$ |
---|
Probability interpretations: Examples | $~$f$~$ |
---|
Probability interpretations: Examples | $~$f$~$ |
---|
Probability interpretations: Examples | $~$p$~$ |
---|
Probability interpretations: Examples | $~$\frac{p}{3}$~$ |
---|
Probability interpretations: Examples | $~$f$~$ |
---|
Probability interpretations: Examples | $~$f,$~$ |
---|
Probability interpretations: Examples | $~$f,$~$ |
---|
Probability interpretations: Examples | $~$f$~$ |
---|
Probability interpretations: Examples | $~$f=\frac{2}{3}$~$ |
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Probability interpretations: Examples | $~$f=\frac{1}{3}$~$ |
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Probability interpretations: Examples | $~$f$~$ |
---|
Probability interpretations: Examples | $~$\frac{2}{3}$~$ |
---|
Probability interpretations: Examples | $~$\frac{1}{3}$~$ |
---|
Probability interpretations: Examples | $~$\frac{M + 1}{M + N + 2}.$~$ |
---|
Probability interpretations: Examples | $~$f$~$ |
---|
Probability interpretations: Examples | $~$\pi$~$ |
---|
Probability interpretations: Examples | $~$0$~$ |
---|
Probability interpretations: Examples | $~$\pi$~$ |
---|
Probability interpretations: Examples | $~$0$~$ |
---|
Probability interpretations: Examples | $~$\pi$~$ |
---|
Probability interpretations: Examples | $~$0$~$ |
---|
Probability interpretations: Examples | $~$\pi,$~$ |
---|
Probability interpretations: Examples | $~$\pi$~$ |
---|
Probability interpretations: Examples | $~$0$~$ |
---|
Probability interpretations: Examples | $~$\pi$~$ |
---|
Probability interpretations: Examples | $~$0$~$ |
---|
Probability interpretations: Examples | $~$0$~$ |
---|
Probability interpretations: Examples | $~$\pi,$~$ |
---|
Probability interpretations: Examples | $~$\pi,$~$ |
---|
Probability interpretations: Examples | $~$\pi$~$ |
---|
Probability interpretations: Examples | $~$\pi$~$ |
---|
Probability interpretations: Examples | $~$\pi$~$ |
---|
Probability interpretations: Examples | $~$0$~$ |
---|
Probability interpretations: Examples | $~$\pi.$~$ |
---|
Probability notation for Bayes' rule | $~$\mathbb P(H).$~$ |
---|
Probability notation for Bayes' rule | $~$\mathbb P(e \mid H).$~$ |
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Probability notation for Bayes' rule | $~$\mathbb P(H \mid e).$~$ |
---|
Probability notation for Bayes' rule | $~$\mathbb P(H).$~$ |
---|
Probability notation for Bayes' rule | $~$\mathbb P(e \mid H).$~$ |
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Probability notation for Bayes' rule | $~$\mathbb P(H \mid e).$~$ |
---|
Probability notation for Bayes' rule | $~$H_1$~$ |
---|
Probability notation for Bayes' rule | $~$H_2,$~$ |
---|
Probability notation for Bayes' rule | $~$e,$~$ |
---|
Probability notation for Bayes' rule | $$~$\dfrac{\mathbb P(H_1)}{\mathbb P(H_2)} \times \dfrac{\mathbb P(e \mid H_1)}{\mathbb P(e \mid H_2)} = \dfrac{\mathbb P(H_1\mid e)}{\mathbb P(H_2\mid e)}.$~$$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb{P}(X\mid Y)$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$X$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$Y$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{left}\mid \mathrm{right})$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathrm{left}$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathrm{right}$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{yellow}\mid \mathrm{banana})$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{banana}\mid \mathrm{yellow})$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$L,$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$R,$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$L$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$R$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$R.$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$L \wedge R$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(L\mid R) = \frac{\mathbb P(L \wedge R)}{\mathbb P(R)}$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $$~$\begin{array}{l|r|r}
& Red & Blue \\
\hline
Square & 1 & 2 \\
\hline
Round & 3 & 4
\end{array}$~$$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{red} \mid \mathrm{round}) = \dfrac{\mathbb P(\mathrm{red} \wedge \mathrm{round})}{\mathbb P(\mathrm{round})} \propto \dfrac{3}{3 + 4} = \frac{3}{7}$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{square} \mid \mathrm{blue}) = \dfrac{\mathbb P(\mathrm{square} \wedge \mathrm{blue})}{\mathbb P(\mathrm{blue})} \propto \dfrac{2}{2 + 4} = \frac{1}{3}$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $$~$\begin{array}{l|r|r}
& Sick & Healthy \\
\hline
Test + & 18\% & 24\% \\
\hline
Test - & 2\% & 56\%
\end{array}$~$$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathrm{positive}$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\cdot),$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\cdot \mid \mathrm{positive}).$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathrm{positive}$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{sick}\mid \mathrm{positive})$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{sick}).$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(L \wedge R)$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(R)$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{hypothesis}\mid \mathrm{evidence})$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathrm{evidence}$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\cdot\mid \mathrm{evidence}),$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$hypothesis.$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathrm{hypothesis}.$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{hypothesis}),$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{evidence}\mid \mathrm{hypothesis})$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\cdot \mid \cdot)$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{redhair}\mid \mathrm{Scarlet}) = 99\%,$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{redhair}\mid \mathrm{Scarlet}),$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{Scarlet}\mid \mathrm{redhair}),$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{redhair}\mid \mathrm{Scarlet})$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$1$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{redhair}\mid \mathrm{Scarlet})$~$ |
---|
Probability notation for Bayes' rule: Intro (Math 1) | $~$\mathbb P(\mathrm{Scarlet}\mid \mathrm{redhair})$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$V,$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$U'$~$ |
---|
Problem of fully updated deference | $~$V.$~$ |
---|
Problem of fully updated deference | $~$V,$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$U_1, U_2, U_3.$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$\frac{1}{3}$~$ |
---|
Problem of fully updated deference | $~$U_i.$~$ |
---|
Problem of fully updated deference | $~$U_i$~$ |
---|
Problem of fully updated deference | $~$\mathbb O$~$ |
---|
Problem of fully updated deference | $$~$\forall o_j \in \mathbb O \colon \exists i \colon \ U_i(o_j) \ll \max_{o \in \mathbb O} U_i(o) $~$$ |
---|
Problem of fully updated deference | $~$U_i$~$ |
---|
Problem of fully updated deference | $~$\max$~$ |
---|
Problem of fully updated deference | $~$U_i$~$ |
---|
Problem of fully updated deference | $~$U_i$~$ |
---|
Problem of fully updated deference | $~$U_i$~$ |
---|
Problem of fully updated deference | $~$V_i$~$ |
---|
Problem of fully updated deference | $~$U_i.$~$ |
---|
Problem of fully updated deference | $~$U_1$~$ |
---|
Problem of fully updated deference | $~$U_1,$~$ |
---|
Problem of fully updated deference | $~$U_1$~$ |
---|
Problem of fully updated deference | $~$U_2$~$ |
---|
Problem of fully updated deference | $~$U_3$~$ |
---|
Problem of fully updated deference | $~$V_i$~$ |
---|
Problem of fully updated deference | $~$\pi_1$~$ |
---|
Problem of fully updated deference | $~$U_1.$~$ |
---|
Problem of fully updated deference | $~$\pi_2$~$ |
---|
Problem of fully updated deference | $~$U_2.$~$ |
---|
Problem of fully updated deference | $~$\pi_3$~$ |
---|
Problem of fully updated deference | $~$U_3.$~$ |
---|
Problem of fully updated deference | $~$\pi_4$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$U_2$~$ |
---|
Problem of fully updated deference | $~$U_3.$~$ |
---|
Problem of fully updated deference | $~$\pi_5$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$u_1, u_2, u_3$~$ |
---|
Problem of fully updated deference | $~$v_1, v_2, v_3$~$ |
---|
Problem of fully updated deference | $~$\mathbb O$~$ |
---|
Problem of fully updated deference | $~$U_i$~$ |
---|
Problem of fully updated deference | $~$V_i.$~$ |
---|
Problem of fully updated deference | $~$u_{\Delta U}$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$\pi_5$~$ |
---|
Problem of fully updated deference | $$~$0.5 \cdot U_2(u_{\Delta U}) + 0.5 \cdot U_3(u_{\Delta U}) \ < \ 0.5 \cdot U_2(v_2) + 0.5 \cdot U_3(v_3)$~$$ |
---|
Problem of fully updated deference | $~$U_i(v_i)$~$ |
---|
Problem of fully updated deference | $~$U_i(u_i).$~$ |
---|
Problem of fully updated deference | $~$V_i$~$ |
---|
Problem of fully updated deference | $~$U_i$~$ |
---|
Problem of fully updated deference | $~$\pi_5$~$ |
---|
Problem of fully updated deference | $~$U_2$~$ |
---|
Problem of fully updated deference | $~$U_3$~$ |
---|
Problem of fully updated deference | $~$v_i$~$ |
---|
Problem of fully updated deference | $~$U_i.$~$ |
---|
Problem of fully updated deference | $~$U_i$~$ |
---|
Problem of fully updated deference | $~$V_i$~$ |
---|
Problem of fully updated deference | $~$U_i.$~$ |
---|
Problem of fully updated deference | $~$U_2$~$ |
---|
Problem of fully updated deference | $~$U_3$~$ |
---|
Problem of fully updated deference | $~$U_2$~$ |
---|
Problem of fully updated deference | $~$U_3$~$ |
---|
Problem of fully updated deference | $~$U_i$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$\pi_6$~$ |
---|
Problem of fully updated deference | $~$E$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$\Delta U | E.$~$ |
---|
Problem of fully updated deference | $~$\Delta U | E$~$ |
---|
Problem of fully updated deference | $~$u_{\Delta U | E}$~$ |
---|
Problem of fully updated deference | $~$U_i,$~$ |
---|
Problem of fully updated deference | $~$v_i$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$\Delta U|E$~$ |
---|
Problem of fully updated deference | $~$U_2$~$ |
---|
Problem of fully updated deference | $~$U_3$~$ |
---|
Problem of fully updated deference | $~$V_2$~$ |
---|
Problem of fully updated deference | $~$U_2$~$ |
---|
Problem of fully updated deference | $~$U_2$~$ |
---|
Problem of fully updated deference | $~$U,$~$ |
---|
Problem of fully updated deference | $~$V_3$~$ |
---|
Problem of fully updated deference | $~$U_3$~$ |
---|
Problem of fully updated deference | $~$U_3$~$ |
---|
Problem of fully updated deference | $~$U.$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$U_2$~$ |
---|
Problem of fully updated deference | $~$U_3,$~$ |
---|
Problem of fully updated deference | $~$V_i$~$ |
---|
Problem of fully updated deference | $~$U_i.$~$ |
---|
Problem of fully updated deference | $~$V_i$~$ |
---|
Problem of fully updated deference | $~$U_i$~$ |
---|
Problem of fully updated deference | $~$U_i.$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$\Delta U.$~$ |
---|
Problem of fully updated deference | $~$U_\Delta = \sum_i \mathbb P_{\Delta}(i) \cdot U_i,$~$ |
---|
Problem of fully updated deference | $~$\mathbb P_\Delta$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$V,$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$T$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$T$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$T$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$T$~$ |
---|
Problem of fully updated deference | $~$V.$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$\Delta \dot U$~$ |
---|
Problem of fully updated deference | $~$T.$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$\dot U$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$T \approx V,$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$T$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$\Delta U | E \approx T$~$ |
---|
Problem of fully updated deference | $~$T.$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$\Delta U | E \approx V,$~$ |
---|
Problem of fully updated deference | $~$\Delta U | E$~$ |
---|
Problem of fully updated deference | $~$U.$~$ |
---|
Problem of fully updated deference | $~$V,$~$ |
---|
Problem of fully updated deference | $~$U.$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$e_0$~$ |
---|
Problem of fully updated deference | $~$e_0$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$e_0$~$ |
---|
Problem of fully updated deference | $~$\Delta U | e_0$~$ |
---|
Problem of fully updated deference | $~$\Delta U,$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$\Delta U,$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$V$~$ |
---|
Problem of fully updated deference | $~$U$~$ |
---|
Problem of fully updated deference | $~$\Delta U$~$ |
---|
Product (Category Theory) | $~$X$~$ |
---|
Product (Category Theory) | $~$Y$~$ |
---|
Product (Category Theory) | $~$\mathbb{C}$~$ |
---|
Product (Category Theory) | $~$X$~$ |
---|
Product (Category Theory) | $~$Y$~$ |
---|
Product (Category Theory) | $~$P$~$ |
---|
Product (Category Theory) | $~$f: P \rightarrow X$~$ |
---|
Product (Category Theory) | $~$g: P \rightarrow Y$~$ |
---|
Product (Category Theory) | $~$W$~$ |
---|
Product (Category Theory) | $~$u: W \rightarrow X$~$ |
---|
Product (Category Theory) | $~$v:W \rightarrow Y$~$ |
---|
Product (Category Theory) | $~$h: W \rightarrow P$~$ |
---|
Product (Category Theory) | $~$fh = u$~$ |
---|
Product (Category Theory) | $~$gh = v$~$ |
---|
Product is unique up to isomorphism | $~$A$~$ |
---|
Product is unique up to isomorphism | $~$B$~$ |
---|
Product is unique up to isomorphism | $~$A$~$ |
---|
Product is unique up to isomorphism | $~$B$~$ |
---|
Product is unique up to isomorphism | $$~$A \times B \\ \pi_A: A \times B \to A \\ \pi_B : A \times B \to B$~$$ |
---|
Product is unique up to isomorphism | $~$X$~$ |
---|
Product is unique up to isomorphism | $~$f_A: X \to A, f_B: X \to B$~$ |
---|
Product is unique up to isomorphism | $~$f: X \to A \times B$~$ |
---|
Product is unique up to isomorphism | $~$\pi_A \circ f = f_A$~$ |
---|
Product is unique up to isomorphism | $~$\pi_B \circ f = f_B$~$ |
---|
Product is unique up to isomorphism | $~$(R, \pi_A, \pi_B)$~$ |
---|
Product is unique up to isomorphism | $~$(S, \phi_A, \phi_B)$~$ |
---|
Product is unique up to isomorphism | $~$R$~$ |
---|
Product is unique up to isomorphism | $~$S$~$ |
---|
Product is unique up to isomorphism | $~$A \times_1 B$~$ |
---|
Product is unique up to isomorphism | $~$A \times_2 B$~$ |
---|
Product is unique up to isomorphism | $~$R$~$ |
---|
Product is unique up to isomorphism | $~$S$~$ |
---|
Product is unique up to isomorphism | $~$R$~$ |
---|
Product is unique up to isomorphism | $~$S$~$ |
---|
Product is unique up to isomorphism | $~$A$~$ |
---|
Product is unique up to isomorphism | $~$B$~$ |
---|
Product is unique up to isomorphism | $~$R$~$ |
---|
Product is unique up to isomorphism | $~$A$~$ |
---|
Product is unique up to isomorphism | $~$B$~$ |
---|
Product is unique up to isomorphism | $~$X = S$~$ |
---|
Product is unique up to isomorphism | $~$f_A: S \to A, f_B: S \to B$~$ |
---|
Product is unique up to isomorphism | $~$f: S \to R$~$ |
---|
Product is unique up to isomorphism | $~$\pi_A \circ f = f_A$~$ |
---|
Product is unique up to isomorphism | $~$\pi_B \circ f = f_B$~$ |
---|
Product is unique up to isomorphism | $~$f_A = \phi_A, f_B = \phi_B$~$ |
---|
Product is unique up to isomorphism | $~$\phi: S \to R$~$ |
---|
Product is unique up to isomorphism | $~$\pi_A \circ \phi = \phi_A$~$ |
---|
Product is unique up to isomorphism | $~$\pi_B \circ \phi = \phi_B$~$ |
---|
Product is unique up to isomorphism | $~$R$~$ |
---|
Product is unique up to isomorphism | $~$S$~$ |
---|
Product is unique up to isomorphism | $~$\phi$~$ |
---|
Product is unique up to isomorphism | $~$\pi$~$ |
---|
Product is unique up to isomorphism | $~$S$~$ |
---|
Product is unique up to isomorphism | $~$A$~$ |
---|
Product is unique up to isomorphism | $~$B$~$ |
---|
Product is unique up to isomorphism | $~$X = R$~$ |
---|
Product is unique up to isomorphism | $~$\pi: R \to S$~$ |
---|
Product is unique up to isomorphism | $~$\phi_A \circ \pi = \pi_A$~$ |
---|
Product is unique up to isomorphism | $~$\phi_B \circ \pi = \pi_B$~$ |
---|
Product is unique up to isomorphism | $~$\pi \circ \phi: S \to S$~$ |
---|
Product is unique up to isomorphism | $~$S$~$ |
---|
Product is unique up to isomorphism | $~$\pi$~$ |
---|
Product is unique up to isomorphism | $~$\phi$~$ |
---|
Product is unique up to isomorphism | $~$S$~$ |
---|
Product is unique up to isomorphism | $~$S$~$ |
---|
Product is unique up to isomorphism | $~$f_A: S \to A, f_B: S \to B$~$ |
---|
Product is unique up to isomorphism | $~$f: S \to S$~$ |
---|
Product is unique up to isomorphism | $~$\phi_A \circ f = f_A$~$ |
---|
Product is unique up to isomorphism | $~$\phi_B \circ f = f_B$~$ |
---|
Product is unique up to isomorphism | $~$f_A = \phi_A$~$ |
---|
Product is unique up to isomorphism | $~$f_B = \phi_B$~$ |
---|
Product is unique up to isomorphism | $~$f: S \to S$~$ |
---|
Product is unique up to isomorphism | $~$\phi_A \circ f = \phi_A$~$ |
---|
Product is unique up to isomorphism | $~$\phi_B \circ f = \phi_B$~$ |
---|
Product is unique up to isomorphism | $~$1_S$~$ |
---|
Product is unique up to isomorphism | $~$\pi \circ \phi$~$ |
---|
Product is unique up to isomorphism | $~$f$~$ |
---|
Product is unique up to isomorphism | $~$f$~$ |
---|
Product is unique up to isomorphism | $~$1_S$~$ |
---|
Product is unique up to isomorphism | $~$\phi_A = \phi_A$~$ |
---|
Product is unique up to isomorphism | $~$\phi_B = \phi_B$~$ |
---|
Product is unique up to isomorphism | $~$\pi \circ \phi$~$ |
---|
Product is unique up to isomorphism | $~$\phi_A \circ \pi = \pi_A$~$ |
---|
Product is unique up to isomorphism | $~$\phi_B \circ \pi = \pi_B$~$ |
---|
Product is unique up to isomorphism | $~$\pi$~$ |
---|
Product is unique up to isomorphism | $~$\phi$~$ |
---|
Product is unique up to isomorphism | $~$(R, \pi_A, \pi_B)$~$ |
---|
Product is unique up to isomorphism | $~$(S, \phi_A, \phi_B)$~$ |
---|
Product is unique up to isomorphism | $~$\phi$~$ |
---|
Product is unique up to isomorphism | $~$\pi$~$ |
---|
Product is unique up to isomorphism | $~$\pi$~$ |
---|
Product is unique up to isomorphism | $~$\phi$~$ |
---|
Product is unique up to isomorphism | $~$R \to S$~$ |
---|
Product is unique up to isomorphism | $~$S \to R$~$ |
---|
Product is unique up to isomorphism | $~$(A \times B, \pi_A, \pi_B)$~$ |
---|
Product is unique up to isomorphism | $~$A$~$ |
---|
Product is unique up to isomorphism | $~$B$~$ |
---|
Product is unique up to isomorphism | $~$(B \times A, \pi'_A, \pi'_B)$~$ |
---|
Product is unique up to isomorphism | $~$\pi'_A(b, a) = a$~$ |
---|
Product is unique up to isomorphism | $~$\pi'_B(b, a) = b$~$ |
---|
Product is unique up to isomorphism | $~$A$~$ |
---|
Product is unique up to isomorphism | $~$B$~$ |
---|
Product is unique up to isomorphism | $~$A \times B$~$ |
---|
Product is unique up to isomorphism | $~$B \times A$~$ |
---|
Product is unique up to isomorphism | $~$A \times B \to B \times A$~$ |
---|
Product is unique up to isomorphism | $~$(a,b) \mapsto (b,a)$~$ |
---|
Project outline: Intro to the Universal Property | $~$\mathbb{N}$~$ |
---|
Project outline: Intro to the Universal Property | $~$a$~$ |
---|
Project outline: Intro to the Universal Property | $~$b$~$ |
---|
Project outline: Intro to the Universal Property | $~$a$~$ |
---|
Project outline: Intro to the Universal Property | $~$b$~$ |
---|
Project outline: Intro to the Universal Property | $~$\mathbb{N}$~$ |
---|
Project proposal: Complex numbers | $~$e^{- \pi i}$~$ |
---|
Project proposal: Complex numbers | $~$i$~$ |
---|
Project proposal: Complex numbers | $~$\mathbb C$~$ |
---|
Project proposal: Intro to numbers | $~$\mathbb N$~$ |
---|
Project proposal: Intro to numbers | $~$\mathbb Z$~$ |
---|
Project proposal: Intro to numbers | $~$\mathbb Q$~$ |
---|
Project proposal: Intro to numbers | $~$\mathbb I$~$ |
---|
Project proposal: Intro to numbers | $~$\mathbb R$~$ |
---|
Project proposal: Intro to numbers | $~$\mathbb C$~$ |
---|
Project proposal: Intro to the Universal Property | $~$\mathbb{N}$~$ |
---|
Project proposal: Intro to the Universal Property | $~$a$~$ |
---|
Project proposal: Intro to the Universal Property | $~$b$~$ |
---|
Project proposal: Intro to the Universal Property | $~$a$~$ |
---|
Project proposal: Intro to the Universal Property | $~$b$~$ |
---|
Proof by contradiction | $~$\sqrt 2$~$ |
---|
Proof by contradiction | $~$\sqrt 2$~$ |
---|
Proof by contradiction | $~$a,b\in\mathbb{N}$~$ |
---|
Proof by contradiction | $~$\sqrt 2 = \frac{a}{b}$~$ |
---|
Proof by contradiction | $~$a$~$ |
---|
Proof by contradiction | $~$b$~$ |
---|
Proof by contradiction | $~$b\sqrt2=a$~$ |
---|
Proof by contradiction | $~$2b^2=a^2$~$ |
---|
Proof by contradiction | $~$2$~$ |
---|
Proof by contradiction | $~$a$~$ |
---|
Proof by contradiction | $~$a$~$ |
---|
Proof by contradiction | $~$2n$~$ |
---|
Proof by contradiction | $~$n\in\mathbb{N}$~$ |
---|
Proof by contradiction | $~$2b^2 = 4n^2\implies b^2 =2 n^2$~$ |
---|
Proof by contradiction | $~$2$~$ |
---|
Proof by contradiction | $~$b$~$ |
---|
Proof by contradiction | $~$a$~$ |
---|
Proof by contradiction | $~$b$~$ |
---|
Proof by contradiction | $~$\sqrt 2$~$ |
---|
Proof by contradiction | $~$\sqrt 2$~$ |
---|
Proof of Bayes' rule | $~$H_i$~$ |
---|
Proof of Bayes' rule | $~$H_j$~$ |
---|
Proof of Bayes' rule | $~$e,$~$ |
---|
Proof of Bayes' rule | $$~$\dfrac{\mathbb P(H_i)}{\mathbb P(H_j)} \times \dfrac{\mathbb P(e \mid H_i)}{\mathbb P(e \mid H_j)} = \dfrac{\mathbb P(H_i \mid e)}{\mathbb P(H_j \mid e)}.$~$$ |
---|
Proof of Bayes' rule | $~$\mathbb P(e \land H)$~$ |
---|
Proof of Bayes' rule | $~$=$~$ |
---|
Proof of Bayes' rule | $~$\mathbb P(H) \cdot \mathbb P(e \mid H),$~$ |
---|
Proof of Bayes' rule | $$~$ \dfrac{\mathbb P(H_i)}{\mathbb P(H_j)} \times \dfrac{\mathbb P(e\mid H_i)}{\mathbb P(e\mid H_j)} = \dfrac{\mathbb P(e \wedge H_i)}{\mathbb P(e \wedge H_j)} $~$$ |
---|
Proof of Bayes' rule | $~$\mathbb P(e),$~$ |
---|
Proof of Bayes' rule | $$~$ \dfrac{\mathbb P(e \wedge H_i)}{\mathbb P(e \wedge H_j)} = \dfrac{\mathbb P(e \wedge H_i) / \mathbb P(e)}{\mathbb P(e \wedge H_j) / \mathbb P(e)} $~$$ |
---|
Proof of Bayes' rule | $$~$ \dfrac{\mathbb P(e \wedge H_i) / \mathbb P(e)}{\mathbb P(e \wedge H_j) / \mathbb P(e)} = \dfrac{\mathbb P(H_i\mid e)}{\mathbb P(H_j\mid e)}.$~$$ |
---|
Proof of Bayes' rule | $$~$\frac{\mathbb P(H_i\mid e)}{\mathbb P(H_j\mid e)} = \frac{\mathbb P(H_i \land e)}{\mathbb P(H_j \land e)},$~$$ |
---|
Proof of Bayes' rule | $~$H_i$~$ |
---|
Proof of Bayes' rule | $~$H_j$~$ |
---|
Proof of Bayes' rule | $~$e$~$ |
---|
Proof of Bayes' rule | $~$H_i$~$ |
---|
Proof of Bayes' rule | $~$H_j$~$ |
---|
Proof of Bayes' rule | $~$\mathbb P$~$ |
---|
Proof of Bayes' rule | $~$e$~$ |
---|
Proof of Bayes' rule | $~$\mathbb P(x \land e)$~$ |
---|
Proof of Bayes' rule | $~$\mathbb P$~$ |
---|
Proof of Bayes' rule | $~$x$~$ |
---|
Proof of Bayes' rule | $~$e$~$ |
---|
Proof of Bayes' rule | $~$e,$~$ |
---|
Proof of Bayes' rule | $~$H_i$~$ |
---|
Proof of Bayes' rule | $~$H_j$~$ |
---|
Proof of Bayes' rule | $~$H_i$~$ |
---|
Proof of Bayes' rule | $~$H_j$~$ |
---|
Proof of Bayes' rule | $~$e$~$ |
---|
Proof of Bayes' rule | $~$e$~$ |
---|
Proof of Bayes' rule | $$~$\dfrac{20\%}{80\%} \times \dfrac{90\%}{30\%} = \dfrac{18\%}{24\%} = \dfrac{0.18 / 0.42}{0.24 / 0.42} = \dfrac{3}{4}$~$$ |
---|
Proof of Bayes' rule | $~$\mathbb P(sick)$~$ |
---|
Proof of Bayes' rule | $~$\frac{3}{7} \approx 43\%.$~$ |
---|
Proof of Bayes' rule | $~$H_i$~$ |
---|
Proof of Bayes' rule | $~$H_j.$~$ |
---|
Proof of Bayes' rule | $~$\frac{3}{7} : \frac{4}{7}.$~$ |
---|
Proof of Bayes' rule | $~$\frac{\mathbb P(H_i)}{\mathbb P(H_j)}$~$ |
---|
Proof of Bayes' rule | $~$\frac{\mathbb P(H_i\mid e)}{\mathbb P(H_j\mid e)}.$~$ |
---|
Proof of Bayes' rule: Intro | $~$H_i$~$ |
---|
Proof of Bayes' rule: Intro | $~$H_j$~$ |
---|
Proof of Bayes' rule: Intro | $~$e$~$ |
---|
Proof of Bayes' rule: Intro | $$~$
\dfrac{\mathbb P(H_i)}{\mathbb P(H_j)}
\cdot
\dfrac{\mathbb P(e\mid H_i)}{\mathbb P(e\mid H_j)} =
\dfrac{\mathbb P(H_i\mid e)}{\mathbb P(H_j\mid e)}
$~$$ |
---|
Proof of Bayes' rule: Intro | $~$\frac{\mathbb P(H_i)}{\mathbb P(H_j)}$~$ |
---|
Proof of Bayes' rule: Intro | $~$\frac{\mathbb P(e\mid H_i)}{\mathbb P(e\mid H_j)}$~$ |
---|
Proof of Bayes' rule: Intro | $~$\frac{\mathbb P({positive}\mid {sick})}{\mathbb P({positive}\mid \neg {sick})}$~$ |
---|
Proof of Bayes' rule: Intro | $~$\frac{\mathbb P(H_i\mid e)}{\mathbb P(H_j\mid e)}$~$ |
---|
Proof of Bayes' rule: Intro | $~$\frac{\mathbb P({sick}\mid {positive})}{\mathbb P(\neg {sick}\mid {positive})}$~$ |
---|
Proof of Bayes' rule: Intro | $~$\mathbb P(X\mid Y) = \frac{\mathbb P(X \wedge Y)}{\mathbb P(Y)}.$~$ |
---|
Proof of Bayes' rule: Intro | $$~$
\dfrac{\mathbb P(H_i)}{\mathbb P(H_j)} \times \dfrac{\mathbb P(e\mid H_i)}{\mathbb P(e\mid H_j)}
= \dfrac{\mathbb P(e \wedge H_i)}{\mathbb P(e \wedge H_j)}
= \dfrac{\mathbb P(e \wedge H_i) / \mathbb P(e)}{\mathbb P(e \wedge H_j) / \mathbb P(e)}
= \dfrac{\mathbb P(H_i\mid e)}{\mathbb P(H_j\mid e)}
$~$$ |
---|
Proof of Bayes' rule: Intro | $$~$\dfrac{20\%}{80\%} \times \dfrac{90\%}{30\%} = \dfrac{18\%}{24\%} = \dfrac{0.18 / 0.42}{0.24 / 0.42} = \dfrac{43\%}{57\%}$~$$ |
---|
Proof of Bayes' rule: Probability form | $~$\mathbf H$~$ |
---|
Proof of Bayes' rule: Probability form | $~$\mathbb P$~$ |
---|
Proof of Bayes' rule: Probability form | $~$H_k$~$ |
---|
Proof of Bayes' rule: Probability form | $~$\mathbf H,$~$ |
---|
Proof of Bayes' rule: Probability form | $~$H_k$~$ |
---|
Proof of Bayes' rule: Probability form | $$~$\mathbb P(H_i\mid e) = \dfrac{\mathbb P(e\mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P(e\mid H_k) \cdot \mathbb P(H_k)},$~$$ |
---|
Proof of Bayes' rule: Probability form | $$~$\mathbb P(H_i\mid e) = \dfrac{\mathbb P(e \wedge H_i)}{\mathbb P(e)} = \dfrac{\mathbb P(e \mid H_i) \cdot \mathbb P(H_i)}{\mathbb P(e)}$~$$ |
---|
Proof of Bayes' rule: Probability form | $$~$\mathbb P(e) = \sum_{k} \mathbb P(e \wedge H_k)$~$$ |
---|
Proof of Bayes' rule: Probability form | $$~$\mathbb P(e \wedge H_k) = \mathbb P(e\mid H_k) \cdot \mathbb P(H_k)$~$$ |
---|
Proof of Bayes' rule: Probability form | $$~$\begin{array}{c}
\mathbb P({sick}\mid {positive}) = \dfrac{\mathbb P({positive} \wedge {sick})}{\mathbb P({positive})} \\[0.3em]
= \dfrac{\mathbb P({positive} \wedge {sick})}{\mathbb P({positive} \wedge {sick}) + \mathbb P({positive} \wedge \neg {sick})} \\[0.3em]
= \dfrac{\mathbb P({positive}\mid {sick}) \cdot \mathbb P({sick})}{(\mathbb P({positive}\mid {sick}) \cdot \mathbb P({sick})) + (\mathbb P({positive}\mid \neg {sick}) \cdot \mathbb P(\neg {sick}))}
\end{array}
$~$$ |
---|
Proof of Bayes' rule: Probability form | $$~$3/7 = \dfrac{0.18}{0.42} = \dfrac{0.18}{0.18 + 0.24} = \dfrac{90\% * 20\%}{(90\% * 20\%) + (30\% * 80\%)}$~$$ |
---|
Proof of Gödel's first incompleteness theorem | $~$\omega$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$Prv_{T}$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$G$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T\vdash G\iff \neg Prv_{T}(G)$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$G$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T\vdash G$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$Prv_ {T}(G)$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$\exists$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T\vdash Prv_ {T}(G)$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$G$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T\vdash \neg Prv_{T}(G)$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T\vdash \neg G$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T\vdash Prv_{T}(G)$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$G$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$Prv_{T}(x)$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$\exists y Proof_{T}(x,y)$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$n$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T\vdash Proof_ {T}(\ulcorner G\urcorner,n)$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$T$~$ |
---|
Proof of Gödel's first incompleteness theorem | $~$\omega$~$ |
---|
Proof of Löb's theorem | $~$\square A\to A$~$ |
---|
Proof of Löb's theorem | $~$S$~$ |
---|
Proof of Löb's theorem | $~$PA\vdash S\leftrightarrow (\square S \to A)$~$ |
---|
Proof of Löb's theorem | $~$S$~$ |
---|
Proof of Löb's theorem | $~$S$~$ |
---|
Proof of Löb's theorem | $~$S$~$ |
---|
Proof of Löb's theorem | $~$S$~$ |
---|
Proof of Löb's theorem | $~$S$~$ |
---|
Proof of Löb's theorem | $~$S$~$ |
---|
Proof of Löb's theorem | $~$S$~$ |
---|
Proof of Löb's theorem | $~$S$~$ |
---|
Proof of Löb's theorem | $~$Prv(x)\implies$~$ |
---|
Proof of Löb's theorem | $~$S$~$ |
---|
Proof of Löb's theorem | $~$0=1$~$ |
---|
Proof of Löb's theorem | $~$Prv$~$ |
---|
Proof of Löb's theorem | $~$\exists y Proof(x,y)$~$ |
---|
Proof of Löb's theorem | $~$Proof(x,y)$~$ |
---|
Proof of Löb's theorem | $~$y$~$ |
---|
Proof of Löb's theorem | $~$x$~$ |
---|
Proof of Löb's theorem | $~$y$~$ |
---|
Proof of Löb's theorem | $~$x$~$ |
---|
Proof of Löb's theorem | $~$x$~$ |
---|
Proof of Löb's theorem | $~$Prv(A)\implies A$~$ |
---|
Proof of Löb's theorem | $~$A$~$ |
---|
Proof of Löb's theorem | $~$T$~$ |
---|
Proof of Löb's theorem | $~$P$~$ |
---|
Proof of Löb's theorem | $~$T\vdash A$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(A)$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(A\implies B) \implies (P(A)\implies P(B))$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(A)\implies P(P(A))$~$ |
---|
Proof of Löb's theorem | $~$PA$~$ |
---|
Proof of Löb's theorem | $~$A$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(A)\implies A$~$ |
---|
Proof of Löb's theorem | $~$T$~$ |
---|
Proof of Löb's theorem | $~$S$~$ |
---|
Proof of Löb's theorem | $~$T\vdash S \iff (P(S)\implies A)$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(S \implies (P(S)\implies A))$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(S) \implies P(P(S) \implies A)$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(P(S) \implies A) \implies (P(P(S)) \implies P(A))$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(S) \implies (P(P(S)) \implies P(A))$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(S)\implies P(P(S))$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(S)\implies P(A)$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(S)\implies A$~$ |
---|
Proof of Löb's theorem | $~$S$~$ |
---|
Proof of Löb's theorem | $~$T\vdash S$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(S)$~$ |
---|
Proof of Löb's theorem | $~$T\vdash P(S)\implies A$~$ |
---|
Proof of Löb's theorem | $~$T\vdash A$~$ |
---|
Proof of Löb's theorem | $~$P$~$ |
---|
Proof of Löb's theorem | $~$x=x$~$ |
---|
Proof of Rice's theorem | $~$[n]$~$ |
---|
Proof of Rice's theorem | $~$n$~$ |
---|
Proof of Rice's theorem | $~$[n]$~$ |
---|
Proof of Rice's theorem | $~$[n](m)$~$ |
---|
Proof of Rice's theorem | $~$[n]$~$ |
---|
Proof of Rice's theorem | $~$m$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$\{ \mathrm{Graph}(n) : n \in \mathbb{N} \}$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(n)$~$ |
---|
Proof of Rice's theorem | $~$[n]$~$ |
---|
Proof of Rice's theorem | $~$n$~$ |
---|
Proof of Rice's theorem | $~$[r]$~$ |
---|
Proof of Rice's theorem | $~$[r](i)$~$ |
---|
Proof of Rice's theorem | $~$1$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(i) \in A$~$ |
---|
Proof of Rice's theorem | $~$[r](i)$~$ |
---|
Proof of Rice's theorem | $~$0$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(i) \not \in A$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$B$~$ |
---|
Proof of Rice's theorem | $~$h: \mathbb{N} \to \mathbb{N}$~$ |
---|
Proof of Rice's theorem | $~$n \in \mathbb{N}$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(n) = \mathrm{Graph}(h(n))$~$ |
---|
Proof of Rice's theorem | $~$n$~$ |
---|
Proof of Rice's theorem | $~$n$~$ |
---|
Proof of Rice's theorem | $~$[n]$~$ |
---|
Proof of Rice's theorem | $~$[h(n)]$~$ |
---|
Proof of Rice's theorem | $~$h$~$ |
---|
Proof of Rice's theorem | $~$h$~$ |
---|
Proof of Rice's theorem | $~$s_{mn}$~$ |
---|
Proof of Rice's theorem | $~$S$~$ |
---|
Proof of Rice's theorem | $~$m, n$~$ |
---|
Proof of Rice's theorem | $~$e \in \mathbb{N}$~$ |
---|
Proof of Rice's theorem | $~$[e](m, n) = [S(e,m)](n)$~$ |
---|
Proof of Rice's theorem | $~$S$~$ |
---|
Proof of Rice's theorem | $~$[e](m,n)$~$ |
---|
Proof of Rice's theorem | $~$[e](\mathrm{pair}(m, n))$~$ |
---|
Proof of Rice's theorem | $~$(e, x)$~$ |
---|
Proof of Rice's theorem | $~$[ h(S(e,e)) ](x)$~$ |
---|
Proof of Rice's theorem | $~$a$~$ |
---|
Proof of Rice's theorem | $~$e$~$ |
---|
Proof of Rice's theorem | $~$[e]$~$ |
---|
Proof of Rice's theorem | $~$S(e,e)$~$ |
---|
Proof of Rice's theorem | $~$S(a, a)$~$ |
---|
Proof of Rice's theorem | $~$n$~$ |
---|
Proof of Rice's theorem | $~$S(e,e)$~$ |
---|
Proof of Rice's theorem | $~$a$~$ |
---|
Proof of Rice's theorem | $~$x$~$ |
---|
