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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$~$

Absent-Minded Driver dilemma

$~$p$~$

Absent-Minded Driver dilemma

$~$p$~$

Absent-Minded Driver dilemma

$~$1 - p$~$

Absent-Minded Driver dilemma

$~$p$~$

Absent-Minded Driver dilemma

$~$1 - p$~$

Absent-Minded Driver dilemma

$~$p^2$~$

Absent-Minded Driver dilemma

$~$0(1-p) + 4(1-p)p + 1p^2$~$

Absent-Minded Driver dilemma

$~$4 -6p$~$

Absent-Minded Driver dilemma

$~$p = \frac{2}{3}$~$

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.$~$

Absent-Minded Driver dilemma

$~$p$~$

Absent-Minded Driver dilemma

$~$q.$~$

Absent-Minded Driver dilemma

$~$1 : q,$~$

Absent-Minded Driver dilemma

$~$\frac{1}{1+q}$~$

Absent-Minded Driver dilemma

$~$\frac{q}{1+q}$~$

Absent-Minded Driver dilemma

$~$p,$~$

Absent-Minded Driver dilemma

$~$4p(1-p) + 1p^2.$~$

Absent-Minded Driver dilemma

$~$p$~$

Absent-Minded Driver dilemma

$~$4(1-p) + 1p.$~$

Absent-Minded Driver dilemma

$~$\frac{1}{1+q}(4p(1-p) + p^2) + \frac{q}{1+q}(4(1-p) + p)$~$

Absent-Minded Driver dilemma

$~$\frac{-6p - 3q + 4}{q+1}$~$

Absent-Minded Driver dilemma

$~$p=\frac{4-3q}{6}.$~$

Absent-Minded Driver dilemma

$~$q$~$

Absent-Minded Driver dilemma

$~$p$~$

Absent-Minded Driver dilemma

$~$q,$~$

Absent-Minded Driver dilemma

$~$p=q=\frac{4}{9}.$~$

Absent-Minded Driver dilemma

$~$\$4\cdot\frac{4}{9}\frac{5}{9} + \$1\cdot\frac{4}{9}\frac{4}{9} \approx \$1.19.$~$

Absent-Minded Driver dilemma

$~$q$~$

Absent-Minded Driver dilemma

$~$p$~$

Absent-Minded Driver dilemma

$~$q$~$

Absent-Minded Driver dilemma

$~$q,$~$

Absent-Minded Driver dilemma

$~$1 : q \cong \frac{1}{1+q} : \frac{q}{1+q}$~$

Absent-Minded Driver dilemma

$~$p,$~$

Absent-Minded Driver dilemma

$~$q$~$

Absent-Minded Driver dilemma

$~$4p(1-q) + 1pq.$~$

Absent-Minded Driver dilemma

$~$p$~$

Absent-Minded Driver dilemma

$~$4(1-p) + 1p.$~$

Absent-Minded Driver dilemma

$~$q,$~$

Absent-Minded Driver dilemma

$~$p$~$

Absent-Minded Driver dilemma

$$~$\frac{1}{1+q}(4p(1-q) + pq) + \frac{q}{1+q}(4(1-p) + p)$~$$

Absent-Minded Driver dilemma

$~$\frac{4 - 6q}{1+q}$~$

Absent-Minded Driver dilemma

$~$p.$~$

Absent-Minded Driver dilemma

$~$q$~$

Absent-Minded Driver dilemma

$~$q$~$

Absent-Minded Driver dilemma

$~$p$~$

Absent-Minded Driver dilemma

$~$4-6q = 0 \implies q=\frac{2}{3}.$~$

Absent-Minded Driver dilemma

$~$p$~$

Absent-Minded Driver dilemma

$~$q$~$

Absent-Minded Driver dilemma

$~$p$~$

Absent-Minded Driver dilemma

$~$p$~$

Absolute Complement

$~$A^\complement$~$

Absolute Complement

$~$A$~$

Absolute Complement

$~$A$~$

Absolute Complement

$~$U$~$

Absolute Complement

$~$A^\complement = U \setminus A$~$

Absolute Complement

$~$A^\complement$~$

Absolute Complement

$~$U$~$

Absolute Complement

$~$A$~$

Ackermann function

$~$A \cdot B = \underbrace{A + A + \ldots A}_{B \text{ copies of } A}$~$

Ackermann function

$~$A^B = \underbrace{A \times A \times \ldots A}_{B \text{ copies of } A}$~$

Ackermann function

$~$A ^ B$~$

Ackermann function

$~$A \uparrow B$~$

Ackermann function

$~$A \uparrow\uparrow B = \underbrace{A^{A^{\ldots^A}}}_{B \text{ copies of } A}$~$

Ackermann function

$~$\uparrow^n$~$

Ackermann function

$~$n$~$

Ackermann function

$~$A \uparrow^2 B = \underbrace{A \uparrow^1 (A \uparrow^1 (\ldots A))}_{B \text{ copies of } A}$~$

Ackermann function

$~$A \uparrow^n B = \underbrace{A \uparrow^{n-1} (A \uparrow^{n-1} (\ldots A))}_{B \text{ copies of } A}$~$

Ackermann function

$~$A(n) = n \uparrow^n n$~$

Ackermann function

$~$A(6)$~$

Ackermann function

$~$A(1)=1$~$

Ackermann function

$~$A(2)=4$~$

Ackermann function

$~$A(3)$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{\text{number}}$~$

Addition of rational numbers (Math 0)

$~$5$~$

Addition of rational numbers (Math 0)

$~$\frac{5}{\text{number}}$~$

Addition of rational numbers (Math 0)

$~$a+b$~$

Addition of rational numbers (Math 0)

$~$a$~$

Addition of rational numbers (Math 0)

$~$b$~$

Addition of rational numbers (Math 0)

$~$\frac{2}{2} + \frac{3}{3} = 2$~$

Addition of rational numbers (Math 0)

$~$\frac{n}{n}$~$

Addition of rational numbers (Math 0)

$~$n$~$

Addition of rational numbers (Math 0)

$~$\frac{5}{3} + \frac{8}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3}$~$

Addition of rational numbers (Math 0)

$~$5+8=13$~$

Addition of rational numbers (Math 0)

$~$\frac{5}{3} + \frac{8}{3} = \frac{13}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{5}{3} + \frac{5}{4}$~$

Addition of rational numbers (Math 0)

$~$\frac{5}{3} + \frac{5}{4}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{4}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{4}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{4}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3} = \frac{4}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{4} = \frac{3}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3} = \frac{4}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{5}{3} = \frac{20}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{5}{4} = \frac{15}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{4}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$5 \times 3 = 15$~$

Addition of rational numbers (Math 0)

$~$\frac{5}{3} + \frac{5}{4}$~$

Addition of rational numbers (Math 0)

$~$\frac{20}{12} + \frac{15}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{35}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{4}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{2}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{5}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{2}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{5}$~$

Addition of rational numbers (Math 0)

$~$2 \times 5 = 10$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{10}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{2}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{10}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{5}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{m}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{n}$~$

Addition of rational numbers (Math 0)

$~$m$~$

Addition of rational numbers (Math 0)

$~$2$~$

Addition of rational numbers (Math 0)

$~$n$~$

Addition of rational numbers (Math 0)

$~$5$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{m \times n}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{n} = \frac{m}{m \times n}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{n}$~$

Addition of rational numbers (Math 0)

$~$m$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{m \times n}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{m} = \frac{n}{m \times n}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{m}$~$

Addition of rational numbers (Math 0)

$~$n$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{m \times n}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{\text{thing}}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{\text{thing}}$~$

Addition of rational numbers (Math 0)

$$~$\frac{1}{m} + \frac{1}{n} = \frac{n}{m \times n} + \frac{m}{m \times n}$~$$

Addition of rational numbers (Math 0)

$$~$\frac{a}{m} + \frac{b}{m} = \frac{a+b}{m}$~$$

Addition of rational numbers (Math 0)

$~$a$~$

Addition of rational numbers (Math 0)

$~$b$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{m}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{n}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{m \times n}$~$

Addition of rational numbers (Math 0)

$~$\frac{5}{4} + \frac{5}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{4}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$3 \times 4 = 12$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{5}{4}$~$

Addition of rational numbers (Math 0)

$~$\frac{15}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{4}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$5 \times 3$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{5}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{20}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{3}$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$5 \times 4 = 20$~$

Addition of rational numbers (Math 0)

$~$\frac{1}{12}$~$

Addition of rational numbers (Math 0)

$~$\frac{15}{12} + \frac{20}{12} = \frac{35}{12}$~$

Addition of rational numbers (Math 0)

$$~$\frac{a}{m} + \frac{b}{n} = \frac{a \times n}{m \times n} + \frac{b \times m}{m \times n} = \frac{a \times n + b \times m}{m \times n}$~$$

Addition of rational numbers (Math 0)

$~$a \times n + b \times m$~$

Addition of rational numbers (Math 0)

$~$a \times n$~$

Addition of rational numbers (Math 0)

$~$b \times m$~$

Addition of rational numbers (Math 0)

$~$a, b, m, n$~$

Addition of rational numbers (Math 0)

$~$m$~$

Addition of rational numbers (Math 0)

$~$n$~$

Addition of rational numbers exercises

$~$\frac{1}{10} + \frac{1}{5}$~$

Addition of rational numbers exercises

$$~$\frac{1}{10} + \frac{1}{5} = \frac{1 \times 5 + 10 \times 1}{10 \times 5} = \frac{5+10}{50} = \frac{15}{50}$~$$

Addition of rational numbers exercises

$~$\frac{3}{10}$~$

Addition of rational numbers exercises

$~$\frac{3}{10}$~$

Addition of rational numbers exercises

$~$\frac{1}{10}$~$

Addition of rational numbers exercises

$~$15$~$

Addition of rational numbers exercises

$~$\frac{1}{50}$~$

Addition of rational numbers exercises

$~$\frac{1}{10}$~$

Addition of rational numbers exercises

$~$\frac{1}{5}$~$

Addition of rational numbers exercises

$~$\frac{1}{5} = \frac{2}{10}$~$

Addition of rational numbers exercises

$~$\frac{1}{10} + \frac{2}{10}$~$

Addition of rational numbers exercises

$~$\frac{3}{10}$~$

Addition of rational numbers exercises

$~$\frac{1}{15} + \frac{1}{10}$~$

Addition of rational numbers exercises

$$~$\frac{1}{10} + \frac{1}{15} = \frac{1 \times 15 + 10 \times 1}{10 \times 15} = \frac{25}{150} = \frac{1}{6}$~$$

Addition of rational numbers exercises

$~$\frac{1}{30}$~$

Addition of rational numbers exercises

$~$\frac{1}{10}$~$

Addition of rational numbers exercises

$~$\frac{1}{15}$~$

Addition of rational numbers exercises

$~$\frac{3}{30} + \frac{2}{30} = \frac{5}{30}$~$

Addition of rational numbers exercises

$~$\frac{5}{30} = \frac{1}{6}$~$

Addition of rational numbers exercises

$~$\frac{25}{150} = \frac{1}{6}$~$

Addition of rational numbers exercises

$~$\frac{1}{10} + \frac{1}{15}$~$

Addition of rational numbers exercises

$$~$\frac{1}{15} + \frac{1}{10} = \frac{1 \times 10 + 15 \times 1}{15 \times 10} = \frac{25}{150} = \frac{1}{6}$~$$

