Debug - All Mathjax (18305)

$X$

$$f: X \rightarrow \bigcup_{Y \in X} Y$$

$X$

$X$

$Y \in X$

$Y$

$f$

$Y$

$f(Y) \in Y$

$$\forall_X \left( \left[\forall_{Y \in X} Y \not= \emptyset \right] \Rightarrow \left[\exists \left( f: X \rightarrow \bigcup_{Y \in X} Y \right) \left(\forall_{Y \in X} \exists_{y \in Y} f(Y) = y \right) \right] \right)$$

$X$

$X$

$Y_1, Y_2, Y_3$

$y_1 \in Y_1, y_2 \in Y_2, y_3 \in Y_3$

$f$

$f(Y_1) = y_1$

$f(Y_2) = y_2$

$f(Y_3) = y_3$

$X$

$X$

$Y_1, Y_2, Y_3, \ldots$

$f$

$Y$

$n$

$n$

$f$

$ax2+bx+c=0$

$ax2+bx+c=0$

$ax2+bx+c=0$

$-\infty$

$+\infty,$

$0$

$1$

$0$

$1$

$\mathbb P(X) + \mathbb P(\lnot X)$

$\lnot X$

$X$

$\aleph_0$

$\{+,\dot,0,1\}$

$2^6 = 64$

$p<0.05$

$10 bet that paid out$

$8$

$4$

$\log_4 8$

$1.5$

$3$

$2$

$log_2 3$

$n \in \mathbb R^{\ge 1},$

$\in \mathbb R^{\ge 0}$

$f(x^y)=yf(x)$

$f(b^n)=nf(b)$

$f(b)=1 \implies f(b^n)=n,$

$n$

$f$

$(\mathbb{R}^{>0},\cdot)$

$(\mathbb{R},+)$

$log$

$\mathbb{R}$

$$s(d) = \textrm{surprise}(d \mid H) = - \log \Pr (d \mid H)$$

$d$

$H$

$s$

$H$

$s$

$H$

$(d \mid H)$

$$\textrm{log-likelihood} = -\textrm{surprise}$$

$d$

$H$

$t(d)$

$t$

$H$

$t$

$t$

$\Pr(d \mid H)$

$H$

$$\Pr(H \mid d) = \Pr(H \mid t(d))$$

$H$

$s$

$d$

$t$

$s$

$H$

$d$

$d$

$H$

$d$

$H$

$d$

$\mathbb P(e \mid GoodDriver)$

$\mathbb P(e \mid BadDriver)$

$\mathbb P(BadDriver)$

$$\aleph_{\alpha} + \aleph_{\alpha} = \aleph_{\alpha} = \aleph_{\alpha} \aleph_{\alpha}$$