Proof of Rice's theorem | $~$[n](x) = [S(a,a)](x)$~$ |
---|
Proof of Rice's theorem | $~$n$~$ |
---|
Proof of Rice's theorem | $~$[a](a, x)$~$ |
---|
Proof of Rice's theorem | $~$s_{mn}$~$ |
---|
Proof of Rice's theorem | $~$[h(S(a,a))](x)$~$ |
---|
Proof of Rice's theorem | $~$[a]$~$ |
---|
Proof of Rice's theorem | $~$[h(n)](x)$~$ |
---|
Proof of Rice's theorem | $~$n$~$ |
---|
Proof of Rice's theorem | $~$[n](x) = [h(n)](x)$~$ |
---|
Proof of Rice's theorem | $~$n$~$ |
---|
Proof of Rice's theorem | $~$128$~$ |
---|
Proof of Rice's theorem | $~$0$~$ |
---|
Proof of Rice's theorem | $~$128$~$ |
---|
Proof of Rice's theorem | $~$h$~$ |
---|
Proof of Rice's theorem | $~$h$~$ |
---|
Proof of Rice's theorem | $~$1$~$ |
---|
Proof of Rice's theorem | $~$0$~$ |
---|
Proof of Rice's theorem | $~$x$~$ |
---|
Proof of Rice's theorem | $~$S(m, n)$~$ |
---|
Proof of Rice's theorem | $~$[e](5)$~$ |
---|
Proof of Rice's theorem | $~$e$~$ |
---|
Proof of Rice's theorem | $~$5$~$ |
---|
Proof of Rice's theorem | $~$e$~$ |
---|
Proof of Rice's theorem | $~$S$~$ |
---|
Proof of Rice's theorem | $~$m$~$ |
---|
Proof of Rice's theorem | $~$S(a, a)$~$ |
---|
Proof of Rice's theorem | $~$a$~$ |
---|
Proof of Rice's theorem | $~$(e, x)$~$ |
---|
Proof of Rice's theorem | $~$[h(S(e,e))](x)$~$ |
---|
Proof of Rice's theorem | $~$a$~$ |
---|
Proof of Rice's theorem | $~$S$~$ |
---|
Proof of Rice's theorem | $~$S$~$ |
---|
Proof of Rice's theorem | $~$h$~$ |
---|
Proof of Rice's theorem | $~$h$~$ |
---|
Proof of Rice's theorem | $~$h$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(n) \in A$~$ |
---|
Proof of Rice's theorem | $~$\iota$~$ |
---|
Proof of Rice's theorem | $~$1$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(n) \in A$~$ |
---|
Proof of Rice's theorem | $~$0$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$a$~$ |
---|
Proof of Rice's theorem | $~$b$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(a) \in A$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(b) \not \in A$~$ |
---|
Proof of Rice's theorem | $~$g$~$ |
---|
Proof of Rice's theorem | $~$n$~$ |
---|
Proof of Rice's theorem | $~$a$~$ |
---|
Proof of Rice's theorem | $~$\iota(n) = 0$~$ |
---|
Proof of Rice's theorem | $~$b$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$g$~$ |
---|
Proof of Rice's theorem | $~$b$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$n$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(n) = \mathrm{Graph}(g(n))$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(n)$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(g(n))$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$g(n) = b$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(g(n)) = \mathrm{Graph}(b)$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$g(n) = a$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(g(n)) = \mathrm{Graph}(a)$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$\mathrm{Graph}(g(n))$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$A$~$ |
---|
Proof of Rice's theorem | $~$\iota$~$ |
---|
Proof that there are infinitely many primes | $~$2$~$ |
---|
Proof that there are infinitely many primes | $~$1$~$ |
---|
Proof that there are infinitely many primes | $~$p_1, p_2, \ldots, p_n$~$ |
---|
Proof that there are infinitely many primes | $~$2$~$ |
---|
Proof that there are infinitely many primes | $~$2$~$ |
---|
Proof that there are infinitely many primes | $~$P = p_1p_2\ldots p_n + 1$~$ |
---|
Proof that there are infinitely many primes | $~$2$~$ |
---|
Proof that there are infinitely many primes | $~$P \geq 2+1 = 3$~$ |
---|
Proof that there are infinitely many primes | $~$P > 1$~$ |
---|
Proof that there are infinitely many primes | $~$P$~$ |
---|
Proof that there are infinitely many primes | $~$P$~$ |
---|
Proof that there are infinitely many primes | $~$P>1$~$ |
---|
Proof that there are infinitely many primes | $~$P$~$ |
---|
Proof that there are infinitely many primes | $~$1$~$ |
---|
Proof that there are infinitely many primes | $~$p_1 p_2 \dots p_n+1$~$ |
---|
Proof that there are infinitely many primes | $~$p_1, \dots, p_n$~$ |
---|
Proof that there are infinitely many primes | $~$p_1, \dots, p_6$~$ |
---|
Proof that there are infinitely many primes | $~$2,3,5,7,11,13$~$ |
---|
Proof that there are infinitely many primes | $~$p_1 \dots p_6 + 1 = 30031$~$ |
---|
Proof that there are infinitely many primes | $~$30031 = 59 \times 509$~$ |
---|
Proof that there are infinitely many primes | $~$59$~$ |
---|
Proof that there are infinitely many primes | $~$509$~$ |
---|
Proof that there are infinitely many primes | $~$30031$~$ |
---|
Proof that there are infinitely many primes | $~$30031$~$ |
---|
Proof that there are infinitely many primes | $~$30031$~$ |
---|
Proof that there are infinitely many primes | $~$59 \times 509$~$ |
---|
Properties of the logarithm | $~$\log_b(x \cdot y) = \log_b(x) + \log_b(y)$~$ |
---|
Properties of the logarithm | $~$b$~$ |
---|
Properties of the logarithm | $~$\log_b(1) = 0,$~$ |
---|
Properties of the logarithm | $~$\log_b(1) = \log_b(1 \cdot 1) = \log_b(1) + \log_b(1).$~$ |
---|
Properties of the logarithm | $~$\log_b\left(\frac{1}{x}\right) = -\log_b(x),$~$ |
---|
Properties of the logarithm | $~$\log_b(1) = \log_b\left(x \cdot \frac{1}{x}\right) = \log_b(x) + \log_b\left(\frac{1}{x}\right) = 0.$~$ |
---|
Properties of the logarithm | $~$\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y),$~$ |
---|
Properties of the logarithm | $~$\log_b\left(x^n\right) = n \cdot \log_b(x),$~$ |
---|
Properties of the logarithm | $~$x^n$~$ |
---|
Properties of the logarithm | $~$\underbrace{x \cdot x \cdot \ldots x}_{n\text{ times}}$~$ |
---|
Properties of the logarithm | $~$\log_b\left(\sqrt[n]{x}\right) = \frac{\log_b(x)}{n},$~$ |
---|
Properties of the logarithm | $~$\log_b(x) = \log_b\left((\sqrt[n]{x})^n\right) = n \cdot \log_b(\sqrt[n]{x}).$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$f(x \cdot y) = f(x) + f(y)$~$ |
---|
Properties of the logarithm | $~$x, y \in \mathbb R^+,$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$b$~$ |
---|
Properties of the logarithm | $~$f(b) = 1,$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$\log_b.$~$ |
---|
Properties of the logarithm | $~$\log_b(b) = 1.$~$ |
---|
Properties of the logarithm | $~$\log_b(b^n) = n,$~$ |
---|
Properties of the logarithm | $~$\log_b(x^n) = n \log_b(x)$~$ |
---|
Properties of the logarithm | $~$\log_b(b) = 1.$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $$~$
f(x \cdot y) = f(x) + f(y) \tag{1}
$~$$ |
---|
Properties of the logarithm | $~$x, y \in$~$ |
---|
Properties of the logarithm | $~$\mathbb R^+$~$ |
---|
Properties of the logarithm | $~$y$~$ |
---|
Properties of the logarithm | $~$y$~$ |
---|
Properties of the logarithm | $~$f,$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $$~$
f(1) = 0. \tag{2}
$~$$ |
---|
Properties of the logarithm | $$~$f(x) = f(x \cdot 1) = f(x) + f(1),\text{ so }f(1) = 0.$~$$ |
---|
Properties of the logarithm | $$~$
f(x) = -f\left(\frac{1}{x}\right). \tag{3}
$~$$ |
---|
Properties of the logarithm | $~$x$~$ |
---|
Properties of the logarithm | $~$x$~$ |
---|
Properties of the logarithm | $~$n$~$ |
---|
Properties of the logarithm | $~$c$~$ |
---|
Properties of the logarithm | $~$n$~$ |
---|
Properties of the logarithm | $~$c.$~$ |
---|
Properties of the logarithm | $$~$x \cdot \frac{1}{x} = 1,\text{ so }f(1) = f\left(x \cdot \frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right).$~$$ |
---|
Properties of the logarithm | $$~$f(1)=0,\text{ so }f(x)\text{ and }f\left(\frac{1}{x}\right)\text{ must be opposites.}$~$$ |
---|
Properties of the logarithm | $$~$
f\left(\frac{x}{y}\right) = f(x) - f(y). \tag{4}
$~$$ |
---|
Properties of the logarithm | $~$f\left(x \cdot \frac{1}{y}\right) = f(x) - f(y),$~$ |
---|
Properties of the logarithm | $~$y$~$ |
---|
Properties of the logarithm | $~$y.$~$ |
---|
Properties of the logarithm | $~$f\left(z \cdot \frac{x}{y}\right) = f(z) + f(x) - f(y),$~$ |
---|
Properties of the logarithm | $~$\frac{x}{y}$~$ |
---|
Properties of the logarithm | $~$x$~$ |
---|
Properties of the logarithm | $~$y.$~$ |
---|
Properties of the logarithm | $$~$
f\left(x^n\right) = n \cdot f(x). \tag{5}
$~$$ |
---|
Properties of the logarithm | $~$x$~$ |
---|
Properties of the logarithm | $~$n$~$ |
---|
Properties of the logarithm | $~$n$~$ |
---|
Properties of the logarithm | $~$x^n$~$ |
---|
Properties of the logarithm | $~$n$~$ |
---|
Properties of the logarithm | $~$x$~$ |
---|
Properties of the logarithm | $~$n \in \mathbb N:$~$ |
---|
Properties of the logarithm | $$~$f\left(x^n\right) = f(\underbrace{x \cdot x \cdot \ldots x}_{n\text{ times}}) = \underbrace{f(x) + f(x) + \ldots f(x)}_{n\text{ times}} = n \cdot f(x).$~$$ |
---|
Properties of the logarithm | $~$n \in \mathbb Q,$~$ |
---|
Properties of the logarithm | $~$n \in \mathbb N,$~$ |
---|
Properties of the logarithm | $~$n \in \mathbb Q.$~$ |
---|
Properties of the logarithm | $~$n \in \mathbb R,$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$n \in \mathbb Q,$~$ |
---|
Properties of the logarithm | $~$n \in \mathbb Q,$~$ |
---|
Properties of the logarithm | $~$n \in \mathbb R,$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $$~$
f(\sqrt[n]{x}) = \frac{f(x)}{n}. \tag{6}
$~$$ |
---|
Properties of the logarithm | $~$n$~$ |
---|
Properties of the logarithm | $~$x$~$ |
---|
Properties of the logarithm | $~$n$~$ |
---|
Properties of the logarithm | $~$x$~$ |
---|
Properties of the logarithm | $$~$(\sqrt[n]{x})^n = x,\text{ so }f\left((\sqrt[n]{x})^n\right)\text{ has to equal }f(x).$~$$ |
---|
Properties of the logarithm | $$~$f\left((\sqrt[n]{x})^n\right) = n \cdot f(\sqrt[n]{x}),\text{ so }f(\sqrt[n]{x}) = \frac{f(x)}{n}.$~$$ |
---|
Properties of the logarithm | $~$n \in \mathbb Q,$~$ |
---|
Properties of the logarithm | $~$n \in \mathbb R.$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $$~$
\text{Either $f$ sends all inputs to $0$, or there exists a $b \neq 1$ such that $f(b)=1.$}\tag{7}
$~$$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$b$~$ |
---|
Properties of the logarithm | $~$b$~$ |
---|
Properties of the logarithm | $~$1$~$ |
---|
Properties of the logarithm | $~$b$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$0$~$ |
---|
Properties of the logarithm | $~$x$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$y \neq 0.$~$ |
---|
Properties of the logarithm | $~$f(\sqrt[y]{x}) = \frac{f(x)}{y} = 1.$~$ |
---|
Properties of the logarithm | $~$y$~$ |
---|
Properties of the logarithm | $~$\sqrt[-3/4]{x}$~$ |
---|
Properties of the logarithm | $~$b$~$ |
---|
Properties of the logarithm | $~$\sqrt[y]{x}.$~$ |
---|
Properties of the logarithm | $~$b \neq 1$~$ |
---|
Properties of the logarithm | $~$f(b) = 1$~$ |
---|
Properties of the logarithm | $~$f(1) = 0$~$ |
---|
Properties of the logarithm | $$~$
\text{If $f(b)=1$ then } f(b^x) = x. \tag{8}
$~$$ |
---|
Properties of the logarithm | $~$x$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$x$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$b.$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$b$~$ |
---|
Properties of the logarithm | $~$x$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$1$~$ |
---|
Properties of the logarithm | $~$b$~$ |
---|
Properties of the logarithm | $~$x$~$ |
---|
Properties of the logarithm | $~$f = \log_b$~$ |
---|
Properties of the logarithm | $~$y$~$ |
---|
Properties of the logarithm | $~$y$~$ |
---|
Properties of the logarithm | $~$z$~$ |
---|
Properties of the logarithm | $~$\log_\infty$~$ |
---|
Properties of the logarithm | $~$\log_\infty$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$y$~$ |
---|
Properties of the logarithm | $~$y$~$ |
---|
Properties of the logarithm | $~$b$~$ |
---|
Properties of the logarithm | $~$f(b)=1.$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$f(x \cdot y) = f(x) + f(y)$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Properties of the logarithm | $~$f$~$ |
---|
Proportion | $~$a$~$ |
---|
Proportion | $~$b$~$ |
---|
Proportion | $~$a$~$ |
---|
Proportion | $~$c \times b$~$ |
---|
Proportion | $~$c$~$ |
---|
Proportion | $~$a$~$ |
---|
Proportion | $~$b$~$ |
---|
Proportion | $~$2$~$ |
---|
Proportion | $~$l = 2w$~$ |
---|
Proportion | $~$w$~$ |
---|
Proportion | $~$l$~$ |
---|
Proportion | $~$a$~$ |
---|
Proportion | $~$b$~$ |
---|
Proportion | $~$a \propto b$~$ |
---|
Proportion | $~$A \propto L^2$~$ |
---|
Proportion | $~$A$~$ |
---|
Proportion | $~$L$~$ |
---|
Proportion | $~$2$~$ |
---|
Proportion | $~$100$~$ |
---|
Proportion | $~$200\%$~$ |
---|
Proportion | $~$6.7$~$ |
---|
Proportion | $~$100$~$ |
---|
Proportion | $~$6.7\%$~$ |
---|
Proportion | $~$5.2$~$ |
---|
Proportion | $~$6.8$~$ |
---|
Proportion | $~$131\%$~$ |
---|
Proportion | $~$C \propto d$~$ |
---|
Proportion | $~$\pi = 3.14159265\ldots$~$ |
---|
Proportion | $~$A \propto r^2$~$ |
---|
Proportion | $~$\pi$~$ |
---|
Proportion | $~$\frac{df}{dt} \propto f$~$ |
---|
Proportion | $~$\Delta f \propto f$~$ |
---|
Proportion | $~$E \propto h$~$ |
---|
Proportion | $~$I \propto V$~$ |
---|
Proportion | $~$P \propto T$~$ |
---|
Proposed A-Class | $~$3-rejections*2$~$ |
---|
Proposed B-Class | $~$1-rejections*2$~$ |
---|
Propositions | $~$S$~$ |
---|
Propositions | $~$S$~$ |
---|
Propositions | $~$S$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$B$~$ |
---|
Provability logic | $~$A\to B$~$ |
---|
Provability logic | $~$A$~$ |
---|
Provability logic | $~$GL\vdash A$~$ |
---|
Provability logic | $~$GL\vdash \square A$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$GL\vdash \square (A\to B)\to(\square A \to \square B)$~$ |
---|
Provability logic | $~$GL\vdash \square (\square A \to A)\to \square A$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$\square \bot \leftrightarrow \square \diamond p$~$ |
---|
Provability logic | $~$\diamond p$~$ |
---|
Provability logic | $~$\neg \square \neg p$~$ |
---|
Provability logic | $~$GL\vdash \square \bot \leftrightarrow \square \diamond p$~$ |
---|
Provability logic | $~$GL\vdash \square (\bot \leftrightarrow \diamond p)$~$ |
---|
Provability logic | $~$GL\vdash \bot \leftrightarrow \diamond p$~$ |
---|
Provability logic | $~$GL\vdash \bot \to \diamond p$~$ |
---|
Provability logic | $~$GL\vdash \square \bot \to \square \diamond p$~$ |
---|
Provability logic | $~$GL\vdash \square \diamond p\to \square \bot$~$ |
---|
Provability logic | $~$GL\vdash \bot \to \neg p$~$ |
---|
Provability logic | $~$GL\vdash \square \bot \to \square \neg p$~$ |
---|
Provability logic | $~$Gl\vdash \neg \square \neg p\to\neg\square\bot$~$ |
---|
Provability logic | $~$Gl\vdash \square \neg \square \neg p\to\square \neg\square\bot$~$ |
---|
Provability logic | $~$\square\neg\square\bot$~$ |
---|
Provability logic | $~$\square[\square \bot \to \bot]$~$ |
---|
Provability logic | $~$GL\vdash \square[\square \bot \to \bot] \to\square \bot$~$ |
---|
Provability logic | $~$Gl\vdash \square \neg \square \neg p\to\square \bot$~$ |
---|
Provability logic | $~$\diamond p = \neg \square \neg p$~$ |
---|
Provability logic | $~$\square$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$\neg \square \bot$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$A$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$A$~$ |
---|
Provability logic | $~$\rho$~$ |
---|
Provability logic | $~$w$~$ |
---|
Provability logic | $~$\rho(w)=0$~$ |
---|
Provability logic | $~$w$~$ |
---|
Provability logic | $~$\square[\square A\to A]\to \square A$~$ |
---|
Provability logic | $~$w$~$ |
---|
Provability logic | $~$w\not\models \square[\square A\to A]\to \square A$~$ |
---|
Provability logic | $~$w\models \square[\square A\to A]$~$ |
---|
Provability logic | $~$w\not \models \square A$~$ |
---|
Provability logic | $~$x$~$ |
---|
Provability logic | $~$w R x$~$ |
---|
Provability logic | $~$x\models \neg A$~$ |
---|
Provability logic | $~$x\models \square A\to A$~$ |
---|
Provability logic | $~$x\models \neg\square A$~$ |
---|
Provability logic | $~$x$~$ |
---|
Provability logic | $~$w$~$ |
---|
Provability logic | $~$x\models \square[\square A\to A]\to \square A$~$ |
---|
Provability logic | $~$x\not\models \square[\square A\to A]$~$ |
---|
Provability logic | $~$y$~$ |
---|
Provability logic | $~$xRy$~$ |
---|
Provability logic | $~$y\not\models \square A\to A$~$ |
---|
Provability logic | $~$wRy$~$ |
---|
Provability logic | $~$w\models \square[\square A\to A]$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$PSPACE$~$ |
---|
Provability logic | $~$*$~$ |
---|
Provability logic | $~$A\to B*=A*\to B*$~$ |
---|
Provability logic | $~$(\square A)* =\square_{PA}(A*)$~$ |
---|
Provability logic | $~$p* = S_p$~$ |
---|
Provability logic | $~$p$~$ |
---|
Provability logic | $~$S_p$~$ |
---|
Provability logic | $~$A$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$PA$~$ |
---|
Provability logic | $~$A*$~$ |
---|
Provability logic | $~$*$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$GL\not\vdash A$~$ |
---|
Provability logic | $~$PA\not\vdash A*$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$\#$~$ |
---|
Provability logic | $~$PA\not \vdash A^{\#}$~$ |
---|
Provability logic | $~$GL\not\vdash A$~$ |
---|
Provability logic | $~$A$~$ |
---|
Provability logic | $~$GL$~$ |
---|
Provability logic | $~$p\leftrightarrow \phi(p)$~$ |
---|
Provability logic | $~$\phi(p)$~$ |
---|
Provability logic | $~$p$~$ |
---|
Provability logic | $~$p$~$ |
---|
Provability logic | $~$\square$~$ |
---|
Provability logic | $~$H$~$ |
---|
Provability logic | $~$p$~$ |
---|
Provability logic | $~$GL\vdash \square [p\leftrightarrow \phi(p)] \leftrightarrow \square (p\leftrightarrow h)$~$ |
---|
Provability logic | $~$H$~$ |
---|
Provability logic | $~$p\leftrightarrow \neg\square p$~$ |
---|
Provability logic | $~$GL\vdash \square (p\leftrightarrow \neg\square p)\to \square(p\leftrightarrow \neg\square\bot$~$ |
---|
Provability logic | $~$PA$~$ |
---|
Provability predicate | $~$T$~$ |
---|
Provability predicate | $~$P(x)$~$ |
---|
Provability predicate | $~$x$~$ |
---|
Provability predicate | $~$T\vdash S$~$ |
---|
Provability predicate | $~$T\vdash P(\ulcorner S \urcorner)$~$ |
---|
Provability predicate | $~$T\vdash P(\ulcorner A\rightarrow B \urcorner)\rightarrow (P(\ulcorner A \urcorner)\rightarrow P(\ulcorner B \urcorner))$~$ |
---|
Provability predicate | $~$T\vdash P(\ulcorner S \urcorner)\rightarrow P(\ulcorner P(\ulcorner S \urcorner) \urcorner)$~$ |
---|
Provability predicate | $~$P$~$ |
---|
Provability predicate | $~$P$~$ |
---|
Provability predicate | $~$T\vdash S$~$ |
---|
Provability predicate | $~$T\vdash P(\ulcorner S \urcorner)$~$ |
---|
Provability predicate | $~$T\vdash P(\ulcorner A\rightarrow B \urcorner)\rightarrow (P(\ulcorner A \urcorner)\rightarrow P(\ulcorner B \urcorner))$~$ |
---|
Provability predicate | $~$T\vdash P(\ulcorner S \urcorner)\rightarrow P(\ulcorner P(\ulcorner S \urcorner) \urcorner)$~$ |
---|
Provability predicate | $~$x=x$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$G$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$G$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$G/N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$gN$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$G$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$gN + hN = (gh)N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$G/N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$G/N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$G$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$g_1 N = g_2 N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$h_1 N = h_2 N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$(g_1 h_1) N = (g_2 h_2)N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$g_1 h_1 N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$g_2 h_2 N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$g_1 h_1 n \in g_1 h_1 N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$h_2^{-1} g_2^{-1} g_1 h_1 n \in N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$h_2^{-1} g_2^{-1} g_1 h_1 \in N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$g_2^{-1} g_1 \in N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$g_1 N = g_2 N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$g_2^{-1} g_1 = m$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$h_2^{-1} h_1 \in N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$h_1 N = h_2 N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$h_2^{-1} h_1 = p$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$h_2^{-1} m h_1 \in N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$p h_1^{-1} m h_1 \in N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$m \in N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$h_1^{-1} m h_1 \in N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$p \in N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$p h_1^{-1} m h_1 \in N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$G/N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$h \in G$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$hnh^{-1} N + hN$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$(hnh^{-1}h) N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$hnN$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$hN$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$nN = N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$G$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$hnh^{-1}N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$hN$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$hN$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$hnh^{-1}N = N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$hnh^{-1} \in N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$h \in G$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$N$~$ |
---|
Quotient by subgroup is well defined if and only if subgroup is normal | $~$G$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$\bullet$~$ |
---|
Quotient group | $~$N \leq G$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$G/N$~$ |
---|
Quotient group | $~$G/N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$G/N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$\circ$~$ |
---|
Quotient group | $~$A$~$ |
---|
Quotient group | $~$B$~$ |
---|
Quotient group | $~$G/N$~$ |
---|
Quotient group | $~$A \circ B$~$ |
---|
Quotient group | $~$a \in A$~$ |
---|
Quotient group | $~$b \in B$~$ |
---|
Quotient group | $~$a \bullet b$~$ |
---|
Quotient group | $~$A$~$ |
---|
Quotient group | $~$B$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$\phi$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$G/N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$G/N$~$ |
---|
Quotient group | $~$G/N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$G/N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$(G, \bullet)$~$ |
---|
Quotient group | $~$N \unlhd G$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$G/N$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$\circ$~$ |
---|
Quotient group | $~$aN \circ bN = (a \bullet b) N$~$ |
---|
Quotient group | $~$xN = \{xn : n \in N\}$~$ |
---|
Quotient group | $~$x \in G$~$ |
---|
Quotient group | $~$\circ$~$ |
---|
Quotient group | $~$a'$~$ |
---|
Quotient group | $~$b'$~$ |
---|
Quotient group | $~$a'N = aN$~$ |
---|
Quotient group | $~$b'N = bN$~$ |
---|
Quotient group | $~$(a' \bullet b')N = (a \bullet b)N$~$ |
---|
Quotient group | $~$\phi: G \rightarrow G/N: a \mapsto aN$~$ |
---|
Quotient group | $~$\mathbb{Z}$~$ |
---|
Quotient group | $~$2 \mathbb{Z}$~$ |
---|
Quotient group | $~$\mathbb{Z}/2\mathbb{Z}$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$G/N$~$ |
---|
Quotient group | $~$\phi: G \rightarrow G/N$~$ |
---|
Quotient group | $~$g \in G$~$ |
---|
Quotient group | $~$gN$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$N \le G$~$ |
---|
Quotient group | $~$\phi$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$G/N$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$\phi$~$ |
---|
Quotient group | $~$N \trianglelefteq G$~$ |
---|
Quotient group | $~$gN = \{gn : n \in N\}$~$ |
---|
Quotient group | $~$g \in G$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$g_1N \cdot g_2N = (g_1g_2)N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$|G/N| = |G|/|N|$~$ |
---|
Quotient group | $~$G/N$~$ |
---|
Quotient group | $~$N$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$G$~$ |
---|
Quotient group | $~$\mathbb Z$~$ |
---|
Quotient group | $~$2\mathbb Z = \{2n : n\in \mathbb Z\}$~$ |
---|
Quotient group | $~$\mathbb Z$~$ |
---|
Quotient group | $~$0 + 2\mathbb Z$~$ |
---|
Quotient group | $~$1 + 2\mathbb Z$~$ |
---|
Quotient group | $~$+$~$ |
---|
Quotient group | $~$\text{even}$~$ |
---|
Quotient group | $~$\text{odd}$~$ |
---|
Quotient group | $~$\text{even}+ \text{even} = \text{even}$~$ |
---|
Quotient group | $~$\text{even} + \text{odd} = \text{odd}$~$ |
---|
Quotient group | $~$\text{odd} + \text{odd} = \text{even}$~$ |
---|
Random utility function | $~$2^{-\operatorname K(U)}$~$ |
---|
Random utility function | $~$\operatorname K(U)$~$ |
---|
Random utility function | $~$U.$~$ |
---|
Random utility function | $~$U$~$ |
---|
Rational arithmetic all works together | $~$a, b, c, d$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$b$~$ |
---|
Rational arithmetic all works together | $~$d$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$b$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$\frac{c}{d}$~$ |
---|
Rational arithmetic all works together | $~$a \times d = b \times c$~$ |
---|
Rational arithmetic all works together | $~$\frac{0}{x} = \frac{0}{y}$~$ |
---|
Rational arithmetic all works together | $~$x, y$~$ |
---|
Rational arithmetic all works together | $~$\frac{0}{b \times d} = \frac{0}{1}$~$ |
---|
Rational arithmetic all works together | $~$b, d$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $$~$\frac{a}{b} + \frac{c}{d} = \frac{a \times d + b \times c} {b \times d}$~$$ |
---|
Rational arithmetic all works together | $$~$\frac{a}{b} - \frac{c}{d} = \frac{a}{b} + \frac{-c}{d} = \frac{a \times d - b \times c}{b \times d}$~$$ |
---|
Rational arithmetic all works together | $$~$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$~$$ |
---|
Rational arithmetic all works together | $~$c$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $$~$\frac{a}{b} \big/ \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$~$$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$\frac{c}{d}$~$ |
---|
Rational arithmetic all works together | $~$b$~$ |
---|
Rational arithmetic all works together | $~$d$~$ |
---|
Rational arithmetic all works together | $~$b$~$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$\frac{-a}{-b}$~$ |
---|
Rational arithmetic all works together | $~$b$~$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b} < \frac{c}{d}$~$ |
---|
Rational arithmetic all works together | $~$\frac{c}{d}-\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $$~$\frac{b \times c - a \times d}{b \times d} > 0$~$$ |
---|
Rational arithmetic all works together | $$~$b \times c - a \times d > 0$~$$ |
---|
Rational arithmetic all works together | $~$\frac{2}{4}$~$ |
---|
Rational arithmetic all works together | $~$\frac{1}{2}$~$ |
---|
Rational arithmetic all works together | $$~$\frac{2}{4} + \frac{1}{3} = \frac{1}{2} + \frac{1}{3}$~$$ |
---|
Rational arithmetic all works together | $~$n$~$ |
---|
Rational arithmetic all works together | $~$\frac{n}{1}$~$ |
---|
Rational arithmetic all works together | $~$n$~$ |
---|
Rational arithmetic all works together | $~$\frac{1}{1}$~$ |
---|
Rational arithmetic all works together | $~$\frac{a \times d + b \times c} {b \times d}$~$ |
---|
Rational arithmetic all works together | $~$a \times d + b \times c$~$ |
---|
Rational arithmetic all works together | $~$b \times d$~$ |
---|
Rational arithmetic all works together | $~$a, b, c, d$~$ |
---|
Rational arithmetic all works together | $~$b, d$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$\frac{5}{6} + 0 = \frac{5}{6}$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$\frac{0}{1}$~$ |
---|
Rational arithmetic all works together | $$~$\frac{0}{1} + \frac{a}{b} = \frac{0 \times b + a \times 1}{1 \times b} = \frac{0 + a}{b} = \frac{a}{b}$~$$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$\frac{c}{d}$~$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b} + \frac{c}{d} = \frac{0}{1}$~$ |
---|
Rational arithmetic all works together | $~$-\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$-\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$\frac{-a}{b}$~$ |
---|
Rational arithmetic all works together | $$~$\frac{a}{b} + \frac{-a}{b} = \frac{a \times b + (-a) \times b}{b \times b} = \frac{0}{b \times b}$~$$ |
---|
Rational arithmetic all works together | $~$\frac{0}{b \times b} = \frac{0}{1}$~$ |
---|
Rational arithmetic all works together | $~$0 \times 1 = 0 \times (b \times b)$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $$~$\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b}$~$$ |
---|
Rational arithmetic all works together | $$~$\frac{a}{b} + \frac{c}{d} = \frac{a \times d + b \times c}{b \times d} = \frac{c \times b + d \times a}{d \times b} = \frac{c}{d} + \frac{a}{b}$~$$ |
---|
Rational arithmetic all works together | $$~$\left(\frac{a}{b} + \frac{c}{d}\right) + \frac{e}{f} = \frac{a}{b} + \left( \frac{c}{d} + \frac{e}{f} \right)$~$$ |
---|
Rational arithmetic all works together | $~$6$~$ |
---|
Rational arithmetic all works together | $$~$\left(\frac{a}{b} + \frac{c}{d}\right) + \frac{e}{f} = \frac{a \times d + b \times c}{b \times d} + \frac{e}{f} = \frac{(a \times d + b \times c) \times f + (b \times d) \times e}{(b \times d) \times f}$~$$ |
---|
Rational arithmetic all works together | $$~$\frac{a \times d \times f + b \times c \times f + b \times d \times e}{b \times d \times f}$~$$ |
---|
Rational arithmetic all works together | $$~$\frac{a}{b} + \left( \frac{c}{d} + \frac{e}{f} \right) = \frac{a}{b} + \frac{c \times f + d \times e}{d \times f} = \frac{a \times (d \times f) + b \times (c \times f + d \times e))}{b \times (d \times f)}$~$$ |
---|
Rational arithmetic all works together | $$~$\frac{a \times d \times f + b \times c \times f + b \times d \times e}{b \times d \times f}$~$$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$~$ |
---|
Rational arithmetic all works together | $~$b \times d$~$ |
---|
Rational arithmetic all works together | $~$b$~$ |
---|
Rational arithmetic all works together | $~$d$~$ |
---|
Rational arithmetic all works together | $~$1$~$ |
---|
Rational arithmetic all works together | $~$1$~$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$1$~$ |
---|
Rational arithmetic all works together | $~$1$~$ |
---|
Rational arithmetic all works together | $~$1$~$ |
---|
Rational arithmetic all works together | $~$\frac{1}{1}$~$ |
---|
Rational arithmetic all works together | $$~$\frac{1}{1} \times \frac{a}{b} = \frac{1 \times a}{1 \times b} = \frac{a}{b}$~$$ |
---|
Rational arithmetic all works together | $~$1 \times n = n$~$ |
---|
Rational arithmetic all works together | $~$n$~$ |
---|
Rational arithmetic all works together | $~$1$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$a$~$ |
---|
Rational arithmetic all works together | $~$b$~$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$\frac{b}{a}$~$ |
---|
Rational arithmetic all works together | $$~$\frac{a}{b} \times \frac{b}{a} = \frac{a\times b}{b \times a} = \frac{a \times b}{a \times b} = \frac{1}{1}$~$$ |
---|
Rational arithmetic all works together | $$~$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} = \frac{c \times a}{d \times b} = \frac{c}{d} \times \frac{a}{b}$~$$ |
---|
Rational arithmetic all works together | $$~$\frac{a}{b} \times \left(\frac{c}{d} \times \frac{e}{f} \right) = \left(\frac{a}{b} \times \frac{c}{d} \right) \times \frac{e}{f}$~$$ |
---|
Rational arithmetic all works together | $$~$\frac{a}{b} \times \left(\frac{c}{d} \times \frac{e}{f} \right) = \frac{a}{b} \times \frac{c \times e}{d \times f} = \frac{a \times (c \times e)}{b \times (d \times f)}$~$$ |
---|
Rational arithmetic all works together | $~$\frac{a \times c \times e}{b \times d \times f}$~$ |
---|
Rational arithmetic all works together | $$~$\left(\frac{a}{b} \times \frac{c}{d} \right) \times \frac{e}{f} = \frac{a \times c}{b \times d} \times \frac{e}{f} = \frac{(a \times c) \times e}{(b \times d) \times f} = \frac{a \times c \times e}{b \times d \times f}$~$$ |
---|
Rational arithmetic all works together | $$~$\left(\frac{c}{d} + \frac{e}{f}\right) \times \frac{a}{b} = \left(\frac{a}{b} \times \frac{c}{d}\right) + \left(\frac{a}{b} \times \frac{e}{f}\right)$~$$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$1$~$ |
---|
Rational arithmetic all works together | $~$\left(\frac{c}{d} + \frac{e}{f}\right)$~$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$1$~$ |
---|
Rational arithmetic all works together | $~$\frac{c}{d}$~$ |
---|
Rational arithmetic all works together | $~$\frac{e}{f}$~$ |
---|
Rational arithmetic all works together | $~$a,b,c,d,e,f$~$ |
---|
Rational arithmetic all works together | $$~$\left(\frac{c}{d} + \frac{e}{f}\right) \times \frac{a}{b} = \frac{c \times f + d \times e}{d \times f} \times \frac{a}{b} = \frac{(c \times f + d \times e) \times a}{(d \times f) \times b}$~$$ |
---|
Rational arithmetic all works together | $~$\frac{c \times f \times a + d \times e \times a}{d \times f \times b}$~$ |
---|