Addition of rational numbers exercises

$~$\frac{1}{10} + \frac{1}{15} = \frac{1}{15} + \frac{1}{10}$~$

Addition of rational numbers exercises

$~$\frac{1}{6}$~$

Addition of rational numbers exercises

$~$\frac{0}{5} + \frac{2}{5}$~$

Addition of rational numbers exercises

$~$5$~$

Addition of rational numbers exercises

$~$\frac{1}{5}$~$

Addition of rational numbers exercises

$~$0$~$

Addition of rational numbers exercises

$~$2$~$

Addition of rational numbers exercises

$~$2$~$

Addition of rational numbers exercises

$~$\frac{2}{5}$~$

Addition of rational numbers exercises

$~$\frac{0}{7} + \frac{2}{5}$~$

Addition of rational numbers exercises

$~$\frac{1}{7}$~$

Addition of rational numbers exercises

$~$\frac{1}{5}$~$

Addition of rational numbers exercises

$~$\frac{2}{5}$~$

Addition of rational numbers exercises

$~$\frac{1}{7}$~$

Addition of rational numbers exercises

$$~$\frac{0}{7} + \frac{2}{5} = \frac{0 \times 5 + 2 \times 7}{5 \times 7} = \frac{0 + 14}{35} = \frac{14}{35}$~$$

Addition of rational numbers exercises

$~$\frac{2}{5}$~$

Addition of rational numbers exercises

$~$\frac{1}{5}$~$

Addition of rational numbers exercises

$~$\frac{1}{5} + \frac{-1}{10}$~$

Addition of rational numbers exercises

$$~$\frac{1}{15} + \frac{-1}{10} = \frac{1 \times 10 + 15 \times (-1)}{15 \times 10} = \frac{10 - 15}{150} = \frac{-5}{150} = \frac{-1}{30}$~$$

Addition of rational numbers exercises

$~$\frac{7}{8}$~$

Addition of rational numbers exercises

$~$\frac{13}{8}$~$

Addition of rational numbers exercises

$~$\frac{a}{b}$~$

Addition of rational numbers exercises

$~$a$~$

Addition of rational numbers exercises

$~$b$~$

Addition of rational numbers exercises

$~$\frac{1}{8}$~$

Addition of rational numbers exercises

$~$\frac{1}{8}$~$

Addition of rational numbers exercises

$~$7$~$

Addition of rational numbers exercises

$~$13$~$

Addition of rational numbers exercises

$~$6$~$

Addition of rational numbers exercises

$~$\frac{6}{8}$~$

Addition of rational numbers exercises

$~$\frac{3}{4}$~$

Addition of rational numbers exercises

$~$\frac{7}{8}$~$

Addition of rational numbers exercises

$~$\frac{13}{7}$~$

Addition of rational numbers exercises

$~$\frac{a}{b}$~$

Addition of rational numbers exercises

$~$a$~$

Addition of rational numbers exercises

$~$b$~$

Addition of rational numbers exercises

$~$\frac{1}{8 \times 7} = \frac{1}{56}$~$

Addition of rational numbers exercises

$~$\frac{1}{8}$~$

Addition of rational numbers exercises

$~$\frac{1}{7}$~$

Addition of rational numbers exercises

$~$\frac{7 \times 7}{7 \times 8} = \frac{49}{56}$~$

Addition of rational numbers exercises

$~$\frac{8 \times 13}{8 \times 7} = \frac{104}{56}$~$

Addition of rational numbers exercises

$~$49$~$

Addition of rational numbers exercises

$~$104$~$

Addition of rational numbers exercises

$~$55$~$

Addition of rational numbers exercises

$~$\frac{1}{56}$~$

Addition of rational numbers exercises

$~$\frac{55}{56}$~$

Advanced agent properties

$~$\mathbb P(Y|X)$~$

Advanced agent properties

$~$X$~$

Advanced agent properties

$~$Y$~$

Advanced agent properties

$~$Y,$~$

Advanced agent properties

$~$X.$~$

Advanced agent properties

$~$X$~$

Advanced agent properties

$~$Y.$~$

Advanced nonagent

$~$\pi_0$~$

Advanced nonagent

$~$\mathbb E [U | \operatorname{do}(\pi_0), HumansObeyPlan]$~$

Advanced nonagent

$~$\mathbb E [U | \operatorname{do}(\pi_0)],$~$

Algebraic field

$~$(R, +, \times)$~$

Algebraic field

$~$R$~$

Algebraic field

$~$1$~$

Algebraic field

$~$0$~$

Algebraic field

$~$r \in R$~$

Algebraic field

$~$x \in R$~$

Algebraic field

$~$xr = rx = 1$~$

Algebraic field

$~$0 \not = 1$~$

Algebraic structure

$~$X$~$

Algebraic structure tree

$~$*$~$

Algebraic structure tree

$~$\circ$~$

Algebraic structure tree

$~$*$~$

Algebraic structure tree

$~$\circ$~$

Algebraic structure tree

$~$\circ$~$

Algebraic structure tree

$~$*$~$

Algebraic structure tree

$~$a \circ (b * c) = (a \circ b) * (a \circ c)$~$

Algebraic structure tree

$~$(a * b) \circ c = (a \circ c) * (b \circ c)$~$

Algebraic structure tree

$~$*$~$

Algebraic structure tree

$~$\circ$~$

Algebraic structure tree

$~$*$~$

Algebraic structure tree

$~$*$~$

Algebraic structure tree

$~$*$~$

Algebraic structure tree

$~$\circ$~$

Algebraic structure tree

$~$\circ$~$

Algebraic structure tree

$~$*$~$

Algebraic structure tree

$~$\circ$~$

Algebraic structure tree

$~$\circ$~$

Algebraic structure tree

$~$*$~$

Algebraic structure tree

$~$\circ$~$

Algebraic structure tree

$~$a \circ (a * b) = a * (a \circ b) = a$~$

Algebraic structure tree

$~$*$~$

Algebraic structure tree

$~$\circ$~$

Algebraic structure tree

$~$\wedge$~$

Algebraic structure tree

$~$\vee$~$

Algorithmic complexity

$~$3\uparrow\uparrow\uparrow3$~$

All you need for SAT Math Here!

$~$\frac{y_2-y_1}{x_2-x_1}=\frac{rise}{run}=tan\theta$~$

All you need for SAT Math Here!

$~$y=mx+b\rightarrow slope=m$~$

All you need for SAT Math Here!

$~$ax+by=c\rightarrow slope=\frac{-a}{b}$~$

All you need for SAT Math Here!

$~$\rightarrow$~$

All you need for SAT Math Here!

$~$\rightarrow$~$

All you need for SAT Math Here!

$~$\rightarrow$~$

All you need for SAT Math Here!

$~$\rightarrow$~$

All you need for SAT Math Here!

$~$y=mx+{b_1}, y=mx+{b_2}, {b_1}\neq {b_2}$~$

All you need for SAT Math Here!

$~$y=mx+{b_1}, y=\frac{-1}{m}x+{b_2}$~$

All you need for SAT Math Here!

$~${a_1}x+{b_1}y={c_1}$~$

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

$$~$ \newcommand{\bP}{\mathbb{P}} $~$$

Example: Dragon Pox

$~$\bP(D) = 0.4$~$

Example: Dragon Pox

$~$\bP(S \mid D) = 0.7$~$

Example: Dragon Pox

$~$\bP(S \mid \neg D) = 0.2$~$

Example: Dragon Pox

$~$(C)$~$

Example: Dragon Pox

$~$(\neg C)$~$

Example: Dragon Pox

$~$(L)$~$

Example: Dragon Pox

$~$(\neg L)$~$

Example: Dragon Pox

$$~$ \begin{align} \bP(L \mid \;\;D,\;\;C) &= 0.4\\ \bP(L \mid \;\;D,\neg C) &= 0.1\\ \bP(L \mid \neg D,\;\;C) &= 0.7\\ \bP(L \mid \neg D,\neg C) &= 0.9 \end{align} $~$$

Example: Dragon Pox

$~$D$~$

Example: Dragon Pox

$~$\bP(D) = 0.4$~$

Example: Dragon Pox

$~$S$~$

Example: Dragon Pox

$~$\bP(S \mid D) = 0.7$~$

Example: Dragon Pox

$~$\bP(S \mid \neg D) = 0.2$~$

Example: Dragon Pox

$~$(C)$~$

Example: Dragon Pox

$~$(\neg C)$~$

Example: Dragon Pox

$~$(L)$~$

Example: Dragon Pox

$~$D$~$

Example: Dragon Pox

$~$C$~$

Example: Dragon Pox

$$~$ \begin{align} \bP(L \mid \;\;D,\;\;C) &= 0.4\\ \bP(L \mid \;\;D,\neg C) &= 0.1\\ \bP(L \mid \neg D,\;\;C) &= 0.7\\ \bP(L \mid \neg D,\neg C) &= 0.9 \end{align} $~$$