$$2^{\aleph_{\alpha}} > \aleph_{\alpha}$$

$\mathbf H$

$H_1, H_2, \ldots$

$\mathbf H,$

$C = AB; c_{ii} = a_{ii} * b_{ii}; ∀ i ≠ j, c_{ij} = 0$

$X_i$

$x_i$

$X_i$

$X_0$

$X_1$

$X_2$

$X_3$

$x_i$

$V$

$U$

$S$

$\le$

$X$

$X \cup X^{-1}$

$r r^{-1}$

$r^{-1} r$

$r r^{-1}$

$r \in X$

$r^{-1} r$

$X \cup X^{-1}$

$(1 : 10^{100})$

$(1 : 10^6)$

$n$

$x$

$x \cdot x \le n$

$x$

$x$

$x$

$x=316$

$x$

$x^2 \le 100000.$

$M$

$N$

$x!$

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

$\Gamma$

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

$x$

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

$x=1$

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

$$1=1$$

$x$

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

$x + 1$

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

$x+1$

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

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

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

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

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

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

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

$(S, \le)$

$S$

$\le$

$S$

$\leq$

$\leq$

$Y$

$X$

$X$

$Y$

$X$

$Y$

$X$

$X.$

$H_{0.55},$

$H_{0.6}$

$H_{0.8}.$

$H_{0.5},$

$A$

$B$

$\bP$

$\bP$

$Y$

$f$

$Y$

$\operatorname{square} : \mathbb R \to \mathbb R$

$\operatorname{square}(x)=x^2$

$\mathbb R$

$\mathbb R$

$\mathbb R$

$\mathbb C$

$Prv(x)$

$x$

$n$

$2^n$

$2^{3,000,000,000,000}$

$2^{3,000,000,000,000}$

$2^\text{3 trillion}$

$a$

$b$

$31a + b$

$31\cdot 30 + 30 = 960$

$\mathcal L(H \mid e) < 0.05$

$H$

$e$

$H$

$H$

$e$

$\mathcal L(H \mid e)$

$e$

$H$

$b$

$n,$

$\log_b(n),$

$b$

$n$

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

$\log_{10}(316) \approx 2.5,$

$316 \approx$

$10 \cdot 10 \cdot \sqrt{10},$

$\sqrt{10}$

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

$f$

$x$

$f(x)$

$1/2$

$f$

$H_{fair},$

$H_{heads}$

$H_{tails}$

$(1/2 : 1/3 : 1/6).$

$(3 : 2 : 1)$

$(2 : 1 : 3).$

$(2 : 3 : 1)$

$(3 : 2 : 1)$

$A$

$B$

$C$

$C$

$\mathcal T$

$B$

$D$

$X$

$Y$

$X$

$Y$

$X \to Y$

$Y^X$

$Y^2$

$Y$

$$1$$

$99\cdot 2=198$

$100$

$LDT$

$198$

$100$

$1$

$0$

$\mathbb P(X_i | \mathbf{pa}_i)$

$X_i$

$x_i$

$\mathbf {pa}_i$

$x_i$

$\mathbf x$

$\bullet$

$G$

$G$

$t = 0$

$4.7 t^2$

$t$

$x$

$n$

$n-1$

$n$

$\log_{10}(x)$

$x;$

$x$

$x$

$P(n)$

$n$

$P(n)$

$n$

$P(m)$

$k \geq m$

$P(k)$

$P(k+1)$

$P(m)$

$P(m+1)$

$P(m+1)$

$P(m+2)$

$-1$

$P$

$P(x)$

$P(X=x)$

$X$

$P$

$s$

$$1 + 2 + \cdots + n = \frac{n(n+1)}{2}$$

$n \ge 1$

$n=1$

$$1 = \frac{1(1+1)}{2} = \frac{2}{2} = 1.$$

$k$

$k\ge1$

$$1 + 2 + \cdots + k = \frac{k(k+1)}{2}$$

$$1 + 2 + \cdots + k + (k+1) = \frac{(k+1)([k+1]+1)}{2}.$$

$k+1$

$$1+2+\cdots + k + (k+1) = \frac{k(k+1)}{2} + k + 1.$$

$$\frac{k(k+1)}{2} + \frac{2(k+1)}{2} = \frac{(k+2)(k+1)}{2} = \frac{(k+1)([k+1]+1)}{2}.$$

$$1 + 2 + \cdots + k + (k+1) = \frac{(k+1)([k+1]+1)}{2}$$

$n$

$k+1$

$\lambda$

$\lambda x.f(x)$

$x$

$f(x)$

$\lambda x.x+1$

$\lambda$

$\lambda x.\lambda y.x+y$

$\lambda xy.x+y$

$\lambda xy$

$\lambda x.\lambda y$

$\newcommand{\rats}{\mathbb{Q}} \newcommand{\Ql}{\rats^\le} \newcommand{\Qr}{\rats^\ge} \newcommand{\Qls}{\rats^<} \newcommand{\Qrs}{\rats^>}$