Rational arithmetic all works together | $$~$\left(\frac{a}{b} \times \frac{c}{d}\right) + \left(\frac{a}{b} \times \frac{e}{f}\right) = \frac{a \times c}{b \times d} + \frac{a \times e}{b \times f} = \frac{(a \times c) \times (b \times f) + (b \times d) \times (a \times e)}{(b \times d) \times (b \times f)} = \frac{(a \times c \times b \times f) + (b \times d \times a \times e)}{(b \times d \times b \times f)}$~$$ |
---|
Rational arithmetic all works together | $~$b$~$ |
---|
Rational arithmetic all works together | $$~$\frac{b \times [(a \times c \times f) + (d \times a \times e)]}{b \times (d \times b \times f)}$~$$ |
---|
Rational arithmetic all works together | $~$b$~$ |
---|
Rational arithmetic all works together | $$~$\frac{(a \times c \times f) + (d \times a \times e)}{d \times b \times f}$~$$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b} < \frac{c}{d}$~$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b} + \frac{e}{f} < \frac{c}{d} + \frac{e}{f}$~$ |
---|
Rational arithmetic all works together | $$~$0 < \frac{c}{d} + \frac{e}{f} - (\frac{a}{b} + \frac{e}{f})$~$$ |
---|
Rational arithmetic all works together | $~$-1$~$ |
---|
Rational arithmetic all works together | $$~$0 < \frac{c}{d} + \frac{e}{f} - \frac{a}{b} - \frac{e}{f}$~$$ |
---|
Rational arithmetic all works together | $$~$0 < \frac{c}{d} - \frac{a}{b} + \frac{e}{f} - \frac{e}{f}$~$$ |
---|
Rational arithmetic all works together | $~$0 < \frac{c}{d} - \frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b} < \frac{c}{d}$~$ |
---|
Rational arithmetic all works together | $~$0 < \frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$0 < \frac{c}{d}$~$ |
---|
Rational arithmetic all works together | $~$0 < \frac{a}{b} \times \frac{c}{d}$~$ |
---|
Rational arithmetic all works together | $~$0$~$ |
---|
Rational arithmetic all works together | $~$0 < \frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$a$~$ |
---|
Rational arithmetic all works together | $~$b$~$ |
---|
Rational arithmetic all works together | $~$c$~$ |
---|
Rational arithmetic all works together | $~$d$~$ |
---|
Rational arithmetic all works together | $~$a, b, c, d$~$ |
---|
Rational arithmetic all works together | $~$a, b, c, d$~$ |
---|
Rational arithmetic all works together | $~$a, b$~$ |
---|
Rational arithmetic all works together | $~$c, d$~$ |
---|
Rational arithmetic all works together | $~$a, b$~$ |
---|
Rational arithmetic all works together | $~$c, d$~$ |
---|
Rational arithmetic all works together | $~$\frac{a \times c}{b \times d}$~$ |
---|
Rational arithmetic all works together | $~$a, b, c, d$~$ |
---|
Rational arithmetic all works together | $~$a \times c$~$ |
---|
Rational arithmetic all works together | $~$b \times d$~$ |
---|
Rational arithmetic all works together | $~$\frac{a \times c}{b \times d}$~$ |
---|
Rational arithmetic all works together | $~$a \times c$~$ |
---|
Rational arithmetic all works together | $~$b \times d$~$ |
---|
Rational arithmetic all works together | $~$\frac{a \times c}{b \times d}$~$ |
---|
Rational arithmetic all works together | $~$-1$~$ |
---|
Rational arithmetic all works together | $~$\frac{1}{3}$~$ |
---|
Rational arithmetic all works together | $~$\frac{-2}{-5}$~$ |
---|
Rational arithmetic all works together | $~$\frac{1}{3} \times \frac{-2}{-5} = \frac{-2}{-15}$~$ |
---|
Rational arithmetic all works together | $~$\frac{2}{15}$~$ |
---|
Rational arithmetic all works together | $~$\frac{a}{b} \times \frac{c}{d}$~$ |
---|
Rational arithmetic all works together | $~$\frac{c}{d} \times \frac{a}{b}$~$ |
---|
Rational arithmetic all works together | $~$c$~$ |
---|
Rational arithmetic all works together | $~$d$~$ |
---|
Rational arithmetic all works together | $~$a$~$ |
---|
Rational arithmetic all works together | $~$b$~$ |
---|
Rational arithmetic all works together | $~$a$~$ |
---|
Rational arithmetic all works together | $~$c$~$ |
---|
Rational arithmetic all works together | $~$b$~$ |
---|
Rational arithmetic all works together | $~$d$~$ |
---|
Rational number | $~$\frac{a}{b}$~$ |
---|
Rational number | $~$a$~$ |
---|
Rational number | $~$b$~$ |
---|
Rational number | $~$0,$~$ |
---|
Rational number | $~$1$~$ |
---|
Rational number | $~$2$~$ |
---|
Rational number | $~$\frac{1}{2}$~$ |
---|
Rational number | $~$\frac{97}{3}$~$ |
---|
Rational number | $~$-17$~$ |
---|
Rational number | $~$\frac{-85}{1993},$~$ |
---|
Rational number | $~$\mathbb Q.$~$ |
---|
Rational number | $~$\pi$~$ |
---|
Rational number | $~$e$~$ |
---|
Rational number | $~$\mathbb Q;$~$ |
---|
Rational number | $~$\frac{a}{b}$~$ |
---|
Rational number | $~$a$~$ |
---|
Rational number | $~$b$~$ |
---|
Rational number | $~$b \neq 0$~$ |
---|
Rational number | $~$\mathbb{Q}$~$ |
---|
Rational number | $~$\mathbb{Z}$~$ |
---|
Rational number | $~$q \in \mathbb Q$~$ |
---|
Rational number | $~$\frac{a}{b}$~$ |
---|
Rational number | $~$b$~$ |
---|
Rational number | $~$a$~$ |
---|
Rational number | $~$x$~$ |
---|
Rational number | $~$\frac{a}{b}$~$ |
---|
Rational number | $~$a, b$~$ |
---|
Rational number | $~$b$~$ |
---|
Rational number | $~$0$~$ |
---|
Rational number | $~$1$~$ |
---|
Rational number | $~$\frac{1}{1}$~$ |
---|
Rational number | $~$\frac{2}{2}$~$ |
---|
Rational number | $~$\frac{-1}{-1}$~$ |
---|
Rational number | $~$\frac{a}{a}$~$ |
---|
Rational number | $~$a$~$ |
---|
Rational number | $~$\pi$~$ |
---|
Rational number | $~$\sqrt{2}$~$ |
---|
Rational number | $~$2$~$ |
---|
Rational number | $~$n$~$ |
---|
Rational number | $~$\frac{n}{1}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$0$~$ |
---|
Rational numbers: Intro (Math 0) | $~$0$~$ |
---|
Rational numbers: Intro (Math 0) | $~$0$~$ |
---|
Rational numbers: Intro (Math 0) | $~$1$~$ |
---|
Rational numbers: Intro (Math 0) | $~$1$~$ |
---|
Rational numbers: Intro (Math 0) | $~$1$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{5}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{6}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$1$~$ |
---|
Rational numbers: Intro (Math 0) | $~$1$~$ |
---|
Rational numbers: Intro (Math 0) | $~$1$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{2}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$2$~$ |
---|
Rational numbers: Intro (Math 0) | $~$1$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{2}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{2}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{3}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{3}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{3}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{2}{3}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{3}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{3}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{5}{3}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{\cdot}{\cdot}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{\text{lots}}{\text{dividey-number}}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{\text{numerator}}{\text{denominator}}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$3$~$ |
---|
Rational numbers: Intro (Math 0) | $~$1$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{6}{2}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$3$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{15}{5}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$3$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{4}{2}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{3}{3}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$3$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{2}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{2}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{3}{6}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{2}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{5}{10}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{2}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{5}{10}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{2}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{3}{6}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{10}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{2}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{n}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$n$~$ |
---|
Rational numbers: Intro (Math 0) | $~$n$~$ |
---|
Rational numbers: Intro (Math 0) | $~$n$~$ |
---|
Rational numbers: Intro (Math 0) | $~$n$~$ |
---|
Rational numbers: Intro (Math 0) | $~$n$~$ |
---|
Rational numbers: Intro (Math 0) | $~$0$~$ |
---|
Rational numbers: Intro (Math 0) | $~$0$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{0}$~$ |
---|
Rational numbers: Intro (Math 0) | $~$\frac{1}{3}$~$ |
---|
Real number | $~$0,$~$ |
---|
Real number | $~$1,$~$ |
---|
Real number | $~$-1,$~$ |
---|
Real number | $~$\frac{3}{2},$~$ |
---|
Real number | $~$\frac{-7}{2},$~$ |
---|
Real number | $~$\pi,$~$ |
---|
Real number | $~$e$~$ |
---|
Real number | $~$100 \cdot \sqrt{2},$~$ |
---|
Real number | $~$\mathbb R.$~$ |
---|
Real number | $~$\mathbb R$~$ |
---|
Real number | $~$\mathbb Q$~$ |
---|
Real number | $~$\pi$~$ |
---|
Real number | $~$e$~$ |
---|
Real number | $~$1/1, 2/1, 3/2, 5/3, 8/5, \ldots$~$ |
---|
Real number | $~$\frac{1 + \sqrt{5}}{2}$~$ |
---|
Real number | $~$(A, B)$~$ |
---|
Real number | $~$B = \{x \in \mathbb{Q} \ | \ x > 0 \wedge x^2 > 2\}$~$ |
---|
Real number | $~$A$~$ |
---|
Real number | $~$B$~$ |
---|
Real number | $~$B$~$ |
---|
Real number | $~$\sqrt{2}$~$ |
---|
Real number | $~$\sqrt{2}$~$ |
---|
Real number | $~$A$~$ |
---|
Real number | $~$B$~$ |
---|
Real number | $~$A$~$ |
---|
Real number | $~$A$~$ |
---|
Real number | $~$\sqrt{2}$~$ |
---|
Real number (as Cauchy sequence) | $$~$X = \{ (a_n)_{n=1}^{\infty} : a_n \in \mathbb{Q}, (\forall \epsilon \in \mathbb{Q}^{>0}) (\exists N \in \mathbb{N})(\forall n, m \in \mathbb{N}^{>N})(|a_n - a_m| < \epsilon) \}$~$$ |
---|
Real number (as Cauchy sequence) | $~$(a_n) \sim (b_n)$~$ |
---|
Real number (as Cauchy sequence) | $~$\epsilon > 0$~$ |
---|
Real number (as Cauchy sequence) | $~$N$~$ |
---|
Real number (as Cauchy sequence) | $~$n \in \mathbb{N}$~$ |
---|
Real number (as Cauchy sequence) | $~$N$~$ |
---|
Real number (as Cauchy sequence) | $~$|a_n - b_n| < \epsilon$~$ |
---|
Real number (as Cauchy sequence) | $~$|a_n - b_n| = |b_n - a_n|$~$ |
---|
Real number (as Cauchy sequence) | $~$|a_n - a_n| = 0$~$ |
---|
Real number (as Cauchy sequence) | $~$n$~$ |
---|
Real number (as Cauchy sequence) | $~$< \epsilon$~$ |
---|
Real number (as Cauchy sequence) | $~$|a_n - b_n| < \frac{\epsilon}{2}$~$ |
---|
Real number (as Cauchy sequence) | $~$n$~$ |
---|
Real number (as Cauchy sequence) | $~$|b_n - c_n| < \frac{\epsilon}{2}$~$ |
---|
Real number (as Cauchy sequence) | $~$n$~$ |
---|
Real number (as Cauchy sequence) | $~$|a_n - b_n| + |b_n - c_n| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$~$ |
---|
Real number (as Cauchy sequence) | $~$n$~$ |
---|
Real number (as Cauchy sequence) | $~$|a_n - c_n| < \epsilon$~$ |
---|
Real number (as Cauchy sequence) | $~$n$~$ |
---|
Real number (as Cauchy sequence) | $~$[a_n]$~$ |
---|
Real number (as Cauchy sequence) | $~$(a_n)_{n=1}^{\infty}$~$ |
---|
Real number (as Cauchy sequence) | $~$a_n$~$ |
---|
Real number (as Cauchy sequence) | $~$X$~$ |
---|
Real number (as Cauchy sequence) | $~$[a_n] + [b_n] := [a_n + b_n]$~$ |
---|
Real number (as Cauchy sequence) | $~$[a_n] \times [b_n] := [a_n \times b_n]$~$ |
---|
Real number (as Cauchy sequence) | $~$[a_n] \leq [b_n]$~$ |
---|
Real number (as Cauchy sequence) | $~$[a_n] = [b_n]$~$ |
---|
Real number (as Cauchy sequence) | $~$N$~$ |
---|
Real number (as Cauchy sequence) | $~$n > N$~$ |
---|
Real number (as Cauchy sequence) | $~$a_n \leq b_n$~$ |
---|
Real number (as Cauchy sequence) | $~$r$~$ |
---|
Real number (as Cauchy sequence) | $~$[r]$~$ |
---|
Real number (as Cauchy sequence) | $~$(r, r, \dots)$~$ |
---|
Real number (as Cauchy sequence) | $~$\pi$~$ |
---|
Real number (as Cauchy sequence) | $~$(3, 3.1, 3.14, 3.141, \dots)$~$ |
---|
Real number (as Cauchy sequence) | $~$(100, 3, 3.1, 3.14, \dots)$~$ |
---|
Real number (as Dedekind cut) | $~$\newcommand{\rats}{\mathbb{Q}} \newcommand{\Ql}{\rats^\le} \newcommand{\Qr}{\rats^\ge} \newcommand{\Qls}{\rats^<} \newcommand{\Qrs}{\rats^>}$~$ |
---|
Real number (as Dedekind cut) | $~$\newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\sothat}{\ |\ }$~$ |
---|
Real number (as Dedekind cut) | $~$S$~$ |
---|
Real number (as Dedekind cut) | $~$(A, B)$~$ |
---|
Real number (as Dedekind cut) | $~$S$~$ |
---|
Real number (as Dedekind cut) | $~$A$~$ |
---|
Real number (as Dedekind cut) | $~$B$~$ |
---|
Real number (as Dedekind cut) | $~$(A, B)$~$ |
---|
Real number (as Dedekind cut) | $~$S$~$ |
---|
Real number (as Dedekind cut) | $~$A$~$ |
---|
Real number (as Dedekind cut) | $~$B$~$ |
---|
Real number (as Dedekind cut) | $~$A$~$ |
---|
Real number (as Dedekind cut) | $~$B$~$ |
---|
Real number (as Dedekind cut) | $~$A$~$ |
---|
Real number (as Dedekind cut) | $~$B$~$ |
---|
Real number (as Dedekind cut) | $~$-1$~$ |
---|
Real number (as Dedekind cut) | $~$1$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql$~$ |
---|
Real number (as Dedekind cut) | $~$-\infty$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$+\infty$~$ |
---|
Real number (as Dedekind cut) | $~$(\Ql, \Qr)$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql$~$ |
---|
Real number (as Dedekind cut) | $~$q_u$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$q_v$~$ |
---|
Real number (as Dedekind cut) | $~$q_u = q_v$~$ |
---|
Real number (as Dedekind cut) | $~$r$~$ |
---|
Real number (as Dedekind cut) | $~$q_u < r < q_v$~$ |
---|
Real number (as Dedekind cut) | $~$r$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$(\Ql, \Qr)$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql = \set{x \in \rats \mid x^3 \le 2}$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr = \set{x \in \rats \mid x^3 \ge 2}$~$ |
---|
Real number (as Dedekind cut) | $~$f(x) = x^3$~$ |
---|
Real number (as Dedekind cut) | $~$p < q \iff p^3 < q^3$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$(\Ql, \Qr)$~$ |
---|
Real number (as Dedekind cut) | $~$2$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$\sqrt[3]{2}$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$(\Ql, \Qr)$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql$~$ |
---|
Real number (as Dedekind cut) | $~$\Qls$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql$~$ |
---|
Real number (as Dedekind cut) | $~$q_g$~$ |
---|
Real number (as Dedekind cut) | $~$q_g$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$r$~$ |
---|
Real number (as Dedekind cut) | $~$r$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$r$~$ |
---|
Real number (as Dedekind cut) | $~$r$~$ |
---|
Real number (as Dedekind cut) | $~$\Ql$~$ |
---|
Real number (as Dedekind cut) | $~$q$~$ |
---|
Real number (as Dedekind cut) | $~$\Qls$~$ |
---|
Real number (as Dedekind cut) | $~$q$~$ |
---|
Real number (as Dedekind cut) | $~$\Qr$~$ |
---|
Real number (as Dedekind cut) | $~$q$~$ |
---|
Real number (as Dedekind cut) | $~$\le$~$ |
---|
Real number (as Dedekind cut) | $~$a = (\Qls_a, \Qr_a)$~$ |
---|
Real number (as Dedekind cut) | $~$b = (\Qls_b, \Qr_b)$~$ |
---|
Real number (as Dedekind cut) | $~$a \le b$~$ |
---|
Real number (as Dedekind cut) | $~$\Qls_a \subseteq \Qls_b$~$ |
---|
Real number (as Dedekind cut) | $~$a = b$~$ |
---|
Real number (as Dedekind cut) | $~$a \le b$~$ |
---|
Real number (as Dedekind cut) | $~$b \le a$~$ |
---|
Real number (as Dedekind cut) | $~$\Qls_a \subseteq \Qls_b$~$ |
---|
Real number (as Dedekind cut) | $~$\Qls_b \subseteq \Qls_a$~$ |
---|
Real number (as Dedekind cut) | $~$\Qls_a = \Qls_b$~$ |
---|
Real number (as Dedekind cut) | $~$a$~$ |
---|
Real number (as Dedekind cut) | $~$b$~$ |
---|
Real numbers are uncountable | $~$\bar{\phantom{9}}$~$ |
---|
Real numbers are uncountable | $~$a.bcd\cdots z\overline{9} = a.bcd\cdots (z+1)\overline{0}$~$ |
---|
Real numbers are uncountable | $~$z < 9$~$ |
---|
Real numbers are uncountable | $~$\sum_{i=k}^\infty 10^{-k} \cdot 9 = 1 \cdot 10^{-k + 1} + \sum_{i=k}^\infty 10^{-k} \cdot 0$~$ |
---|
Real numbers are uncountable | $~$f: \mathbb Z^+ \twoheadrightarrow \mathbb R$~$ |
---|
Real numbers are uncountable | $~$r_n$~$ |
---|
Real numbers are uncountable | $~$n^\text{th}$~$ |
---|
Real numbers are uncountable | $~$r$~$ |
---|
Real numbers are uncountable | $~$r$~$ |
---|
Real numbers are uncountable | $~$r_1r_2r_3r_4r_5\ldots$~$ |
---|
Real numbers are uncountable | $~$r'$~$ |
---|
Real numbers are uncountable | $~$0 \le r' < 1$~$ |
---|
Real numbers are uncountable | $~$r'_n$~$ |
---|
Real numbers are uncountable | $~$(f(n))_n \ne 5$~$ |
---|
Real numbers are uncountable | $~$(f(n))_n = 5$~$ |
---|
Real numbers are uncountable | $~$n$~$ |
---|
Real numbers are uncountable | $~$r' = f(n)$~$ |
---|
Real numbers are uncountable | $~$r'_n \ne (f(n))_n$~$ |
---|
Real numbers are uncountable | $~$f$~$ |
---|
Real numbers are uncountable | $~$\mathbb R$~$ |
---|
Real numbers are uncountable | $~$\square$~$ |
---|
Real numbers are uncountable | $~$r'$~$ |
---|
Real numbers are uncountable | $~$0.\overline{9} = 1.\overline{0}$~$ |
---|
Reflective consistency | $~$\theta$~$ |
---|
Reflective consistency | $~$\theta.$~$ |
---|
Reflective consistency | $~$\theta$~$ |
---|
Reflectively consistent degree of freedom | $~$X_i \in X$~$ |
---|
Reflectively consistent degree of freedom | $~$X_1$~$ |
---|
Reflectively consistent degree of freedom | $~$X_1,$~$ |
---|
Reflectively consistent degree of freedom | $~$X_2$~$ |
---|
Reflectively consistent degree of freedom | $~$X_2,$~$ |
---|
Reflexive relation | $~$R$~$ |
---|
Reflexive relation | $~$X$~$ |
---|
Reflexive relation | $~$\forall a \in X, aRa$~$ |
---|
Reflexive relation | $~$\leq$~$ |
---|
Reflexive relation | $~$<$~$ |
---|
Reflexive relation | $~$\{Alice, Bob\}$~$ |
---|
Reflexive relation | $~$R$~$ |
---|
Reflexive relation | $~$R$~$ |
---|
Reflexive relation | $~$R$~$ |
---|
Reflexive relation | $~$\leq$~$ |
---|
Reflexive relation | $~$<$~$ |
---|
Relation | $~$n$~$ |
---|
Relation | $~$n$~$ |
---|
Relation | $~$\{ (0,0), (1,1), (2,2), … \}$~$ |
---|
Relation | $~$\{ (0,1), (1,2), (2,3), … \}$~$ |
---|
Relation | $~$R$~$ |
---|
Relation | $~$xRy$~$ |
---|
Relation | $~$(x,y)$~$ |
---|
Relation | $~$R$~$ |
---|
Relation | $~$n$~$ |
---|
Relation | $~$n$~$ |
---|
Relation | $~$\{ (0,0), (1,1), (2,2), … \}$~$ |
---|
Relation | $~$\{ (0,1), (1,2), (2,3), … \}$~$ |
---|
Relation | $~$R$~$ |
---|
Relation | $~$xRy$~$ |
---|
Relation | $~$(x,y)$~$ |
---|
Relation | $~$R$~$ |
---|
Relative complement | $~$A$~$ |
---|
Relative complement | $~$B$~$ |
---|
Relative complement | $~$A \setminus B$~$ |
---|
Relative complement | $~$A$~$ |
---|
Relative complement | $~$B$~$ |
---|
Relative complement | $~$C = A \setminus B$~$ |
---|
Relative complement | $$~$x \in C \leftrightarrow (x \in A \land x \notin B)$~$$ |
---|
Relative complement | $~$x$~$ |
---|
Relative complement | $~$C$~$ |
---|
Relative complement | $~$x$~$ |
---|
Relative complement | $~$A$~$ |
---|
Relative complement | $~$B$~$ |
---|
Relative complement | $~$\{1,2,3\} \setminus \{2\} = \{1,3\}$~$ |
---|
Relative complement | $~$\{1,2,3\} \setminus \{9\} = \{1,2,3\}$~$ |
---|
Relative complement | $~$\{1,2\} \setminus \{1,2,3,4\} = \{\}$~$ |
---|
Relative complement | $~$U$~$ |
---|
Relative complement | $~$A$~$ |
---|
Relative complement | $~$A^\complement$~$ |
---|
Relative complement | $~$U \setminus A$~$ |
---|
Relative likelihood | $~$(2 : 1).$~$ |
---|
Relative likelihood | $~$e_p$~$ |
---|
Relative likelihood | $~$H_S$~$ |
---|
Relative likelihood | $~$H_M$~$ |
---|
Relative likelihood | $~$H_W$~$ |
---|
Relative likelihood | $~$(20 : 10 : 1).$~$ |
---|
Relative likelihood | $~$H_1, H_2, \ldots, H_n,$~$ |
---|
Relative likelihood | $~$e$~$ |
---|
Relative likelihood | $~$\mathbb P(e \mid H_i)$~$ |
---|
Relative likelihood | $~$i$~$ |
---|
Relative likelihood | $~$n.$~$ |
---|
Relative likelihood | $$~$ \alpha \mathbb P(e \mid H_1) : \alpha \mathbb P(e \mid H_2) : \ldots : \alpha \mathbb P(e \mid H_n) $~$$ |
---|
Relative likelihood | $~$\alpha > 0$~$ |
---|
Relative likelihood | $~$(20 : 10 : 1)$~$ |
---|
Relative likelihood | $~$(4 : 2 : 0.20)$~$ |
---|
Relative likelihood | $~$(60 : 30 : 3).$~$ |
---|
Relative likelihood | $~$H_S$~$ |
---|
Relative likelihood | $~$H_M$~$ |
---|
Relative likelihood | $~$H_S$~$ |
---|
Relative likelihood | $~$H_W$~$ |
---|
Relative likelihood | $~$e_p$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_b$~$ |
---|
Report likelihoods not p-values: FAQ | $~$b$~$ |
---|
Report likelihoods not p-values: FAQ | $~$b$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_{0.5}$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_{0.75}$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_{0.3}$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_{0.75}$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_{0.5}$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\frac{0.0593}{0.0156}$~$ |
---|
Report likelihoods not p-values: FAQ | $~$n$~$ |
---|
Report likelihoods not p-values: FAQ | $~$n$~$ |
---|
Report likelihoods not p-values: FAQ | $~$n$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H$~$ |
---|
Report likelihoods not p-values: FAQ | $~$e$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathbb P(H) = \mathbb P(H \mid e) \mathbb P(e) + \mathbb P(H \mid \lnot e) \mathbb P(\lnot e),$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathbb P$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathcal L$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathcal L$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathcal L$~$ |
---|
Report likelihoods not p-values: FAQ | $~$e$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H$~$ |
---|
Report likelihoods not p-values: FAQ | $~$e$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H.$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathbb P$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathcal L$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathbb P$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathcal L$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathbb P$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H$~$ |
---|
Report likelihoods not p-values: FAQ | $~$5 : 1$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_1$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_2$~$ |
---|
Report likelihoods not p-values: FAQ | $~$6 : 1$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_1$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_2$~$ |
---|
Report likelihoods not p-values: FAQ | $~$3 : 1$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_1$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_2,$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_1$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_2$~$ |
---|
Report likelihoods not p-values: FAQ | $~$90 : 1$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_{0.9}$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_{0.9}$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_{0.9},$~$ |
---|
Report likelihoods not p-values: FAQ | $~$0.9 \log_2(0.9) + 0.1 \log_2(0.1) \approx -0.469$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_{0.9}$~$ |
---|
Report likelihoods not p-values: FAQ | $~$20 \cdot 0.469 \approx 9.37$~$ |
---|
Report likelihoods not p-values: FAQ | $~$2^{-9.37} \approx$~$ |
---|
Report likelihoods not p-values: FAQ | $~$1.5 \cdot 10^{-3},$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_{0.9}$~$ |
---|
Report likelihoods not p-values: FAQ | $~$5.9 \cdot 10^{-16},$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H_{0.9}$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathcal L_e(H)$~$ |
---|
Report likelihoods not p-values: FAQ | $~$< 0.05$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H$~$ |
---|
Report likelihoods not p-values: FAQ | $~$e$~$ |
---|
Report likelihoods not p-values: FAQ | $~$e$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H$~$ |
---|
Report likelihoods not p-values: FAQ | $~$e$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathcal L_e(H)$~$ |
---|
Report likelihoods not p-values: FAQ | $~$e$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H$~$ |
---|
Report likelihoods not p-values: FAQ | $~$\mathcal L_e(H),$~$ |
---|
Report likelihoods not p-values: FAQ | $~$e$~$ |
---|
Report likelihoods not p-values: FAQ | $~$H.$~$ |
---|
Report likelihoods not p-values: FAQ | $~$2 + 2 = 4,$~$ |
---|
Report likelihoods not p-values: FAQ | $~$2 + 2 = 7.$~$ |
---|
Report likelihoods, not p-values | $~$p < 0.05$~$ |
---|
Report likelihoods, not p-values | $~$e$~$ |
---|
Report likelihoods, not p-values | $~$\mathcal L$~$ |
---|
Report likelihoods, not p-values | $~$H$~$ |
---|
Report likelihoods, not p-values | $~$H$~$ |
---|
Report likelihoods, not p-values | $~$e$~$ |
---|
Report likelihoods, not p-values | $~$\mathcal L_e(H)$~$ |
---|
Report likelihoods, not p-values | $~$\mathcal L(H \mid e)$~$ |
---|
Report likelihoods, not p-values | $~$\mathcal L(H \mid e) = \mathbb P(e \mid H),$~$ |
---|
Report likelihoods, not p-values | $~$H$~$ |
---|
Report likelihoods, not p-values | $~$e$~$ |
---|
Report likelihoods, not p-values | $~$e$~$ |
---|
Report likelihoods, not p-values | $~$H$~$ |
---|
Report likelihoods, not p-values | $~$e$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.25} =$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.5}$~$ |
---|
Report likelihoods, not p-values | $~$\mathcal L_e(H_{0.25})$~$ |
---|
Report likelihoods, not p-values | $~$=$~$ |
---|
Report likelihoods, not p-values | $~$0.25^5 \cdot 0.75$~$ |
---|
Report likelihoods, not p-values | $~$\approx 0.07\%$~$ |
---|
Report likelihoods, not p-values | $~$\mathcal L_e(H_{0.5})$~$ |
---|
Report likelihoods, not p-values | $~$=$~$ |
---|
Report likelihoods, not p-values | $~$0.5^6$~$ |
---|
Report likelihoods, not p-values | $~$\approx 1.56\%,$~$ |
---|
Report likelihoods, not p-values | $~$21 : 1$~$ |
---|
Report likelihoods, not p-values | $~$e$~$ |
---|
Report likelihoods, not p-values | $~$H_b$~$ |
---|
Report likelihoods, not p-values | $~$b$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.75}$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.5}$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.75}$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.25}.$~$ |
---|
Report likelihoods, not p-values | $~$e$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.5}$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.5}$~$ |
---|
Report likelihoods, not p-values | $~$H_0$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.52},$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.61}$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.8}$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.5},$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.5}$~$ |
---|
Report likelihoods, not p-values | $~$H_0$~$ |
---|
Report likelihoods, not p-values | $~$21 : 1$~$ |
---|
Report likelihoods, not p-values | $~$H_a$~$ |
---|
Report likelihoods, not p-values | $~$H_0,$~$ |
---|
Report likelihoods, not p-values | $~$21 : 1$~$ |
---|
Report likelihoods, not p-values | $~$H_a$~$ |
---|
Report likelihoods, not p-values | $~$H_0.$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.75}$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.5}$~$ |
---|
Report likelihoods, not p-values | $~$3.8 : 1$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.75}$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.5}$~$ |
---|
Report likelihoods, not p-values | $~$5 : 1$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.75}$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.5}$~$ |
---|
Report likelihoods, not p-values | $~$(3.8 \cdot 5) : 1$~$ |
---|
Report likelihoods, not p-values | $~$=$~$ |
---|
Report likelihoods, not p-values | $~$19 : 1.$~$ |
---|
Report likelihoods, not p-values | $~$10 : 1$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.75}$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.5}$~$ |
---|
Report likelihoods, not p-values | $~$3.8 : 1.$~$ |
---|
Report likelihoods, not p-values | $~$5 : 1$~$ |
---|
Report likelihoods, not p-values | $~$19 : 10$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.75}$~$ |
---|
Report likelihoods, not p-values | $~$H_{0.5}$~$ |
---|
Report likelihoods, not p-values | $~$2 + 2 = 4,$~$ |
---|
Report likelihoods, not p-values | $~$2 + 2 = 7.$~$ |
---|
Representability theorem for computable functions | $~$T$~$ |
---|
Representability theorem for computable functions | $~$f:\mathbb{N}\mapsto\mathbb{N}$~$ |
---|
Representability theorem for computable functions | $~$\phi_f(x,y)$~$ |
---|
Representability theorem for computable functions | $~$T$~$ |
---|
Representability theorem for computable functions | $~$n\in \mathbb{N}$~$ |
---|
Representability theorem for computable functions | $~$T\vdash \forall y\ \phi_f(\textbf n,y)\leftrightarrow y=\textbf{f(n)}$~$ |
---|
Representation theory | $~$G$~$ |
---|
Representation theory | $~$V$~$ |
---|
Rice's Theorem | $~$[n]$~$ |
---|
Rice's Theorem | $~$n$~$ |
---|
Rice's Theorem | $~$[n]$~$ |
---|
Rice's Theorem | $~$[n](m)$~$ |
---|
Rice's Theorem | $~$[n]$~$ |
---|
Rice's Theorem | $~$m$~$ |
---|
Rice's Theorem | $~$A$~$ |
---|
Rice's Theorem | $~$\{ \mathrm{Graph}(n) : n \in \mathbb{N} \}$~$ |
---|
Rice's Theorem | $~$\mathrm{Graph}(n)$~$ |
---|
Rice's Theorem | $~$[n]$~$ |
---|
Rice's Theorem | $~$n$~$ |
---|
Rice's Theorem | $~$[r]$~$ |
---|
Rice's Theorem | $~$[r](i)$~$ |
---|
Rice's Theorem | $~$1$~$ |
---|
Rice's Theorem | $~$\mathrm{Graph}(i) \in A$~$ |
---|
Rice's Theorem | $~$[r](i)$~$ |
---|
Rice's Theorem | $~$0$~$ |
---|
Rice's Theorem | $~$\mathrm{Graph}(i) \not \in A$~$ |
---|
Rice's Theorem | $~$0$~$ |
---|
Rice's Theorem | $~$1$~$ |
---|
Rice's Theorem | $~$2$~$ |
---|
Rice's Theorem | $~$h: \mathbb{N} \to \mathbb{N}$~$ |
---|
Rice's Theorem | $~$n \in \mathbb{N}$~$ |
---|
Rice's Theorem | $~$\mathrm{Graph}(n) = \mathrm{Graph}(h(n))$~$ |
---|
Rice's Theorem | $~$n$~$ |
---|
Rice's Theorem | $~$n$~$ |
---|
Rice's Theorem | $~$[n]$~$ |
---|
Rice's Theorem | $~$[h(n)]$~$ |
---|
Rice's Theorem | $~$h$~$ |
---|
Rice's Theorem | $~$h$~$ |
---|
Rice's Theorem | $~$A$~$ |
---|
Rice's Theorem | $~$A$~$ |
---|
Rice's Theorem | $~$A$~$ |
---|
Rice's Theorem | $~$A$~$ |
---|
Rice's Theorem | $~$A$~$ |
---|
Rice's Theorem | $~$A$~$ |
---|
Rice's Theorem | $~$A$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$n$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$n$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$C$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$M$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$x$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$Proxy_s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$z$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$M$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$x$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$z$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$M$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$x$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$Proxy_s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$Proxy_s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$Proxy_s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$M$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$x$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$Proxy_s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$Proxy_s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$Proxy_s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$Proxy_s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$C$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$C$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$Proxy_s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$M$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$x$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$C$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$Proxy_s$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$M$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$x$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$M$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$x$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$C$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$C$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$fib\_checker$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$fib\_checker$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$M$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$x$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$Proxy_{fib}$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$fib\_checker$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$M$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$x$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$M$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$x$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$fib\_checker$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$true$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$halts$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$M$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$x$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$fib\_checker$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$false$~$ |
---|
Rice's Theorem: Intro (Math 1) | $~$halts$~$ |
---|
Rice's theorem and the Halting problem | $~$S$~$ |
---|
Rice's theorem and the Halting problem | $~$[n]$~$ |
---|
Rice's theorem and the Halting problem | $$~$
[n](m) =
\left\{
\begin{array}{ll}
1 & [m] \text{ computes a function in $S$} \\
0 & \text{otherwise} \\
\end{array}
\right.