Example: Dragon Pox

$~$\bP(L \mid D,C) > \bP(L \mid D,\neg C)$~$

Example: Dragon Pox

$~$\neg D$~$

Example: Dragon Pox

$~$\bP(L \mid \neg D,C) < \bP(L \mid \neg D,\neg C)$~$

Exchange rates between digits

$~$n$~$

Exchange rates between digits

$~$b$~$

Exchange rates between digits

$~$\log_b(n).$~$

Exchange rates between digits

$~$2^\text{3,000,000,000,000}$~$

Exchange rates between digits

$~$n$~$

Exchange rates between digits

$~$2^n$~$

Exchange rates between digits

$~$2^4=16$~$

Exchange rates between digits

$~$2^6 < 101 < 2^7$~$

Exchange rates between digits

$~$2^{12} < 8000 < 2^{13}$~$

Exchange rates between digits

$~$2^{13} < 15,000 < 2^{14}$~$

Exchange rates between digits

$~$x$~$

Exchange rates between digits

$~$n$~$

Exchange rates between digits

$~$n$~$

Exchange rates between digits

$~$3n$~$

Exchange rates between digits

$~$10^n > 2^{3n}$~$

Exchange rates between digits

$~$n$~$

Exchange rates between digits

$~$x$~$

Exchange rates between digits

$~$3n$~$

Exchange rates between digits

$~$n$~$

Exchange rates between digits

$~$10^n$~$

Exchange rates between digits

$~$2^3$~$

Exchange rates between digits

$~$2^{3n}$~$

Exchange rates between digits

$~$n$~$

Exchange rates between digits

$~$2^{3(n-1)}$~$

Exchange rates between digits

$~$n \ge 11,$~$

Exchange rates between digits

$~$x$~$

Exchange rates between digits

$~$10^{10} < 2^{35}.$~$

Exchange rates between digits

$~$2^{33} < 10^{10} < 2^{34},$~$

Exchange rates between digits

$~$2^{332} < 10^{100} < 2^{333},$~$

Exchange rates between digits

$~$p$~$

Exchange rates between digits

$~$2^p > 10$~$

Exchange rates between digits

$~$2^p < 10$~$

Exchange rates between digits

$~$p$~$

Exchange rates between digits

$~$2^p = 10,$~$

Exchange rates between digits

$~$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$~$

Exchange rates between digits

$~$2 + 2 + 2 + 2 + 2 = 10.$~$

Exchange rates between digits

$~$p$~$

Exchange rates between digits

$~$p$~$

Exchange rates between digits

$~$2^p = 10$~$

Exchange rates between digits

$~$\log_2(10),$~$

Exchange rates between digits

$~$\log_b(x)$~$

Exchange rates between digits

$~$x$~$

Exchange rates between digits

$~$b$~$

Exchange rates between digits

$~$\log_2(6) \approx 2.58$~$

Exchange rates between digits

$~$2^2 < 6 < 2^3$~$

Exchange rates between digits

$~$2^{25} < 6^{10} < 2^{26}$~$

Exchange rates between digits

$~$2^{258} < 6^{100} < 2^{259}.$~$

Exchange rates between digits

$~$\log_2(6)$~$

Exchange rates between digits

$~$\log_b(x)$~$

Exchange rates between digits

$~$b$~$

Exchange rates between digits

$~$x$~$

Exchange rates between digits

$~$b$~$

Exchange rates between digits

$~$x$~$

Exchange rates between digits

$~$b$~$

Exchange rates between digits

$~$x$~$

Exchange rates between digits

$~$x$~$

Exchange rates between digits

$~$b$~$

Exchange rates between digits

$~$b$~$

Exchange rates between digits

$~$x$~$

Exchange rates between digits

$~$\log_b(x)$~$

Exchange rates between digits

$~$x$~$

Exchange rates between digits

$~$b$~$

Exchange rates between digits

$~$\log_x(b) = \frac{1}{\log_b(x)}$~$

Exchange rates between digits

$~$x$~$

Exchange rates between digits

$~$b$~$

Exchange rates between digits

$~$b$~$

Exchange rates between digits

$~$x$~$

Exchange rates between digits

$~$\log_{1.5}(2.5)$~$

Existence Proof of Logical Inductor

$~$\overline{\mathbb{P}}$~$

Existence Proof of Logical Inductor

$~$\overline{D}$~$

Existence Proof of Logical Inductor

$~$\overline{T}$~$

Existence Proof of Logical Inductor

$~$\overline{\mathbb{P}}$~$

Existence Proof of Logical Inductor

$~$\overline{D}$~$

Existence Proof of Logical Inductor

$~$\overline{LIA}$~$

Existence Proof of Logical Inductor

$~$\overline{D}$~$

Existence Proof of Logical Inductor

$~$-b$~$

Existence Proof of Logical Inductor

$~$-b$~$

Existence Proof of Logical Inductor

$~$\overline{T}$~$

Existence Proof of Logical Inductor

$~$n$~$

Existence Proof of Logical Inductor

$~$\mathbb{P}_n$~$

Existence Proof of Logical Inductor

$~$T_n(\mathbb{P}_{\leq n})$~$

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])})}$~$

Existence Proof of Logical Inductor

$~$\mathbb{V}^{\text{fix}}$~$

Existence Proof of Logical Inductor

$~$\phi$~$

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])})}$~$

Existence Proof of Logical Inductor

$~$\mathcal{V}' \to \mathcal{V}'$~$

Existence Proof of Logical Inductor

$~$\mathcal{V}'$~$

Existence Proof of Logical Inductor

$~$[0,1]^{S'}$~$

Existence Proof of Logical Inductor

$~$x$~$

Existence Proof of Logical Inductor

$~$f(x)=x$~$

Existence Proof of Logical Inductor

$~$\text{fix}\mathbb{V}(\phi)$~$

Existence Proof of Logical Inductor

$~$T$~$

Existence Proof of Logical Inductor

$~$\mathbb{P}$~$

Existence Proof of Logical Inductor

$~$T$~$

Existence Proof of Logical Inductor

$~$1-2^{-n}$~$

Existence Proof of Logical Inductor

$~$2^{-n}$~$

Existence Proof of Logical Inductor

$~$B(n,b, T_n, \mathbb{P}_{\leq n-1})$~$

Existence Proof of Logical Inductor

$~$(n-1)$~$

Existence Proof of Logical Inductor

$~$m<n$~$

Expected value

$~$V = x_{1},$~$

Expected value

$~$V = x_{2}, …, $~$

Expected value

$~$V = x_{k}$~$

Expected value

$~$P(x_{i})$~$

Expected value

$~$V = x_{i}$~$

Expected value

$$~$\sum_{i=1}^{k}x_{i}P(x_{i})$~$$

Expected value

$~$x \in \mathbb{R}$~$

Expected value

$~$P(x)$~$

Expected value

$~$\lim_{dx \to 0}$~$

Expected value

$~$x<V<(x+dx)$~$

Expected value

$~$dx$~$

Expected value

$$~$\int_{-∞}^{∞}xP(x)dx$~$$

Explicit Bayes as a counter for 'worrying'

$~$\mathbb P(\text{cancel}|\text{desirable})$~$

Explicit Bayes as a counter for 'worrying'

$~$\mathbb P(\text{cancel}|\text{undesirable})$~$

Exponential

$~$b$~$

Exponential

$~$x$~$

Exponential

$~$b^x,$~$

Exponential

$~$b$~$

Exponential

$~$x$~$

Exponential

$~$10^3$~$

Exponential

$~$10 \cdot 10 \cdot 10 = 1000$~$

Exponential

$~$2^4=16,$~$

Exponential

$~$2 \cdot 2 \cdot 2 \cdot 2 = 16.$~$

Exponential

$~$x$~$

Exponential

$~$10^{1/2}$~$

Exponential

$~$n$~$

Exponential

$~$n$~$

Exponential

$~$n \approx 3.16,$~$

Exponential

$~$n \cdot n \approx 10.$~$

Exponential

$~$f(x) = c \times a^x$~$

Exponential

$~$c$~$

Exponential

$~$a$~$

Exponential

$~$1.02$~$

Exponential

$~$f(x) = 100 \times 1.02^x$~$

Exponential

$~$x$~$

Exponential

$~$x$~$

Exponential

$~$f(x) = 1 \times 2^x$~$

Exponential

$~$f(x) = f(x-1) \times 1.02$~$

Exponential

$~$\Delta f(x) = f(x+1) - f(x) = 0.02 \times f(x)$~$

Exponential

$~$f(x) = f(x-1) + 0.02 \times f(0)$~$

Exponential

$~$f(0)$~$

Exponential

$~$f(x)$~$

Exponential notation for function spaces

$~$X$~$

Exponential notation for function spaces

$~$Y$~$

Exponential notation for function spaces

$~$X$~$

Exponential notation for function spaces

$~$Y$~$

Exponential notation for function spaces

$~$X \to Y$~$

Exponential notation for function spaces

$~$Y^X$~$

Exponential notation for function spaces

$~$Y^3$~$

Exponential notation for function spaces

$~$Y$~$

Exponential notation for function spaces

$~$f : X \to Y$~$

Exponential notation for function spaces

$~$X$~$

Exponential notation for function spaces

$~$Y$~$

Exponential notation for function spaces

$~$Y$~$

Exponential notation for function spaces

$~$X$~$

Exponential notation for function spaces

$~$Y^n$~$

Exponential notation for function spaces

$~$n$~$

Exponential notation for function spaces

$~$Y$~$

Exponential notation for function spaces

$~$|X| = n$~$

Exponential notation for function spaces

$~$Y^X \cong Y^n$~$

Exponential notation for function spaces

$~$Z^{X \times Y} \cong (Z^X)^Y$~$

Exponential notation for function spaces

$~$Z^{X + Y} \cong Z^X \times Z^Y$~$

Exponential notation for function spaces

$~$Z^1 \cong Z$~$

Exponential notation for function spaces

$~$1$~$

Exponential notation for function spaces

$~$Z$~$

Exponential notation for function spaces

$~$Z$~$

Exponential notation for function spaces

$~$Z^0 \cong 1$~$

Exponential notation for function spaces

$~$0$~$

Exponential notation for function spaces

$~$Y^X$~$

Exponential notation for function spaces

$~$\text{Hom}_{\mathcal{C}}(X, Y)$~$

Exponential notation for function spaces

$~$\mathcal{C}$~$

Extensionality Axiom

$$~$ \forall A \forall B : ( \forall x : (x \in A \iff x \in B) \Rightarrow A=B)$~$$

Extensionality Axiom

$~$\{1,2\} = \{2,1\}$~$

Extensionality Axiom

$~$1$~$

Extensionality Axiom

$~$2$~$

Extensionality Axiom

$~$5$~$

Extensionality Axiom

$~$73$~$

Extraordinary claims require extraordinary evidence

$~$(1 : 9 ) \times (3 : 1) \ = \ (3 : 9) \ \cong \ (1 : 3)$~$

Extraordinary claims require extraordinary evidence

$~$X$~$

Extraordinary claims require extraordinary evidence

$~$X$~$

Extraordinary claims require extraordinary evidence

$~$X.$~$

Extraordinary claims require extraordinary evidence

$$~$\text{Likelihood ratio} = \dfrac{\text{Probability of seeing the evidence, assuming the claim is true}}{\text{Probability of seeing the evidence, assuming the claim is false}}$~$$

Extraordinary claims require extraordinary evidence

$~$10^{100}$~$

Extraordinary claims require extraordinary evidence

$~$10^{94}$~$

Extraordinary claims require extraordinary evidence

$~$(10^{94} : 1)$~$

Extraordinary claims require extraordinary evidence

$~$10^{-94}$~$

Extraordinary claims require extraordinary evidence

$~$(1 : 10^{100})$~$

Extraordinary claims require extraordinary evidence

$~$(1 : 10^6)$~$

Factorial

$~$5!$~$

Factorial

$~$1*2*3*4*5$~$

Factorial

$~$n$~$

Factorial

$~$n!=\prod_{i=1}^{n}i$~$

Factorial

$~$0! = 1$~$

Factorial

$~$n!$~$

Factorial

$~$n$~$

Factorial

$~$A$~$

Factorial

$~$B$~$

Factorial

$~$C$~$

Factorial

$$~$ABC$~$$

Factorial

$$~$ACB$~$$

Factorial

$$~$BAC$~$$

Factorial

$$~$BCA$~$$

Factorial

$$~$CAB$~$$

Factorial

$$~$CBA$~$$

Factorial

$~$6$~$

Factorial

$~$3$~$

Factorial

$~$6 = 3*2*1 = 3!$~$

Factorial

$~$1$~$

Factorial

$~$n$~$

Factorial

$~$n+1$~$

Factorial

$~$1$~$

Factorial

$$~$A$~$$

Factorial

$$~$1 = \prod_{i=1}^{1}i = 1!$~$$

Factorial

$~$\{A_{1},A_{2},…,A_{n},A_{n+1}\}$~$

Factorial

$~$n+1$~$

Factorial

$~$A_{n+1}$~$

Factorial

$~$n$~$

Factorial

$~$n$~$

Factorial

$~$n!$~$

Factorial

$~$A_{n+1}$~$

Factorial

$~$A_{n+1}$~$

Factorial

$~$n$~$

Factorial

$~$n$~$

Factorial

$~$n!$~$

Factorial

$~$A_{n+1}$~$

Factorial

$~$n!$~$

Factorial

$~$A_{n+1}$~$

Factorial

$~$n!*(n+1)$~$

Factorial

$~$(n+1)!$~$

Factorial

$~$x!$~$

Factorial

$$~$x! = \Gamma (x+1),$~$$

Factorial

$~$\Gamma $~$

Factorial

$$~$\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\mathrm{d} t$~$$

Factorial

$~$x$~$

Factorial

$$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$

Factorial

$~$x=1$~$

Factorial

$$~$\prod_{i=1}^{1}i = \int_{0}^{\infty}t^{1}e^{-t}\mathrm{d} t$~$$

Factorial

$$~$1=1$~$$

Factorial

$~$x$~$

Factorial

$$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$

Factorial

$~$x + 1$~$

Factorial

$$~$\prod_{i=1}^{x+1}i = \int_{0}^{\infty}t^{x+1}e^{-t}\mathrm{d} t$~$$

Factorial

$~$x+1$~$

Factorial

$$~$\prod_{i=1}^{x}i = \int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$