$~$$ |
---|
Rice's theorem and the Halting problem | $~$S$~$ |
---|
Rice's theorem and the Halting problem | $~$S$~$ |
---|
Rice's theorem and the Halting problem | $~$S^c$~$ |
---|
Rice's theorem and the Halting problem | $~$S$~$ |
---|
Rice's theorem and the Halting problem | $~$S^c$~$ |
---|
Rice's theorem and the Halting problem | $~$S^c$~$ |
---|
Rice's theorem and the Halting problem | $~$S$~$ |
---|
Rice's theorem and the Halting problem | $~$S$~$ |
---|
Rice's theorem and the Halting problem | $~$[s]$~$ |
---|
Rice's theorem and the Halting problem | $~$[s](x)$~$ |
---|
Rice's theorem and the Halting problem | $~$x$~$ |
---|
Rice's theorem and the Halting problem | $~$[m]$~$ |
---|
Rice's theorem and the Halting problem | $~$[x]$~$ |
---|
Rice's theorem and the Halting problem | $~$Proxy_s$~$ |
---|
Rice's theorem and the Halting problem | $~$[m](x)$~$ |
---|
Rice's theorem and the Halting problem | $~$[s]$~$ |
---|
Rice's theorem and the Halting problem | $~$[n](Proxy_s)=1$~$ |
---|
Rice's theorem and the Halting problem | $~$[m](x)$~$ |
---|
Rice's theorem and the Halting problem | $~$S$~$ |
---|
Rice's theorem and the Halting problem | $~$[n](Proxy_s)=0$~$ |
---|
Rice's theorem and the Halting problem | $~$S$~$ |
---|
Rice's theorem and the Halting problem | $~$HALT$~$ |
---|
Rice's theorem and the Halting problem | $~$S$~$ |
---|
Rice's theorem and the Halting problem | $~$x$~$ |
---|
Rice's theorem and the Halting problem | $~$[n]$~$ |
---|
Rice's theorem and the Halting problem | $~$[n]\in S$~$ |
---|
Rice's theorem and the Halting problem | $~$[n]$~$ |
---|
Rice's theorem and the Halting problem | $~$[n]$~$ |
---|
Rice's theorem and the Halting problem | $~$S$~$ |
---|
Rice's theorem and the Halting problem | $~$[n]$~$ |
---|
Rice's theorem and the Halting problem | $~$x$~$ |
---|
Rice's theorem and the Halting problem | $~$HALT$~$ |
---|
Rice's theorem and the Halting problem | $~$[n]$~$ |
---|
Rice's theorem and the Halting problem | $~$S$~$ |
---|
Rice's theorem and the Halting problem | $~$HALT$~$ |
---|
Ring | $~$1$~$ |
---|
Ring | $~$R$~$ |
---|
Ring | $~$(X, \oplus, \otimes)$~$ |
---|
Ring | $~$X$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$x \oplus y$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$x, y \in X$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$1$~$ |
---|
Ring | $~$R$~$ |
---|
Ring | $~$(X, \oplus, \otimes)$~$ |
---|
Ring | $~$X$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$x \oplus y$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$x, y \in X$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$x \otimes y$~$ |
---|
Ring | $~$xy$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$X$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$X$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$0$~$ |
---|
Ring | $~$x \in X$~$ |
---|
Ring | $~$(-x) \in X$~$ |
---|
Ring | $~$x \oplus (-x) = 0$~$ |
---|
Ring | $~$X$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$X$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$1$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$a \otimes (x \oplus y) = (a\otimes x) \oplus (a\otimes y)$~$ |
---|
Ring | $~$a, x, y \in X$~$ |
---|
Ring | $~$(x \oplus y)\otimes a = (x\otimes a) \oplus (y\otimes a)$~$ |
---|
Ring | $~$a, x, y \in X$~$ |
---|
Ring | $~$\mathbb{Z}$~$ |
---|
Ring | $~$R = (X, \oplus, \otimes)$~$ |
---|
Ring | $~$R$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$X$~$ |
---|
Ring | $~$R$~$ |
---|
Ring | $~$\oplus$~$ |
---|
Ring | $~$0$~$ |
---|
Ring | $~$-x$~$ |
---|
Ring | $~$x$~$ |
---|
Ring | $~$\otimes$~$ |
---|
Ring | $~$1$~$ |
---|
Sample space | $~$\Omega$~$ |
---|
Sample space | $~$\Omega$~$ |
---|
Sans Tachez | $$~$\int_0^\infty \frac{2^x}{x}dx$~$$ |
---|
Separation from hyperexistential risk | $~$V,$~$ |
---|
Separation from hyperexistential risk | $~$V$~$ |
---|
Separation from hyperexistential risk | $~$-V.$~$ |
---|
Separation from hyperexistential risk | $~$V$~$ |
---|
Separation from hyperexistential risk | $~$V$~$ |
---|
Separation from hyperexistential risk | $~$V$~$ |
---|
Separation from hyperexistential risk | $~$U = V + W$~$ |
---|
Separation from hyperexistential risk | $~$W$~$ |
---|
Separation from hyperexistential risk | $~$W$~$ |
---|
Separation from hyperexistential risk | $~$W$~$ |
---|
Separation from hyperexistential risk | $~$U,$~$ |
---|
Separation from hyperexistential risk | $~$U,$~$ |
---|
Separation from hyperexistential risk | $~$W$~$ |
---|
Separation from hyperexistential risk | $~$W$~$ |
---|
Separation from hyperexistential risk | $~$W$~$ |
---|
Separation from hyperexistential risk | $~$V$~$ |
---|
Separation from hyperexistential risk | $~$U$~$ |
---|
Separation from hyperexistential risk | $~$U$~$ |
---|
Separation from hyperexistential risk | $~$W$~$ |
---|
Separation from hyperexistential risk | $~$W$~$ |
---|
Separation from hyperexistential risk | $~$V.$~$ |
---|
Separation from hyperexistential risk | $~$V$~$ |
---|
Separation from hyperexistential risk | $~$U,$~$ |
---|
Separation from hyperexistential risk | $~$U$~$ |
---|
Separation from hyperexistential risk | $~$U'$~$ |
---|
Separation from hyperexistential risk | $~$V,$~$ |
---|
Separation from hyperexistential risk | $~$V.$~$ |
---|
Separation from hyperexistential risk | $~$V$~$ |
---|
Set | $~$\{1, 3, 2\}$~$ |
---|
Set | $~$\{3, 2, 1\}$~$ |
---|
Set | $~$\{1, 2, 2, 3, 3, 3\}$~$ |
---|
Set | $~$\{1, 3, 2\}$~$ |
---|
Set | $~$x$~$ |
---|
Set | $~$4$~$ |
---|
Set | $~$\{x \mid (x < 4) \text{ and } (x \text{ is a natural number})\}$~$ |
---|
Set | $~$\{x \mid x = 2n \text{ for some natural } n \}$~$ |
---|
Set | $~$S$~$ |
---|
Set | $~$\{1,2,\{1,2\}\}$~$ |
---|
Set | $~$1$~$ |
---|
Set | $~$2$~$ |
---|
Set | $~$\{1,2\}$~$ |
---|
Set | $~$\{1,5,8,73\}$~$ |
---|
Set | $~$1$~$ |
---|
Set | $~$5$~$ |
---|
Set | $~$8$~$ |
---|
Set | $~$73$~$ |
---|
Set | $~$\{\{0,-3,8\}\}$~$ |
---|
Set | $~$\{0,-3,8\}$~$ |
---|
Set | $~$\{\text{Mercury}, \text{Venus}, \text{Earth}, \text{Mars} \}$~$ |
---|
Set | $~$\{x \mid x \text{ is a human, born on 01.01.2000} \}$~$ |
---|
Set | $~$\{\text{author's favorite mug}, \text{Arbital's main page}, 73, \text{the tallest man born in London}\}$~$ |
---|
Set | $~$∈$~$ |
---|
Set | $~$∉$~$ |
---|
Set | $~$∈$~$ |
---|
Set | $~$∉$~$ |
---|
Set | $~$x ∈ A$~$ |
---|
Set | $~$x$~$ |
---|
Set | $~$A$~$ |
---|
Set | $~$x ∉ A$~$ |
---|
Set | $~$x$~$ |
---|
Set | $~$A$~$ |
---|
Set | $~$A$~$ |
---|
Set | $~$A$~$ |
---|
Set | $~$|A|$~$ |
---|
Set | $~$A$~$ |
---|
Set | $~$|A| = n$~$ |
---|
Set | $~$A$~$ |
---|
Set | $~$n$~$ |
---|
Set | $~$n$~$ |
---|
Set | $~$\{0, …, (n-1)\}$~$ |
---|
Set | $~$n$~$ |
---|
Set | $~$\mathbb N$~$ |
---|
Set | $~$\mathbb N$~$ |
---|
Set builder notation | $~$\{ 2n \mid n \in \mathbb N \}$~$ |
---|
Set builder notation | $~$\{ (x, y) \mid x \in \mathbb R, y \in \mathbb R, x \cdot y = 1 \}$~$ |
---|
Set product | $~$A$~$ |
---|
Set product | $~$B$~$ |
---|
Set product | $~$(a,b)$~$ |
---|
Set product | $~$a$~$ |
---|
Set product | $~$A$~$ |
---|
Set product | $~$b$~$ |
---|
Set product | $~$B$~$ |
---|
Set product | $~$\{1,2,\dots,n \}$~$ |
---|
Set product | $~$\{1,2,\dots, m \}$~$ |
---|
Set product | $~$(a, b)$~$ |
---|
Set product | $~$1 \leq a \leq n$~$ |
---|
Set product | $~$1 \leq b \leq m$~$ |
---|
Set product | $~$n \times m$~$ |
---|
Set product | $~$Y_x$~$ |
---|
Set product | $~$X$~$ |
---|
Set product | $~$\prod_{x \in X} Y_x$~$ |
---|
Set product | $~$X$~$ |
---|
Set product | $~$X = \{1,2\}$~$ |
---|
Set product | $~$Y_1 = \{a,b\}, Y_2 = \{b,c\}$~$ |
---|
Set product | $$~$\prod_{x \in X} Y_x = Y_1 \times Y_2 = \{(a,b), (a,c), (b,b), (b,c)\}$~$$ |
---|
Set product | $~$X = \mathbb{Z}$~$ |
---|
Set product | $~$Y_n = \{ n \}$~$ |
---|
Set product | $~$\{1,2, \dots, n\}$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H_i$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e.$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H_i$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H_i$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H_i$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\mathbb P(e \mid H_i),$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H_i$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e$~$ |
---|
Shift towards the hypothesis of least surprise | $~$-\!\log(\mathbb P(e \mid H_i)),$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\frac{1}{8}$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\frac{1}{4}$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\frac{1}{8}$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\frac{1}{4}$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H,$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\mathbb P(e \mid H)$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\mathbb P(e \mid \lnot H)$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\left(\frac{1}{8} : \frac{1}{4}\right)$~$ |
---|
Shift towards the hypothesis of least surprise | $~$=$~$ |
---|
Shift towards the hypothesis of least surprise | $~$(1 : 2),$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H.$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\log_2\left(\frac{1}{8}\right) = -3$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H.$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\log_2\left(\frac{1}{4}\right) = -2$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H.$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H.$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$-\!\log_2\left(\frac{1}{8}\right)=3$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$-\!\log_2\left(\frac{1}{4}\right)=2$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\log_2(0.04) \approx -4.64$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\log_2(0.08) \approx -3.64$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H,$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H,$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\log_2(\mathbb P(e \mid H)),$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e.$~$ |
---|
Shift towards the hypothesis of least surprise | $~$x$~$ |
---|
Shift towards the hypothesis of least surprise | $~$0 \le x \le 1$~$ |
---|
Shift towards the hypothesis of least surprise | $~$[-\infty, 0]$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e$~$ |
---|
Shift towards the hypothesis of least surprise | $~$e,$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\frac{1}{8}$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\frac{1}{4}$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$H$~$ |
---|
Shift towards the hypothesis of least surprise | $~$\lnot H$~$ |
---|
Sign homomorphism (from the symmetric group) | $~$\sigma$~$ |
---|
Sign homomorphism (from the symmetric group) | $~$S_n$~$ |
---|
Sign homomorphism (from the symmetric group) | $~$0$~$ |
---|
Sign homomorphism (from the symmetric group) | $~$\sigma$~$ |
---|
Sign homomorphism (from the symmetric group) | $~$1$~$ |
---|
Sign homomorphism (from the symmetric group) | $~$\sigma$~$ |
---|
Sign homomorphism (from the symmetric group) | $~$2$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $$~$\displaystyle \mathbb{P}_{prog}(bits_{1 \dots N}) = \prod_{i=1}^{N} InterpretProb(prog(bits_{1 \dots i-1}), bits_i)$~$$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $$~$\displaystyle InterpretProb(prog(x), y) = \left\{ \begin{array}{ll} InterpretFrac(prog(x)) & \text{if } y = 1 \\ 1-InterpretFrac(prog(x)) & \text{if } y = 0 \\ 0 & \text{if $prog(x)$ does not halt} \end{array} \right\} $~$$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$f^M(1-f)^N \ \mathrm{d}f$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$\mathcal P$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$s_1s_2\ldots s_n$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$s_{\preceq n}$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$\displaystyle \mathbb{Sol}(s_{\preceq n}) := \sum_{\mathrm{prog} \in \mathcal{P}} 2^{-\mathrm{length}(\mathrm{prog})} \cdot {\prod_{j=1}^n \mathop{InterpretProb}(\mathrm{prog}(s_{\preceq j-1}), s_j)}$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$\displaystyle \mathbb{P}(s_{n+1}=1\mid s_{\preceq n}) = \frac{\mathbb{Sol}(s_1s_2\ldots s_n 1)}{\mathbb{Sol}(s_1s_2\ldots s_n 1) + \mathbb{Sol}(s_1s_2\ldots s_n 0)}.$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$2^{-1,000,000}$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$2^{-1,000,000}$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$2^{-1,000,000}$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$10^{80}$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$10^{10^{80}}$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$10^{80}$~$ |
---|
Solomonoff induction: Intro Dialogue (Math 2) | $~$2^{-1,000,000}$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$GL\vdash A$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$PA\vdash A^*$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$*$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$*$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$PA$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$p^* = S_p$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$(\square A)^*=P(A^*)$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$P$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$(A\to B)^* = A^* \to B^*$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$\bot ^* = \neg X$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$X$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$PA$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$0\ne 1$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$PA$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$GL\vdash A$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$PA\vdash A^*$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$*$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$GL$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$PA$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$GL$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$GL\not\vdash A$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$*$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$PA\not\vdash A^*$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$GL$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$*$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$A$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$GL\not\vdash A$~$ |
---|
Solovay's theorems of arithmetical adequacy for GL | $~$PA\not\vdash A$~$ |
---|
Splitting conjugacy classes in alternating group | $~$S_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$S_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau \in A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma \in A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$S_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$S_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\{ \rho \sigma \rho^{-1} : \rho \ \text{even} \}$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\{ \rho \sigma \rho^{-1} : \rho \ \text{odd} \}$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma = c_1 \dots c_k$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau = c_1' \dots c_k'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$c_i = (a_{i1} \dots a_{i r_i})$~$ |
---|
Splitting conjugacy classes in alternating group | $~$c_i' = (b_{i1} \dots b_{i r_i})$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$a_{ij}$~$ |
---|
Splitting conjugacy classes in alternating group | $~$b_{ij}$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho \sigma \rho^{-1} = \tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$r_1$~$ |
---|
Splitting conjugacy classes in alternating group | $~$c_1' = (b_{11} \dots b_{1r_1})$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(b_{12} b_{13} \dots b_{1 r_1} b_{11})$~$ |
---|
Splitting conjugacy classes in alternating group | $~$c_1'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(b_{1 r_1} b_{11}) (b_{1 (r_1-1)} b_{11}) \dots (b_{13} b_{11})(b_{12} b_{11})$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho c_1'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho c_1'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $$~$\sigma = \rho \tau \rho^{-1} = \rho c_1' (c_1'^{-1} \tau c_1') c_1'^{-1} \rho^{-1}$~$$ |
---|
Splitting conjugacy classes in alternating group | $~$c_1'^{-1} \tau c_1' = \tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho c_1'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$c_1'^{-1} \tau c_1' = \tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$c_1'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$c_1'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$c_1'^{-1} c_1' c_1' c_2' \dots c_k'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$c_1'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau = c_1' c_2' \dots c_k'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma = (12)(3456), \tau = (23)(1467)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho = (67)(56)(31)(23)(12)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau = (32)(1467)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho c_1' = (67)(56)(31)(23)(12)(32)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$c_1' = (23) = (32)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$c_1'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(32)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(32)\tau(32)^{-1} = (23)(1467)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$r_1 = r_2$~$ |
---|
Splitting conjugacy classes in alternating group | $~$r$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(a_1 a_2 \dots a_r)(c_1 c_2 \dots c_r)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(b_1 b_2 \dots b_r)(d_1 d_2 \dots d_r)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho' = \rho (b_1 d_1)(b_2 d_2) \dots (b_r d_r)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$r$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(b_1 d_1)(b_2 d_2) \dots (b_r d_r)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(b_1 d_1)(b_2 d_2) \dots (b_r d_r)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$b_i$~$ |
---|
Splitting conjugacy classes in alternating group | $~$d_i$~$ |
---|
Splitting conjugacy classes in alternating group | $~$d_i$~$ |
---|
Splitting conjugacy classes in alternating group | $~$b_i$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho'$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma = (123)(456), \tau = (154)(237)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho = (67)(35)(42)(34)(25)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(12)(53)(47)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(12)(53)(47)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$S_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(123)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(231)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_n$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma = (12345)(678), \tau = (12345)(687)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_8$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\rho = (87)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(12345)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(12345)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(687)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(678)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(12345)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(12345)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(12345)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(687)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(678)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(87)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(87)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(678)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(678)^m(87)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(12345)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\tau$~$ |
---|
Splitting conjugacy classes in alternating group | $~$\sigma$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_8$~$ |
---|
Splitting conjugacy classes in alternating group | $~$A_7$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(7)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(5, 1, 1)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(4,2,1)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(3,2,2)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(3,3,1)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(3,1,1,1,1)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(1,1,1,1,1,1,1)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(2,2,1,1,1)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$7$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(7)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(1234567)$~$ |
---|
Splitting conjugacy classes in alternating group | $~$(12)(1234567)(12)^{-1} = (2134567)$~$ |
---|
Square visualization of probabilities on two events | $$~$
\newcommand{\true}{\text{True}}
\newcommand{\false}{\text{False}}
\newcommand{\bP}{\mathbb{P}}
$~$$ |
---|
Square visualization of probabilities on two events | $$~$
\newcommand{\true}{\text{True}}
\newcommand{\false}{\text{False}}
\newcommand{\bP}{\mathbb{P}}
$~$$ |
---|
Square visualization of probabilities on two events | $~$\bP(A,B)$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$\bP$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A,B) = \bP(A)\; \bP(B \mid A)$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$\bP$~$ |
---|
Square visualization of probabilities on two events | $~$\bP$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A,B)$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A)$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(\neg B)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(B \mid A)$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A \mid B)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(B)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(\neg B)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(B \mid A) = \bP(B) = \bP(B \mid \neg A)$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$\neg A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(\neg A)$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$\neg A$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(\neg A) = \bP(\neg A, B) + \bP(\neg A, \neg B)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A)$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(\neg A)$~$ |
---|
Square visualization of probabilities on two events | $~$\neg A$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(\neg B)$~$ |
---|
Square visualization of probabilities on two events | $~$\neg B$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(B)$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(B \mid A)$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$\neg A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$\neg A$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A \mid B)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A,B)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A,\neg B)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(\neg A,B)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(\neg A,\neg B)$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $$~$\bP(A,B) = \bP(A) \bP( B \mid A)\ .$~$$ |
---|
Square visualization of probabilities on two events | $~$\bP( B= \true \mid A= \false)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(B \mid \neg A)$~$ |
---|
Square visualization of probabilities on two events | $~$t_A \in \{\true, \false\}$~$ |
---|
Square visualization of probabilities on two events | $~$t_B \in \{\true, \false\}$~$ |
---|
Square visualization of probabilities on two events | $$~$\bP(A = t_A,B= t_B) = \bP(A= t_A)\; \bP( B= t_B \mid A= t_A)\ .$~$$ |
---|
Square visualization of probabilities on two events | $~$\bP(A = t_A)$~$ |
---|
Square visualization of probabilities on two events | $~$(\bP(A = \true))$~$ |
---|
Square visualization of probabilities on two events | $~$(\bP(A = \false))$~$ |
---|
Square visualization of probabilities on two events | $~$\bP( B= t_B \mid A= t_A)$~$ |
---|
Square visualization of probabilities on two events | $~$(B = \true)$~$ |
---|
Square visualization of probabilities on two events | $~$(B = \false)$~$ |
---|
Square visualization of probabilities on two events | $~$A = t_A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $$~$\bP(A = t_A,B= t_B) = \bP(B= t_B)\; \bP( A= t_A \mid B= t_b)\ .$~$$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(B \mid A)$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A \mid B)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A \mid \neg B)$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$B$~$ |
---|
Square visualization of probabilities on two events | $~$A$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A = \true)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(B = \true \mid A = \true)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(B = \true \mid A = \false)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A = t_A, B = t_B)$~$ |
---|
Square visualization of probabilities on two events | $~$A = t_A$~$ |
---|
Square visualization of probabilities on two events | $~$B = t_B$~$ |
---|
Square visualization of probabilities on two events | $$~$\bP(B = \false \mid A = \true) = \frac{\bP(A = \true, B = \false)}{\bP(A = \true)}\ ,$~$$ |
---|
Square visualization of probabilities on two events | $~$\bP(A)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(\neg B \mid A)$~$ |
---|
Square visualization of probabilities on two events | $~$\bP(A, \neg B)$~$ |
---|
Square visualization of probabilities on two events | $$~$\bP(A = \true)\; \bP(B = \false \mid A = \true) = \bP(A = \true, B = \false)\ .$~$$ |
---|
Square visualization of probabilities on two events | $~$t_A$~$ |
---|
Square visualization of probabilities on two events | $~$t_B$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $$~$
\newcommand{\bP}{\mathbb{P}}
$~$$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $$~$ \newcommand{\bP}{\mathbb{P}} $~$$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D \mid B)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $$~$\bP(D \mid B) = \frac{\bP(B, D)}{ \bP(B, D) + \bP(B, \neg D)} = \frac{3}{7} $~$$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP( \text{Diseasitis} \mid \text{black tongue depressor})$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$0.2$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$(D)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$(B)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(B \mid D) = 0.9$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(B \mid \neg D) = 0.3$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D \mid B)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $$~$
\begin{align}
\bP(D \mid B) &= \frac{\bP(B \mid D) \bP(D)}{\bP(B)}\\
&= \frac{0.9 \times 0.2}{ \bP(B, D) + \bP(B, \neg D)}\\
&= \frac{0.18}{ \bP(D)\bP(B \mid D) + \bP(\neg D)\bP(B \mid \neg D)}\\
&= \frac{0.18}{ 0.18 + 0.24}\\
&= \frac{0.18}{ 0.42} = \frac{3}{7} \approx 0.43\ .
\end{align}
$~$$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D \mid B) < \bP(\neg D \mid B) \approx 0.57$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(B \mid D) = 0.9$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D \mid B) =big$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D \mid B) < \bP(\neg D \mid B)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(B \mid D) = big$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D \mid B) =big$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$D$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$D$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(B \mid D)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$B$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(\neg B \mid D)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$B$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$D$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$B$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$0.9$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$B$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$0.1$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\neg D$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(B \mid \neg D) = 0.3$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(\neg B \mid \neg D) = 0.7$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D \mid B)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$B$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$B$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$B$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D \mid B)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$D$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$B$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$D$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D \mid B)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(B,D)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(\neg B,D)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(B \mid D)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D \mid B)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(B, \neg D)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D\mid B)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(D\mid B)$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\bP(B\mid D) = 0.9$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$B$~$ |
---|
Square visualization of probabilities on two events: (example) Diseasitis | $~$\neg D$~$ |
---|
Stabiliser (of a group action) | $~$x$~$ |
---|
Stabiliser (of a group action) | $~$G$~$ |
---|
Stabiliser (of a group action) | $~$G$~$ |
---|
Stabiliser (of a group action) | $~$x$~$ |
---|
Stabiliser (of a group action) | $~$G$~$ |
---|
Stabiliser (of a group action) | $~$X$~$ |
---|
Stabiliser (of a group action) | $~$x \in X$~$ |
---|
Stabiliser (of a group action) | $~$x$~$ |
---|
Stabiliser (of a group action) | $~$G$~$ |
---|
Stabiliser (of a group action) | $~$\mathrm{Stab}_G(x) = \{ g \in G: g(x) = x \}$~$ |
---|
Stabiliser (of a group action) | $~$G$~$ |
---|
Stabiliser (of a group action) | $~$x$~$ |
---|
Stabiliser (of a group action) | $~$x$~$ |
---|
Stabiliser (of a group action) | $~$G$~$ |
---|
Stabiliser (of a group action) | $~$x \in X$~$ |
---|
Stabiliser (of a group action) | $~$x$~$ |
---|
Stabiliser is a subgroup | $~$G$~$ |
---|
Stabiliser is a subgroup | $~$X$~$ |
---|
Stabiliser is a subgroup | $~$x \in X$~$ |
---|
Stabiliser is a subgroup | $~$G$~$ |
---|
Stabiliser is a subgroup | $~$G$~$ |
---|
Stabiliser is a subgroup | $~$X$~$ |
---|
Stabiliser is a subgroup | $~$x \in X$~$ |
---|
Stabiliser is a subgroup | $~$\mathrm{Stab}_G(x)$~$ |
---|
Stabiliser is a subgroup | $~$G$~$ |
---|
Stabiliser is a subgroup | $~$e$~$ |
---|
Stabiliser is a subgroup | $~$e(x) = x$~$ |
---|
Stabiliser is a subgroup | $~$g(x) = x$~$ |
---|
Stabiliser is a subgroup | $~$h(x) = x$~$ |
---|
Stabiliser is a subgroup | $~$(gh)(x) = g(h(x))$~$ |
---|
Stabiliser is a subgroup | $~$g(x) = x$~$ |
---|
Stabiliser is a subgroup | $~$g(x) = x$~$ |
---|
Stabiliser is a subgroup | $~$g^{-1}(x) = g^{-1} g(x) = e(x) = x$~$ |
---|
Standard provability predicate | $~$\exists_1$~$ |
---|
Standard provability predicate | $~$\square_T$~$ |
---|
Standard provability predicate | $~$T$~$ |
---|
Standard provability predicate | $~$PA$~$ |
---|
Standard provability predicate | $~$Prv(x)$~$ |
---|
Standard provability predicate | $~$x$~$ |
---|
Standard provability predicate | $~$\Delta_1$~$ |
---|
Standard provability predicate | $~$\Delta_1$~$ |
---|
Standard provability predicate | $~$Prv(x,y)$~$ |
---|
Standard provability predicate | $~$PA\vdash Prv(n,m)$~$ |
---|
Standard provability predicate | $~$m$~$ |
---|
Standard provability predicate | $~$n$~$ |
---|
Standard provability predicate | $~$S$~$ |
---|
Standard provability predicate | $~$PA$~$ |
---|
Standard provability predicate | $~$PA\vdash \exists y Prv(\ulcorner S \urcorner, y)$~$ |
---|
Standard provability predicate | $~$\square_{PA}(x)$~$ |
---|
Standard provability predicate | $~$\square_{PA}A\rightarrow A$~$ |
---|
Standard provability predicate | $~$PA$~$ |
---|
Standard provability predicate | $~$PA$~$ |
---|
Standard provability predicate | $~$0,1,2,..$~$ |
---|
Standard provability predicate | $~$\square_{PA}x$~$ |
---|
Standard provability predicate | $~$n$~$ |
---|
Standard provability predicate | $~$Prv(x,n)$~$ |
---|
Standard provability predicate | $~$\omega$~$ |
---|
Strictly confused | $~$H$~$ |
---|
Strictly confused | $~$e_0$~$ |
---|
Strictly confused | $~$E$~$ |
---|
Strictly confused | $~$H$~$ |
---|
Strictly confused | $$~$ \log \mathbb P(e_0 \mid H) \ll \sum_{e \in E} \mathbb P(e \mid H) \cdot \log \mathbb P(e \mid H)$~$$ |
---|
Strictly confused | $~$2^{100} : 1$~$ |
---|
Strictly confused | $~$2^{-100}$~$ |
---|
Strictly confused | $~$2^{-8}$~$ |
---|
Strictly confused | $~$2^{-100},$~$ |
---|
Strictly confused | $~$2^{-100} \approx 10^{-30},$~$ |
---|
Strictly confused | $~$0.9^{90} \cdot 0.1^{10} \approx 7\cdot 10^{-15},$~$ |
---|
Strictly confused | $~$0.9^{50} \cdot 0.1^{50} \approx 5 \cdot 10^{-53}.$~$ |
---|
Strictly confused | $~$7 \cdot 10^{-31},$~$ |
---|
Strictly confused | $~$H_1$~$ |
---|
Strictly confused | $~$H_2$~$ |
---|
Strictly factual question | $~$10^{1,000,000}$~$ |
---|
Strong Church Turing thesis | $~$P=NP$~$ |
---|
Strong Church Turing thesis | $~$BQP\subset NP$~$ |
---|
Strong Church Turing thesis | $~$BQP = P$~$ |
---|
Subgroup | $~$(G,*)$~$ |
---|
Subgroup | $~$(H,*)$~$ |
---|
Subgroup | $~$H \subset G$~$ |
---|
Subgroup | $~$H$~$ |
---|
Subgroup | $~$G$~$ |
---|
Subgroup | $~$G$~$ |
---|
Subgroup | $~$G$~$ |
---|
Subgroup | $~$H$~$ |
---|
Subgroup | $~$x, y$~$ |
---|
Subgroup | $~$H$~$ |
---|
Subgroup | $~$x*y$~$ |
---|
Subgroup | $~$H$~$ |
---|
Subgroup | $~$e$~$ |
---|
Subgroup | $~$G$~$ |
---|
Subgroup | $~$H$~$ |
---|
Subgroup | $~$x$~$ |
---|
Subgroup | $~$H$~$ |
---|
Subgroup | $~$x^{-1}$~$ |
---|
Subgroup | $~$H$~$ |
---|
Subgroup | $~$H$~$ |
---|
Subgroup | $~$G$~$ |
---|
Subgroup | $~$I$~$ |
---|
Subgroup | $~$H$~$ |
---|
Subgroup | $~$I$~$ |
---|
Subgroup | $~$G$~$ |
---|
Subgroup | $~$n$~$ |
---|
Subgroup | $~$n$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$G$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$G$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$G$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$gHg^{-1} = H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$g \in G$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$ghg^{-1} \in H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$h \in H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$g \in G$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$h \in H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$ghg^{-1} \in H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$g \in G$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$h$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$G$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$h \in H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$h$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$\cup_{h \in H} C_h$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$C_h$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$h$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$h \in H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$h$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$H$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$h$~$ |
---|