Factorial

$$~$(x+1)\prod_{i=1}^{x}i = (x+1)\int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$

Factorial

$$~$\prod_{i=1}^{x+1}i = (x+1)\int_{0}^{\infty}t^{x}e^{-t}\mathrm{d} t$~$$

Factorial

$$~$= 0+\int_{0}^{\infty}(x+1)t^{x}e^{-t}\mathrm{d} t$~$$

Factorial

$$~$= \left (-t^{x+1}e^{-t}) \right]_{0}^{\infty}+\int_{0}^{\infty}(x+1)t^{x}e^{-t}\mathrm{d} t$~$$

Factorial

$$~$= \left (-t^{x+1}e^{-t}) \right]_{0}^{\infty}-\int_{0}^{\infty}(x+1)t^{x}(-e^{-t})\mathrm{d} t$~$$

Factorial

$$~$=\int_{0}^{\infty}t^{x+1}e^{-t}\mathrm{d} t$~$$

Factorial

$~$n$~$

Factorial

$~$n$~$

Factorial

$~$1,2,3$~$

Factorial

$~$1,2,3$~$

Factorial

$~$1,3,2$~$

Factorial

$~$1$~$

Factorial

$~$2$~$

Factorial

$~$3$~$

Factorial

$~$6$~$

Factorial

$~$1,2,3$~$

Factorial

$~$1,3,2$~$

Factorial

$~$2,1,3$~$

Factorial

$~$2,3,1$~$

Factorial

$~$3,1,2$~$

Factorial

$~$3,2,1$~$

Factorial

$~$1$~$

Factorial

$~$2$~$

Factorial

$~$3$~$

Factorial

$~$6$~$

Factorial

$~$24$~$

Factorial

$~$1,2,3,4$~$

Factorial

$~$1,2,4,3$~$

Factorial

$~$1,3,2,4$~$

Factorial

$~$1,3,4,2$~$

Factorial

$~$1,4,2,3$~$

Factorial

$~$1,4,3,2$~$

Factorial

$~$2,1,3,4$~$

Factorial

$~$24$~$

Factorial

$~$6$~$

Factorial

$~$6$~$

Factorial

$~$1$~$

Factorial

$~$6$~$

Factorial

$~$2$~$

Factorial

$~$6$~$

Factorial

$~$3$~$

Factorial

$~$6$~$

Factorial

$~$4$~$

Factorial

$~$24$~$

Factorial

$~$120$~$

Factorial

$~$24$~$

Factorial

$~$24$~$

Factorial

$~$1$~$

Factorial

$~$24$~$

Factorial

$~$2$~$

Factorial

$~$24$~$

Factorial

$~$3$~$

Factorial

$~$24$~$

Factorial

$~$4$~$

Factorial

$~$24$~$

Factorial

$~$5$~$

Factorial

$~$120$~$

Factorial

$~$5$~$

Factorial

$~$4$~$

Factorial

$~$n$~$

Factorial

$~$n-1$~$

Factorial

$~$n$~$

Factorial

$~$n-1$~$

Factorial

$~$n$~$

Factorial

$~$n$~$

Factorial

$~$1$~$

Factorial

$~$2$~$

Factorial

$~$n$~$

Factorial

$~$n$~$

Factorial

$~$n-1$~$

Factorial

$~$n-1$~$

Factorial

$~$5!$~$

Factorial

$~$120$~$

Factorial

$~$4!$~$

Factorial

$~$n!$~$

Factorial

$~$n$~$

Factorial

$~$n$~$

Factorial

$~$5! = 5 \times 4!$~$

Factorial

$~$4! = 4 \times 3!$~$

Factorial

$~$5$~$

Factorial

$~$n-1$~$

Factorial

$~$n \times n - 1!$~$

Factorial

$~$(n \times n)-1!$~$

Factorial

$$~$n! = n \times (n-1)!$~$$

Factorial

$~$n! = n \times (n-1)!$~$

Factorial

$~$(n-1)! = (n-1) \times (n-2)!$~$

Factorial

$~$(n-2)! = (n-2) \times (n-3)!$~$

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,$~$

Generalized associative law

$~$[x, y, z, \ldots]$~$

Generalized associative law

$~$\chi,$~$

Generalized associative law

$~$f(\alpha, \chi)$~$

Generalized associative law

$~$[a, b, c, \ldots, x, y, z, \ldots]:$~$

Generalized associative law

$~$f$~$

Generalized associative law

$~$f$~$

Generalized associative law

$~$f$~$

Generalized associative law

$~$f$~$

Generalized associative law

$~$f_n : X^n \to X$~$

Generalized associative law

$~$n \ge 0,$~$

Generalized associative law

$~$0_X$~$

Generalized associative law

$~$X$~$

Generalized associative law

$~$f_0$~$

Generalized associative law

$~$0_X$~$

Generalized associative law

$~$f.$~$

Generalized element

$~$X$~$

Generalized element

$~$x : A \to X$~$

Generalized element

$~$X$~$

Generalized element

$~$A$~$

Generalized element

$~$x$~$

Generalized element

$~$I$~$

Generalized element

$~$*$~$

Generalized element

$~$I = \{*\}$~$

Generalized element

$~$X$~$

Generalized element

$~$X$~$

Generalized element

$~$I$~$

Generalized element

$~$X$~$

Generalized element

$~$x$~$

Generalized element

$~$X$~$

Generalized element

$~$I$~$

Generalized element

$~$X$~$

Generalized element

$~$f(i) = x$~$

Generalized element

$~$i \in I$~$

Generalized element

$~$f$~$

Generalized element

$~$x$~$

Generalized element

$~$f : I \to X$~$

Generalized element

$~$*$~$

Generalized element

$~$I$~$

Generalized element

$~$f(*)$~$

Generalized element

$~$X$~$

Generalized element

$~$X$~$

Generalized element

$~$I$~$

Generalized element

$~$I \to X$~$

Generalized element

$~$A$~$

Generalized element

$~$n$~$

Generalized element

$~$A$~$

Generalized element

$~$X$~$

Generalized element

$~$n$~$

Generalized element

$~$X$~$

Generalized element

$~$1$~$

Generalized element

$~$1$~$

Generalized element

$~$\mathbb{Z}$~$

Generalized element

$~$\mathbb{Z}$~$

Generalized element

$~$A$~$

Generalized element

$~$A$~$

Generalized element

$~$\text{Set} \times \text{Set}$~$

Generalized element

$~$(X,Y)$~$

Generalized element

$~$(2^A, 2^{X + B})$~$

Generalized element

$~$(2^{Y + A}, 2^{B})$~$

Generalized element

$~$(X,Y)$~$

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}$~$

Generalized element

$~$(X,Y)$~$

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}$~$

Generalized element

$~$X$~$

Generalized element

$~$Y$~$

Generalized element

$~$(0,1)$~$

Generalized element

$~$(1,0)$~$

Generalized element

$~$x$~$

Generalized element

$~$A$~$

Generalized element

$~$X$~$

Generalized element

$~$f$~$

Generalized element

$~$X$~$

Generalized element

$~$Y$~$

Generalized element

$~$f(x) := f\circ x$~$

Generalized element

$~$A$~$

Generalized element

$~$Y$~$

Generalized element

$~$f(xu) = f(x) u$~$

Geometric product

$~$e^{\text{I}\theta}$~$

Geometric product

$~$n$~$

Geometric product

$~$|a|^2 + |b|^2 = |a+b|^2$~$

Geometric product

$~$(a+b)^2 = a^2 + ab + ba + b^2$~$

Geometric product

$~$ab$~$

Geometric product

$~$ba$~$

Geometric product

$~$2ab$~$

Geometric product

$~$a^2 = |a|^2$~$

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|$~$

Geometric product

$~$\frac{1}{a}=\frac{a}{|a|^2}$~$

Geometric product

$~$a^{-1}$~$

Geometric product

$~$ae^{\text{I}\pi/2} = b$~$

Geometric product

$~$e^{\text{I}\pi/2} = \frac{ab}{|a|^2}$~$

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$~$

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)$~$

Geometric product

$~$k$~$

Geometry of vectors: direction

$~$\mathbf a$~$

Geometry of vectors: direction

$~$\mathbf b$~$

Geometry of vectors: direction

$~$\mathbf x$~$

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$~$

Geometry of vectors: direction

$~$\mathbf I$~$

Geometry of vectors: direction

$~$(\mathbf {x},\mathbf{y})$~$

Geometry of vectors: direction

$~$(\mathbf{y},\mathbf{z})$~$

Geometry of vectors: direction

$~$(\mathbf{x},\mathbf{z})$~$

Geometry of vectors: direction

$~$(\mathbf {x},\mathbf{y})$~$

Geometry of vectors: direction

$~$\mathbf w$~$

Geometry of vectors: direction

$~$\mathbf x$~$

Geometry of vectors: direction

$~$\mathbf y$~$

Geometry of vectors: direction

$~$\mathbf z$~$

Geometry of vectors: direction

$~$(\mathbf{w},\mathbf{x})$~$

Geometry of vectors: direction

$~$(\mathbf{w},\mathbf{y})$~$

Geometry of vectors: direction

$~$(\mathbf {w},\mathbf{z})$~$

Geometry of vectors: direction

$~$(\mathbf{x},\mathbf{y})$~$

Geometry of vectors: direction

$~$(\mathbf{x},\mathbf{z})$~$

Geometry of vectors: direction

$~$(\mathbf{y},\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$~$

Geometry of vectors: direction

$~$\mathbf a$~$

Geometry of vectors: direction

$~$\pi$~$

Geometry of vectors: direction

$~$3.14$~$

Geometry of vectors: direction

$~$\pi$~$

Geometry of vectors: direction

$~$\frac{\pi}{2}$~$

Geometry of vectors: direction

$~$\frac{\pi}{2}$~$

Geometry of vectors: direction

$~$0$~$

Geometry of vectors: direction

$~$\pi$~$

Geometry of vectors: direction

$~$\frac{\pi}{4}$~$

Geometry of vectors: direction

$~$R$~$

Geometry of vectors: direction

$~$\mathbf B$~$

Geometry of vectors: direction

$~$e$~$

Geometry of vectors: direction

$~$R = e^{\mathbf B}$~$

Goodhart's Curse

$~$V$~$

Goodhart's Curse

$~$V$~$

Goodhart's Curse

$~$U$~$

Goodhart's Curse

$~$V,$~$

Goodhart's Curse

$~$U$~$

Goodhart's Curse

$~$V,$~$

Goodhart's Curse

$~$U$~$

Goodhart's Curse

$~$U$~$

Goodhart's Curse

$~$V.$~$

Goodhart's Curse

$~$U$~$

Goodhart's Curse

$~$U-V$~$

Goodhart's Curse

$~$\|U - V\|$~$

Graham's number

$~$f(x) = 3\uparrow^n 3$~$

Graham's number

$~$f^n(x) = \underbrace{f(f(f(\cdots f(f(x)) \cdots ))}_{n\text{ applications of }f}$~$

Graham's number

$~$f^{64}(4).$~$

Greatest common divisor

$~$a$~$

Greatest common divisor

$~$b$~$

Greatest common divisor

$~$a$~$

Greatest common divisor

$~$b$~$

Greatest common divisor

$~$a$~$

Greatest common divisor

$~$b$~$

Greatest common divisor

$~$c$~$

Greatest common divisor

$~$c \mid a$~$

Greatest common divisor

$~$c \mid b$~$

Greatest common divisor

$~$d \mid a$~$

Greatest common divisor

$~$d \mid b$~$

Greatest common divisor

$~$d \mid c$~$

Greatest common divisor

$~$a$~$

Greatest common divisor

$~$b$~$

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$~$

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$~$

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$~$

Greatest lower bound in a poset

$~$P$~$

Greatest lower bound in a poset

$~$\leq$~$

Greatest lower bound in a poset

$~$x$~$

Greatest lower bound in a poset

$~$y$~$

Greatest lower bound in a poset

$~$P$~$

Greatest lower bound in a poset

$~$z \in P$~$

Greatest lower bound in a poset

$~$x$~$

Greatest lower bound in a poset

$~$y$~$

Greatest lower bound in a poset

$~$z \leq x$~$

Greatest lower bound in a poset

$~$z \leq y$~$

Greatest lower bound in a poset

$~$z \in P$~$

Greatest lower bound in a poset

$~$x$~$

Greatest lower bound in a poset

$~$y$~$

Greatest lower bound in a poset

$~$z$~$

Greatest lower bound in a poset

$~$x$~$

Greatest lower bound in a poset

$~$y$~$

Greatest lower bound in a poset

$~$w$~$

Greatest lower bound in a poset

$~$x$~$

Greatest lower bound in a poset

$~$y$~$

Greatest lower bound in a poset

$~$w \leq z$~$

Group

$~$120^\circ$~$

Group

$~$240^\circ$~$

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

$~$G$~$

Group

$~$(X, \bullet)$~$

Group

$~$X$~$

Group

$~$\bullet$~$

Group

$~$x, y$~$

Group

$~$X$~$

Group

$~$x \bullet y$~$

Group

$~$X$~$

Group

$~$x \bullet y$~$

Group

$~$xy$~$

Group

$~$x(yz) = (xy)z$~$

Group

$~$x, y, z \in X$~$

Group

$~$e$~$

Group

$~$xe=ex=x$~$

Group

$~$x \in X$~$

Group

$~$x$~$

Group

$~$X$~$

Group

$~$x^{-1} \in X$~$

Group

$~$xx^{-1}=x^{-1}x=e$~$

Group

$~$120^\circ$~$

Group

$~$240^\circ$~$

Group

$~$G$~$

Group

$~$(X, \bullet)$~$

Group

$~$X$~$

Group

$~$X$~$

Group

$~$G$~$

Group

$~$\bullet : G \times G \to G$~$

Group

$~$x \bullet y$~$

Group

$~$xy$~$

Group

$~$\bullet$~$

Group

$~$x, y$~$

Group

$~$X$~$

Group

$~$x \bullet y$~$

Group

$~$X$~$

Group

$~$x \bullet y$~$

Group

$~$xy$~$

Group

$~$e$~$

Group

$~$xe=ex=x$~$

Group

$~$x \in X$~$

Group

$~$x$~$

Group

$~$X$~$

Group

$~$x^{-1} \in X$~$

Group

$~$xx^{-1}=x^{-1}x=e$~$

Group

$~$x(yz) = (xy)z$~$

Group

$~$x, y, z \in X$~$

Group

$~$\bullet$~$

Group

$~$\bullet$~$

Group

$~$G \times G \to G$~$

Group

$~$e$~$

Group

$~$G$~$

Group

$~$\bullet$~$

Group

$~$e$~$

Group

$~$x$~$

Group

$~$\bullet$~$

Group

$~$x$~$

Group

$~$e$~$

Group

$~$z$~$

Group

$~$ze = ez = z.$~$

Group

$~$e$~$

Group

$~$G$~$

Group

$~$e$~$

Group

$~$e$~$

Group

$~$e$~$

Group

$~$1$~$

Group

$~$1_G$~$

Group

$~$\bullet$~$

Group

$~$X$~$

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)$~$

Group action induces homomorphism to the symmetric group

$~$\mathrm{Sym}(X)$~$

Group action induces homomorphism to the symmetric group

$~$\rho(gh)(x) = \rho(gh, x)$~$

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))$~$

Group action induces homomorphism to the symmetric group

$~$\rho(g)$~$

Group action induces homomorphism to the symmetric group

$~$\rho(g)(\rho(h)(x))$~$

Group action induces homomorphism to the symmetric group

$~$\rho(h)$~$

Group conjugate

$~$x, y$~$

Group conjugate

$~$G$~$

Group conjugate

$~$h \in G$~$

Group conjugate

$~$hxh^{-1} = y$~$

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$~$

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}$~$

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}$~$

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$~$

Introduction to Logical Decision Theory for Analytic Philosophers

$~$C$~$

Introduction to Logical Decision Theory for Analytic Philosophers

$~$C$~$

Introduction to Logical Decision Theory for Analytic Philosophers

$~$\mathcal T$~$

Introduction to Logical Decision Theory for Analytic Philosophers

$~$B$~$

Introduction to Logical Decision Theory for Analytic Philosophers

$~$D$~$

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.$~$

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)$~$

Logarithm

$~$\log_a(n) = \frac{\log_b(n)}{\log_b(a)}$~$

Logarithm

$~$\log_{10}(100)=2,$~$

Logarithm

$~$\log_b(\cdot)$~$

Logarithm

$~$b^{\ \cdot}.$~$

Logarithm

$~$\log_b(n) = x$~$

Logarithm

$~$b^x = n,$~$

Logarithm

$~$\log_b(b^x)=x$~$

Logarithm

$~$b^{\log_b(n)}=n.$~$

Logarithm

$~$e$~$

Logarithm

$~$e$~$

Logarithm

$~$x$~$

Logarithm

$~$\ln(x)$~$

Logarithm

$~$x$~$

Logarithm

$~$\frac{x}{\ln(x)},$~$

Logarithm

$~$n$~$

Logarithm

$~$\log_2(n),$~$

Logarithm base 1

$~$\log_b(1)=0$~$

Logarithm base 1

$~$b$~$

Logarithm base 1

$~$\log_b(b)=1$~$

Logarithm base 1

$~$b$~$

Logarithm base 1

$~$\log_1.$~$

Logarithm base 1

$~$\log(1 \cdot 1) = \log(1) + \log(1)$~$

Logarithm base 1

$~$x > 1$~$

Logarithm base 1

$~$\infty$~$

Logarithm base 1

$~$0 < x < 1$~$

Logarithm base 1

$~$-\infty,$~$

Logarithm base 1

$~$\log_1,$~$

Logarithm base 1

$~$\log_b(1)=0$~$

Logarithm base 1

$~$b$~$

Logarithm base 1

$~$\log_b(b)=1$~$

Logarithm base 1

$~$b$~$

Logarithm base 1

$~$\log_1.$~$

Logarithm base 1

$~$\log(1 \cdot 1) = \log(1) + \log(1)$~$

Logarithm base 1

$~$x > 1$~$

Logarithm base 1

$~$\infty$~$

Logarithm base 1

$~$0 < x < 1$~$

Logarithm base 1

$~$-\infty,$~$

Logarithm base 1

$~$1$~$

Logarithm base 1

$~$\log_1$~$

Logarithm base 1

$~$\mathbb R^+$~$

Logarithm base 1

$~$\mathbb R \cup \{ \infty, -\infty \}.$~$

Logarithm base 1

$~$1$~$

Logarithm base 1

$~$\mathbb R$~$

Logarithm base 1

$~$x > 1$~$

Logarithm base 1

$~$\{ \infty \}$~$

Logarithm base 1

$~$0 < x < 1$~$

Logarithm base 1

$~$\{ -\infty \}$~$

Logarithm base 1

$~$=$~$

Logarithm base 1

$~$\in$~$

Logarithm base 1

$~$1^{\{\infty\}}$~$

Logarithm base 1

$~$(1, \infty)$~$

Logarithm base 1

$~$1^{\{-\infty\}}$~$

Logarithm base 1

$~$(0, 1)$~$

Logarithm base 1

$~$0 \in \log_1(1)$~$

Logarithm base 1

$~$1 \in \log_1(1)$~$

Logarithm base 1

$~$\log_1(1) + \log_1(1) \in \log_1(1 \cdot 1)$~$

Logarithm base 1

$~$7 \in log_1(1^7)$~$

Logarithm base 1

$~$15 \in 1^{\log_1(15)}$~$

Logarithm tutorial overview

$~$\log_{10}(\text{2,310,426})$~$

Logarithm tutorial overview

$$~$\underbrace{\text{2,310,426}}_\text{7 digits}$~$$

Logarithm tutorial overview

$~$\log_{10}$~$

Logarithm tutorial overview

$~$\log_6(52).$~$

Logarithm tutorial overview

$~$f$~$

Logarithm tutorial overview

$~$f(x \cdot y) = f(x) + f(y)$~$

Logarithm tutorial overview

$~$x, y \in$~$

Logarithm tutorial overview

$~$\mathbb R^+$~$

Logarithm tutorial overview

$~$f$~$

Logarithm tutorial overview

$~$\log_e,$~$

Logarithm: Examples

$~$\log_{10}(100)=2.$~$

Logarithm: Examples

$~$\log_2(4)=2.$~$

Logarithm: Examples

$~$\log_2(3)\approx 1.58.$~$

Logarithm: Exercises

$~$\log_{10}(4321)$~$

Logarithmic identities

$~$b^{\log_b(n)} = \log_b(b^n) = n.$~$

Logarithmic identities

$~$\log_b(1) = 0$~$

Logarithmic identities

$~$\log_b(b) = 1$~$

Logarithmic identities

$~$\log_b(x\cdot y) = log_b(x) + \log_b(y).$~$

Logarithmic identities

$~$\log_b(x^n) = n\log_b(x).$~$

Logarithmic identities

$~$x^{\log_b(y)} = y^{\log_b(x)}.$~$

Logarithmic identities

$~$\log_b(n) = \frac{\log_a(n)}{\log_a(b)}$~$

Logarithmic identities

$~$\log_b(n)$~$

Logarithmic identities

$~$b$~$

Logarithmic identities

$~$n.$~$

Logarithmic identities

$~$b$~$

Logarithmic identities

$~$b^{\log_b(n)} = \log_b(b^n) = n.$~$

Logarithmic identities

$~$\log_b(1) = 0$~$

Logarithmic identities

$~$\log_b(b) = 1$~$

Logarithmic identities

$~$\log_b(x\cdot y) = log_b(x) + \log_b(y).$~$

Logarithmic identities

$~$\log_b(x^n) = n\log_b(x).$~$

Logarithmic identities

$~$x^{\log_b(y)} = y^{\log_b(x)}.$~$

Logarithmic identities

$~$\log_a(n) = \frac{\log_b(n)}{\log_b(a)}$~$

Logarithms invert exponentials

$~$\log_b(\cdot)$~$

Logarithms invert exponentials

$~$b^{(\cdot)}.$~$

Logarithms invert exponentials

$~$\log_b(n) = x$~$

Logarithms invert exponentials

$~$b^x = n,$~$

Logarithms invert exponentials

$~$\log_b(b^x)=x$~$

Logarithms invert exponentials

$~$b^{\log_b(n)}=n.$~$

Logarithms invert exponentials

$~$\log_2(2^3) = 3$~$

Logarithms invert exponentials

$~$2^{\log_2(8)} = 8.$~$

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$~$

Logical Induction (incomplete)