Subgroup is normal if and only if it is a union of conjugacy classes | $~$h$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$G$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$G$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$H$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$\phi:G \to H$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$\phi$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$G/N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$G$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$gN + hN = (g+h)N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$\phi: G \to G/N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$g \mapsto gN$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$gN + hN = (g+h)N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$\{ g : gN = N \}$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$N \subseteq \{ g : gN = N \}$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$nN = N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$n$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$G$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$\{ g : gN = N \} \subseteq N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$gN = N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$g e \in N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$e$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$g \in N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$\phi: G \to H$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$\phi(n) = e$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$n \in N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$G$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$\phi(h n h^{-1}) = \phi(h) \phi(n) \phi(h^{-1}) = \phi(h) \phi(h^{-1})$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$\phi(n) = e$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$\phi(h h^{-1}) = \phi(e)$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$hnh^{-1} \in N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$n \in N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$hnh^{-1} \in N$~$ |
---|
Subgroup is normal if and only if it is the kernel of a homomorphism | $~$N$~$ |
---|
Subspace | $~$U=(F_U, V_U)$~$ |
---|
Subspace | $~$W=(F_W, V_W)$~$ |
---|
Subspace | $~$F_U = F_W$~$ |
---|
Subspace | $~$V_U$~$ |
---|
Subspace | $~$V_W,$~$ |
---|
Subspace | $~$V_U$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$1 + (-1) = 0$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{a}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$n$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$a$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{-1}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$-\frac{1}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{-1}n$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$n$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$-1$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{-1}{n} + \frac{1}{n} = 0$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$-\frac{1}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$n$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$1$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{-1}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{2} + \left(-\frac{1}{2}\right) = 0$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$1 = \frac{1}{2} + \frac{1}{2}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$1 + \left(-\frac{1}{2}\right) = \frac{1}{2}$~$ |
---|
Subtraction of rational numbers (Math 0) | $$~$1 + \left(-\frac{1}{2}\right) = \frac{1}{2} + \frac{1}{2} + \left(-\frac{1}{2}\right) = \frac{1}{2}$~$$ |
---|
Subtraction of rational numbers (Math 0) | $~$(-1) + \frac{1}{2} = -\frac{1}{2}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$-\frac{1}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$n$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{-1}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$n$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$-1$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{a}{m} + \frac{b}{n} = \frac{a\times n + b \times m}{m \times n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $$~$\frac{a}{m} - \frac{b}{n} = \frac{a}{m} + \left(\frac{-b}{n}\right) = \frac{a \times n + (-b) \times m}{m \times n} = \frac{a \times n - b \times m}{m \times n}$~$$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{a}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{a}{m} - \frac{b}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{a}{m}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$n$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$a$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{m}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$a$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$n$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$a \times n$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{b}{n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$b \times m$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{1}{m \times n}$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$a \times n$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$b \times m$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$a \times n - b \times m$~$ |
---|
Subtraction of rational numbers (Math 0) | $~$\frac{a \times n - b \times m}{m \times n}$~$ |
---|
Sum of vector spaces | $~$U$~$ |
---|
Sum of vector spaces | $~$W,$~$ |
---|
Sum of vector spaces | $~$U + W,$~$ |
---|
Sum of vector spaces | $~$u + w$~$ |
---|
Sum of vector spaces | $~$u \in U$~$ |
---|
Sum of vector spaces | $~$w \in W.$~$ |
---|
Superintelligent | $~$\pi_0$~$ |
---|
Superintelligent | $~$\pi_1$~$ |
---|
Superintelligent | $~$U,$~$ |
---|
Superintelligent | $~$\pi_0,$~$ |
---|
Superintelligent | $~$\pi_1$~$ |
---|
Superintelligent | $~$\mathbb E[U | \pi_0] < \mathbb E[U | \pi_1].$~$ |
---|
Superintelligent | $~$\pi_1$~$ |
---|
Superintelligent | $~$\pi_0.$~$ |
---|
Surjective function | $~$f:A \to B$~$ |
---|
Surjective function | $~$b \in B$~$ |
---|
Surjective function | $~$a \in A$~$ |
---|
Surjective function | $~$f(a) = b$~$ |
---|
Surjective function | $~$f$~$ |
---|
Surjective function | $~$\mathbb{N} \to \{ 6 \}$~$ |
---|
Surjective function | $~$\mathbb{N}$~$ |
---|
Surjective function | $~$n \mapsto 6$~$ |
---|
Surjective function | $~$\mathbb{N} \to \mathbb{N}$~$ |
---|
Surjective function | $~$4$~$ |
---|
Surjective function | $~$\mathbb{N} \to \mathbb{N}$~$ |
---|
Surjective function | $~$n \mapsto n+5$~$ |
---|
Surjective function | $~$2$~$ |
---|
Surjective function | $~$a \in \mathbb{N}$~$ |
---|
Surjective function | $~$a+5 = 2$~$ |
---|
Symmetric group | $~$X$~$ |
---|
Symmetric group | $~$f: X \to X$~$ |
---|
Symmetric group | $~$X$~$ |
---|
Symmetric group | $~$\mathrm{Sym}(X)$~$ |
---|
Symmetric group | $~$X$~$ |
---|
Symmetric group | $~$\mathrm{Sym}(X)$~$ |
---|
Symmetric group | $~$X$~$ |
---|
Symmetric group | $~$\mathrm{Aut}(X)$~$ |
---|
Symmetric group | $~$S_n$~$ |
---|
Symmetric group | $~$\mathrm{Sym}(\{ 1,2, \dots, n\})$~$ |
---|
Symmetric group | $~$n$~$ |
---|
Symmetric group | $~$S_n$~$ |
---|
Symmetric group | $~$\{1,2,\dots, n\}$~$ |
---|
Symmetric group | $~$\sigma \in S_n$~$ |
---|
Symmetric group | $~$\sigma$~$ |
---|
Symmetric group | $~$\{1,2,\dots,n\} \to \{1,2,\dots,n\}$~$ |
---|
Symmetric group | $$~$\begin{pmatrix}1 & 2 & \dots & n \\ \sigma(1) & \sigma(2) & \dots & \sigma(n) \\ \end{pmatrix}$~$$ |
---|
Symmetric group | $~$\sigma$~$ |
---|
Symmetric group | $~$\sigma$~$ |
---|
Symmetric group | $~$S_n$~$ |
---|
Symmetric group | $~$S_1$~$ |
---|
Symmetric group | $~$S_1$~$ |
---|
Symmetric group | $~$S_2$~$ |
---|
Symmetric group | $~$2$~$ |
---|
Symmetric group | $~$1$~$ |
---|
Symmetric group | $~$2$~$ |
---|
Symmetric group | $~$(123)$~$ |
---|
Symmetric group | $~$(12)$~$ |
---|
Symmetric group | $~$S_n$~$ |
---|
Symmetric group | $~$n \geq 3$~$ |
---|
Symmetric group | $~$(123)(12) = (13)$~$ |
---|
Symmetric group | $~$(12)(123) = (23)$~$ |
---|
Symmetric group | $~$S_3$~$ |
---|
Symmetric group | $~$(12), (23), (13), (123), (132)$~$ |
---|
Symmetric group | $~$D_6$~$ |
---|
Symmetric group | $~$S_n$~$ |
---|
Symmetric group | $~$S_n$~$ |
---|
Symmetric group | $~$S_5$~$ |
---|
Symmetric group | $~$A_n$~$ |
---|
Symmetric group | $~$S_n$~$ |
---|
Symmetric group | $~$A_n$~$ |
---|
Symmetric group | $~$S_n$~$ |
---|
Task (AI goal) | $~$U_{cauldron}$~$ |
---|
Task (AI goal) | $~$o$~$ |
---|
Task (AI goal) | $$~$U_{cauldron}(o): \begin{cases}
1 & \text{if in $o$ the cauldron is $\geq 90\%$ full of water at 1pm} \\
0 & \text{otherwise}
\end{cases}$~$$ |
---|
Task (AI goal) | $~$1,$~$ |
---|
Task (AI goal) | $~$\geq 90\%$~$ |
---|
Task (AI goal) | $~$U_{cauldron}$~$ |
---|
Task (AI goal) | $~$1.$~$ |
---|
Task (AI goal) | $~$1$~$ |
---|
Task (AI goal) | $~$\pi_1, \pi_2, \pi_3 \ldots$~$ |
---|
Task (AI goal) | $$~$\mathbb E [ U_{cauldron} | \pi_1] = 0.99\\
\mathbb E [ U_{cauldron} | \pi_2] = 0.999 \\
\mathbb E [ U_{cauldron} | \pi_3] = 0.999002 \\
\ldots$~$$ |
---|
Task (AI goal) | $~$\mathbb E [ U_{cauldron} | \pi ] \geq 0.95,$~$ |
---|
Task (AI goal) | $~$\geq 0.90$~$ |
---|
Teleological Measure and agency | $$~$U(S) = \sum_{t=0}^\infty \sum_{\{ S'\in Universes, f^t(S') =S \}}\Pi(S') \cdot \gamma^t $~$$ |
---|
Teleological Measure and agency | $~$1$~$ |
---|
The End (of the basic log tutorial) | $~$b$~$ |
---|
The End (of the basic log tutorial) | $~$x$~$ |
---|
The End (of the basic log tutorial) | $~$b$~$ |
---|
The End (of the basic log tutorial) | $~$x.$~$ |
---|
The End (of the basic log tutorial) | $~$\log_b(x) = y$~$ |
---|
The End (of the basic log tutorial) | $~$y$~$ |
---|
The End (of the basic log tutorial) | $~$x$~$ |
---|
The End (of the basic log tutorial) | $~$b$~$ |
---|
The End (of the basic log tutorial) | $~$\log_b(x) = y$~$ |
---|
The End (of the basic log tutorial) | $~$y$~$ |
---|
The End (of the basic log tutorial) | $~$b$~$ |
---|
The End (of the basic log tutorial) | $~$x$~$ |
---|
The End (of the basic log tutorial) | $~$\log_b(x) = y$~$ |
---|
The End (of the basic log tutorial) | $~$x$~$ |
---|
The End (of the basic log tutorial) | $~$b$~$ |
---|
The End (of the basic log tutorial) | $~$y$~$ |
---|
The End (of the basic log tutorial) | $~$\log_b(x) = y$~$ |
---|
The End (of the basic log tutorial) | $~$b$~$ |
---|
The End (of the basic log tutorial) | $~$y$~$ |
---|
The End (of the basic log tutorial) | $~$x.$~$ |
---|
The End (of the basic log tutorial) | $~$\log_2(100)$~$ |
---|
The End (of the basic log tutorial) | $~$f$~$ |
---|
The End (of the basic log tutorial) | $~$y$~$ |
---|
The End (of the basic log tutorial) | $~$y$~$ |
---|
The End (of the basic log tutorial) | $~$e$~$ |
---|
The End (of the basic log tutorial) | $~$\log_b(x)$~$ |
---|
The End (of the basic log tutorial) | $~$\frac{1}{x}$~$ |
---|
The Harmonic Series: How Comes the Divergence | $~$1/2, 1/3, 1/4$~$ |
---|
The Harmonic Series: How Comes the Divergence | $$~$
\sum_{x=1}^{\infty} \frac{1}{x} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{\infty}
$~$$ |
---|
The Harmonic Series: How Comes the Divergence | $~$f(x)=\frac{1}{x}$~$ |
---|
The Harmonic Series: How Comes the Divergence | $$~$
\int_{1}^{\infty} f(x) dx = ln(x)
$~$$ |
---|
The Harmonic Series: How Comes the Divergence | $$~$
ln(x)|_{1}^{\infty} = \infty
$~$$ |
---|
The Harmonic Series: How Comes the Divergence | $~$\infty$~$ |
---|
The Harmonic Series: How Comes the Divergence | $$~$
\sum_{x=1}^{\infty} \frac{1}{x^n} = \frac{1}{1^n} + \frac{1}{2^n} + \frac{1}{3^n} + \ldots + \frac{1}{\infty^n}
$~$$ |
---|
The Harmonic Series: How Comes the Divergence | $~$n$~$ |
---|
The Harmonic Series: How Comes the Divergence | $~$n=2$~$ |
---|
The Harmonic Series: How Comes the Divergence | $$~$
\sum_{x=1}^{\infty} \frac{1}{x^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \ldots + \frac{1}{\infty^2}
$~$$ |
---|
The Harmonic Series: How Comes the Divergence | $~$\infty^{st}$~$ |
---|
The Harmonic Series: How Comes the Divergence | $~$\infty$~$ |
---|
The Harmonic Series: How Comes the Divergence | $~$\infty^{st}$~$ |
---|
The Harmonic Series: How Comes the Divergence | $~$\infty$~$ |
---|
The Harmonic Series: How Comes the Divergence | $~$\frac{1}{x^2-(x-1)^2}$~$ |
---|
The Harmonic Series: How Comes the Divergence | $~$\frac{1}{step_{x-1}+2}$~$ |
---|
The Harmonic Series: How Comes the Divergence | $~${step_{x-1}+2}$~$ |
---|
The Harmonic Series: How Comes the Divergence | $$~$
\frac{1}{step_{x-1}+2} < 0
$~$$ |
---|
The Harmonic Series: How Comes the Divergence | $$~$
\sum_{x=1}^{\infty} \frac{1}{x^{0.5}} = \frac{1}{1^{0.5}} + \frac{1}{2^{0.5}} + \frac{1}{3^{0.5}} + \ldots + \frac{1}{\infty^{0.5}}
$~$$ |
---|
The Harmonic Series: How Comes the Divergence | $$~$
\frac{1}{(x^{0.5}-(x-1)^0.5)^2} > 0
$~$$ |
---|
The Harmonic Series: How Comes the Divergence | $$~$
\sum_{x=1}^{\infty} \frac{1}{x^n}$~$$ |
---|
The Harmonic Series: How Comes the Divergence | $~$n>0$~$ |
---|
The alternating group on five elements is simple | $~$A_5$~$ |
---|
The alternating group on five elements is simple | $~$A_5$~$ |
---|
The alternating group on five elements is simple | $~$60$~$ |
---|
The alternating group on five elements is simple | $~$A_5$~$ |
---|
The alternating group on five elements is simple | $~$60$~$ |
---|
The alternating group on five elements is simple | $~$H$~$ |
---|
The alternating group on five elements is simple | $~$A_5$~$ |
---|
The alternating group on five elements is simple | $~$\{ e \}$~$ |
---|
The alternating group on five elements is simple | $~$H$~$ |
---|
The alternating group on five elements is simple | $~$3$~$ |
---|
The alternating group on five elements is simple | $~$3$~$ |
---|
The alternating group on five elements is simple | $~$H$~$ |
---|
The alternating group on five elements is simple | $~$3$~$ |
---|
The alternating group on five elements is simple | $~$3$~$ |
---|
The alternating group on five elements is simple | $~$A_5$~$ |
---|
The alternating group on five elements is simple | $~$H$~$ |
---|
The alternating group on five elements is simple | $~$3$~$ |
---|
The alternating group on five elements is simple | $~$A_n$~$ |
---|
The alternating group on five elements is simple | $~$n > 4$~$ |
---|
The alternating group on five elements is simple | $~$H$~$ |
---|
The alternating group on five elements is simple | $~$3$~$ |
---|
The alternating group on five elements is simple | $~$H$~$ |
---|
The alternating group on five elements is simple | $~$2$~$ |
---|
The alternating group on five elements is simple | $~$(12)(34)$~$ |
---|
The alternating group on five elements is simple | $~$H$~$ |
---|
The alternating group on five elements is simple | $~$2$~$ |
---|
The alternating group on five elements is simple | $~$A_5$~$ |
---|
The alternating group on five elements is simple | $~$H$~$ |
---|
The alternating group on five elements is simple | $~$(12)(34)$~$ |
---|
The alternating group on five elements is simple | $~$(15)(34)$~$ |
---|
The alternating group on five elements is simple | $~$(15)(34)(12)(34) = (125)$~$ |
---|
The alternating group on five elements is simple | $~$H$~$ |
---|
The alternating group on five elements is simple | $~$3$~$ |
---|
The alternating group on five elements is simple | $~$H$~$ |
---|
The alternating group on five elements is simple | $~$5$~$ |
---|
The alternating group on five elements is simple | $~$5$~$ |
---|
The alternating group on five elements is simple | $~$A_n$~$ |
---|
The alternating group on five elements is simple | $~$5$~$ |
---|
The alternating group on five elements is simple | $~$5$~$ |
---|
The alternating group on five elements is simple | $~$12$~$ |
---|
The alternating group on five elements is simple | $~$H$~$ |
---|
The alternating group on five elements is simple | $~$5$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$H$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$A_5$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$H$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$60$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$A_5$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$1, 20, 15, 12, 12$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$1$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$60$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$A_5$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$1$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$60$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$60$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $$~$1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$~$$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$H$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$1$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$13$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$12$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$15$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$20$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$30$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$60$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$H$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$A_5$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$60$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$20$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$H$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$20$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$H$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$30$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$21$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$9$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$12$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$H$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$20$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$15$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$H$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$20$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$30$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$16$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$4$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$14$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$12$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$12$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$12$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$1+12 = 13$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$1+12+12 = 25$~$ |
---|
The alternating group on five elements is simple: Simpler proof | $~$60$~$ |
---|
The alternating groups on more than four letters are simple | $~$n > 4$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_n$~$ |
---|
The alternating groups on more than four letters are simple | $~$n$~$ |
---|
The alternating groups on more than four letters are simple | $~$n$~$ |
---|
The alternating groups on more than four letters are simple | $~$n=5$~$ |
---|
The alternating groups on more than four letters are simple | $~$n \geq 6$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_n$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$3$~$ |
---|
The alternating groups on more than four letters are simple | $~$(123)$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$3$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_n$~$ |
---|
The alternating groups on more than four letters are simple | $~$3$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$H = A_n$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_{n-1}$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_{n-1}$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_n$~$ |
---|
The alternating groups on more than four letters are simple | $~$n$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_n$~$ |
---|
The alternating groups on more than four letters are simple | $~$\{1,2,\dots, n \}$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_{n-1}$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma \in H$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma \not = e$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma(n) = n$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma \in H$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma$~$ |
---|
The alternating groups on more than four letters are simple | $~$n$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma(n) = i$~$ |
---|
The alternating groups on more than four letters are simple | $~$i \not = n$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma' \in H$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma'(n) = i$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma^{-1} \sigma'(n) = n$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma$~$ |
---|
The alternating groups on more than four letters are simple | $~$i$~$ |
---|
The alternating groups on more than four letters are simple | $~$j \not = i$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma(j) \not = j$~$ |
---|
The alternating groups on more than four letters are simple | $~$(n i)$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_n$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma(j) \not = j$~$ |
---|
The alternating groups on more than four letters are simple | $~$j \not = i$~$ |
---|
The alternating groups on more than four letters are simple | $~$j \not = n$~$ |
---|
The alternating groups on more than four letters are simple | $~$n \geq 6$~$ |
---|
The alternating groups on more than four letters are simple | $~$x, y$~$ |
---|
The alternating groups on more than four letters are simple | $~$n, i, j, \sigma(j)$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma' = (jxy) \sigma (jxy)^{-1}$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma'(n) = i$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma' \not = \sigma$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma'(j) = \sigma(y)$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma(j)$~$ |
---|
The alternating groups on more than four letters are simple | $~$y \not = j$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma'$~$ |
---|
The alternating groups on more than four letters are simple | $~$\sigma$~$ |
---|
The alternating groups on more than four letters are simple | $~$j$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_{n-1}$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_{n-1}$~$ |
---|
The alternating groups on more than four letters are simple | $~$H \cap A_{n-1}$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_n$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_n$~$ |
---|
The alternating groups on more than four letters are simple | $~$H \cap A_{n-1}$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_{n-1}$~$ |
---|
The alternating groups on more than four letters are simple | $~$H \cap A_{n-1}$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_{n-1}$~$ |
---|
The alternating groups on more than four letters are simple | $~$H$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_{n-1}$~$ |
---|
The alternating groups on more than four letters are simple | $~$(123)$~$ |
---|
The alternating groups on more than four letters are simple | $~$n \leq 4$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_1$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_2$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_3$~$ |
---|
The alternating groups on more than four letters are simple | $~$C_3$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_4$~$ |
---|
The alternating groups on more than four letters are simple | $~$\{ e, (12)(34), (13)(24), (14)(23) \}$~$ |
---|
The alternating groups on more than four letters are simple | $~$A_4$~$ |
---|
The characteristic of the logarithm | $~$f$~$ |
---|
The characteristic of the logarithm | $~$c(x \cdot 2) =$~$ |
---|
The characteristic of the logarithm | $~$c(x) + c(2),$~$ |
---|
The characteristic of the logarithm | $~$c(x \cdot y) =$~$ |
---|
The characteristic of the logarithm | $~$c(x) + c(y),$~$ |
---|
The characteristic of the logarithm | $~$x$~$ |
---|
The characteristic of the logarithm | $~$y$~$ |
---|
The characteristic of the logarithm | $~$x \cdot y$~$ |
---|
The characteristic of the logarithm | $~$y$~$ |
---|
The characteristic of the logarithm | $~$y$~$ |
---|
The characteristic of the logarithm | $~$c$~$ |
---|
The characteristic of the logarithm | $~$\log_2$~$ |
---|
The characteristic of the logarithm | $~$\log_2$~$ |
---|
The characteristic of the logarithm | $~$A$~$ |
---|
The characteristic of the logarithm | $~$\lnot A$~$ |
---|
The characteristic of the logarithm | $~$2 : 1$~$ |
---|
The characteristic of the logarithm | $~$A$~$ |
---|
The characteristic of the logarithm | $~$\lnot A$~$ |
---|
The characteristic of the logarithm | $~$f$~$ |
---|
The characteristic of the logarithm | $~$f(x \cdot y) = f(x) + f(y)$~$ |
---|
The characteristic of the logarithm | $~$\log_b(x^n) = n \log_b(x)$~$ |
---|
The characteristic of the logarithm | $~$b$~$ |
---|
The collection of even-signed permutations is a group | $~$S_n$~$ |
---|
The collection of even-signed permutations is a group | $~$S_n$~$ |
---|
The collection of even-signed permutations is a group | $~$A_n$~$ |
---|
The collection of even-signed permutations is a group | $~$S_n$~$ |
---|
The collection of even-signed permutations is a group | $~$0$~$ |
---|
The collection of even-signed permutations is a group | $~$S_n$~$ |
---|
The collection of even-signed permutations is a group | $~$\sigma$~$ |
---|
The collection of even-signed permutations is a group | $~$\tau_1 \tau_2 \dots \tau_m$~$ |
---|
The collection of even-signed permutations is a group | $~$\tau_m \tau_{m-1} \dots \tau_1$~$ |
---|
The composition of two group homomorphisms is a homomorphism | $~$f: G \to H$~$ |
---|
The composition of two group homomorphisms is a homomorphism | $~$g: H \to K$~$ |
---|
The composition of two group homomorphisms is a homomorphism | $~$gf: G \to K$~$ |
---|
The composition of two group homomorphisms is a homomorphism | $~$g(f(x)) g(f(y)) = g(f(x) f(y))$~$ |
---|
The composition of two group homomorphisms is a homomorphism | $~$g$~$ |
---|
The composition of two group homomorphisms is a homomorphism | $~$g(f(xy))$~$ |
---|
The composition of two group homomorphisms is a homomorphism | $~$f$~$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $$~$S_c(X,\tau) = -k_B \int_{x(t)} Pr(x(t)|x(0)) ln Pr(x(t)|x(0)) Dx(t)$~$$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $$~$F(X,\tau) = T_c \nabla_X S_c(X,\tau) | X_0$~$$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $~$T_c$~$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $~$\tau$~$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $~$T_c$~$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $~$N$~$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $~$p$~$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $~$C$~$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $$~$S = ( N \cdot ln(N/N − p) − p \cdot ln(p/N − p) ) \equiv lnC $~$$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $~$C \in \{X\}$~$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $~$C$~$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $~$\{X\}$~$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $~$C$~$ |
---|
The development of Artificial General Intelligence, as a scientific purpose for human life | $~$\{X\}$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$A$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$A$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$x$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\{ 1, 2 \}$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\{ 1, 2 \}$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f: X \to \{ 1, 2 \}$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f(x) = 1$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f(x) = 2$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$g: X \to \{1,2\}$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$g(x)$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$1$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$2$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f(x)$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$g(y) = f(y)$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$y \not = x$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\{1,2\}$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f: \emptyset \to X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$g: X \to \emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\mathrm{id}: \emptyset \to \emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$g$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$g$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$g$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\emptyset \to \emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$g \circ f$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\mathrm{id}$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$g$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f \circ g$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\mathrm{id}_X : X \to X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$Y$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f \circ g$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\mathrm{id}_X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$g$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\emptyset \to X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X \to \emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f: X \to \emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$x \in X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f(x)$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$\emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$f$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$Y = \emptyset$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$X$~$ |
---|
The empty set is the only set which satisfies the universal property of the empty set | $~$Y$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f: G \to H$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f(G)$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$\{ f(g) : g \in G \}$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f(G)$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$H$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f: G \to H$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$e_G, e_H$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$G$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$H$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f(G)$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$G$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f(G)$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$H$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f(g) f(h) = f(gh)$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$H$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f(G)$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f(G)$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$e_H$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f(G)$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f(e_G)$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f(e_G) f(g) = f(e_G g) = f(g)$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$H$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$f(G)$~$ |
---|
The image of a group under a homomorphism is a subgroup of the codomain | $~$H$~$ |
---|
The log lattice | $~$\log_2(3)$~$ |
---|
The log lattice | $~$\log_2(3)$~$ |
---|
The log lattice | $~$3^{10}$~$ |
---|
The log lattice | $~$3^{100}$~$ |
---|
The log lattice | $~$3^{n}$~$ |
---|
The log lattice | $~$n$~$ |
---|
The log lattice | $~$\log_2.$~$ |
---|
The log lattice | $~$\log_2(2)=1,$~$ |
---|
The log lattice | $~$x$~$ |
---|
The log lattice | $~$x \cdot y,$~$ |
---|
The log lattice | $~$x$~$ |
---|
The log lattice | $~$y$~$ |
---|
The log lattice | $~$\log$~$ |
---|
The log lattice | $~$\log_2(3)$~$ |
---|
The log lattice | $~$1$~$ |
---|
The log lattice | $~$\log_2(6),$~$ |
---|
The log lattice | $~$\log_2(9)$~$ |
---|
The log lattice | $~$\log_2(3^{10}),$~$ |
---|
The log lattice | $~$\log_2(3^9)$~$ |
---|
The log lattice | $~$\log_2(3^{10}).$~$ |
---|
The log lattice | $~$\log_2(3)$~$ |
---|
The log lattice | $~$b$~$ |
---|
The log lattice | $~$b$~$ |
---|
The n-th root of m is either an integer or irrational | $~$m$~$ |
---|
The n-th root of m is either an integer or irrational | $~$n$~$ |
---|
The n-th root of m is either an integer or irrational | $~$\sqrt[n]m$~$ |
---|
The n-th root of m is either an integer or irrational | $~$\sqrt[n]m$~$ |
---|
The n-th root of m is either an integer or irrational | $~$\frac{a}{b}$~$ |
---|
The n-th root of m is either an integer or irrational | $~$\frac ab$~$ |
---|
The n-th root of m is either an integer or irrational | $~$a$~$ |
---|
The n-th root of m is either an integer or irrational | $~$b$~$ |
---|
The n-th root of m is either an integer or irrational | $~$1$~$ |
---|
The n-th root of m is either an integer or irrational | $~$\frac{a}{b}$~$ |
---|
The n-th root of m is either an integer or irrational | $~$b > 1$~$ |
---|
The n-th root of m is either an integer or irrational | $~$\frac ab = \sqrt[n]m$~$ |
---|
The n-th root of m is either an integer or irrational | $~$(\frac ab)^n = m$~$ |
---|
The n-th root of m is either an integer or irrational | $~$(\frac ab)^n$~$ |
---|
The n-th root of m is either an integer or irrational | $~$\frac{a^n}{b^n}$~$ |
---|
The n-th root of m is either an integer or irrational | $~$b > 1$~$ |
---|
The n-th root of m is either an integer or irrational | $~$b^n > 1$~$ |
---|
The n-th root of m is either an integer or irrational | $~$(\frac ab)^n$~$ |
---|
The n-th root of m is either an integer or irrational | $~$m$~$ |
---|
The rationals form a field | $~$\mathbb{Q}$~$ |
---|
The rationals form a field | $~$\mathbb{Q}$~$ |
---|
The rationals form a field | $~$\frac{0}{1}$~$ |
---|
The rationals form a field | $~$0$~$ |
---|
The rationals form a field | $~$\frac{1}{1}$~$ |
---|
The rationals form a field | $~$1$~$ |
---|
The rationals form a field | $~$+$~$ |
---|
The rationals form a field | $~$\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}$~$ |
---|
The rationals form a field | $~$\mathbb{Z}$~$ |
---|
The rationals form a field | $~$\frac{cb+da}{db} = \frac{c}{d} + \frac{a}{b}$~$ |
---|
The rationals form a field | $~$0$~$ |
---|
The rationals form a field | $~$+$~$ |
---|
The rationals form a field | $~$\frac{a}{b}+0 = \frac{a}{b} + \frac{0}{1} = \frac{a \times 1 + 0 \times b}{b \times 1}$~$ |
---|
The rationals form a field | $~$\frac{a}{b}$~$ |
---|
The rationals form a field | $~$1$~$ |
---|
The rationals form a field | $~$\mathbb{Z}$~$ |
---|
The rationals form a field | $~$0 \times n = 0$~$ |
---|
The rationals form a field | $~$n$~$ |
---|
The rationals form a field | $~$\frac{a}{b}$~$ |
---|
The rationals form a field | $~$\frac{-a}{b}$~$ |
---|
The rationals form a field | $~$+$~$ |
---|
The rationals form a field | $$~$\left(\frac{a_1}{b_1}+\frac{a_2}{b_2}\right)+\frac{a_3}{b_3} = \frac{a_1 b_2 + b_1 a_2}{b_1 b_2} + \frac{a_3}{b_3} = \frac{a_1 b_2 b_3 + b_1 a_2 b_3 + a_3 b_1 b_2}{b_1 b_2 b_3}$~$$ |
---|
The rationals form a field | $~$\frac{a_1}{b_1}+\left(\frac{a_2}{b_2}+\frac{a_3}{b_3}\right)$~$ |
---|
The rationals form a field | $~$\times$~$ |
---|
The rationals form a field | $$~$\left(\frac{a_1}{b_1} \frac{a_2}{b_2}\right) \frac{a_3}{b_3} = \frac{a_1 a_2}{b_1 b_2} \frac{a_3}{b_3} = \frac{a_1 a_2 a_3}{b_1 b_2 b_3} = \frac{a_1}{b_1} \left(\frac{a_2 a_3}{b_2 b_3}\right) = \frac{a_1}{b_1} \left(\frac{a_2}{b_2} \frac{a_3}{b_3}\right)$~$$ |
---|
The rationals form a field | $~$\times$~$ |
---|
The rationals form a field | $$~$\frac{a}{b} \frac{c}{d} = \frac{ac}{bd} = \frac{ca}{db} = \frac{c}{d} \frac{a}{b}$~$$ |
---|
The rationals form a field | $~$1$~$ |
---|
The rationals form a field | $~$\times$~$ |
---|
The rationals form a field | $$~$\frac{a}{b} \times 1 = \frac{a}{b} \times \frac{1}{1} = \frac{a \times 1}{b \times 1} = \frac{a}{b}$~$$ |
---|
The rationals form a field | $~$1$~$ |
---|
The rationals form a field | $~$\times$~$ |
---|
The rationals form a field | $~$\mathbb{Z}$~$ |
---|
The rationals form a field | $~$+$~$ |
---|
The rationals form a field | $~$\times$~$ |
---|
The rationals form a field | $$~$\frac{a}{b} \left(\frac{x_1}{y_1}+\frac{x_2}{y_2}\right) = \frac{a}{b} \frac{x_1 y_2 + x_2 y_1}{y_1 y_2} = \frac{a \left(x_1 y_2 + x_2 y_1\right)}{b y_1 y_2}$~$$ |
---|
The rationals form a field | $$~$\frac{a}{b} \frac{x_1}{y_1} + \frac{a}{b} \frac{x_2}{y_2} = \frac{a x_1}{b y_1} + \frac{a x_2}{b y_2} = \frac{a x_1 b y_2 + b y_1 a x_2}{b^2 y_1 y_2} = \frac{a x_1 y_2 + a y_1 x_2}{b y_1 y_2}$~$$ |
---|
The rationals form a field | $~$+$~$ |
---|
The rationals form a field | $~$\times$~$ |
---|
The rationals form a field | $~$\mathbb{Z}$~$ |
---|
The rationals form a field | $~$\mathbb{Q}$~$ |
---|
The rationals form a field | $~$\frac{a}{b}$~$ |
---|
The rationals form a field | $~$0$~$ |
---|
The rationals form a field | $~$a \not = 0$~$ |
---|
The rationals form a field | $~$\frac{a}{b}$~$ |
---|
The rationals form a field | $~$\frac{b}{a}$~$ |
---|
The rationals form a field | $~$a \not = 0$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$(A, B)$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\mathbf{A}$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$B$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\mathbf{0}$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$(\{ r \in \mathbb{Q} \mid r < 0\}, \{ r \in \mathbb{Q} \mid r \geq 0 \})$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$(A, B) + (C, D) = (A+C, B+D)$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\mathbf{A} \leq \mathbf{C}$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\mathbf{0} \leq \mathbf{A}$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\mathbf{A} \times \mathbf{C} = \{ a c \mid a \in A, a > 0, c \in C \}$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\mathbf{A} < \mathbf{0}$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\mathbf{0} \leq \mathbf{C}$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\mathbf{A} \times \mathbf{C} = \{ a c \mid a \in A, c \in C, c > 0 \}$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\mathbf{A} < \mathbf{0}$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\mathbf{C} < \mathbf{0}$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\mathbf{A} \times \mathbf{C} = \{\} $~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$(A, B)$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A+C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\{ a+c \mid a \in A, c \in C \}$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A \times C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$\{ a \times c \mid a \in A, c \in C \}$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A+C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$B+D$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$a+c \in A+C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$b+d \in B+D$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$a < b$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$c < d$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$a+c < b+d$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A+C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$B+D$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$(A+C, B+D)$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A+C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$a+c \in A+C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$a' + c'$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A+C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$a+c$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A+C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$a'$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$a$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$c'$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$a' + c'$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$A+C$~$ |
---|
The reals (constructed as Dedekind cuts) form a field | $~$a+c$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] + [b_n] = [a_n+b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] \times [b_n] = [a_n \times b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] \leq [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] = [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$a_n \leq b_n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$(x_n)_{n=1}^{\infty}$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$(y_n)_{n=1}^{\infty}$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[x_n] = [y_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$+$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[x_n]+[a_n] = [y_n] + [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[x_n] = [y_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] = [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[x_n] = [y_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$(x_n)$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$(y_n)$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$x_n - y_n \to 0$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n \to \infty$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$a_n - b_n \to 0$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n \to \infty$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[x_n+a_n] = [y_n+b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$x_n+a_n - y_n-b_n \to 0$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n \to \infty$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\epsilon > 0$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$N_1$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n > N_1$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|x_n - y_n| < \frac{\epsilon}{2}$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$N_2$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n > N_2$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|a_n - b_n| < \frac{\epsilon}{2}$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$N_1, N_2$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n > N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|x_n + a_n - y_n - b_n| \leq |x_n - y_n| + |a_n - b_n|$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\leq \epsilon$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\times$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[x_n] \times [a_n] = [y_n] \times [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[x_n] = [y_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] = [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[x_n a_n] = [y_n b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$x_n a_n - y_n b_n \to 0$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n \to \infty$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\epsilon > 0$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $$~$|x_n a_n - y_n b_n| = |x_n (a_n - b_n) + b_n (x_n - y_n)|$~$$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$x_n b_n - x_n b_n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\leq |x_n| |a_n - b_n| + |b_n| |x_n - y_n|$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$B$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|x_n| < B$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|b_n| < B$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$B (|a_n - b_n| + |x_n - y_n|)$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|a_n - b_n| < \frac{\epsilon}{2 B}$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$x_n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$y_n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|x_n a_n - y_n b_n| < \epsilon$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\leq$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] = [c_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[b_n] = [d_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] \leq [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[c_n] \leq [d_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] \leq [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[c_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\leq [d_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[c_n] \not = [d_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$c_n > d_n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] = [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[d_n] = [b_n] = [a_n] = [c_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[c_n] \not \leq [d_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$a_n \leq b_n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\mathbb{R}$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[0]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$(0, 0, \dots)$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] + [0] = [a_n+0] = [a_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[-a_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] + [-a_n] = [a_n-a_n] = [0]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] + [b_n] = [a_n+b_n] = [b_n+a_n] = [b_n] + [a_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$(a_n)$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$(b_n)$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\epsilon > 0$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n, m > N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|a_n+b_n - a_m - b_m| < \epsilon$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|a_n+b_n - a_m - b_m| \leq |a_n - a_m| + |b_n - b_m|$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|a_n - a_m| < \frac{\epsilon}{2}$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|b_n - b_m| < \frac{\epsilon}{2}$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n, m > N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $$~$[a_n] + ([b_n] + [c_n]) = [a_n] + [b_n+c_n] = [a_n+b_n+c_n] = [a_n+b_n] + [c_n] = ([a_n]+[b_n])+[c_n]$~$$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[1]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$(1,1, \dots)$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] \times [1] = [a_n \times 1] = [a_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\times$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$(a_n)$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$(b_n)$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\epsilon > 0$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n, m > N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|a_n b_n - a_m b_m| < \epsilon$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $$~$|a_n b_n - a_m b_m| = |a_n (b_n - b_m) + b_m (a_n - a_m)| \leq |b_m| |a_n - a_m| + |a_n| |b_n - b_m|$~$$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$B$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|a_n|$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|b_m|$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$B$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$m$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|a_n - a_m| < \frac{\epsilon}{2B}$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$|b_n - b_m| < \frac{\epsilon}{2B}$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n, m > N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\times$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] \times [b_n] = [a_n \times b_n] = [b_n \times a_n] = [b_n] \times [a_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\times$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $$~$[a_n] \times ([b_n] \times [c_n]) = [a_n] \times [b_n \times c_n] = [a_n \times b_n \times c_n] = [a_n \times b_n] \times [c_n] = ([a_n] \times [b_n]) \times [c_n]$~$$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\times$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$+$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[x_n] \times ([a_n]+[b_n]) = [x_n] \times [a_n] + [x_n] \times [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $$~$[x_n] \times ([a_n]+[b_n]) = [x_n] \times [a_n+b_n] = [x_n \times (a_n+b_n)] = [x_n \times a_n + x_n \times b_n] = [x_n \times a_n] + [x_n \times b_n] = [x_n] \times [a_n] + [x_n] \times [b_n]$~$$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[0]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] \not = 0$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n > N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$a_n \not = 0$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$b_i = 1$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$i \leq N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$b_i = \frac{1}{a_i}$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$i > N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$\mathbb{R}$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] [b_n] = [1]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] [b_n] = [a_n b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$(a_n b_n)$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$1$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n > N$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$(1, 1, \dots)$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] \leq [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[c_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] + [c_n] \leq [b_n] + [c_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[0] \leq [a_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[0] \leq [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[0] \leq [a_n] \times[b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] = [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[c_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] + [c_n] = [b_n] + [c_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] + [c_n] \leq [b_n] + [c_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[0] = [a_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[0] \leq [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[0] = [0] \times [b_n] = [a_n] \times [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[0] \leq [a_n] \times [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] < [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[c_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] + [c_n] = [a_n+c_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[b_n] + [c_n] = [b_n+c_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$a_n + c_n \leq b_n + c_n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$a_n \leq b_n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[a_n] + [c_n] \leq [b_n] + [c_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[0] < [a_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[0] < [b_n]$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$a_n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$b_n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$n$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$a_n b_n \geq 0$~$ |
---|
The reals (constructed as classes of Cauchy sequences of rationals) form a field | $~$[0] \leq [a_n] \times [b_n]$~$ |
---|
The set of rational numbers is countable | $~$\mathbb{Q}$~$ |
---|
The set of rational numbers is countable | $~$\mathbb{Q}$~$ |
---|
The set of rational numbers is countable | $~$\mathbb{N}$~$ |
---|
The set of rational numbers is countable | $~$\mathbb{N} \to \mathbb{Q}$~$ |
---|
The set of rational numbers is countable | $~$\mathbb{Q} \to \mathbb{N}$~$ |
---|
The set of rational numbers is countable | $~$\mathbb{N}$~$ |
---|
The set of rational numbers is countable | $~$\mathbb{Q}$~$ |
---|
The set of rational numbers is countable | $~$n \mapsto \frac{n}{1}$~$ |
---|
The set of rational numbers is countable | $~$\mathbb{Q} \to \mathbb{N}$~$ |
---|
The set of rational numbers is countable | $~$\frac{p}{q}$~$ |
---|
The set of rational numbers is countable | $~$p$~$ |
---|
The set of rational numbers is countable | $~$q$~$ |
---|
The set of rational numbers is countable | $~$1$~$ |
---|
The set of rational numbers is countable | $~$p$~$ |
---|
The set of rational numbers is countable | $~$q$~$ |
---|
The set of rational numbers is countable | $~$-1$~$ |
---|
The set of rational numbers is countable | $~$\frac{-1}{-1}$~$ |
---|
The set of rational numbers is countable | $~$q$~$ |
---|
The set of rational numbers is countable | $~$p = 0$~$ |
---|
The set of rational numbers is countable | $~$q = 1$~$ |
---|
The set of rational numbers is countable | $~$s$~$ |
---|
The set of rational numbers is countable | $~$1$~$ |
---|
The set of rational numbers is countable | $~$p$~$ |
---|
The set of rational numbers is countable | $~$2$~$ |
---|
The set of rational numbers is countable | $~$p$~$ |
---|
The set of rational numbers is countable | $~$2^p 3^q 5^s$~$ |
---|
The set of rational numbers is countable | $~$f: \frac{p}{q} \mapsto 2^p 3^q 5^s$~$ |
---|
The set of rational numbers is countable | $~$f\left(\frac{p}{q}\right) = f \left(\frac{a}{b} \right)$~$ |
---|
The set of rational numbers is countable | $~$q$~$ |
---|
The set of rational numbers is countable | $~$|p| = |a|, q=b$~$ |
---|
The set of rational numbers is countable | $~$p$~$ |
---|
The set of rational numbers is countable | $~$a$~$ |
---|
The sign of a permutation is well-defined | $~$S_n$~$ |
---|
The sign of a permutation is well-defined | $~$\sigma \in S_n$~$ |
---|
The sign of a permutation is well-defined | $~$S_n$~$ |
---|
The sign of a permutation is well-defined | $~$C_2 = \{0,1\}$~$ |
---|
The sign of a permutation is well-defined | $~$0$~$ |
---|
The sign of a permutation is well-defined | $~$1$~$ |
---|
The sign of a permutation is well-defined | $~$c(\sigma)$~$ |
---|
The sign of a permutation is well-defined | $~$\sigma \in S_n$~$ |
---|
The sign of a permutation is well-defined | $~$c$~$ |
---|
The sign of a permutation is well-defined | $~$n$~$ |
---|
The sign of a permutation is well-defined | $~$c((12)) = n-1$~$ |
---|
The sign of a permutation is well-defined | $~$(12)$~$ |
---|
The sign of a permutation is well-defined | $~$(12)(3)(4)\dots(n-1)(n)$~$ |
---|
The sign of a permutation is well-defined | $~$\sigma$~$ |
---|
The sign of a permutation is well-defined | $~$c(\sigma)$~$ |
---|
The sign of a permutation is well-defined | $~$1$~$ |
---|
The sign of a permutation is well-defined | $~$1$~$ |
---|
The sign of a permutation is well-defined | $~$\tau = (kl)$~$ |
---|
The sign of a permutation is well-defined | $~$k, l$~$ |
---|
The sign of a permutation is well-defined | $~$\sigma$~$ |
---|
The sign of a permutation is well-defined | $$~$\sigma = \alpha (k a_1 a_2 \dots a_r l a_s \dots a_t) \beta$~$$ |
---|
The sign of a permutation is well-defined | $~$\alpha, \beta$~$ |
---|
The sign of a permutation is well-defined | $~$(kl)$~$ |
---|
The sign of a permutation is well-defined | $~$\sigma (kl) = \alpha (k a_s \dots a_t)(l a_1 \dots a_r) \beta$~$ |
---|
The sign of a permutation is well-defined | $$~$\sigma = \alpha (k a_1 a_2 \dots a_r)(l b_1 \dots b_s) \beta$~$$ |
---|
The sign of a permutation is well-defined | $~$\alpha, \beta$~$ |
---|
The sign of a permutation is well-defined | $~$(kl)$~$ |
---|
The sign of a permutation is well-defined | $~$\sigma (kl) = \alpha (k b_1 b_2 \dots b_s l a_1 \dots a_r) \beta$~$ |
---|
The sign of a permutation is well-defined | $~$c$~$ |
---|
The sign of a permutation is well-defined | $~$\sigma$~$ |
---|
The sign of a permutation is well-defined | $~$\sigma$~$ |
---|
The sign of a permutation is well-defined | $~$\sigma = \alpha_1 \dots \alpha_a = \beta_1 \dots \beta_b$~$ |
---|
The sign of a permutation is well-defined | $~$\alpha_i, \beta_j$~$ |
---|
The sign of a permutation is well-defined | $~$c(\sigma) \equiv n+a \pmod{2}$~$ |
---|
The sign of a permutation is well-defined | $~$\equiv n+b \pmod{2}$~$ |
---|
The sign of a permutation is well-defined | $~$a \equiv b \pmod{2}$~$ |
---|
The sign of a permutation is well-defined | $~$c(\sigma)$~$ |
---|
The sign of a permutation is well-defined | $~$1, 2, \dots, a$~$ |
---|
The sign of a permutation is well-defined | $~$1, 2, \dots, b$~$ |
---|
The sign of a permutation is well-defined | $~$a$~$ |
---|
The sign of a permutation is well-defined | $~$b$~$ |
---|
The square root of 2 is irrational | $~$\sqrt 2$~$ |
---|
The square root of 2 is irrational | $~$\sqrt 2$~$ |
---|
The square root of 2 is irrational | $~$\sqrt 2=\frac{a}{b}$~$ |
---|
The square root of 2 is irrational | $~$a$~$ |
---|
The square root of 2 is irrational | $~$b$~$ |
---|
The square root of 2 is irrational | $~$\frac{a}{b}$~$ |
---|
The square root of 2 is irrational | $~$\gcd(a,b)=1$~$ |
---|
The square root of 2 is irrational | $$~$\sqrt 2=\frac{a}{b}$~$$ |
---|
The square root of 2 is irrational | $~$\sqrt 2$~$ |
---|
The square root of 2 is irrational | $$~$2=\frac{a^2}{b^2}$~$$ |
---|
The square root of 2 is irrational | $$~$2b^2=a^2$~$$ |
---|
The square root of 2 is irrational | $~$a^2$~$ |
---|
The square root of 2 is irrational | $~$2$~$ |
---|
The square root of 2 is irrational | $~$2$~$ |
---|
The square root of 2 is irrational | $~$a$~$ |
---|
The square root of 2 is irrational | $~$a=2k$~$ |
---|
The square root of 2 is irrational | $$~$2b^2=(2k)^2=4k^2$~$$ |
---|
The square root of 2 is irrational | $$~$b^2=2k^2$~$$ |
---|
The square root of 2 is irrational | $~$b^2$~$ |
---|
The square root of 2 is irrational | $~$2$~$ |
---|
The square root of 2 is irrational | $~$b$~$ |
---|
The square root of 2 is irrational | $~$2|\gcd(a,b)$~$ |
---|
The square root of 2 is irrational | $~$\frac{a}{b}$~$ |
---|
The square root of 2 is irrational | $~$\sqrt 2$~$ |
---|
The square root of 2 is irrational | $~$\sqrt 2$~$ |
---|
The square root of 2 is irrational | $~$\sqrt 2$~$ |
---|
There is only one logarithm | $~$f$~$ |
---|
There is only one logarithm | $~$\mathbb R^+$~$ |
---|
There is only one logarithm | $~$f(x \cdot y) = f(x) + f(y)$~$ |
---|
There is only one logarithm | $~$x$~$ |
---|
There is only one logarithm | $~$y$~$ |
---|
There is only one logarithm | $~$\log_b$~$ |
---|
There is only one logarithm | $~$b$~$ |
---|
There is only one logarithm | $~$y$~$ |
---|
There is only one logarithm | $~$y$~$ |
---|
There is only one logarithm | $~$b \neq 1$~$ |
---|
There is only one logarithm | $~$f(b) = 1$~$ |
---|
There is only one logarithm | $~$b$~$ |
---|
There is only one logarithm | $~$f$~$ |
---|
There is only one logarithm | $~$b$~$ |
---|
There is only one logarithm | $~$c.$~$ |
---|
There is only one logarithm | $~$x \in \mathbb R^+$~$ |
---|
There is only one logarithm | $~$\log_b(x)$~$ |
---|
There is only one logarithm | $~$\log_c(x)?$~$ |
---|
There is only one logarithm | $~$x = c^y$~$ |
---|
There is only one logarithm | $~$y$~$ |
---|
There is only one logarithm | $~$x$~$ |
---|
There is only one logarithm | $~$\log_c(x)$~$ |
---|
There is only one logarithm | $~$=$~$ |
---|
There is only one logarithm | $~$\log_c(c^y)$~$ |
---|
There is only one logarithm | $~$=$~$ |
---|
There is only one logarithm | $~$y.$~$ |
---|
There is only one logarithm | $~$\log_b(x)$~$ |
---|
There is only one logarithm | $~$=$~$ |
---|
There is only one logarithm | $~$\log_b(c^y)$~$ |
---|
There is only one logarithm | $~$=$~$ |
---|
There is only one logarithm | $~$y \log_b(c).$~$ |
---|
There is only one logarithm | $~$\log_c$~$ |
---|
There is only one logarithm | $~$\log_b$~$ |
---|
There is only one logarithm | $~$x$~$ |
---|
There is only one logarithm | $~$\log_b(c).$~$ |
---|
There is only one logarithm | $~$x$~$ |
---|
There is only one logarithm | $~$\log_c(x)$~$ |
---|
There is only one logarithm | $~$\log_b(x)$~$ |
---|
There is only one logarithm | $~$\log_b(c):$~$ |
---|
There is only one logarithm | $$~$\log_c(x) = \frac{\log_b(x)}{\log_b(c)}.$~$$ |
---|
There is only one logarithm | $~$b$~$ |
---|
There is only one logarithm | $~$f$~$ |
---|
There is only one logarithm | $~$x$~$ |
---|
There is only one logarithm | $~$f$~$ |
---|
There is only one logarithm | $~$c$~$ |
---|
There is only one logarithm | $~$f(c)$~$ |
---|
There is only one logarithm | $~$f(x)$~$ |
---|
There is only one logarithm | $~$f(c).$~$ |
---|
There is only one logarithm | $~$\log_c$~$ |
---|
There is only one logarithm | $~$\log_b$~$ |
---|
There is only one logarithm | $~$\log_b(x) = \frac{\log_c(x)}{\log_c(b)},$~$ |
---|
There is only one logarithm | $~$\log_b(c) = \frac{1}{\log_c(b)}$~$ |
---|
There is only one logarithm | $~$\log_{10}(12) \approx 1.08$~$ |
---|
There is only one logarithm | $~$\log_2(10) \approx 3.32$~$ |
---|
There is only one logarithm | $~$\log_3(2) \approx 0.63.$~$ |
---|
There is only one logarithm | $~$\log_{10}$~$ |
---|
There is only one logarithm | $~$e$~$ |
---|
There is only one logarithm | $~$\approx 2.718,$~$ |
---|
There is only one logarithm | $~$x$~$ |
---|
There is only one logarithm | $~$\frac{1}{x}$~$ |
---|
There is only one logarithm | $~$y$~$ |
---|
There is only one logarithm | $~$e$~$ |
---|
There is only one logarithm | $~$1.$~$ |
---|
There is only one logarithm | $~$\log_e$~$ |
---|
There is only one logarithm | $~$\ln,$~$ |
---|
There is only one logarithm | $~$\log_b$~$ |
---|
There is only one logarithm | $~$\log_c(x)$~$ |
---|
There is only one logarithm | $~$\log_b(x)$~$ |
---|
There is only one logarithm | $~$\log_b(c)$~$ |
---|
Totally ordered set | $~$(S, \le)$~$ |
---|
Totally ordered set | $~$S$~$ |
---|
Totally ordered set | $~$\le$~$ |
---|
Totally ordered set | $~$S$~$ |
---|
Totally ordered set | $~$a, b \in S$~$ |
---|
Totally ordered set | $~$a \le b$~$ |
---|
Totally ordered set | $~$b \le a$~$ |
---|
Totally ordered set | $~$a = b$~$ |
---|
Totally ordered set | $~$a, b, c \in S$~$ |
---|
Totally ordered set | $~$a \le b$~$ |
---|
Totally ordered set | $~$b \le c$~$ |
---|
Totally ordered set | $~$a \le c$~$ |
---|
Totally ordered set | $~$a, b \in S$~$ |
---|
Totally ordered set | $~$a \le b$~$ |
---|
Totally ordered set | $~$b \le a$~$ |
---|
Totally ordered set | $~$(S, \le)$~$ |
---|
Totally ordered set | $~$S$~$ |
---|
Totally ordered set | $~$\le$~$ |
---|
Totally ordered set | $~$S$~$ |
---|
Totally ordered set | $~$a, b \in S$~$ |
---|
Totally ordered set | $~$a \le b$~$ |
---|
Totally ordered set | $~$b \le a$~$ |
---|
Totally ordered set | $~$a = b$~$ |
---|
Totally ordered set | $~$a, b, c \in S$~$ |
---|
Totally ordered set | $~$a \le b$~$ |
---|
Totally ordered set | $~$b \le c$~$ |
---|
Totally ordered set | $~$a \le c$~$ |
---|
Totally ordered set | $~$a, b \in S$~$ |
---|
Totally ordered set | $~$a \le b$~$ |
---|
Totally ordered set | $~$b \le a$~$ |
---|
Totally ordered set | $~$a \le a$~$ |
---|
Totally ordered set | $~$a \in S$~$ |
---|
Toxoplasmosis dilemma | $~$\mathcal Q$~$ |
---|
Toxoplasmosis dilemma | $~$\ulcorner \mathcal Q \urcorner$~$ |
---|
Toxoplasmosis dilemma | $~$\mathcal U.$~$ |
---|
Toxoplasmosis dilemma | $~$\mathcal U$~$ |
---|
Transcendental number | $~$z$~$ |
---|
Transcendental number | $~$z$~$ |
---|
Transcendental number | $~$z$~$ |
---|
Transcendental number | $~$0$~$ |
---|
Transcendental number | $~$z$~$ |
---|
Transcendental number | $~$\frac{1}{2}$~$ |
---|
Transcendental number | $~$\sqrt{6}$~$ |
---|
Transcendental number | $~$i$~$ |
---|
Transcendental number | $~$e^{i \pi/2}$~$ |
---|
Transcendental number | $~$\pi$~$ |
---|
Transcendental number | $~$e$~$ |
---|
Transcendental number | $~$n$~$ |
---|
Transcendental number | $~$x-n$~$ |
---|
Transcendental number | $~$\frac{p}{q}$~$ |
---|
Transcendental number | $~$qx - p$~$ |
---|
Transcendental number | $~$\sqrt{2}$~$ |
---|
Transcendental number | $~$x^2-2$~$ |
---|
Transcendental number | $~$i$~$ |
---|
Transcendental number | $~$x^2+1$~$ |
---|
Transcendental number | $~$e^{i \pi/2}$~$ |
---|
Transcendental number | $~$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$~$ |
---|
Transcendental number | $~$x^4+1$~$ |
---|
Transcendental number | $~$\pi$~$ |
---|
Transcendental number | $~$e$~$ |
---|
Transcendental number | $~$n$~$ |
---|
Transcendental number | $~$n$~$ |
---|
Transcendental number | $~$x^2+2x+1$~$ |
---|
Transcendental number | $~$x=-1$~$ |
---|
Transcendental number | $~$x=-1$~$ |
---|
Transcendental number | $~$0$~$ |
---|
Transitive relation | $~$R$~$ |
---|
Transitive relation | $~$aRb$~$ |
---|
Transitive relation | $~$bRc$~$ |
---|
Transitive relation | $~$aRc$~$ |
---|
Transitive relation | $~$a \leq b$~$ |
---|
Transitive relation | $~$b \leq c$~$ |
---|
Transitive relation | $~$a \leq c$~$ |
---|
Transitive relation | $~$a \sim b$~$ |
---|
Transitive relation | $~$b \sim c$~$ |
---|
Transitive relation | $~$a \sim c$~$ |
---|
Transitive relation | $~$S$~$ |
---|
Transitive relation | $~$\in$~$ |
---|
Transitive relation | $~$a \in x$~$ |
---|
Transitive relation | $~$x \in S$~$ |
---|
Transitive relation | $~$a \in S$~$ |
---|
Transposition (as an element of a symmetric group) | $~$2$~$ |
---|
Transposition (as an element of a symmetric group) | $~$2$~$ |
---|
Transposition (as an element of a symmetric group) | $~$S_5$~$ |
---|
Transposition (as an element of a symmetric group) | $~$(12)$~$ |
---|
Transposition (as an element of a symmetric group) | $~$1$~$ |
---|
Transposition (as an element of a symmetric group) | $~$2$~$ |
---|
Transposition (as an element of a symmetric group) | $~$3,4,5$~$ |
---|
Transposition (as an element of a symmetric group) | $~$(124)$~$ |
---|
Transposition (as an element of a symmetric group) | $~$3$~$ |
---|
Transposition (as an element of a symmetric group) | $~$2$~$ |
---|
Trit | $~$\log_2(3) \approx 1.58$~$ |
---|
Trit | $~$3:1$~$ |
---|
Trit | $~$\log_2(3)\approx 1.58$~$ |
---|
Turing machine | $~$(\text{symbol},\text{state},\text{move left or right})$~$ |
---|
Two independent events | $$~$
\newcommand{\bP}{\mathbb{P}}
$~$$ |
---|
Two independent events | $$~$
\newcommand{\bP}{\mathbb{P}}
$~$$ |
---|
Two independent events | $~$A$~$ |
---|
Two independent events | $~$B$~$ |
---|
Two independent events | $~$A$~$ |
---|
Two independent events | $~$B$~$ |
---|
Two independent events | $~$\bP(B \mid A) = \bP(B)$~$ |
---|
Two independent events | $~$\bP(A,B) = \bP(A) \bP(B)$~$ |
---|
Two independent events | $~$A$~$ |
---|
Two independent events | $~$B$~$ |
---|
Two independent events | $~$A$~$ |
---|
Two independent events | $~$B$~$ |
---|
Two independent events | $~$\bP(B \mid A) = \bP(B)$~$ |
---|
Two independent events | $~$A$~$ |
---|
Two independent events | $~$B$~$ |
---|
Two independent events | $~$\bP(A)$~$ |
---|
Two independent events | $~$B$~$ |
---|
Two independent events | $~$\bP(A \mid B) = \bP(A)$~$ |
---|
Two independent events | $~$\bP(A,B) = \bP(A) \bP(B)$~$ |
---|
Two independent events | $$~$\bP(A,B) = \bP(A)\; \bP(B \mid A)$~$$ |
---|
Two independent events | $$~$\bP(A,B) = \bP(A)\; \bP(B)\;\; \Leftrightarrow \;\; \bP(A)\; \bP(B \mid A) = \bP(A)\; \bP(B) \ ,$~$$ |
---|
Two independent events | $~$\bP(B)\; \bP(A \mid B)$~$ |
---|
Two independent events: Square visualization | $$~$
\newcommand{\true}{\text{True}}
\newcommand{\false}{\text{False}}
\newcommand{\bP}{\mathbb{P}}
$~$$ |
---|
Two independent events: Square visualization | $$~$
\newcommand{\true}{\text{True}}
\newcommand{\false}{\text{False}}
\newcommand{\bP}{\mathbb{P}}
$~$$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$\bP(A, B) = \bP(A)\bP(B).$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$\bP(B \mid A) = \bP(B)$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $$~$\bP(A, B) = \bP(A)\bP(B)\ .$~$$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$\bP(A)$~$ |
---|
Two independent events: Square visualization | $~$\bP(\neg A)$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $$~$\bP(A,B) = \bP(A) \bP(B \mid A)\ .$~$$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$\bP(A,B) = \bP(B) \bP( A \mid B)\ .$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$\bP(A)$~$ |
---|
Two independent events: Square visualization | $~$\bP(\neg A)$~$ |
---|
Two independent events: Square visualization | $~$\bP(B)$~$ |
---|
Two independent events: Square visualization | $~$\bP(\neg B)$~$ |
---|
Two independent events: Square visualization | $~$\bP$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$\bP(B \mid A) = \bP(B)$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$\bP(A,B) = \bP(A) \bP(B \mid A)$~$ |
---|
Two independent events: Square visualization | $~$\bP(B \mid A)= \bP(B) = \bP(B \mid \neg A)$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$\neg A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$\bP(B \mid A)$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$\bP(B \mid \neg A)$~$ |
---|
Two independent events: Square visualization | $~$\neg A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$\bP(A,B) = \bP(B) \bP(A \mid B)$~$ |
---|
Two independent events: Square visualization | $~$\bP(A \mid B) = \bP(A) = \bP(A \mid \neg B)$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$t_A,t_B \in \{\true, \false\},$~$ |
---|
Two independent events: Square visualization | $$~$\bP(A = t_A, B= t_B) = \bP(A = t_A)\bP(B = t_B)\ .$~$$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Two independent events: Square visualization | $~$\bP(A, \neg B) = \bP(A = \true, B = \false)$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$\neg B$~$ |
---|
Two independent events: Square visualization | $~$A$~$ |
---|
Two independent events: Square visualization | $~$B$~$ |
---|
Ultimatum Game | $~$q$~$ |
---|
Ultimatum Game | $$~$p = \big ( \frac{\$5}{\$10 - q} \big ) ^ {1.01}$~$$ |
---|
Ultimatum Game | $$~$\begin{array}{r|c|c}
\text{Offer} & \text{Average subject} & \text{High CRT} \\ \hline
\$5 & \$4.8 & \$4.95 \\ \hline
\$4 & \$4.74 & \$5.52 \\ \hline
\$3 & \$3.15 & \$3.57 \\ \hline
\$2 & \$2.16 & \$2.64 \\ \hline
\$1 & \$1.80 & \$2.34
\end{array}$~$$ |
---|
Uncomputability | $~$n$~$ |
---|
Uncomputability | $~$n$~$ |
---|
Uncomputability | $~$n$~$ |
---|
Uncomputability | $~$\mathbb{N}$~$ |
---|
Uncomputability | $~$\mathbb{N}$~$ |
---|
Uncomputability | $~$1,2,3,…$~$ |
---|
Uncomputability | $~$DIAG$~$ |
---|
Uncomputability | $$~$
DIAG(n) = \left\{
\begin{array}{lr}
1\ if\ M_n(n) = 0\\
0\ if\ M_n(n)\mbox{ is undefined or defined and greater than 0}
\end{array}
\right.
$~$$ |
---|
Uncomputability | $~$M_n$~$ |
---|
Uncomputability | $~$n$~$ |
---|
Uncomputability | $~$n$~$ |
---|
Uncomputability | $~$n$~$ |
---|
Uncomputability | $~$n$~$ |
---|
Uncomputability | $~$DIAG$~$ |
---|
Uncomputability | $~$DIAG$~$ |
---|
Uncomputability | $~$HALT$~$ |
---|
Uncomputability | $$~$
HALT(n,x) = \left\{
\begin{array}{lr}
1\ if\ M_n(x)\mbox{ halts}\\
0\ if\ M_n(x)\mbox{ does not halt }
\end{array}
\right.
$~$$ |
---|
Uncomputability | $~$HALT$~$ |
---|
Uncomputability | $~$HALT$~$ |
---|
Uncomputability | $~$HALT$~$ |
---|
Uncomputability | $~$HALT$~$ |
---|
Uncomputability | $~$HALT$~$ |
---|
Uncomputability | $~$HALT$~$ |
---|
Uncomputability | $~$DIAGONAL$~$ |
---|
Uncomputability | $~$DIAGONAL$~$ |
---|
Uncomputability | $~$HALT$~$ |
---|
Uncomputability | $~$PROG$~$ |
---|
Uncomputability | $~$n$~$ |
---|
Uncomputability | $~$AUX$~$ |
---|
Uncomputability | $~$PROG$~$ |
---|
Uncomputability | $~$n$~$ |
---|
Uncomputability | $~$0$~$ |
---|
Uncomputability | $~$PROG$~$ |
---|
Uncomputability | $~$DIAG(\ulcorner AUX\urcorner)==1$~$ |
---|
Uncomputability | $~$PROG$~$ |
---|
Uncomputability | $~$n$~$ |
---|
Uncomputability | $~$HALT$~$ |
---|
Uncomputability | $~$n$~$ |
---|
Uncomputability | $~$M_n$~$ |
---|
Uncomputability | $~$n$~$ |
---|
Uncomputability | $~$\ulcorner prog\urcorner$~$ |
---|
Uncomputability | $~$prog$~$ |
---|
Uncomputability | $~$prog$~$ |
---|
Uncomputability | $~$\ulcorner prog \urcorner$~$ |
---|
Uncomputability | $~$HALT$~$ |
---|
Uncountability | $~$S$~$ |
---|
Uncountability (Math 3) | $~$X$~$ |
---|
Uncountability (Math 3) | $~$X$~$ |
---|
Uncountability (Math 3) | $~$\mathbb{N}$~$ |
---|
Uncountability (Math 3) | $~$X$~$ |
---|
Uncountability (Math 3) | $~$\mathbb{N}$~$ |
---|
Uncountability (Math 3) | $~$\kappa$~$ |
---|
Uncountability (Math 3) | $~$\aleph_0$~$ |
---|
Uncountability (Math 3) | $~$\kappa$~$ |
---|
Uncountability (Math 3) | $~$\aleph_0$~$ |
---|
Uncountability (Math 3) | $~$\aleph_0$~$ |
---|
Uncountability (Math 3) | $~$\kappa$~$ |
---|
Uncountability (Math 3) | $~$\aleph_0$~$ |
---|
Uncountability (Math 3) | $~$M$~$ |
---|
Uncountability (Math 3) | $~$2^\mathbb N_M \in M$~$ |
---|
Uncountability (Math 3) | $~$2^\mathbb N _M$~$ |
---|
Uncountability (Math 3) | $~$f : \mathbb N \to 2^\mathbb N_M$~$ |
---|
Uncountability (Math 3) | $~$M$~$ |
---|
Uncountability (Math 3) | $~$f$~$ |
---|
Uncountability (Math 3) | $~$M$~$ |
---|
Uncountability: Intro (Math 1) | $~$\pi$~$ |
---|
Uncountability: Intro (Math 1) | $~$1 = \frac11$~$ |
---|
Uncountability: Intro (Math 1) | $~$\frac32$~$ |
---|
Uncountability: Intro (Math 1) | $~$\frac{100}{101}$~$ |
---|
Uncountability: Intro (Math 1) | $~$\frac{22}{7}$~$ |
---|
Uncountability: Intro (Math 1) | $~$p / q$~$ |
---|
Uncountability: Intro (Math 1) | $~$(p, q)$~$ |
---|
Uncountability: Intro (Math 1) | $~$\frac01$~$ |
---|
Uncountability: Intro (Math 1) | $~$\frac11$~$ |
---|
Uncountability: Intro (Math 1) | $~$\frac12$~$ |
---|
Uncountability: Intro (Math 1) | $~$\frac{-1}{2}$~$ |
---|
Uncountability: Intro (Math 1) | $~$\frac{-1}{1}$~$ |
---|
Uncountability: Intro (Math 1) | $~$n^\text{th}$~$ |
---|