$~$\mathbb{P}_{LIL}(\sigma)>C*\mathbb{P}_{USM}(\sigma)$~$

Logical Induction (incomplete)

$~$\prod_{1}$~$

Logical Induction (incomplete)

$~$C*poly(n)$~$

Logical Induction (incomplete)

$~$\mathcal{O}(n^{100})$~$

Logical Induction (incomplete)

$~$2^{2^{n}}$~$

Logical Induction (incomplete)

$~$1.16*10^{77}$~$

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}$~$

Logical Inductor Notation and Definitions

$~$\alpha:\overline{\mathbb{V}}\rightarrow\mathbb{R}$~$

Logical Inductor Notation and Definitions

$~$\overline{\mathbb{V}}$~$

Logical Inductor Notation and Definitions

$~$\phi,\psi$~$

Logical Inductor Notation and Definitions

$~$\mathcal{S}$~$

Logical Inductor Notation and Definitions

$~$X$~$

Logical Inductor Notation and Definitions

$~$\forall x: X(x)\rightarrow x>3$~$

Logical Inductor Notation and Definitions

$~$\mathcal{U}$~$

Logical Inductor Notation and Definitions

$~$\overline{\phi}$~$

Logical Inductor Notation and Definitions

$~$\overline{X}$~$

Logical Inductor Notation and Definitions

$~$\overline{D}$~$

Logical Inductor Notation and Definitions

$~$D_{n}$~$

Logical Inductor Notation and Definitions

$~$D_{\infty}$~$

Logical Inductor Notation and Definitions

$~$\mathbb{V}$~$

Logical Inductor Notation and Definitions

$~$\mathcal{S}\rightarrow[0,1]$~$

Logical Inductor Notation and Definitions

$~$\mathbb{P}$~$

Logical Inductor Notation and Definitions

$~$\mathbb{P}(\phi)$~$

Logical Inductor Notation and Definitions

$~$\phi$~$

Logical Inductor Notation and Definitions

$~$\overline{\mathbb{P}}$~$

Logical Inductor Notation and Definitions

$~$\mathbb{P}_{n}$~$

Logical Inductor Notation and Definitions

$~$\mathbb{P}_{n}(\phi)$~$

Logical Inductor Notation and Definitions

$~$\phi$~$

Logical Inductor Notation and Definitions

$~$n$~$

Logical Inductor Notation and Definitions

$~$\mathbb{P}_{\infty}$~$

Logical Inductor Notation and Definitions

$~$\mathbb{W}$~$

Logical Inductor Notation and Definitions

$~$\mathbb{W}(\phi)$~$

Logical Inductor Notation and Definitions

$~$\phi$~$

Logical Inductor Notation and Definitions

$~$\mathbb{W}$~$

Logical Inductor Notation and Definitions

$~$\mathcal{PC}(\Gamma)$~$

Logical Inductor Notation and Definitions

$~$\Gamma$~$

Logical Inductor Notation and Definitions

$~$\Gamma$~$

Logical Inductor Notation and Definitions

$~$\Gamma$~$

Logical Inductor Notation and Definitions

$~$\mathbb{W}(X)$~$

Logical Inductor Notation and Definitions

$~$X$~$

Logical Inductor Notation and Definitions

$~$\mathbb(W)$~$

Logical Inductor Notation and Definitions

$~$X$~$

Logical Inductor Notation and Definitions

$~$X=\frac{1}{\sqrt{2}}+3\epsilon$~$

Logical Inductor Notation and Definitions

$~$\overline{\mathbb{V}}\rightarrow\mathbb{R}$~$

Logical Inductor Notation and Definitions

$~$\overline{\mathbb{V}}$~$

Logical Inductor Notation and Definitions

$~$\mathcal{F}$~$

Logical Inductor Notation and Definitions

$~$\phi^{*n}$~$

Logical Inductor Notation and Definitions

$~$\mathbb{P}_{n}(\phi)$~$

Logical Inductor Notation and Definitions

$~$\phi$~$

Logical Inductor Notation and Definitions

$~$\overline{\mathbb{P}}$~$

Logical Inductor Notation and Definitions

$~$\xi$~$

Logical Inductor Notation and Definitions

$~$\alpha^{\dagger}$~$

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(-,-)$~$

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}}$~$

Logical Inductor Notation and Definitions

$~$\mathcal{EF}$~$

Logical Inductor Notation and Definitions

$~$\alpha^{\dagger}_{n}$~$

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}$~$

Multiplication of rational numbers (Math 0)

$~$\frac{a}{n}$~$

Multiplication of rational numbers (Math 0)

$~$n$~$

Multiplication of rational numbers (Math 0)

$~$\frac{1}{n}$~$

Multiplication of rational numbers (Math 0)

$~$a$~$

Multiplication of rational numbers (Math 0)

$~$a$~$

Multiplication of rational numbers (Math 0)

$~$\frac{1}{n}$~$

Multiplication of rational numbers (Math 0)

$~$1$~$

Multiplication of rational numbers (Math 0)

$~$\frac{a}{n}$~$

Multiplication of rational numbers (Math 0)

$~$\frac{1}{n} \times a$~$

Multiplication of rational numbers (Math 0)

$~$a$~$

Multiplication of rational numbers (Math 0)

$~$n$~$

Multiplication of rational numbers (Math 0)

$~$\frac{1}{n}$~$

Multiplication of rational numbers (Math 0)

$~$a$~$

Multiplication of rational numbers (Math 0)

$~$1$~$

Multiplication of rational numbers (Math 0)

$~$\frac{a}{n} = a \times \frac{1}{n}$~$

Multiplication of rational numbers (Math 0)

$~$a$~$

Multiplication of rational numbers (Math 0)

$~$\frac{a}{1}$~$

Multiplication of rational numbers (Math 0)

$~$\frac{a}{n}$~$

Multiplication of rational numbers (Math 0)

$~$\frac{a}{1} \times \frac{1}{n}$~$

Multiplication of rational numbers (Math 0)

$~$\frac{a}{1} \times \frac{1}{n} = \frac{a \times 1}{1 \times n} = \frac{a}{n}$~$

Multiplication of rational numbers (Math 0)

$~$0$~$

Multiplication of rational numbers (Math 0)

$~$a$~$

Multiplication of rational numbers (Math 0)

$~$0$~$

Multiplication of rational numbers (Math 0)

$~$\frac{c}{d}$~$

Multiplication of rational numbers (Math 0)

$~$\frac{a}{b} \times \frac{c}{d} = 1$~$

Multiplication of rational numbers (Math 0)

$~$1$~$

Multiplication of rational numbers (Math 0)

$~$0$~$

Multiplication of rational numbers (Math 0)

$~$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$~$

Multiplication of rational numbers (Math 0)

$~$1$~$

Multiplication of rational numbers (Math 0)

$~$a \times c$~$

Multiplication of rational numbers (Math 0)

$~$b \times d$~$

Multiplication of rational numbers (Math 0)

$~$c = b$~$

Multiplication of rational numbers (Math 0)

$~$d = a$~$

Multiplication of rational numbers (Math 0)

$~$\frac{a \times b}{b \times a} = \frac{a \times b}{a \times b} = \frac{1}{1} = 1$~$

Multiplication of rational numbers (Math 0)

$~$\frac{b}{a}$~$

Multiplication of rational numbers (Math 0)

$~$\frac{a}{b}$~$

Multiplication of rational numbers (Math 0)

$~$a$~$

Multiplication of rational numbers (Math 0)

$~$0$~$

Multiplication of rational numbers (Math 0)

$~$\frac{b}{a}$~$

Multiplication of rational numbers (Math 0)

$~$a$~$

Mutually exclusive and exhaustive

$~$X$~$

Mutually exclusive and exhaustive

$~$\forall i: \forall j: i \neq j \implies \mathbb{P}(X_i \wedge X_j) = 0.$~$

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).$~$

Mutually exclusive and exhaustive

$~$X,$~$

Mutually exclusive and exhaustive

$~$1$~$

Mutually exclusive and exhaustive

$~$X_i$~$

Mutually exclusive and exhaustive

$~$1$~$

Mutually exclusive and exhaustive

$~$\mathbb{P}(X_1 \vee X_2 \vee \dots \vee X_N) = 1.$~$

Mutually exclusive and exhaustive

$~$\displaystyle \sum_i \mathbb{P}(X_i) = 1.$~$

Natural number

$~$\mathbb N.$~$

Natural number

$~$\mathbb N.$~$

Natural number

$~$2 + 3 = 5$~$

Natural number

$~$2 \cdot 3 = 6$~$

Natural numbers: Intro to Number Sets

$~$0$~$

Natural numbers: Intro to Number Sets

$~$0$~$

Natural numbers: Intro to Number Sets

$~$0$~$

Natural numbers: Intro to Number Sets

$~$1$~$

Natural numbers: Intro to Number Sets

$~$2$~$

Natural numbers: Intro to Number Sets

$~$3$~$

Natural numbers: Intro to Number Sets

$~$\{0, 1, 2, 3, \ldots\}$~$

Natural numbers: Intro to Number Sets

$~$4$~$

Natural numbers: Intro to Number Sets

$~$5$~$

Natural numbers: Intro to Number Sets

$~$n$~$

Natural numbers: Intro to Number Sets

$~$n'$~$

Natural numbers: Intro to Number Sets

$~$n$~$

Natural numbers: Intro to Number Sets

$~$2'$~$

Natural numbers: Intro to Number Sets

$~$3$~$

Natural numbers: Intro to Number Sets

$~$59'$~$

Natural numbers: Intro to Number Sets

$~$60$~$

Natural numbers: Intro to Number Sets

$~$9287'$~$

Natural numbers: Intro to Number Sets

$~$9288$~$

Natural numbers: Intro to Number Sets

$~$\prime$~$

Natural numbers: Intro to Number Sets

$~$2'''$~$

Natural numbers: Intro to Number Sets

$~$5$~$

Natural numbers: Intro to Number Sets

$~$0$~$

Natural numbers: Intro to Number Sets

$~$1$~$

Natural numbers: Intro to Number Sets

$~$1$~$

Natural numbers: Intro to Number Sets

$~$0$~$

Natural numbers: Intro to Number Sets

$~$n < m$~$

Natural numbers: Intro to Number Sets

$~$m$~$

Natural numbers: Intro to Number Sets

$~$n''''^\ldots$~$

Natural numbers: Intro to Number Sets

$~$\prime$~$

Natural numbers: Intro to Number Sets

$~$5 = 2'''$~$

Natural numbers: Intro to Number Sets

$~$2 < 5$~$

Natural numbers: Intro to Number Sets

$~$a < b$~$

Natural numbers: Intro to Number Sets

$~$b < c$~$

Natural numbers: Intro to Number Sets

$~$a < c$~$

Natural numbers: Intro to Number Sets

$~$a < b$~$

Natural numbers: Intro to Number Sets

$~$b < c$~$

Natural numbers: Intro to Number Sets

$~$c < a$~$

Natural numbers: Intro to Number Sets

$~$2 < 4$~$

Natural numbers: Intro to Number Sets

$~$4 < 6$~$

Natural numbers: Intro to Number Sets

$~$2 < 6$~$

Natural numbers: Intro to Number Sets

$~$a < b$~$

Natural numbers: Intro to Number Sets

$~$b = a''''^\ldots$~$

Natural numbers: Intro to Number Sets

$~$\prime$~$

Natural numbers: Intro to Number Sets

$~$b < c$~$

Natural numbers: Intro to Number Sets

$~$c = b''''^\ldots$~$

Natural numbers: Intro to Number Sets

$~$\prime$~$

Natural numbers: Intro to Number Sets

$~$c = b''''^\ldots = (a''''^\ldots)''''^\ldots$~$

Natural numbers: Intro to Number Sets

$~$a''''^\ldots$~$

Natural numbers: Intro to Number Sets

$~$\prime$~$

Natural numbers: Intro to Number Sets

$~$4 + 3$~$

Natural numbers: Intro to Number Sets

$~$4 = 0''''$~$

Natural numbers: Intro to Number Sets

$~$4 + 3 = 3'''' = 7$~$

Natural numbers: Intro to Number Sets

$~$4 \times 3$~$

Natural numbers: Intro to Number Sets

$~$4 = 0''''$~$

Natural numbers: Intro to Number Sets

$~$4 \times 3 = 0 + 3 + 3 + 3 + 3 = 12$~$

Natural numbers: Intro to Number Sets

$~$\{1, 3, 5, 7, 9,\ldots\}$~$

Natural numbers: Intro to Number Sets

$~$3$~$

Natural numbers: Intro to Number Sets

$~$5$~$

Natural numbers: Intro to Number Sets

$~$3$~$

Natural numbers: Intro to Number Sets

$~$3$~$

Natural numbers: Intro to Number Sets

$~$6$~$

Natural numbers: Intro to Number Sets

$~$a$~$

Natural numbers: Intro to Number Sets

$~$c$~$

Natural numbers: Intro to Number Sets

$~$b$~$

Natural numbers: Intro to Number Sets

$~$a + b = c$~$

Natural numbers: Intro to Number Sets

$~$3$~$

Natural numbers: Intro to Number Sets

$~$2$~$

Natural numbers: Intro to Number Sets

$~$2$~$

Natural numbers: Intro to Number Sets

$~$3$~$

Natural numbers: Intro to Number Sets

$~$n$~$

Natural numbers: Intro to Number Sets

$~$n + 3 = 2$~$

Natural numbers: Intro to Number Sets

$~$a$~$

Natural numbers: Intro to Number Sets

$~$c$~$

Natural numbers: Intro to Number Sets

$~$b$~$

Natural numbers: Intro to Number Sets

$~$a \times b = c$~$

Natural numbers: Intro to Number Sets

$~$3$~$

Natural numbers: Intro to Number Sets

$~$2$~$

Natural numbers: Intro to Number Sets

$~$n$~$

Natural numbers: Intro to Number Sets

$~$n \times 2 = 3$~$

Natural numbers: Intro to Number Sets

$~$1$~$

Nearest unblocked strategy

$~$X$~$

Nearest unblocked strategy

$~$P$~$

Nearest unblocked strategy

$~$X,$~$

Nearest unblocked strategy

$~$X'$~$

Nearest unblocked strategy

$~$X$~$

Nearest unblocked strategy

$~$P.$~$

Nearest unblocked strategy

$~$i,$~$

Nearest unblocked strategy

$~$U_i,$~$

Nearest unblocked strategy

$~$X_i$~$

Nearest unblocked strategy

$~$P_i$~$

Nearest unblocked strategy

$~$U_i$~$

Nearest unblocked strategy

$~$U_{i+1},$~$

Nearest unblocked strategy

$~$X_i^*$~$

Nearest unblocked strategy

$~$U_{i+1}$~$

Nearest unblocked strategy

$~$X_{i+1}$~$

Nearest unblocked strategy

$~$X_i$~$

Nearest unblocked strategy

$~$P_i,$~$

Nearest unblocked strategy

$~$P_{i+1}$~$

Nearest unblocked strategy

$~$X,$~$

Nearest unblocked strategy

$~$X'$~$

Nearest unblocked strategy

$~$X.$~$

Nearest unblocked strategy

$~$X$~$

Nearest unblocked strategy

$~$G$~$

Nearest unblocked strategy

$~$X$~$

Nearest unblocked strategy

$~$X'$~$

Nearest unblocked strategy

$~$G.$~$

Negation of propositions

$~$S$~$

Negation of propositions

$~$Q$~$

Negation of propositions

$~$P$~$

Negation of propositions

$~$Q$~$

Negation of propositions

$~$Q$~$

Negation of propositions

$~$P$~$

Negation of propositions

$~$ Q \equiv \neg P$~$

Negation of propositions

$~$P$~$

Negation of propositions

$~$\neg P$~$

Negation of propositions

$~$P$~$

Negation of propositions

$~$\neg P$~$

Normal subgroup

$~$N$~$

Normal subgroup

$~$G$~$

Normal subgroup

$~$h \in G$~$

Normal subgroup

$~$\{ h n h^{-1} : n \in N \} = N$~$

Normal subgroup

$~$hNh^{-1} = N$~$

Normal subgroup

$~$G$~$

Normal subgroup

$~$G$~$

Normal subgroup

$~$G$~$

Normal subgroup

$~$H$~$

Normal subgroup

$~$f$~$

Normal subgroup

$~$f$~$

Normal subgroup

$~$f$~$

Normal system of provability logic

$~$L\vdash A$~$

Normal system of provability logic

$~$L\vdash \square A$~$

Normal system of provability logic

$~$L\vdash A\rightarrow B$~$

Normal system of provability logic

$~$L\vdash A$~$

Normal system of provability logic

$~$L\vdash B$~$

Normal system of provability logic

$~$\square(A\rightarrow B)\rightarrow (\square A \rightarrow \square B)$~$

Normal system of provability logic

$~$L\vdash F(p)$~$

Normal system of provability logic

$~$L\vdash F(H)$~$

Normal system of provability logic

$~$H$~$

Normal system of provability logic

$~$L\vdash \square(A_1\wedge … \wedge A_n)\leftrightarrow (\square A_1 \wedge … \wedge \square A_n)$~$

Normal system of provability logic

$~$L\vdash A\rightarrow B$~$

Normal system of provability logic

$~$L\vdash \square A \rightarrow \square B$~$

Normal system of provability logic

$~$L\vdash \diamond A \rightarrow \diamond B$~$

Normal system of provability logic

$~$L\vdash \diamond A \wedge \square B \rightarrow \diamond (A\wedge B)$~$

Normal system of provability logic

$~$L\vdash A\leftrightarrow B$~$

Normal system of provability logic

$~$F(p)$~$

Normal system of provability logic

$~$p$~$

Normal system of provability logic

$~$L\vdash F(A)\leftrightarrow F(B)$~$

Normalization (probability)

$~$3:2 \cong \frac{3}{3+2} : \frac{2}{3+2} = 0.6 : 0.4.$~$

Normalization (probability)

$~$\frac{3}{3+2} : \frac{2}{3+2} = 0.6 : 0.4.$~$

Normalization (probability)

$~$m : n$~$

Normalization (probability)

$~$\frac{m}{m+n} : \frac{n}{m+n},$~$

Normalization (probability)

$~$\frac{1}{m+n}$~$

Normalization (probability)

$~$\mathbb{O}(H)$~$

Normalization (probability)

$~$H$~$

Normalization (probability)

$~$\mathbb{P}(H)$~$

Normalization (probability)

$~$\mathbb{O}(H)$~$

Normalization (probability)

$~$\mathbb{O}(H).$~$

Normalization (probability)

$~$\mathbb{P}(H_i) = \frac{\mathbb{O}(H_i)}{\sum_i \mathbb{O}(H_i)}$~$

Normalization (probability)

$~$\mathbb{O}(x)$~$

Normalization (probability)

$~$X$~$

Normalization (probability)

$~$\mathbb{P}(x)$~$

Normalization (probability)

$~$\mathbb{O}(x)$~$

Normalization (probability)

$~$\mathbb{P}(x) = \frac{\mathbb{O}(x)}{\int \mathbb{O}(x) \operatorname{d}x}$~$

Normalization (probability)

$~$\mathbb{P}(H) \propto \mathbb{O}(H) \implies \mathbb{P}(H) = \frac{\mathbb{O}(H)}{\sum \mathbb{O}(H)}$~$

Note on 1x1 Convolutions

$$~$ \vec{y_{n}}=(\mathbf{W_n}^T \times \vec{y_{n-1}} + \vec{b_n})+\mathbf{W_{new}}^T (\mathbf{W_n}^T \times \vec{y_{n-1}} + \vec{b_n}) $~$$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$B$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$B$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$B$~$

Object identity via interactions

$~$\{1\}$~$

Object identity via interactions

$~$\{1\} \to A$~$

Object identity via interactions

$~$\{1\} \to B$~$

Object identity via interactions

$~$\{1\}$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$f$~$

Object identity via interactions

$~$1$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$\{1\}$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$A = \{ 5, 6 \}$~$

Object identity via interactions

$~$\{1\} \to A$~$

Object identity via interactions

$~$f: 1 \mapsto 5$~$

Object identity via interactions

$~$g: 1 \mapsto 6$~$

Object identity via interactions

$~$5 \in A$~$

Object identity via interactions

$~$f$~$

Object identity via interactions

$~$6 \in A$~$

Object identity via interactions

$~$g$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$\{ f, g \}$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$A$~$

Object identity via interactions

$~$B$~$

Object identity via interactions

$~$\{1\} \to B$~$

Odds

$~$2:3 = \frac{2}{3+2}:\frac{3}{3+2} = 0.4:0.6.$~$

Odds

$~$7:9$~$

Odds form to probability form

$~$H_i$~$

Odds form to probability form

$~$H_j,$~$

Odds form to probability form

$$~$\frac{\mathbb P(H_i \mid e)}{\mathbb P(H_j \mid e)} = \frac{\mathbb P(H_i)}{\mathbb P(H_j)} \times \frac{\mathbb P(e \mid H_i)}{\mathbb P(e \mid H_j)} \tag{1}.$~$$

Odds form to probability form

$~$H_1,H_2,H_3,\ldots$~$

Odds form to probability form

$$~$\mathbb P(H_i\mid e) = \frac{\mathbb P(e\mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P(e\mid H_k) \cdot \mathbb P(H_k)} \tag{2}.$~$$