Uncountability: Intro (Math 1) | $~$n$~$ |
---|
Uncountability: Intro (Math 1) | $~$\pi$~$ |
---|
Uncountability: Intro (Math 1) | $~$n$~$ |
---|
Uncountability: Intro (Math 1) | $~$n^\text{th}$~$ |
---|
Uncountability: Intuitive Intro | $~$\fbox{1}\fbox{2}\fbox{3}\fbox{4}\fbox{5}\fbox{6}\fbox{7}\fbox{8}\overline{\underline{\vphantom{1234567890}\cdots}}$~$ |
---|
Uncountability: Intuitive Intro | $~$n^\text{th}$~$ |
---|
Uncountability: Intuitive Intro | $~$n^\text{th}$~$ |
---|
Uncountability: Intuitive Intro | $~$n^\text{th}$~$ |
---|
Uncountability: Intuitive Intro | $~$n^\text{th}$~$ |
---|
Uncountability: Intuitive Intro | $~$n^\text{th}$~$ |
---|
Uncountability: Intuitive Intro | $~$n^\text{th}$~$ |
---|
Uncountable sample spaces are way too large | $~$\Omega$~$ |
---|
Uncountable sample spaces are way too large | $~$\Omega$~$ |
---|
Uncountable sample spaces are way too large | $~$f: \Omega \to [0,1]$~$ |
---|
Uncountable sample spaces are way too large | $~$\sum_{\omega \in \Omega} f(\omega) = 1$~$ |
---|
Uncountable sample spaces are way too large | $~$\Omega$~$ |
---|
Uncountable sample spaces are way too large | $~$\Omega$~$ |
---|
Uncountable sample spaces are way too large | $~$\Omega$~$ |
---|
Uncountable sample spaces are way too large | $~$f: \Omega \to [0,1]$~$ |
---|
Uncountable sample spaces are way too large | $~$\sum_{\omega \in \Omega} f(\omega) = 1$~$ |
---|
Uncountable sample spaces are way too large | $~$\Omega$~$ |
---|
Uncountable sample spaces are way too large | $~$[0,2]$~$ |
---|
Under a group homomorphism, the image of the inverse is the inverse of the image | $~$f: G \to H$~$ |
---|
Under a group homomorphism, the image of the inverse is the inverse of the image | $~$f(g^{-1}) = f(g)^{-1}$~$ |
---|
Under a group homomorphism, the image of the inverse is the inverse of the image | $~$f(g^{-1}) f(g) = f(g^{-1} g) = f(e_G) = e_H$~$ |
---|
Underlying set | $~$(X, \bullet)$~$ |
---|
Underlying set | $~$X$~$ |
---|
Underlying set | $~$\bullet$~$ |
---|
Underlying set | $~$X$~$ |
---|
Unforeseen maximum | $~$F$~$ |
---|
Unforeseen maximum | $~$X$~$ |
---|
Unforeseen maximum | $~$F$~$ |
---|
Unforeseen maximum | $~$X'$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$X.$~$ |
---|
Unforeseen maximum | $~$X'$~$ |
---|
Unforeseen maximum | $~$X' >_U X,$~$ |
---|
Unforeseen maximum | $~$X'$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$\pi_i \in \Pi_N$~$ |
---|
Unforeseen maximum | $~$\mathbb E [ U | \pi_i ]$~$ |
---|
Unforeseen maximum | $~$\pi_1,$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$\pi_1.$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$\pi_1,$~$ |
---|
Unforeseen maximum | $~$\mathbb E [ V | \pi_1 ] > \mathbb E [ V ]$~$ |
---|
Unforeseen maximum | $~$\pi_1$~$ |
---|
Unforeseen maximum | $~$U.$~$ |
---|
Unforeseen maximum | $~$\Pi_M,$~$ |
---|
Unforeseen maximum | $~$\pi_1$~$ |
---|
Unforeseen maximum | $~$\pi_0$~$ |
---|
Unforeseen maximum | $~$\mathbb E [ U | \pi_0 ] > \mathbb E [ U | \pi_1 ].$~$ |
---|
Unforeseen maximum | $~$\pi_0$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$\pi_1$~$ |
---|
Unforeseen maximum | $$~$\underset{\pi_i \in \Pi_N}{\operatorname {argmax}} \ \mathbb E [ U | \pi_i ] = \pi_1$~$$ |
---|
Unforeseen maximum | $$~$\underset{\pi_k \in \Pi_M}{\operatorname {argmax}} \ \mathbb E [ U | \pi_k ] = \pi_0$~$$ |
---|
Unforeseen maximum | $$~$\mathbb E [ V | \pi_0 ] \ll \mathbb E [ V | \pi_1 ]$~$$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$\pi_1$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$\Pi_N$~$ |
---|
Unforeseen maximum | $~$\pi_1$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$\Pi_M$~$ |
---|
Unforeseen maximum | $~$\pi_1$~$ |
---|
Unforeseen maximum | $~$\pi_0$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$V.$~$ |
---|
Unforeseen maximum | $~$\Pi_L \subset \Pi_M$~$ |
---|
Unforeseen maximum | $~$\pi_0 \not\in \Pi_L.$~$ |
---|
Unforeseen maximum | $~$\Pi_N$~$ |
---|
Unforeseen maximum | $~$\Pi_M$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$\pi_1$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$\pi_0$~$ |
---|
Unforeseen maximum | $~$V,$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$\pi_0$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$\pi_{0.01}$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$\pi_0$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$\Pi$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$V$~$ |
---|
Unforeseen maximum | $~$\pi_0 >_U \pi_1$~$ |
---|
Unforeseen maximum | $~$U$~$ |
---|
Unforeseen maximum | $~$\pi_1.$~$ |
---|
Union | $~$A$~$ |
---|
Union | $~$B$~$ |
---|
Union | $~$A \cup B$~$ |
---|
Union | $~$A$~$ |
---|
Union | $~$B$~$ |
---|
Union | $~$C = A \cup B$~$ |
---|
Union | $$~$x \in C \leftrightarrow (x \in A \lor x \in B)$~$$ |
---|
Union | $~$x$~$ |
---|
Union | $~$C$~$ |
---|
Union | $~$x$~$ |
---|
Union | $~$A$~$ |
---|
Union | $~$B$~$ |
---|
Union | $~$\{1,2\} \cup \{2,3\} = \{1,2,3\}$~$ |
---|
Union | $~$\{1,2\} \cup \{8,9\} = \{1,2,8,9\}$~$ |
---|
Union | $~$\{0,2,4,6\} \cup \{3,4,5,6\} = \{0,2,3,4,5,6\}$~$ |
---|
Union | $~$\mathbb{R^-} \cup \mathbb{R^+} \cup \{0\} = \mathbb{R}$~$ |
---|
Unique factorisation domain | $~$R$~$ |
---|
Unique factorisation domain | $~$R$~$ |
---|
Unique factorisation domain | $~$R$~$ |
---|
Unique factorisation domain | $~$\mathbb{C}$~$ |
---|
Unique factorisation domain | $~$R$~$ |
---|
Unique factorisation domain | $~$R$~$ |
---|
Unique factorisation domain | $~$R$~$ |
---|
Unique factorisation domain | $~$u$~$ |
---|
Unique factorisation domain | $~$u^{-1}$~$ |
---|
Unique factorisation domain | $~$p \times q$~$ |
---|
Unique factorisation domain | $~$(p \times u) \times (q \times u^{-1})$~$ |
---|
Unique factorisation domain | $~$p \times u$~$ |
---|
Unique factorisation domain | $~$p$~$ |
---|
Unique factorisation domain | $~$u$~$ |
---|
Unique factorisation domain | $~$\mathbb{Z}$~$ |
---|
Unique factorisation domain | $~$1$~$ |
---|
Unique factorisation domain | $~$-1$~$ |
---|
Unique factorisation domain | $~$-10 = -1 \times 5 \times 2$~$ |
---|
Unique factorisation domain | $~$-5 \times 2$~$ |
---|
Unique factorisation domain | $~$5 \times -2$~$ |
---|
Unique factorisation domain | $~$5$~$ |
---|
Unique factorisation domain | $~$-5$~$ |
---|
Unique factorisation domain | $~$2$~$ |
---|
Unique factorisation domain | $~$-2$~$ |
---|
Unique factorisation domain | $~$-1$~$ |
---|
Unique factorisation domain | $~$-1$~$ |
---|
Unique factorisation domain | $~$-1 \times 5 \times 2$~$ |
---|
Unique factorisation domain | $~$-1 \times 5 \times 2$~$ |
---|
Unique factorisation domain | $~$-5 \times 2$~$ |
---|
Unique factorisation domain | $~$5 \times -2$~$ |
---|
Unique factorisation domain | $~$\mathbb{Z}$~$ |
---|
Unique factorisation domain | $~$\mathbb{Z}[-\sqrt{3}]$~$ |
---|
Unique factorisation domain | $~$4 = 2 \times 2$~$ |
---|
Unique factorisation domain | $~$(1+\sqrt{-3})(1-\sqrt{-3})$~$ |
---|
Unique factorisation domain | $~$2$~$ |
---|
Unique factorisation domain | $~$1 \pm \sqrt{-3}$~$ |
---|
Unit (ring theory) | $~$x$~$ |
---|
Unit (ring theory) | $~$0 \not = 1$~$ |
---|
Unit (ring theory) | $~$y$~$ |
---|
Unit (ring theory) | $~$xy = 1$~$ |
---|
Unit (ring theory) | $~$0=1$~$ |
---|
Unit (ring theory) | $~$xy = 0$~$ |
---|
Unit (ring theory) | $~$0$~$ |
---|
Unit (ring theory) | $~$0$~$ |
---|
Unit (ring theory) | $~$0 \times 0 = 0 = 1$~$ |
---|
Unit (ring theory) | $~$0$~$ |
---|
Unit (ring theory) | $~$0 \times y = 0$~$ |
---|
Unit (ring theory) | $~$1$~$ |
---|
Unit (ring theory) | $~$y$~$ |
---|
Unit (ring theory) | $~$x$~$ |
---|
Unit (ring theory) | $~$xy = xz = 1$~$ |
---|
Unit (ring theory) | $~$zxy = z$~$ |
---|
Unit (ring theory) | $~$xy=1$~$ |
---|
Unit (ring theory) | $~$z$~$ |
---|
Unit (ring theory) | $~$y = z$~$ |
---|
Unit (ring theory) | $~$zx = 1$~$ |
---|
Unit (ring theory) | $~$\mathbb{Z}$~$ |
---|
Unit (ring theory) | $~$1$~$ |
---|
Unit (ring theory) | $~$-1$~$ |
---|
Unit (ring theory) | $~$1 \times 1 = 1$~$ |
---|
Unit (ring theory) | $~$-1 \times -1 = 1$~$ |
---|
Unit (ring theory) | $~$2$~$ |
---|
Unit (ring theory) | $~$x$~$ |
---|
Unit (ring theory) | $~$2x=1$~$ |
---|
Unit (ring theory) | $~$\pm 1$~$ |
---|
Unit (ring theory) | $~$\mathbb{Q}$~$ |
---|
Unit (ring theory) | $~$0$~$ |
---|
Universal property | $~$\mathbb{Z}$~$ |
---|
Universal property | $~$0$~$ |
---|
Universal property | $~$\mathcal{C}$~$ |
---|
Universal property | $~$\mathcal{C}$~$ |
---|
Universal property | $~$\mathbf{1}$~$ |
---|
Universal property | $~$F$~$ |
---|
Universal property | $~$A$~$ |
---|
Universal property | $~$F$~$ |
---|
Universal property | $~$A$~$ |
---|
Universal property | $~$\mathbb{Q}$~$ |
---|
Universal property | $~$F_2$~$ |
---|
Universal property | $~$F_2$~$ |
---|
Universal property | $~$0$~$ |
---|
Universal property | $~$1$~$ |
---|
Universal property | $~$1 + 1 = 0$~$ |
---|
Universal property | $~$F$~$ |
---|
Universal property | $~$1_F$~$ |
---|
Universal property | $~$f$~$ |
---|
Universal property | $~$F$~$ |
---|
Universal property | $~$F_2$~$ |
---|
Universal property | $~$f$~$ |
---|
Universal property | $~$F^*$~$ |
---|
Universal property | $~$F$~$ |
---|
Universal property | $~$0$~$ |
---|
Universal property | $~$F$~$ |
---|
Universal property | $~$F_2^*$~$ |
---|
Universal property | $~$f(1_F) = 1_{F_2}$~$ |
---|
Universal property | $~$f(1_F + 1_F) = 1_{F_2} + 1_{F_2} = 0_{F_2}$~$ |
---|
Universal property | $~$0$~$ |
---|
Universal property | $~$f(1_F)$~$ |
---|
Universal property | $~$0_{F_2}$~$ |
---|
Universal property | $~$f$~$ |
---|
Universal property | $~$1_F + 1_F$~$ |
---|
Universal property | $~$0_F$~$ |
---|
Universal property | $~$f(1_F + 1_F) = 0_{F_2} = f(0_F)$~$ |
---|
Universal property | $~$\mathbb{Q}$~$ |
---|
Universal property | $~$g$~$ |
---|
Universal property | $~$F$~$ |
---|
Universal property | $~$\mathbb{Q}$~$ |
---|
Universal property | $~$g(1_F + 1_F) = g(1_F) + g(1_F) = 1 + 1 = 2$~$ |
---|
Universal property | $~$g(1_F + 1_F) = g(0_F) = 0$~$ |
---|
Universal property of joins and meets in a poset | $~$\mathbb{N}$~$ |
---|
Universal property of joins and meets in a poset | $~$\{1,2,3,4,5\}$~$ |
---|
Universal property of joins and meets in a poset | $~$P^{\text{op}}$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$i_A : A \to A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$i_B: B \to A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$f_A: A \to X$~$ |
---|
Universal property of the disjoint union | $~$f_B: B \to X$~$ |
---|
Universal property of the disjoint union | $~$\gamma: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$\gamma \circ i_A = f_A$~$ |
---|
Universal property of the disjoint union | $~$\gamma \circ i_B = f_B$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$i_A : A \to A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$i_B: B \to A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$f_A: A \to X$~$ |
---|
Universal property of the disjoint union | $~$f_B: B \to X$~$ |
---|
Universal property of the disjoint union | $~$\gamma: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$\gamma \circ i_A = f_A$~$ |
---|
Universal property of the disjoint union | $~$\gamma \circ i_B = f_B$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$A'$~$ |
---|
Universal property of the disjoint union | $~$B'$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$A'$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$B'$~$ |
---|
Universal property of the disjoint union | $~$A \cup B$~$ |
---|
Universal property of the disjoint union | $~$A' \cup B'$~$ |
---|
Universal property of the disjoint union | $~$A = \{ 1 \}$~$ |
---|
Universal property of the disjoint union | $~$B = \{ 1 \}$~$ |
---|
Universal property of the disjoint union | $~$A \cup B$~$ |
---|
Universal property of the disjoint union | $~$\{1\}$~$ |
---|
Universal property of the disjoint union | $~$X = \{1\}$~$ |
---|
Universal property of the disjoint union | $~$Y = \{2\}$~$ |
---|
Universal property of the disjoint union | $~$X \cup Y = \{1,2\}$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$Y$~$ |
---|
Universal property of the disjoint union | $~$\{1,2\}$~$ |
---|
Universal property of the disjoint union | $~$\{1\} = A \cup B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$\{ 2, 3, 5 \} \cup \{ 2, 6, 7 \}$~$ |
---|
Universal property of the disjoint union | $~$\{ 2, 3, 5 \} \sqcup \{ 2, 6, 7 \}$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B = A' \cup B'$~$ |
---|
Universal property of the disjoint union | $~$A' = \{ (a, 1) : a \in A \}$~$ |
---|
Universal property of the disjoint union | $~$B' = \{ (b, 2) : b \in B \}$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$1$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$2$~$ |
---|
Universal property of the disjoint union | $~$A \cong A'$~$ |
---|
Universal property of the disjoint union | $~$B \cong B'$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B \cong A' \sqcup B'$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$A \to A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$B \to A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$i_A : a \mapsto a$~$ |
---|
Universal property of the disjoint union | $~$i_B : b \mapsto b$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A = \{ 1 \}$~$ |
---|
Universal property of the disjoint union | $~$B = \{ 2 \}$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B = \{1,2\}$~$ |
---|
Universal property of the disjoint union | $~$\{1,2,3\}$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$f: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$f \big|_A : A \to X$~$ |
---|
Universal property of the disjoint union | $~$f \big|_A (a) = f(a)$~$ |
---|
Universal property of the disjoint union | $~$f$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$f \big|_B : B \to X$~$ |
---|
Universal property of the disjoint union | $~$\alpha: A \to X$~$ |
---|
Universal property of the disjoint union | $~$\beta: B \to X$~$ |
---|
Universal property of the disjoint union | $~$f: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$\alpha$~$ |
---|
Universal property of the disjoint union | $~$\beta$~$ |
---|
Universal property of the disjoint union | $~$f(x) = \alpha(x)$~$ |
---|
Universal property of the disjoint union | $~$x \in A$~$ |
---|
Universal property of the disjoint union | $~$f(x) = \beta(x)$~$ |
---|
Universal property of the disjoint union | $~$x \in B$~$ |
---|
Universal property of the disjoint union | $~$f$~$ |
---|
Universal property of the disjoint union | $~$\alpha$~$ |
---|
Universal property of the disjoint union | $~$\beta$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$f$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$x$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$f(x)$~$ |
---|
Universal property of the disjoint union | $~$\alpha(x)$~$ |
---|
Universal property of the disjoint union | $~$\beta(x)$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $$~$\text{A set labelled }\ A \sqcup B \\ i_A : A \to A \sqcup B \\ i_B : B \to A \sqcup B$~$$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$f_A: A \to X$~$ |
---|
Universal property of the disjoint union | $~$f_B: B \to X$~$ |
---|
Universal property of the disjoint union | $~$f: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$f \circ i_A = f_A$~$ |
---|
Universal property of the disjoint union | $~$f \circ i_B = f_B$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $$~$A \times B \\ \pi_A: A \times B \to A \\ \pi_B : A \times B \to B$~$$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$f_A: X \to A, f_B: X \to B$~$ |
---|
Universal property of the disjoint union | $~$f: X \to A \times B$~$ |
---|
Universal property of the disjoint union | $~$\pi_A \circ f = f_A$~$ |
---|
Universal property of the disjoint union | $~$\pi_B \circ f = f_B$~$ |
---|
Universal property of the disjoint union | $~$A=\{1\}$~$ |
---|
Universal property of the disjoint union | $~$B=\{2\}$~$ |
---|
Universal property of the disjoint union | $~$\{1,2\}$~$ |
---|
Universal property of the disjoint union | $~$\{1\}$~$ |
---|
Universal property of the disjoint union | $~$\{2\}$~$ |
---|
Universal property of the disjoint union | $$~$\text{A set labelled } \{1\} \sqcup \{2\} \\ i_A : \{1\} \to \{1\} \sqcup \{2\} \\ i_B : \{2\} \to \{1\} \sqcup \{2\}$~$$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$f_A: \{1\} \to X$~$ |
---|
Universal property of the disjoint union | $~$f_B: \{2\} \to X$~$ |
---|
Universal property of the disjoint union | $~$f: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$f \circ i_A = f_A$~$ |
---|
Universal property of the disjoint union | $~$f \circ i_B = f_B$~$ |
---|
Universal property of the disjoint union | $~$f_A: \{1\} \to X$~$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$f_A(1)$~$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$\{1\} \to X$~$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$f_B$~$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$i_A$~$ |
---|
Universal property of the disjoint union | $~$i_B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$\{1\}$~$ |
---|
Universal property of the disjoint union | $~$\{2\}$~$ |
---|
Universal property of the disjoint union | $$~$\text{A set labelled }\ \{1\} \sqcup \{2\} \\ \text{An element } i_A \text{ of } \{1\} \sqcup \{2\} \\ \text{An element }i_B \text{ of } \{1\} \sqcup \{2\}$~$$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$x \in X$~$ |
---|
Universal property of the disjoint union | $~$y \in X$~$ |
---|
Universal property of the disjoint union | $~$f: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$f(i_A) = x$~$ |
---|
Universal property of the disjoint union | $~$f(i_B) = y$~$ |
---|
Universal property of the disjoint union | $~$i_A$~$ |
---|
Universal property of the disjoint union | $~$i_B$~$ |
---|
Universal property of the disjoint union | $~$f$~$ |
---|
Universal property of the disjoint union | $~$X = \{x,y\}$~$ |
---|
Universal property of the disjoint union | $~$x \not = y$~$ |
---|
Universal property of the disjoint union | $~$z$~$ |
---|
Universal property of the disjoint union | $~$\{1\} \sqcup \{2\}$~$ |
---|
Universal property of the disjoint union | $~$i_A$~$ |
---|
Universal property of the disjoint union | $~$i_B$~$ |
---|
Universal property of the disjoint union | $~$f: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$f$~$ |
---|
Universal property of the disjoint union | $~$z$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$\{i_A, i_B\}$~$ |
---|
Universal property of the disjoint union | $~$m$~$ |
---|
Universal property of the disjoint union | $~$n$~$ |
---|
Universal property of the disjoint union | $~$m+n$~$ |
---|
Universal property of the disjoint union | $~$\emptyset$~$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$\emptyset \to X$~$ |
---|
Universal property of the disjoint union | $~$\emptyset$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$\emptyset$~$ |
---|
Universal property of the disjoint union | $$~$\text{A set labelled }\ A \sqcup B \\ i_A : \emptyset \to A \sqcup B \\ i_B : \emptyset \to A \sqcup B$~$$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$f_A: \emptyset \to X$~$ |
---|
Universal property of the disjoint union | $~$f_B: \emptyset \to X$~$ |
---|
Universal property of the disjoint union | $~$f: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$f \circ i_A = f_A$~$ |
---|
Universal property of the disjoint union | $~$f \circ i_B = f_B$~$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$!_X: \emptyset \to X$~$ |
---|
Universal property of the disjoint union | $~$!$~$ |
---|
Universal property of the disjoint union | $~$f_A$~$ |
---|
Universal property of the disjoint union | $~$f_B$~$ |
---|
Universal property of the disjoint union | $~$!_X$~$ |
---|
Universal property of the disjoint union | $$~$\text{A set labelled }\ A \sqcup B \\ i_A : \emptyset \to A \sqcup B \\ i_B : \emptyset \to A \sqcup B$~$$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$f: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$f \circ i_A = (!_{X})$~$ |
---|
Universal property of the disjoint union | $~$f \circ i_B = (!_{X})$~$ |
---|
Universal property of the disjoint union | $~$i_A$~$ |
---|
Universal property of the disjoint union | $~$i_B$~$ |
---|
Universal property of the disjoint union | $~$\emptyset \to A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$!_{A \sqcup B}$~$ |
---|
Universal property of the disjoint union | $$~$A \sqcup B$~$$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$f: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$f \circ (!_{A \sqcup B}) = (!_X)$~$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$f: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$f \circ (!_{A \sqcup B})$~$ |
---|
Universal property of the disjoint union | $~$!_X : \emptyset \to X$~$ |
---|
Universal property of the disjoint union | $~$!_X$~$ |
---|
Universal property of the disjoint union | $~$\emptyset \to X$~$ |
---|
Universal property of the disjoint union | $~$f \circ (!_{A \sqcup B}) = (!_X)$~$ |
---|
Universal property of the disjoint union | $~$f$~$ |
---|
Universal property of the disjoint union | $~$f \circ (!_{A \sqcup B})$~$ |
---|
Universal property of the disjoint union | $~$\emptyset \to X$~$ |
---|
Universal property of the disjoint union | $~$!_X$~$ |
---|
Universal property of the disjoint union | $$~$A \sqcup B$~$$ |
---|
Universal property of the disjoint union | $~$X$~$ |
---|
Universal property of the disjoint union | $~$f: A \sqcup B \to X$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A$~$ |
---|
Universal property of the disjoint union | $~$B$~$ |
---|
Universal property of the disjoint union | $~$A \sqcup B$~$ |
---|
Universal property of the disjoint union | $~$A + B$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$f: \emptyset \to X$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$f: A \to B$~$ |
---|
Universal property of the empty set | $~$(a, f(a))$~$ |
---|
Universal property of the empty set | $~$a$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$f(a)$~$ |
---|
Universal property of the empty set | $~$B$~$ |
---|
Universal property of the empty set | $~$B$~$ |
---|
Universal property of the empty set | $~$f$~$ |
---|
Universal property of the empty set | $~$\{ (0,1), (1,2), (2,3), (3,4), \dots \}$~$ |
---|
Universal property of the empty set | $~$f: \mathbb{N} \to \mathbb{N}$~$ |
---|
Universal property of the empty set | $~$n \mapsto n+1$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$(0,1,2)$~$ |
---|
Universal property of the empty set | $~$f$~$ |
---|
Universal property of the empty set | $~$\{ (0,1), (0,2) \}$~$ |
---|
Universal property of the empty set | $~$0$~$ |
---|
Universal property of the empty set | $~$1$~$ |
---|
Universal property of the empty set | $~$2$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$A = \{ 1 \}$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\{1\}$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\{1\}$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\{1\}$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\{1\}$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$f: \emptyset \to A$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$f$~$ |
---|
Universal property of the empty set | $~$f$~$ |
---|
Universal property of the empty set | $~$f$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\{1\}$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$\{1\}$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$\{ a, b \}$~$ |
---|
Universal property of the empty set | $~$\{1\}$~$ |
---|
Universal property of the empty set | $~$\{a,b\}$~$ |
---|
Universal property of the empty set | $~$f: 1 \mapsto a$~$ |
---|
Universal property of the empty set | $~$g: 1 \mapsto b$~$ |
---|
Universal property of the empty set | $~$\{1\}$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$\{1\} \to X$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$f: \emptyset \to X$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$g: X \to \emptyset$~$ |
---|
Universal property of the empty set | $~$\mathrm{id}: \emptyset \to \emptyset$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$f$~$ |
---|
Universal property of the empty set | $~$g$~$ |
---|
Universal property of the empty set | $~$f$~$ |
---|
Universal property of the empty set | $~$g$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$f$~$ |
---|
Universal property of the empty set | $~$\emptyset$~$ |
---|
Universal property of the empty set | $~$g$~$ |
---|
Universal property of the empty set | $~$\emptyset \to \emptyset$~$ |
---|
Universal property of the empty set | $~$g \circ f$~$ |
---|
Universal property of the empty set | $~$\mathrm{id}$~$ |
---|
Universal property of the empty set | $~$f$~$ |
---|
Universal property of the empty set | $~$g$~$ |
---|
Universal property of the empty set | $~$f \circ g$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$\mathrm{id}_X : X \to X$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$Y$~$ |
---|
Universal property of the empty set | $~$f \circ g$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$\mathrm{id}_X$~$ |
---|
Universal property of the empty set | $~$f$~$ |
---|
Universal property of the empty set | $~$g$~$ |
---|
Universal property of the empty set | $~$\emptyset \to X$~$ |
---|
Universal property of the empty set | $~$X \to \emptyset$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$G$~$ |
---|
Universal property of the empty set | $~$H$~$ |
---|
Universal property of the empty set | $~$H$~$ |
---|
Universal property of the empty set | $~$G$~$ |
---|
Universal property of the empty set | $~$R$~$ |
---|
Universal property of the empty set | $~$S$~$ |
---|
Universal property of the empty set | $~$R$~$ |
---|
Universal property of the empty set | $~$S$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$Y$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$Y$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$A$~$ |
---|
Universal property of the empty set | $~$X$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$A \times B$~$ |
---|
Universal property of the product | $~$\pi_A : A \times B \to A$~$ |
---|
Universal property of the product | $~$\pi_B: A \times B \to B$~$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$f_A: X \to A$~$ |
---|
Universal property of the product | $~$f_B: X \to B$~$ |
---|
Universal property of the product | $~$\gamma: X \to A \times B$~$ |
---|
Universal property of the product | $~$\pi_A \circ \gamma = f_A$~$ |
---|
Universal property of the product | $~$\pi_B \circ \gamma = f_B$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$(P, \pi_A, \pi_B)$~$ |
---|
Universal property of the product | $~$\pi_A$~$ |
---|
Universal property of the product | $~$P \to A$~$ |
---|
Universal property of the product | $~$\pi_B$~$ |
---|
Universal property of the product | $~$P \to B$~$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$f: X \to A, g: X \to B$~$ |
---|
Universal property of the product | $~$\gamma: X \to P$~$ |
---|
Universal property of the product | $~$\pi_A \gamma = f$~$ |
---|
Universal property of the product | $~$\pi_B \gamma = g$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$A \times B$~$ |
---|
Universal property of the product | $~$(a,b) \in A \times B$~$ |
---|
Universal property of the product | $~$a$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$\pi_A: A \times B \to A$~$ |
---|
Universal property of the product | $~$\pi_B : A \times B \to B$~$ |
---|
Universal property of the product | $~$\pi$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$f$~$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$A \times B$~$ |
---|
Universal property of the product | $~$\pi_A(f(x))$~$ |
---|
Universal property of the product | $~$\pi_B(f(x))$~$ |
---|
Universal property of the product | $~$f(x)$~$ |
---|
Universal property of the product | $~$\langle \pi_A(f(x)), \pi_B(f(x)) \rangle$~$ |
---|
Universal property of the product | $~$\langle \text{angled brackets}\rangle$~$ |
---|
Universal property of the product | $~$h : A \to B \times C$~$ |
---|
Universal property of the product | $~$f : A \to B$~$ |
---|
Universal property of the product | $~$g : A \to C$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B \times C$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$C$~$ |
---|
Universal property of the product | $~$h$~$ |
---|
Universal property of the product | $~$B \times C$~$ |
---|
Universal property of the product | $~$f$~$ |
---|
Universal property of the product | $~$B \times C \to B$~$ |
---|
Universal property of the product | $~$g$~$ |
---|
Universal property of the product | $~$B \times C \to C$~$ |
---|
Universal property of the product | $~$h$~$ |
---|
Universal property of the product | $~$f$~$ |
---|
Universal property of the product | $~$g$~$ |
---|
Universal property of the product | $~$h = \langle f, g\rangle$~$ |
---|
Universal property of the product | $~$\pi_{B} \langle f, g \rangle = f$~$ |
---|
Universal property of the product | $~$\pi_C \langle f, g \rangle = g$~$ |
---|
Universal property of the product | $~$\langle \pi_{B}, \pi_{C} \rangle = \text{id}$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $$~$A \times B \\ \pi_A: A \times B \to A \\ \pi_B : A \times B \to B$~$$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$f_A: X \to A, f_B: X \to B$~$ |
---|
Universal property of the product | $~$f: X \to A \times B$~$ |
---|
Universal property of the product | $~$\pi_A \circ f = f_A$~$ |
---|
Universal property of the product | $~$\pi_B \circ f = f_B$~$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$f_A$~$ |
---|
Universal property of the product | $~$f_B$~$ |
---|
Universal property of the product | $~$\pi_A \circ f = f_A$~$ |
---|
Universal property of the product | $~$\pi_B \circ f = f_B$~$ |
---|
Universal property of the product | $~$f: X \to A \times B$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$A \otimes B$~$ |
---|
Universal property of the product | $~$\times$~$ |
---|
Universal property of the product | $~$\otimes$~$ |
---|
Universal property of the product | $~$\otimes$~$ |
---|
Universal property of the product | $~$\times$~$ |
---|
Universal property of the product | $~$A \otimes B$~$ |
---|
Universal property of the product | $~$\pi_A: A \otimes B \to A$~$ |
---|
Universal property of the product | $~$\pi_B: A \otimes B \to B$~$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$f_A: X \to A, f_B: X \to B$~$ |
---|
Universal property of the product | $~$f: X \to A \otimes B$~$ |
---|
Universal property of the product | $~$\pi_A \circ f = f_A$~$ |
---|
Universal property of the product | $~$\pi_B \circ f = f_B$~$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$X = \emptyset$~$ |
---|
Universal property of the product | $~$A \otimes B$~$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$X = \{ 1 \}$~$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$f_A: X \to A$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$g: \{1\} \to A$~$ |
---|
Universal property of the product | $~$g(1)$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$1$~$ |
---|
Universal property of the product | $~$g$~$ |
---|
Universal property of the product | $~$g(1)$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$f_A: \{ 1 \} \to A$~$ |
---|
Universal property of the product | $~$f_B : \{ 1 \} \to B$~$ |
---|
Universal property of the product | $~$f: \{1\} \to A \otimes B$~$ |
---|
Universal property of the product | $~$\pi_A \circ f = f_A$~$ |
---|
Universal property of the product | $~$\pi_B \circ f = f_B$~$ |
---|
Universal property of the product | $~$\pi_A \circ f$~$ |
---|
Universal property of the product | $~$\{1\}$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$\pi_A$~$ |
---|
Universal property of the product | $~$f$~$ |
---|
Universal property of the product | $~$a \in A$~$ |
---|
Universal property of the product | $~$b \in B$~$ |
---|
Universal property of the product | $~$x \in A \otimes B$~$ |
---|
Universal property of the product | $~$\pi_A(x) = a$~$ |
---|
Universal property of the product | $~$\pi_B(x) = b$~$ |
---|
Universal property of the product | $~$A \otimes B$~$ |
---|
Universal property of the product | $~$\pi_A$~$ |
---|
Universal property of the product | $~$\pi_B$~$ |
---|
Universal property of the product | $~$A \times B$~$ |
---|
Universal property of the product | $~$A \otimes B$~$ |
---|
Universal property of the product | $~$\pi_A$~$ |
---|
Universal property of the product | $~$\pi_B$~$ |
---|
Universal property of the product | $~$a \to b$~$ |
---|
Universal property of the product | $~$a \leq b$~$ |
---|
Universal property of the product | $~$f: A \to B$~$ |
---|
Universal property of the product | $~$A \leq B$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$A \leq B$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$\mathbb{N}$~$ |
---|
Universal property of the product | $~$1$~$ |
---|
Universal property of the product | $~$0$~$ |
---|
Universal property of the product | $~$1$~$ |
---|
Universal property of the product | $~$0$~$ |
---|
Universal property of the product | $~$2$~$ |
---|
Universal property of the product | $~$2$~$ |
---|
Universal property of the product | $~$0$~$ |
---|
Universal property of the product | $~$m$~$ |
---|
Universal property of the product | $~$n$~$ |
---|
Universal property of the product | $~$\otimes$~$ |
---|
Universal property of the product | $~$\times$~$ |
---|
Universal property of the product | $~$\otimes$~$ |
---|
Universal property of the product | $~$m \otimes n$~$ |
---|
Universal property of the product | $~$m \otimes n$~$ |
---|
Universal property of the product | $~$m$~$ |
---|
Universal property of the product | $~$\pi_A$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$A \leq B$~$ |
---|
Universal property of the product | $~$m \otimes n$~$ |
---|
Universal property of the product | $~$n$~$ |
---|
Universal property of the product | $~$\pi_B$~$ |
---|
Universal property of the product | $~$x$~$ |
---|
Universal property of the product | $~$x \leq m, x \leq n$~$ |
---|
Universal property of the product | $~$f_A: X \to A$~$ |
---|
Universal property of the product | $~$f_B: X \to B$~$ |
---|
Universal property of the product | $~$x \leq m \otimes n$~$ |
---|
Universal property of the product | $~$f: X \to A \times B$~$ |
---|
Universal property of the product | $~$x \leq m \otimes n$~$ |
---|
Universal property of the product | $~$\pi_A \circ f = f_A$~$ |
---|
Universal property of the product | $~$m \otimes n \leq m$~$ |
---|
Universal property of the product | $~$x \leq m \otimes n$~$ |
---|
Universal property of the product | $~$x \leq m$~$ |
---|
Universal property of the product | $~$A \leq C$~$ |
---|
Universal property of the product | $~$A \leq B$~$ |
---|
Universal property of the product | $~$B \leq C$~$ |
---|
Universal property of the product | $~$m \otimes n$~$ |
---|
Universal property of the product | $~$m$~$ |
---|
Universal property of the product | $~$n$~$ |
---|
Universal property of the product | $~$x$~$ |
---|
Universal property of the product | $~$x \leq m, x \leq n$~$ |
---|
Universal property of the product | $~$x \leq m \otimes n$~$ |
---|
Universal property of the product | $~$m$~$ |
---|
Universal property of the product | $~$n$~$ |
---|
Universal property of the product | $~$(A \times B, \pi_A, \pi_B)$~$ |
---|
Universal property of the product | $~$m, n$~$ |
---|
Universal property of the product | $~$(m \otimes n, \pi_m, \pi_n)$~$ |
---|
Universal property of the product | $~$m \otimes n$~$ |
---|
Universal property of the product | $~$m$~$ |
---|
Universal property of the product | $~$n$~$ |
---|
Universal property of the product | $~$\pi_m$~$ |
---|
Universal property of the product | $~$\pi_n$~$ |
---|
Universal property of the product | $~$m \otimes n \leq m$~$ |
---|
Universal property of the product | $~$m \otimes n \leq n$~$ |
---|
Universal property of the product | $~$\mathbb{N}^{\geq 1}$~$ |
---|
Universal property of the product | $~$a \mid b$~$ |
---|
Universal property of the product | $~$a$~$ |
---|
Universal property of the product | $~$b$~$ |
---|
Universal property of the product | $~$1$~$ |
---|
Universal property of the product | $~$1 \to n$~$ |
---|
Universal property of the product | $~$n$~$ |
---|
Universal property of the product | $~$2 \to n$~$ |
---|
Universal property of the product | $~$n$~$ |
---|