Odds form to probability form

$~$H_1,H_2,H_3,\ldots$~$

Odds form to probability form

$~$H_i$~$

Odds form to probability form

$~$\lnot H_i$~$

Odds form to probability form

$~$H_1,H_2,H_3,\ldots$~$

Odds form to probability form

$~$H_i.$~$

Odds form to probability form

$~$\lnot H_i$~$

Odds form to probability form

$~$H_j$~$

Odds form to probability form

$$~$\frac{\mathbb P(H_i \mid e)}{\mathbb P(\lnot H_i \mid e)} = \frac{\mathbb P(H_i) \cdot \mathbb P(e \mid H_i)}{\mathbb P(\lnot H_i)\cdot \mathbb P(e \mid \lnot H_i)}.$~$$

Odds form to probability form

$~$\mathbb P(\lnot H_i)\cdot \mathbb P(e \mid \lnot H_i)$~$

Odds form to probability form

$~$\lnot H_i$~$

Odds form to probability form

$~$\lnot H_i$~$

Odds form to probability form

$~$e.$~$

Odds form to probability form

$~$\lnot H_i$~$

Odds form to probability form

$~$\mathbb P(H_k) \cdot \mathbb P(e \mid H_k)$~$

Odds form to probability form

$~$H_k$~$

Odds form to probability form

$~$H_i.$~$

Odds form to probability form

$$~$\frac{\mathbb P(H_i \mid e)}{\mathbb P(\lnot H_i \mid e)} = \frac{\mathbb P(e \mid H_i) \cdot \mathbb P(H_i)}{\sum_{k \neq i} \mathbb P(e \mid H_k) \cdot \mathbb P(H_k)}.$~$$

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,$~$

Odds form to probability form

$~$3 : 4$~$

Odds form to probability form

$$~$\mathbb P(H_i\mid e) = \frac{\mathbb P(e\mid H_i) \cdot \mathbb P(H_i)}{\sum_k \mathbb P(e\mid H_k) \cdot \mathbb P(H_k)}.$~$$

Odds: Introduction

$~$(x : y)$~$

Odds: Introduction

$~$\alpha$~$

Odds: Introduction

$~$(\alpha x : \alpha y).$~$

Odds: Introduction

$~$(r : b : g)$~$

Odds: Introduction

$~$(p_r : p_b : p_g)$~$

Odds: Introduction

$~$p_r + p_g + p_b$~$

Odds: Introduction

$~$(p_r : p_b : p_g)$~$

Odds: Introduction

$~$(r : b : g)$~$

Odds: Introduction

$~$(1 : 2 : 1)$~$

Odds: Introduction

$~$\frac{1}{4} : \frac{2}{4} : \frac{1}{4},$~$

Odds: Introduction

$~$(r : b)$~$

Odds: Introduction

$~$(a : b : c \ldots)$~$

Odds: Introduction

$~$(a + b + c \ldots)$~$

Odds: Introduction

$$~$\left(\frac{a}{a + b + c \ldots} : \frac{b}{a + b + c \ldots} : \frac{c}{a + b + c \ldots}\ldots\right)$~$$

Odds: Introduction

$~$\mathbb P(R) = \frac{1}{4}$~$

Odds: Introduction

$~$\mathbb P(B) = \frac{1}{2}.$~$

Odds: Introduction

$~$\mathbb P(R) : \mathbb P(B) = \left(\frac{1}{4} : \frac{1}{2}\right),$~$

Odds: Introduction

$~$\left(\frac{\mathbb P(R)}{\mathbb P(B)} : 1\right)$~$

Odds: Introduction

$~$\frac{\mathbb P(R)}{\mathbb P(B)}$~$

Odds: Introduction

$~$\frac{\mathbb P(R)}{\mathbb P(B)} = \frac{1}{2},$~$

Odds: Introduction

$~$\frac{\mathbb P(R)}{\mathbb P(B)}$~$

Odds: Introduction

$~$\left(\frac{\mathbb P(R)}{\mathbb P(B)} : 1\right).$~$

Odds: Introduction

$~$x$~$

Odds: Introduction

$~$y$~$

Odds: Introduction

$~$\frac{x}{y}.$~$

Odds: Introduction

$~$\frac{x}{y}$~$

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

$$~$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots$~$$

Primer on Infinite Series

$$~$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots = 1$~$$

Primer on Infinite Series

$~$\frac{1}{2}$~$

Primer on Infinite Series

$$~$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots = 1$~$$

Primer on Infinite Series

$$~$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots \approx \text{ (is approximately equal to) }1$~$$

Primer on Infinite Series

$$~$\frac{1}{2}+\frac{1}{4}+\frac{1}{8} \approx 1$~$$

Primer on Infinite Series

$$~$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32} \approx 1$~$$

Primer on Infinite Series

$$~$ \text{Circumference }(C) = \pi \cdot \text{Diameter }(D) = 2\pi \cdot \text{Radius }(r)$~$$

Primer on Infinite Series

$$~$ \text{Area of a Circle}(A) = \pi \cdot r^2$~$$

Primer on Infinite Series

$~$\pi$~$

Primer on Infinite Series

$~$\frac{C}{D}$~$

Primer on Infinite Series

$~$\pi$~$

Primer on Infinite Series

$~$\frac{C}{D} = \pi$~$

Primer on Infinite Series

$~$C=\pi D$~$

Primer on Infinite Series

$$~$\text{Area of polygon with }n \text{ sides} = \text{sum of } n \text{ triangles}$~$$

Primer on Infinite Series

$~$= \frac{1}{2} \cdot \text{base}\cdot \text{height}$~$

Primer on Infinite Series

$~$s$~$

Primer on Infinite Series

$~$h$~$

Primer on Infinite Series

$$~$\text{Area of polygon with }n \text{ sides} = \underbrace{\frac{1}{2}sh+\frac{1}{2}sh+\ldots+\frac{1}{2}sh}_{n \text{ times}}$~$$

Primer on Infinite Series

$~$\frac{1}{2}h$~$

Primer on Infinite Series

$$~$\text{Area of polygon with }n \text{ sides} = \frac{1}{2}h(\underbrace{s+s+\ldots+s}_{n \text{ times}})$~$$

Primer on Infinite Series

$~$n$~$

Primer on Infinite Series

$~$P$~$

Primer on Infinite Series

$$~$\text{Area of polygon} = \frac{1}{2}hP$~$$

Primer on Infinite Series

$$~$ \text{Area of polygon} \rightarrow \text{Area of a Circle}(A)$~$$

Primer on Infinite Series

$$~$h \rightarrow r$~$$

Primer on Infinite Series

$$~$P \rightarrow C$~$$

Primer on Infinite Series

$$~$ A = \frac{1}{2}rC = \frac{1}{2}r(\pi D) = \frac{1}{2}r(\pi(2r))=\pi r^2$~$$

Primer on Infinite Series

$$~$\text{infinite sum} = \text{finite number}$~$$

Principal ideal domain

$~$I$~$

Principal ideal domain

$~$i \in I$~$

Principal ideal domain

$~$\langle i \rangle = I$~$

Principal ideal domain

$~$I$~$

Principal ideal domain

$~$i$~$

Principal ideal domain

$~$R$~$

Principal ideal domain

$~$R$~$

Principal ideal domain

$~$\mathbb{Z}$~$

Principal ideal domain

$~$\{ 0 \}$~$

Principal ideal domain

$~$0$~$

Principal ideal domain

$~$1$~$

Principal ideal domain

$~$F[X]$~$

Principal ideal domain

$~$F$~$

Principal ideal domain

$~$\mathbb{Z}[i]$~$

Principal ideal domain

$~$\mathbb{Z}[X]$~$

Principal ideal domain

$~$\langle 2, X \rangle$~$

Principal ideal domain

$~$\mathbb{Z}_6$~$

Principal ideal domain

$~$3 \times 2 = 0$~$

Principal ideal domain

$~$\mathbb{Z}[\frac{1}{2} (1+\sqrt{-19})]$~$

Principal ideal domain

$~$\mathbb{Z}[X]$~$

Prior probability

$~$H$~$

Prior probability

$~$\mathbb P(H)$~$

Prior probability

$~$e$~$

Prior probability

$~$\mathbb P(H\mid e).$~$

Prior probability

$~$\mathbb P (H\mid I_0)$~$

Prior probability

$~$H$~$

Prior probability

$~$I_0$~$

Prior probability

$~$\mathbb P(X)$~$

Prior probability

$~$e_0$~$

Prior probability

$~$e_1$~$

Prior probability

$~$\mathbb P(H\mid e_0)$~$

Prior probability

$~$\mathbb P(H\mid e_1 \wedge e_0).$~$

Prisoner's Dilemma

$~$D$~$

Prisoner's Dilemma

$~$C$~$

Prisoner's Dilemma

$~$(p_1, p_2)$~$

Prisoner's Dilemma

$$~$\begin{array}{r|c|c} & D_2 & C_2 \\ \hline D_1 & (\$1, \$1) & (\$3, \$0) \\ \hline C_1 & (\$0, \$3) & (\$2, \$2) \end{array}$~$$

Prisoner's Dilemma

$~$(o_1, o_2)$~$

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$~$

Prisoner's Dilemma

$~$C,$~$

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)$~$

Probability

$~$X$~$

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)$~$

Probability

$~$\mathbb{P}(X \wedge Y)$~$

Probability

$~$\mathbb{P}(X \vee Y)$~$

Probability

$~$\mathbb{P}(X|Y) := \frac{\mathbb{P}(X \wedge Y}{\mathbb{P}(Y)}$~$

Probability

$~$\mathbb{P}(X|Y)$~$

Probability

$~$\mathbb{P}(yellow|banana)$~$

Probability

$~$\mathbb{P}(banana|yellow)$~$

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)

$~$\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$~$

Probability distribution (countable sample space)

$~$\mathbb{P}(x) = r$~$

Probability distribution (countable sample space)

$~$\Omega$~$

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(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}$~$

Probability interpretations: Examples

$~$f=\frac{1}{3}$~$

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).$~$

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).$~$

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$~$

ε-ROI Lemma

$~$\displaystyle\sum_{i>n}\beta_{i}\alpha_{i}\le\frac{1}{2}$~$

ε-ROI Lemma

$$~$\le 0+\sum_{n<i<N}\beta_{i}\alpha_{i}\le\frac{1}{2}$~$$

ε-ROI Lemma

$$~$\alpha_{k}\beta_{k}=\alpha_{k}\left(1-\sum_{i<k}open(i,k)\beta_{i}\alpha_{i}\right)$~$$

ε-ROI Lemma

$~$\displaystyle\sum_{i>N}open(i,N)\beta_{i}\alpha_{i}\le\frac{1}{2}$~$

ε-ROI Lemma

$~$k>N$~$

ε-ROI Lemma

$~$k>N$~$

ε-ROI Lemma

$~$\alpha_{k}>\delta$~$

ε-ROI Lemma

$$~$\alpha_{k}\left(1-\sum_{i<k}open(i,k)\beta_{i}\alpha_{i}\right)\ge\alpha_{k}(1-\frac{1}{2})\ge\frac{\delta}{2}$~$$

ε-ROI Lemma

$$~$\sum_{k}\alpha_{k}\beta_{k}=\infty$~$$

ε-ROI Lemma

$~$\displaystyle\lim_{n\to\infty}\sum_{guaranteed, k\le n}\frac{\varepsilon}{3}\beta_{k}\alpha_{k}$~$

ε-ROI Lemma

$~$\displaystyle\lim_{k\to\infty}\alpha_{k}\not=0$~$

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}\not=0$~$