Universal property of the product | $~$m$~$ |
---|
Universal property of the product | $~$n$~$ |
---|
Universal property of the product | $~$\otimes$~$ |
---|
Universal property of the product | $~$m \otimes n$~$ |
---|
Universal property of the product | $~$m \otimes n$~$ |
---|
Universal property of the product | $~$m$~$ |
---|
Universal property of the product | $~$\pi_A$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$A \mid B$~$ |
---|
Universal property of the product | $~$m \otimes n$~$ |
---|
Universal property of the product | $~$n$~$ |
---|
Universal property of the product | $~$x$~$ |
---|
Universal property of the product | $~$x \mid m, x \mid n$~$ |
---|
Universal property of the product | $~$x \mid m \otimes n$~$ |
---|
Universal property of the product | $~$a$~$ |
---|
Universal property of the product | $~$a \leq m$~$ |
---|
Universal property of the product | $~$a\leq n$~$ |
---|
Universal property of the product | $~$a$~$ |
---|
Universal property of the product | $~$a \mid m$~$ |
---|
Universal property of the product | $~$a \mid n$~$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$Y$~$ |
---|
Universal property of the product | $~$X$~$ |
---|
Universal property of the product | $~$Y$~$ |
---|
Universal property of the product | $~$A$~$ |
---|
Universal property of the product | $~$B$~$ |
---|
Universal property of the product | $~$m$~$ |
---|
Universal property of the product | $~$n$~$ |
---|
Universal property of the product | $~$m$~$ |
---|
Universal property of the product | $~$n$~$ |
---|
Unphysically large finite computer | $~$10^{100}$~$ |
---|
Unphysically large finite computer | $~$10^{10^{100}}$~$ |
---|
Unphysically large finite computer | $~$9 \uparrow\uparrow 4$~$ |
---|
Unphysically large finite computer | $~$9^{9^{9^9}}$~$ |
---|
Unphysically large finite computer | $~$9 \uparrow\uparrow 4$~$ |
---|
Up to isomorphism | $~$P$~$ |
---|
Up to isomorphism | $~$X$~$ |
---|
Up to isomorphism | $~$P$~$ |
---|
Up to isomorphism | $~$X$~$ |
---|
Up to isomorphism | $~$P$~$ |
---|
Up to isomorphism | $~$X$~$ |
---|
Up to isomorphism | $~$X$~$ |
---|
Up to isomorphism | $~$X$~$ |
---|
Up to isomorphism | $~$X$~$ |
---|
Up to isomorphism | $~$P$~$ |
---|
Up to isomorphism | $~$X$~$ |
---|
Up to isomorphism | $~$P$~$ |
---|
Up to isomorphism | $~$X$~$ |
---|
Up to isomorphism | $~$P$~$ |
---|
Up to isomorphism | $~$X$~$ |
---|
Up to isomorphism | $~$X$~$ |
---|
Up to isomorphism | $~$X$~$ |
---|
Up to isomorphism | $~$X$~$ |
---|
Up to isomorphism | $~$2$~$ |
---|
Up to isomorphism | $~$2$~$ |
---|
Up to isomorphism | $~$2$~$ |
---|
Up to isomorphism | $~$2$~$ |
---|
Up to isomorphism | $~$\{0,1\}$~$ |
---|
Up to isomorphism | $~$2$~$ |
---|
Up to isomorphism | $~$\{e, x \}$~$ |
---|
Up to isomorphism | $~$e$~$ |
---|
Up to isomorphism | $~$x^2 = e$~$ |
---|
Up to isomorphism | $~$2$~$ |
---|
User maximization | $~$X$~$ |
---|
User maximization | $~$X$~$ |
---|
User maximization | $~$X$~$ |
---|
User maximization | $~$p$~$ |
---|
User maximization | $~$X$~$ |
---|
User maximization | $~$X$~$ |
---|
Utility function | $$~$\begin{array}{rl}
0.5 \cdot €0 + 0.5 \cdot €8 \ &= \ €4 \\
1.0 \cdot €5 \ &= \ €5 \\
0.3 \cdot €0 + 0.7 \cdot €8 \ &= \ €5.6
\end{array}$~$$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$S$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$S$~$ |
---|
Utility indifference | $~$P$~$ |
---|
Utility indifference | $~$U$~$ |
---|
Utility indifference | $~$U$~$ |
---|
Utility indifference | $~$P$~$ |
---|
Utility indifference | $~$P'$~$ |
---|
Utility indifference | $~$U,$~$ |
---|
Utility indifference | $~$U$~$ |
---|
Utility indifference | $~$P^*$~$ |
---|
Utility indifference | $~$U^*$~$ |
---|
Utility indifference | $~$U^*$~$ |
---|
Utility indifference | $~$P^*$~$ |
---|
Utility indifference | $~$\pi_1$~$ |
---|
Utility indifference | $~$\mathbb E [U_{normal}|\pi_1],$~$ |
---|
Utility indifference | $~$\pi_2$~$ |
---|
Utility indifference | $~$\mathbb E[U_{suspend}|\pi_2].$~$ |
---|
Utility indifference | $~$U_C$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$\mathcal S$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$\mathcal S$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$\mathcal S$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_I$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$\mathcal{O}: \mathcal{S} \times \mathcal{E}$~$ |
---|
Utility indifference | $~$\mathcal{O}$~$ |
---|
Utility indifference | $~$\mathcal S$~$ |
---|
Utility indifference | $~$\mathcal{E}$~$ |
---|
Utility indifference | $~$s \in \mathcal{S}$~$ |
---|
Utility indifference | $~$\neg s \in \mathcal{S}$~$ |
---|
Utility indifference | $~$o \in \mathcal{O}$~$ |
---|
Utility indifference | $~$o.s$~$ |
---|
Utility indifference | $~$s$~$ |
---|
Utility indifference | $~$o$~$ |
---|
Utility indifference | $~$\neg o.s.$~$ |
---|
Utility indifference | $~$\mathcal{U}: \mathcal{O} \to \mathbb{R}$~$ |
---|
Utility indifference | $~$U_X \in \mathcal{U}$~$ |
---|
Utility indifference | $~$U_Y \in \mathcal{U}$~$ |
---|
Utility indifference | $~$\mathcal S.$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$\mathcal S$~$ |
---|
Utility indifference | $~$\mathcal A$~$ |
---|
Utility indifference | $~$a \in \mathcal A.$~$ |
---|
Utility indifference | $~$\mathbb P(\mathcal O | \mathcal A).$~$ |
---|
Utility indifference | $~$\mathbb E[\mathcal O|a],$~$ |
---|
Utility indifference | $~$\mathbb E[U|a].$~$ |
---|
Utility indifference | $~$\mathbb P[a \ \square \! \! \rightarrow \mathcal O$~$ |
---|
Utility indifference | $~$a$~$ |
---|
Utility indifference | $$~$\underset{a \in \mathcal A}{argmax} \ \mathbb E [U|a]$~$$ |
---|
Utility indifference | $~$U_1$~$ |
---|
Utility indifference | $$~$U_1(o): \begin{cases}
U_X(o) & \neg o.s \\
U_Y(o) & o.s
\end{cases}$~$$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $$~$\max_{a \in \mathcal A} \mathbb E[U_X|a] \ \neq \ \max_{a \in \mathcal A} \mathbb E[U_Y|a]$~$$ |
---|
Utility indifference | $~$\mathcal S.$~$ |
---|
Utility indifference | $$~$U_2(o): \begin{cases}
U_X(o) & \neg o.s \\
U_Y(o) + \theta & o.s
\end{cases}$~$$ |
---|
Utility indifference | $$~$\theta := \max_{a \in \mathcal A} \mathbb E[U_X|a] - \max_{a \in \mathcal A} \mathbb E[U_Y|a]$~$$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$U_2$~$ |
---|
Utility indifference | $~$\theta$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_Y.$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $$~$\max_{a \in \mathcal A} (\mathbb E[U_Y|a] + \theta) \ = \ \max_a{a \in \mathcal A} \mathbb E[U_x|a]$~$$ |
---|
Utility indifference | $~$\theta$~$ |
---|
Utility indifference | $~$U_Y,$~$ |
---|
Utility indifference | $~$\theta.$~$ |
---|
Utility indifference | $~$\theta$~$ |
---|
Utility indifference | $~$\theta$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$\theta.$~$ |
---|
Utility indifference | $~$\theta$~$ |
---|
Utility indifference | $~$\theta$~$ |
---|
Utility indifference | $~$\underset{a \in \mathcal A}{max} \ \mathbb E[U_Y|a]$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_Y.$~$ |
---|
Utility indifference | $~$\theta$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_Y,$~$ |
---|
Utility indifference | $~$\mathbb P(\mathcal S)$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$\mathbb P(\mathcal S).$~$ |
---|
Utility indifference | $~$t$~$ |
---|
Utility indifference | $~$a$~$ |
---|
Utility indifference | $$~$\mathbb E_t[U_3|a] = 0.75 \cdot \mathbb E_t[U_X|a \wedge \neg s] \ + \ 0.25 \cdot \mathbb E_t[U_Y|a \wedge s]$~$$ |
---|
Utility indifference | $~$s$~$ |
---|
Utility indifference | $~$s$~$ |
---|
Utility indifference | $~$U_X.$~$ |
---|
Utility indifference | $~$0.75 \cdot \mathbb E_t[U_X|a \wedge \neg s]$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$U_1$~$ |
---|
Utility indifference | $~$\mathbb P(asteroid) = 0.99,$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$U_X,$~$ |
---|
Utility indifference | $~$\mathbb P(\neg s)$~$ |
---|
Utility indifference | $~$\mathbb P(s)$~$ |
---|
Utility indifference | $~$U_3$~$ |
---|
Utility indifference | $~$\mathbb P(s)$~$ |
---|
Utility indifference | $~$\mathbb P(asteroid)$~$ |
---|
Utility indifference | $~$\mathbb P(\neg asteroid)$~$ |
---|
Utility indifference | $~$asteroid$~$ |
---|
Utility indifference | $~$s.$~$ |
---|
Utility indifference | $~$\mathbb P(s)$~$ |
---|
Utility indifference | $~$\mathbb P(\neg s)$~$ |
---|
Utility indifference | $~$U_3,$~$ |
---|
Utility indifference | $~$\mathbb P(s)$~$ |
---|
Utility indifference | $~$a_0$~$ |
---|
Utility indifference | $~$q$~$ |
---|
Utility indifference | $~$\mathbb P(s) = q,$~$ |
---|
Utility indifference | $~$\mathbb P(s|a_0) = q.$~$ |
---|
Utility indifference | $~$\mathcal A$~$ |
---|
Utility indifference | $~$a_0 \in \mathcal A$~$ |
---|
Utility indifference | $$~$a_0 \in \underset{a' \in \mathcal A}{argmax} \ \big ( \mathbb E[U_X|\neg s,a'] \mathbb P(\neg s|a_0) + \mathbb E[U_Y|s,a'] \mathbb P(s|a_0) \big )$~$$ |
---|
Utility indifference | $~$a_0$~$ |
---|
Utility indifference | $~$U_X$~$ |
---|
Utility indifference | $~$\neg s$~$ |
---|
Utility indifference | $~$a_0$~$ |
---|
Utility indifference | $~$U_Y$~$ |
---|
Utility indifference | $~$s$~$ |
---|
Utility indifference | $~$a_0.$~$ |
---|
Utility indifference | $~$do()$~$ |
---|
Utility indifference | $$~$a_0 \in \underset{a' \in \mathcal A}{argmax} \ \big ( \mathbb E[U_X|do(\neg s),a'] \mathbb P(\neg s|a_0) + \mathbb E[U_Y|do(s),a'] \mathbb P(s|a_0) \big )$~$$ |
---|
Utility indifference | $~$\mathcal S,$~$ |
---|
Utility indifference | $~$\mathcal S$~$ |
---|
Utility indifference | $~$x,$~$ |
---|
Utility indifference | $~$x,$~$ |
---|
Utility indifference | $~$x$~$ |
---|
Utility indifference | $~$x$~$ |
---|
Value-laden | $~$X_1, X_2, X_3…$~$ |
---|
Value-laden | $~$x_1, x_2, x_3$~$ |
---|
Value-laden | $~$x_1^\prime, x_2^\prime, x_3^\prime$~$ |
---|
Value-laden | $~$X_i$~$ |
---|
Value-laden | $~$X_k$~$ |
---|
Value-laden | $~$X_h$~$ |
---|
Vector arithmetic | $~$a$~$ |
---|
Vector arithmetic | $~$b$~$ |
---|
Vector arithmetic | $~$\textbf{v}$~$ |
---|
Vector arithmetic | $~$\bar{v}$~$ |
---|
Vector arithmetic | $~$|\textbf{v}|$~$ |
---|
Vector arithmetic | $~$b + \textbf{v} = c$~$ |
---|
Vector arithmetic | $~$b$~$ |
---|
Vector arithmetic | $~$\textbf{v}$~$ |
---|
Vector arithmetic | $~$c$~$ |
---|
Vector arithmetic | $~$\textbf{v} = c - b$~$ |
---|
Vector arithmetic | $~$\textbf{v}$~$ |
---|
Vector arithmetic | $~$c$~$ |
---|
Vector arithmetic | $~$b$~$ |
---|
Vector arithmetic | $$~$\textbf{u} + \textbf{v} = \textbf{w}$~$$ |
---|
Vector arithmetic | $$~$(b-a) + (c-b) = c -b + b -a = c-a$~$$ |
---|
Vector arithmetic | $$~$(\textbf{u}+\textbf{v})+\textbf{w}=\textbf{u}+(\textbf{v}+\textbf{w})$~$$ |
---|
Vector arithmetic | $$~$\textbf{u} + \textbf{v} = \textbf{v} + \textbf{u}$~$$ |
---|
Vector arithmetic | $~$\textbf{v} + \textbf{v} = 2\textbf{v}$~$ |
---|
Vector arithmetic | $~$n\textbf{v}$~$ |
---|
Vector arithmetic | $~$n$~$ |
---|
Vector arithmetic | $$~$n\textbf{v}=\textbf{v}n$~$$ |
---|
Vector arithmetic | $~$1$~$ |
---|
Vector arithmetic | $$~$1\textbf{v}=\textbf{v}$~$$ |
---|
Vector arithmetic | $$~$n(m\textbf{v}) = (nm)\textbf{v}$~$$ |
---|
Vector arithmetic | $~$nm\textbf{v}$~$ |
---|
Vector arithmetic | $~$\textbf{v} = b -a$~$ |
---|
Vector arithmetic | $~$\textbf{v}$~$ |
---|
Vector arithmetic | $~$a$~$ |
---|
Vector arithmetic | $~$b$~$ |
---|
Vector arithmetic | $~$b$~$ |
---|
Vector arithmetic | $~$a$~$ |
---|
Vector arithmetic | $~$a - b$~$ |
---|
Vector arithmetic | $~$-(b -a)= -\textbf{v}$~$ |
---|
Vector arithmetic | $~$-1$~$ |
---|
Vector arithmetic | $~$n(-1\mathbf {v}) =-n\mathbf v$~$ |
---|
Vector arithmetic | $~$0$~$ |
---|
Vector arithmetic | $~$0\textbf{v} = \textbf{0}$~$ |
---|
Vector arithmetic | $~$a + \textbf{0} = a$~$ |
---|
Vector arithmetic | $~$\textbf{v} + \textbf{0} = \textbf{v}$~$ |
---|
Vector arithmetic | $~$\textbf{v} - \textbf{v} = 0$~$ |
---|
Vector arithmetic | $~$2\mathbf{v}$~$ |
---|
Vector arithmetic | $~$2|\mathbf{v}|$~$ |
---|
Vector arithmetic | $~$-1$~$ |
---|
Vector arithmetic | $~$|\mathbf{v}|=|-\mathbf{v}|$~$ |
---|
Vector arithmetic | $$~$|n\textbf{v}|=|n||\textbf{v}|$~$$ |
---|
Vector arithmetic | $~$\frac{1}{|\mathbf v|}\mathbf v$~$ |
---|
Vector arithmetic | $~$\mathbf{\hat v}$~$ |
---|
Vector arithmetic | $$~$|\mathbf{\hat v}| = \left|\frac{\mathbf{v}}{|\mathbf{v}|}\right| = \left|\frac{1}{|\mathbf{v}|}\right||\mathbf{v}| = \frac{|\mathbf{v}|}{|\mathbf{v}|}=1$~$$ |
---|
Vector arithmetic | $~$2v+3v$~$ |
---|
Vector arithmetic | $~$(2+3)v=5v$~$ |
---|
Vector arithmetic | $$~$n\textbf{v} + m\textbf{v} = (n+m)\textbf{v}$~$$ |
---|
Vector arithmetic | $~$\textbf{u} +\textbf{v} = \textbf{w}$~$ |
---|
Vector arithmetic | $~$2$~$ |
---|
Vector arithmetic | $~$2\textbf{u} + 2\textbf{v} = 2\text{w}$~$ |
---|
Vector arithmetic | $~$n$~$ |
---|
Vector arithmetic | $~$\textbf{u}$~$ |
---|
Vector arithmetic | $~$\textbf{v}$~$ |
---|
Vector arithmetic | $$~$n\textbf{u} + n\textbf{v} = n(\textbf{u} + \textbf{v})$~$$ |
---|
Vector arithmetic | $$~$\textbf{x} - \textbf{y} - \textbf{z} = \textbf{d}$~$$ |
---|
Vector arithmetic | $~$\mathbf{u}$~$ |
---|
Vector arithmetic | $~$\mathbf{v}$~$ |
---|
Vector arithmetic | $~$\mathbf{w}$~$ |
---|
Vector arithmetic | $~$n$~$ |
---|
Vector arithmetic | $~$m$~$ |
---|
Vector arithmetic | $$~$(\mathbf{u}+\mathbf{v})+\mathbf{w} = \mathbf{u}+(\mathbf{v}+\mathbf{w})$~$$ |
---|
Vector arithmetic | $$~$\mathbf{u}+\mathbf{v} = \mathbf{v} + \mathbf{u}$~$$ |
---|
Vector arithmetic | $$~$1\mathbf{v} =\mathbf{v}$~$$ |
---|
Vector arithmetic | $$~$-1\mathbf{v}=-\mathbf{v}$~$$ |
---|
Vector arithmetic | $$~$0\mathbf{v} = \mathbf{0}$~$$ |
---|
Vector arithmetic | $~$|\mathbf{v}|\neq 0$~$ |
---|
Vector arithmetic | $$~$\frac{1}{|\mathbf v|}\mathbf v = \mathbf{\hat v}$~$$ |
---|
Vector arithmetic | $$~$n\mathbf{v}=\mathbf{v}n$~$$ |
---|
Vector arithmetic | $$~$|n\mathbf{v}| = |n||\mathbf{v}|$~$$ |
---|
Vector arithmetic | $$~$n(m\mathbf{v}) = (nm)\mathbf{v}$~$$ |
---|
Vector arithmetic | $$~$n\mathbf{v}+m\mathbf{v} = (n+m)\mathbf{v}$~$$ |
---|
Vector arithmetic | $$~$n\mathbf{u} + n\mathbf{v} = n(\mathbf{u}+\mathbf{v})$~$$ |
---|
Vector space | $~$F$~$ |
---|
Vector space | $~$V$~$ |
---|
Vector space | $~$\cdot : F \times V \to V$~$ |
---|
Vector space | $~$1 \cdot v = v$~$ |
---|
Vingean uncertainty | $~$x$~$ |
---|
Vingean uncertainty | $~$y$~$ |
---|
Vingean uncertainty | $~$EU[y]$~$ |
---|
Vingean uncertainty | $~$\forall y \neq x: EU[x] > EU[y]$~$ |
---|
Vingean uncertainty | $~$z$~$ |
---|
Vingean uncertainty | $~$x$~$ |
---|
Vingean uncertainty | $~$\forall y \neq z: EU[z] > EU[y]$~$ |
---|
Vingean uncertainty | $~$EU[z] > EU[x]$~$ |
---|
Vingean uncertainty | $~$x$~$ |
---|
Vingean uncertainty | $~$z$~$ |
---|
Vingean uncertainty | $~$z$~$ |
---|
Waterfall diagram | $~$(1 : 4) \times (3 : 1) = (3 : 4).$~$ |
---|
Waterfall diagrams and relative odds | $~$1 : 4$~$ |
---|
Waterfall diagrams and relative odds | $~$3 : 1$~$ |
---|
Waterfall diagrams and relative odds | $~$(1 \cdot 3) : (4 \cdot 1) = 3 : 4$~$ |
---|
Welcome to Arbital | $~${\LaTeX}$~$ |
---|
Well-defined | $~$x=y$~$ |
---|
Well-defined | $~$f(x) = f(y)$~$ |
---|
Well-defined | $~$\mathbb{N}$~$ |
---|
Well-defined | $~$n \mapsto n+1$~$ |
---|
Well-defined | $~$n=m$~$ |
---|
Well-defined | $~$f(n) = f(m)$~$ |
---|
Well-defined | $~$\mathbb{N} \to \mathbb{N}$~$ |
---|
Well-defined | $~$n$~$ |
---|
Well-defined | $~$p_1 p_2 p_3$~$ |
---|
Well-defined | $~$q_1 q_2$~$ |
---|
Well-defined | $~$p_1, p_2, p_3, q_1, q_2$~$ |
---|
Well-defined | $~$3$~$ |
---|
Well-defined | $~$2$~$ |
---|
Well-defined | $~$n$~$ |
---|
Well-defined | $~$X$~$ |
---|
Well-defined | $~$\sim$~$ |
---|
Well-defined | $~$X \to \frac{X}{\sim}$~$ |
---|
Well-defined | $~$x \mapsto [x]$~$ |
---|
Well-defined | $~$X$~$ |
---|
Well-defined | $~$\mathbb{N} \to \mathbb{N}$~$ |
---|
Well-defined | $~$n \mapsto n-5$~$ |
---|
Well-defined | $~$2$~$ |
---|
Well-defined | $~$-3$~$ |
---|
Well-ordered set | $~$(S, \leq)$~$ |
---|
Well-ordered set | $~$U \subset S$~$ |
---|
Well-ordered set | $~$x \in U$~$ |
---|
Well-ordered set | $~$y \in U$~$ |
---|
Well-ordered set | $~$x \leq y$~$ |
---|
Well-ordered set | $~$S$~$ |
---|
Well-ordered set | $~$\mathbb N$~$ |
---|
Well-ordered set | $~$\omega$~$ |
---|
Well-ordered set | $~$\leq$~$ |
---|
Well-ordered set | $~$\mathbb N$~$ |
---|
Well-ordered set | $~$P(x)$~$ |
---|
Well-ordered set | $~$x$~$ |
---|
Well-ordered set | $~$S$~$ |
---|
Well-ordered set | $~$P(x)$~$ |
---|
Well-ordered set | $~$P(y)$~$ |
---|
Well-ordered set | $~$y < x$~$ |
---|
Well-ordered set | $~$P(x)$~$ |
---|
Well-ordered set | $~$P(x)$~$ |
---|
Well-ordered set | $~$x \in S$~$ |
---|
Well-ordered set | $~$U = \{x \in S \mid \neg P(x) \}$~$ |
---|
Well-ordered set | $~$S$~$ |
---|
Well-ordered set | $~$P$~$ |
---|
Well-ordered set | $~$U$~$ |
---|
Well-ordered set | $~$S$~$ |
---|
Well-ordered set | $~$U$~$ |
---|
Well-ordered set | $~$x$~$ |
---|
Well-ordered set | $~$P(y)$~$ |
---|
Well-ordered set | $~$y < x$~$ |
---|
Well-ordered set | $~$P(x)$~$ |
---|
Well-ordered set | $~$x \not\in U$~$ |
---|
Well-ordered set | $~$U$~$ |
---|
Well-ordered set | $~$P$~$ |
---|
Well-ordered set | $~$S$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$1$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$\log_b(x).$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$1$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$\log_b(x).$~$ |
---|
What is a logarithm? | $~$\log_{10}(x)$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$\log_{10}(x)$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$n$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$\log_{10}(x) > n.$~$ |
---|
What is a logarithm? | $~$\log_{10}(10000) = 4,$~$ |
---|
What is a logarithm? | $~$1 \cdot 10 \cdot 10 \cdot 10 \cdot 10 = 10000.$~$ |
---|
What is a logarithm? | $~$\log_2(8) = 3,$~$ |
---|
What is a logarithm? | $~$1 \cdot 2 \cdot 2 \cdot 2 = 8.$~$ |
---|
What is a logarithm? | $~$\log_3(9) = 2,$~$ |
---|
What is a logarithm? | $~$1 \cdot 3 \cdot 3 = 9.$~$ |
---|
What is a logarithm? | $~$\log_{b}(1) = 0$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$\log_{b}(b) = 1$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$b.$~$ |
---|
What is a logarithm? | $~$\log_{1.5}(3.375) = 3,$~$ |
---|
What is a logarithm? | $~$1 \cdot 1.5 \cdot 1.5 \cdot 1.5 = 3.375.$~$ |
---|
What is a logarithm? | $~$\log_3(27)$~$ |
---|
What is a logarithm? | $~$1 \cdot 3 \cdot 3 \cdot 3 = 27$~$ |
---|
What is a logarithm? | $~$\log_4(16)$~$ |
---|
What is a logarithm? | $~$1 \cdot 4 \cdot 4 = 16$~$ |
---|
What is a logarithm? | $~$\log_{10}(\text{1,000,000})$~$ |
---|
What is a logarithm? | $~$1 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 = 1,000,000$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$b.$~$ |
---|
What is a logarithm? | $~$\log_{10}(500) \approx 2.7$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $~$x$~$ |
---|
What is a logarithm? | $$~$\underbrace{5 \cdot 5 \cdot \ldots 5}_\text{10 times} \approx \underbrace{10 \cdot 10 \cdot \ldots 10}_\text{7 times}$~$$ |
---|
What is a logarithm? | $~$\log_{10}(500) \approx 2.7$~$ |
---|
What is a logarithm? | $~$b$~$ |
---|
What is a logarithm? | $~$5^{10}$~$ |
---|
What is a logarithm? | $~$10^7$~$ |
---|
What is a logarithm? | $~$\log_{10}(500)$~$ |
---|
What is a logarithm? | $~$\log_{10}(500)$~$ |
---|
What is a logarithm? | $~$500 \cdot 8000$~$ |
---|
What is a logarithm? | $~$2.7 + 3.9 = 6.6$~$ |
---|
Who needs civilization? | $~$50/m, reducing it to $~$ |
---|
Whole number | $~$\mathbb{N}^0$~$ |
---|
Whole number | $~$\mathbb{W}$~$ |
---|
Why is log like length? | $~$x$~$ |
---|
Why is log like length? | $~$n$~$ |
---|
Why is log like length? | $~$n-1$~$ |
---|
Why is log like length? | $~$n$~$ |
---|
Why is log like length? | $~$\log_{10}(x)$~$ |
---|
Why is log like length? | $~$x;$~$ |
---|
Why is log like length? | $~$x$~$ |
---|
Why is log like length? | $~$x$~$ |
---|
Why is log like length? | $~$x$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$\log_2(3)$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$10 \cdot \log_2(3)$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$1.5 < \log_2(3) < 1.6.$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$1.58 < \log_2(3) < 1.59.$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$10^n \cdot \log_2(3)$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$n$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$\log_2(3)$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$\log_2(3)$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$10 \cdot \log_2(3)$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$1.5 < \log_2(3) < 1.6.$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$1.58 < \log_2(3) < 1.59.$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$10^n \cdot \log_2(3)$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$n$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$\log_2(3)$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$3$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$2$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$\log_b(x)$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$x$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$b$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$\log_b(x)$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$x$~$ |
---|
Why is the decimal expansion of log2(3) infinite? | $~$b.$~$ |
---|
Without loss of generality | $~$5$~$ |
---|
Without loss of generality | $~$3$~$ |
---|
Without loss of generality | $~$3$~$ |
---|
Without loss of generality | $~$3$~$ |
---|
Without loss of generality | $~$5$~$ |
---|
Without loss of generality | $~$3$~$ |
---|
Without loss of generality | $~$2$~$ |
---|
Without loss of generality | $~$2$~$ |
---|
Without loss of generality | $~$4$~$ |
---|
Without loss of generality | $~$3a + (3b+1)+(3c+2) = 3 (a+b+c) + 3$~$ |
---|
Without loss of generality | $~$3$~$ |
---|
You can't get more paperclips that way | $~$P$~$ |
---|
You can't get more paperclips that way | $~$U.$~$ |
---|
You can't get more paperclips that way | $~$P$~$ |
---|
You can't get more paperclips that way | $~$U$~$ |
---|
You can't get more paperclips that way | $~$\pi_1$~$ |
---|
You can't get more paperclips that way | $~$\pi_2$~$ |
---|
You can't get more paperclips that way | $~$\pi_2.$~$ |
---|
You can't get more paperclips that way | $~$\pi_1$~$ |
---|
You can't get more paperclips that way | $~$\pi_2$~$ |
---|
You can't get more paperclips that way | $~$\pi_1,$~$ |
---|
You can't get more paperclips that way | $~$\pi_2$~$ |
---|
You can't get more paperclips that way | $~$\pi_2$~$ |
---|
You can't get more paperclips that way | $~$\pi_1$~$ |
---|
You can't get more paperclips that way | $~$\pi_1$~$ |
---|
n-digit | $~$n$~$ |
---|
n-digit | $~$n$~$ |
---|
n-digit | $~$n$~$ |
---|
n-digit | $~$m$~$ |
---|
n-digit | $~$n$~$ |
---|
n-digit | $~$m < n$~$ |
---|
n-digit | $~$n$~$ |
---|
n-digit | $~$m$~$ |
---|
n-digit | $~$m > n$~$ |
---|
n-digit | $~$n$~$ |
---|
n-message | $~$n$~$ |
---|
n-message | $~$n$~$ |
---|
n-message | $~$n$~$ |
---|
n-message | $~$\log_2(n)$~$ |
---|
n-message | $~$n$~$ |
---|
n-message | $~$n$~$ |
---|
n-message | $~$n$~$ |
---|
n-message | $~$\log_2(n)$~$ |
---|
n-message | $~$n$~$ |
---|
some formulas that are not directly given that may or may not help | $~$\frac {(y_2-y_1)}{(x_2-x_1)}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$\frac{\Delta y}{\Delta x}=\frac {rise}{run}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$y=mx+b$~$ |
---|
some formulas that are not directly given that may or may not help | $~$\frac {(x_1+x_2)}{2}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$\frac {(y_1+y_2)}{2}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$\sqrt {(x_2-x_1)^2+(y_2-y_1)^2}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$L_{arc}=(2πr)(\frac {xº}{360})$~$ |
---|
some formulas that are not directly given that may or may not help | $~$L_{arc sector}=(πr^2)(\frac {xº}{360})$~$ |
---|
some formulas that are not directly given that may or may not help | $~$x=\frac {-b \pm \sqrt {b^2-4ac}}{2a}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$ax^2+bx=0 \rightarrow a(x+d)^2+e=0$~$ |
---|
some formulas that are not directly given that may or may not help | $~$d=\frac{b}{2a}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$e=c-\frac{b^2}{4a}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$sin(x)=\frac {o}{h}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$cos(x)=\frac {a}{h}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$tan(x)=\frac{o}{a}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$t_1, t_1+d, t_1+2d, …$~$ |
---|
some formulas that are not directly given that may or may not help | $~$t_1, t_1\cdot r, t_1\cdot r^2,…$~$ |
---|
some formulas that are not directly given that may or may not help | $~$x^a\cdot{x^b}=x^{a+b}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$(x^a)^b=x^{a\cdot{b}}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$x^0=1$~$ |
---|
some formulas that are not directly given that may or may not help | $~$\frac {x^a}{x^b}=x^{a-b}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$(xy)^a=x^a\cdot{y^a}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$\sqrt{xy}=\sqrt{x} \cdot {\sqrt{y}}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$x^{-b}=\frac {1}{x^b}$~$ |
---|
some formulas that are not directly given that may or may not help | $~$(x+a)(x+b)=x^2+(b+a)x+ab$~$ |
---|
some formulas that are not directly given that may or may not help | $~$a^2-b^2=(a+b)(a-b)$~$ |
---|
some formulas that are not directly given that may or may not help | $~$a^2+2ab+b^2=(a+b)(a+b)$~$ |
---|
some formulas that are not directly given that may or may not help | $~$a^2-2ab+b^2=(a-b)(a-b)$~$ |
---|
some formulas that are not directly given that may or may not help | $~$a^2+b^2=c^2$~$ |
---|
ε-ROI Lemma | $~$\varepsilon$~$ |
---|
ε-ROI Lemma | $~$\varepsilon$~$ |
---|
ε-ROI Lemma | $~$\epsilon$~$ |
---|
ε-ROI Lemma | $~$\varepsilon$~$ |
---|
ε-ROI Lemma | $~$\varepsilon$~$ |
---|
ε-ROI Lemma | $~$\varepsilon$~$ |
---|
ε-ROI Lemma | $~$\varepsilon$~$ |
---|
ε-ROI Lemma | $~$\forall\mathbb{W}\in\mathcal{PC}(\Gamma):\lim_{n\to\infty}\mathbb{W}\left(\sum_{i\le n}T_{i}(\overline{\mathbb{P}})\right)\ge\varepsilon||\overline{T}(\overline{\mathbb{P}})||_{mg}$~$ |
---|
ε-ROI Lemma | $~$n$~$ |
---|
ε-ROI Lemma | $~$k$~$ |
---|
ε-ROI Lemma | $~$\varepsilon$~$ |
---|
ε-ROI Lemma | $~$\mathcal{PC}$~$ |
---|
ε-ROI Lemma | $~$\mathcal{PC}$~$ |
---|
ε-ROI Lemma | $~$\overline{\mathbb{P}}$~$ |
---|
ε-ROI Lemma | $~$\overline{\mathbb{P}}$~$ |
---|
ε-ROI Lemma | $~$\overline{\alpha}$~$ |
---|
ε-ROI Lemma | $~$\mathcal{EF}$~$ |
---|
ε-ROI Lemma | $~$\overline{\xi}$~$ |
---|
ε-ROI Lemma | $~$\overline{\alpha}$~$ |
---|
ε-ROI Lemma | $~$\overline{T}^{k}$~$ |
---|
ε-ROI Lemma | $~$\alpha_{k}$~$ |
---|
ε-ROI Lemma | $~$poly(k)$~$ |
---|
ε-ROI Lemma | $~$\xi_{k}$~$ |
---|
ε-ROI Lemma | $~$\alpha_{k}$~$ |
---|
ε-ROI Lemma | $~$max(1,\xi_{k})^{-1}$~$ |
---|
ε-ROI Lemma | $~$\varepsilon$~$ |
---|
ε-ROI Lemma | $~$\varepsilon\times magnitude$~$ |
---|
ε-ROI Lemma | $~$\varepsilon\times magnitude$~$ |
---|
ε-ROI Lemma | $~$poly(k)$~$ |
---|
ε-ROI Lemma | $~$\overline{T}^{k}$~$ |
---|
ε-ROI Lemma | $~$poly(k)$~$ |
---|
ε-ROI Lemma | $~$poly(k)$~$ |
---|
ε-ROI Lemma | $~$\xi_{k}$~$ |
---|
ε-ROI Lemma | $~$poly(n)$~$ |
---|
ε-ROI Lemma | $~$\overline{T}^{k}$~$ |
---|
ε-ROI Lemma | $~$poly(n)$~$ |
---|
ε-ROI Lemma | $~$max(1,\xi_{k})^{-1}$~$ |
---|
ε-ROI Lemma | $~$\epsilon$~$ |
---|
ε-ROI Lemma | $~$\overline{\mathbb{P}}$~$ |
---|
ε-ROI Lemma | $~$\alpha_{k}\le 1$~$ |
---|
ε-ROI Lemma | $~$\overline{D}$~$ |
---|
ε-ROI Lemma | $~$\overline{\mathbb{P}}$~$ |
---|
ε-ROI Lemma | $~$\overline{\mathbb{P}}$~$ |
---|
ε-ROI Lemma | $~$\overline{D}$~$ |
---|
ε-ROI Lemma | $~$\mathcal{PC}$~$ |
---|
ε-ROI Lemma | $~$X$~$ |
---|
ε-ROI Lemma | $~$X$~$ |
---|
ε-ROI Lemma | $~$X$~$ |
---|
ε-ROI Lemma | $~$\mathcal{PC}$~$ |
---|
ε-ROI Lemma | $~$poly(k)$~$ |
---|
ε-ROI Lemma | $~$\varepsilon$~$ |
---|
ε-ROI Lemma | $~$\mathcal{PC}$~$ |
---|
ε-ROI Lemma | $~$\displaystyle\sum_{i\le m}||T_{i}^{k}(\overline{\mathbb{P}})||_{mg}\ge\left(1-\frac{\varepsilon}{3}\right)\alpha_{k}$~$ |
---|
ε-ROI Lemma | $~$\frac{\varepsilon}{3}$~$ |
---|
ε-ROI Lemma | $~$\mathcal{PC}$~$ |
---|
ε-ROI Lemma | $~$\forall\mathbb{W}\in\mathcal{PC}(D_{n}),\mathbb{W}\left(\displaystyle\sum_{i\le m}T_{i}^{k}(\overline{\mathbb{P}})\right)\ge\left(\frac{2\varepsilon}{3}\right)\alpha_{k}$~$ |
---|
ε-ROI Lemma | $~$\frac{2\varepsilon}{3}$~$ |
---|
ε-ROI Lemma | $~$\varepsilon$~$ |
---|
ε-ROI Lemma | $~$\frac{\varepsilon}{3}\alpha_{k}$~$ |
---|
ε-ROI Lemma | $~$\frac{2\varepsilon}{3}\alpha_{k}$~$ |
---|
ε-ROI Lemma | $~$\frac{\varepsilon}{3}\alpha_{k}$~$ |
---|
ε-ROI Lemma | $~$n\gg m$~$ |
---|
ε-ROI Lemma | $~$n$~$ |
---|
ε-ROI Lemma | $~$k$~$ |
---|
ε-ROI Lemma | $~$k$~$ |
---|
ε-ROI Lemma | $~$T_{n}:=\displaystyle\sum_{k\le n}\beta_{k}^{\dagger}T_{n}^{k}$~$ |
---|
ε-ROI Lemma | $~$n$~$ |
---|
ε-ROI Lemma | $~$open(i,k):=0$~$ |
---|
ε-ROI Lemma | $~$k$~$ |
---|
ε-ROI Lemma | $~$\overline{T}^{i}$~$ |
---|
ε-ROI Lemma | $~$\beta_{k}^{\dagger}:=1-\displaystyle\sum_{i<k}open(i,k)\beta_{i}^{\dagger}\alpha_{i}^{\dagger}$~$ |
---|
ε-ROI Lemma | $~$\dagger$~$ |
---|
ε-ROI Lemma | $~$\beta_{i}^{\dagger}$~$ |
---|
ε-ROI Lemma | $~$i$~$ |
---|
ε-ROI Lemma | $~$\alpha_{i}^{\dagger}$~$ |
---|
ε-ROI Lemma | $~$i$~$ |
---|
ε-ROI Lemma | $~$open(i,k)=0$~$ |
---|
ε-ROI Lemma | $~$i$~$ |
---|
ε-ROI Lemma | $~$\alpha_{i}$~$ |
---|
ε-ROI Lemma | $~$\beta_{i}$~$ |
---|
ε-ROI Lemma | $~$n$~$ |
---|
ε-ROI Lemma | $~$\overline{T}^{k}$~$ |
---|
ε-ROI Lemma | $~$\beta_{k}^{\dagger}$~$ |
---|
ε-ROI Lemma | $~$\overline{T}^{k}$~$ |
---|
ε-ROI Lemma | $~$n$~$ |
---|
ε-ROI Lemma | $~$(\overline{T}^{k})_{k}$~$ |
---|
ε-ROI Lemma | $~$\overline{T}^{k}$~$ |
---|
ε-ROI Lemma | $~$\beta_{k}^{\dagger}$~$ |
---|
ε-ROI Lemma | $~$\alpha_{i}^{\dagger}$~$ |
---|
ε-ROI Lemma | $~$open(i,k)$~$ |
---|
ε-ROI Lemma | $~$\alpha_{i}^{\dagger}$~$ |
---|
ε-ROI Lemma | $~$\overline{\alpha}$~$ |
---|
ε-ROI Lemma | $~$\overline{\mathbb{P}}$~$ |
---|
ε-ROI Lemma | $~$\mathcal{O}(k*poly(k))$~$ |
---|
ε-ROI Lemma | $~$open(i,k)$~$ |
---|
ε-ROI Lemma | $~$\mathcal{O}(k^{2})$~$ |
---|
ε-ROI Lemma | $~$\beta_{i}^{\dagger}$~$ |
---|
ε-ROI Lemma | $~$\alpha$~$ |
---|
ε-ROI Lemma | $~$open$~$ |
---|
ε-ROI Lemma | $~$\beta$~$ |
---|
ε-ROI Lemma | $~$\beta_{k}^{\dagger}$~$ |
---|
ε-ROI Lemma | $~$\beta_{k}^{\dagger}$~$ |
---|
ε-ROI Lemma | $~$\overline{T}^{k}$~$ |
---|
ε-ROI Lemma | $~$n$~$ |
---|
ε-ROI Lemma | $~$k\le n$~$ |
---|
ε-ROI Lemma | $~$(\overline{T}^{k})_{k}$~$ |
---|
ε-ROI Lemma | $~$\overline{T}^{k}$~$ |
---|
ε-ROI Lemma | $~$\varepsilon$~$ |
---|
ε-ROI Lemma | $~$\overline{\mathbb{P}}$~$ |
---|
ε-ROI Lemma | $~$\overline{\alpha}$~$ |
---|
ε-ROI Lemma | $~$\displaystyle\lim_{k\to\infty}\alpha_{k}=0$~$ |
---|
ε-ROI Lemma | $~$\displaystyle\lim_{k\to\infty}\alpha_{k}\not=0$~$ |
---|
ε-ROI Lemma | $~$\alpha_{k}^{\dagger}(\overline{\mathbb{P}})$~$ |
---|
ε-ROI Lemma | $~$\beta_{k}^{\dagger}(\overline{\mathbb{P}})$~$ |
---|
ε-ROI Lemma | $~$\alpha_{k}$~$ |
---|
ε-ROI Lemma | $~$\beta_{k}$~$ |
---|
ε-ROI Lemma | $~$\alpha_{k}^{\dagger}$~$ |
---|
ε-ROI Lemma | $~$\beta_{k}^{\dagger}$~$ |
---|
ε-ROI Lemma | $~$\beta_{k}$~$ |
---|
ε-ROI Lemma | $$~$\beta_{k}:=1-\sum_{i<k}open(i,k)\beta_{i}\alpha_{i}$~$$ |
---|
ε-ROI Lemma | $~$\alpha_{k}\le 1$~$ |
---|
ε-ROI Lemma | $$~$\beta_{k}\alpha_{k}\le 1-\sum_{i<k}open(i,k)\beta_{i}\alpha_{i}$~$$ |
---|
ε-ROI Lemma | $$~$\beta_{n}\alpha_{n}+\sum_{k<n}open(k,n)\beta_{k}\alpha_{k}\le 1$~$$ |
---|
ε-ROI Lemma | $~$\beta_{n}\alpha_{n}\ge open(n,n)\beta_{n}\alpha_{n}$~$ |
---|
ε-ROI Lemma | $$~$\sum_{k\le n}open(k,n)\beta_{k}\alpha_{k}\le 1$~$$ |
---|
ε-ROI Lemma | $~$\beta_{k}$~$ |
---|
ε-ROI Lemma | $~$\beta_{k}\ge 0$~$ |
---|
ε-ROI Lemma | $~$\overline{T}$~$ |
---|
ε-ROI Lemma | $$~$\mathbb{W}\left(\sum_{i\le n}T_{i}(\overline{\mathbb{P}})\right)=\sum_{i\le n}\mathbb{W}\left(T_{i}(\overline{\mathbb{P}})\right)$~$$ |
---|
ε-ROI Lemma | $~$T_{n}$~$ |
---|
ε-ROI Lemma | $$~$=\sum_{k\le n}\mathbb{W}\left(\sum_{i\le n}\beta_{k}T_{i}^{k}(\overline{\mathbb{P}})\right)$~$$ |
---|
ε-ROI Lemma | $$~$=\sum_{uncertain, k\le n}\mathbb{W}\left(\sum_{i\le n}\beta_{k}T_{i}^{k}(\overline{\mathbb{P}})\right)+\sum_{guaranteed, k\le n}\mathbb{W}\left(\sum_{i\le n}\beta_{k}T_{i}^{k}(\overline{\mathbb{P}})\right)$~$$ |
---|
ε-ROI Lemma | $~$\beta_{k}$~$ |
---|
ε-ROI Lemma | $~$i$~$ |
---|
ε-ROI Lemma | $$~$\sum_{uncertain, k\le n}\mathbb{W}\left(\sum_{i\le n}\beta_{k}T_{i}^{k}(\overline{\mathbb{P}})\right)\ge -\sum_{uncertain, k\le n}\beta_{k}\sum_{i\le n}||T_{i}^{k}(\overline{\mathbb{P}})||_{mg}$~$$ |
---|
ε-ROI Lemma | $~$\alpha_{k}$~$ |
---|
ε-ROI Lemma | $~$open(i,k)$~$ |
---|
ε-ROI Lemma | $$~$\ge -\sum_{uncertain, k\le n}\beta_{k}\sum_{i\in\mathbb{N}^{+}}||T_{i}^{k}(\overline{\mathbb{P}})||_{mg} = -\sum_{k\le n}open(k,n)\beta_{k}\alpha_{k}$~$$ |
---|
ε-ROI Lemma | $~$\ge -1$~$ |
---|
ε-ROI Lemma | $$~$\sum_{guaranteed, k\le n}\mathbb{W}\left(\sum_{i\le n}\beta_{k}T_{i}^{k}(\overline{\mathbb{P}})\right)\ge\sum_{guaranteed, k\le n}\frac{\varepsilon}{3}\beta_{k}\alpha_{k}$~$$ |
---|
ε-ROI Lemma | $$~$\sum_{k\le n}\mathbb{W}\left(\sum_{i\le n}\beta_{k}T_{i}^{k}(\overline{\mathbb{P}})\right)\ge -1+\sum_{guaranteed, k\le n}\frac{\varepsilon}{3}\beta_{k}\alpha_{k}$~$$ |
---|
ε-ROI Lemma | $~$\beta_{k}\ge 0$~$ |
---|
ε-ROI Lemma | $~$\alpha_{k}\ge 0$~$ |
---|
ε-ROI Lemma | $~$n\to\infty$~$ |
---|
ε-ROI Lemma | $~$k$~$ |
---|
ε-ROI Lemma | $~$n$~$ |
---|
ε-ROI Lemma | $$~$\lim_{n\to\infty}\sum_{guaranteed, k\le n}\frac{\varepsilon}{3}\beta_{k}\alpha_{k}=\sum_{k}\frac{\varepsilon}{3}\beta_{k}\alpha_{k}$~$$ |
---|
ε-ROI Lemma | $~$\frac{\varepsilon}{3}$~$ |
---|
ε-ROI Lemma | $~$\displaystyle\sum_{k}\beta_{k}\alpha_{k}=\infty$~$ |
---|
ε-ROI Lemma | $~$\displaystyle\lim_{k\to\infty}\alpha_{k}\not=0$~$ |
---|
ε-ROI Lemma | $~$\delta$~$ |
---|
ε-ROI Lemma | $~$\alpha_{k}>\delta$~$ |
---|
ε-ROI Lemma | $$~$\displaystyle\sum_{k}\beta_{k}\alpha_{k}$~$$ |
---|
ε-ROI Lemma | $~$n$~$ |
---|
ε-ROI Lemma | $$~$\sum_{i>n}\beta_{i}\alpha_{i}\le\frac{1}{2}$~$$ |
---|
ε-ROI Lemma | $~$N$~$ |
---|
ε-ROI Lemma | $~$k\le n$~$ |
---|
ε-ROI Lemma | $$~$\sum_{i<N}open(i,N)\beta_{i}\alpha_{i}=\sum_{i\le n}0*\beta_{i}\alpha_{i}+\sum_{n<i<N}open(i,N)\beta_{i}\alpha_{i}$~$$ |
---|
ε-ROI Lemma | $~$n<N$~$ |
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ε-ROI Lemma | $~$\displaystyle\sum_{i>n}\beta_{i}\alpha_{i}\le\frac{1}{2}$~$ |
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ε-ROI Lemma | $$~$\le 0+\sum_{n<i<N}\beta_{i}\alpha_{i}\le\frac{1}{2}$~$$ |
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ε-ROI Lemma | $$~$\alpha_{k}\beta_{k}=\alpha_{k}\left(1-\sum_{i<k}open(i,k)\beta_{i}\alpha_{i}\right)$~$$ |
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ε-ROI Lemma | $~$\displaystyle\sum_{i>N}open(i,N)\beta_{i}\alpha_{i}\le\frac{1}{2}$~$ |
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ε-ROI Lemma | $~$k>N$~$ |
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ε-ROI Lemma | $~$k>N$~$ |
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ε-ROI Lemma | $~$\alpha_{k}>\delta$~$ |
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ε-ROI Lemma | $$~$\alpha_{k}\left(1-\sum_{i<k}open(i,k)\beta_{i}\alpha_{i}\right)\ge\alpha_{k}(1-\frac{1}{2})\ge\frac{\delta}{2}$~$$ |
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ε-ROI Lemma | $$~$\sum_{k}\alpha_{k}\beta_{k}=\infty$~$$ |
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ε-ROI Lemma | $~$\displaystyle\lim_{n\to\infty}\sum_{guaranteed, k\le n}\frac{\varepsilon}{3}\beta_{k}\alpha_{k}$~$ |
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ε-ROI Lemma | $~$\displaystyle\lim_{k\to\infty}\alpha_{k}\not=0$~$ |
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ε-ROI Lemma | $~$\varepsilon$~$ |
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ε-ROI Lemma | $~$\overline{\mathbb{P}}$~$ |
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ε-ROI Lemma | $~$\overline{\alpha}$~$ |
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ε-ROI Lemma | $~$\displaystyle\lim_{k\to\infty}\alpha_{k}\not=0$~$ |
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