$\newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\sothat}{\ |\ }$

$S$

$2^6 < 101 < 2^7$

$\lambda x.(\lambda y.(x+y))$

$(\lambda x.(\lambda y.(x+y)))$

$f\ x\ y$

$f$

$x$

$y$

$(f\ x)\ y$

$f\ (x\ y)$

$\lambda$

$\lambda x.\lambda y.x+y$

$\lambda x.(\lambda y.(x+y))$

$(\lambda x.\lambda y.x)+y$

$\lambda x.(\lambda y.x)+y$

$\lambda$

$\lambda xy.x+y$

$\lambda x.\lambda y.x+y$

$x$

$n$

$n-1$

$n$

$\log_{10}(x)$

$x;$

$x$

$x$

$\log_{10}(12) \approx 1.08$

$\log_2(10) \approx 3.32$

$2 : 1$

$8 : 1,$

$2 : 1$

$4 : 1.$

$(a_1 a_2 \dots a_k)$

$(a_k a_{k-1} \dots a_1)$

$0.999\dotsc=1$

$A \cdot B = A + A + A$

$B$

$\mathbb P$

$\operatorname{d}\!f$

$\operatorname{d}\!f$

$X$

$\mathbb P(X)$

$X.$

$f$

$X$

$I$

$Y$

$I$

$(X, \bullet)$

$X$

$\bullet$

$X$

${^-3}$

${^-1}$

${^-4}$

$(1 : 16)$

$\mathbb{N}$

$\omega$

$d$

$d$

$S$

$$d: S \times S \to [0, \infty)$$

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

$x$

$n$

$n-1$

$n$

$\log_{10}(x)$

$x;$

$x$

$x$

$x$

$A = \{ \{ 1,2 \}, \{ 3,4 \}, 1, 2, 3, 4 \}$

$x = \{1,2\}$

$a = 2$

$a \in x$

$x \in A$

$a \in A$

$B = \{ \{ 1,2 \}, \{ 3,4 \} \}$

$y = \{1,2\}$

$b = 2$

$b \in y$

$y \in B$

$b \notin B$

$3^{10}$

$n^k$

$\log_2(6) + \log_2(10) + 3\log_2(2) \approx 8.9$

$2*3^6 = 432,$

$\log_2(2) + 3*\log_2(6) \approx 8.75$

$\mathbb P({positive}\mid {HIV}) = .997$

$\mathbb P({negative}\mid \neg {HIV}) = .998$

$\mathbb P({positive} \mid \neg {HIV}) = .002.$

$1 : 100,000$

$500 : 1.$

$\mathbb P(sick \mid blackened)$

$\mathbb P(sick \wedge blackened)$

$\mathbb P(blackened)$

$PA$

$\square_{PA}$

$PA$

$PA$

$A$

$\square_{PA}(\ulcorener A\urcorner$

$A$

$PA$

$x$

$x$

$n$

$c$

$n$

$c.$

${\bf \hat v}$

$$|\mathbf{\hat v}| = \left|\frac{\mathbf{v}}{|\mathbf{v}|}\right| = \left|\frac{1}{|\mathbf{v}|}\right||\mathbf{v}| = \frac{|\mathbf{v}|}{|\mathbf{v}|}=1$$

$\hat{\mathbf v} = \frac{1}{| \mathbf v |}\mathbf v = \frac{\mathbf v}{| \mathbf v |}$

$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$

$\frac{1}{2}$

$G_0 \xrightarrow{f_1} G_1 \xrightarrow{f_2} G_2 \xrightarrow{f_3} \cdots \xrightarrow{f_n} G_n$

$\text{im}(f_k) = \text{ker}(f_{k+1})$

$0 \le k < n$

$n\times n$

$A$

$a_{i,j}$

$\det(A) = \sum_{\sigma\in S_n}\text{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma_i}$

$S_n$

$n$

$\log_b(x)$

$b$

$x$

$K$

$O$

$\mathbb P(X_i | \mathbf{pa}_i)$

$X_i$

$x_i$

$\mathbf {pa}_i$

$x_i$

$\mathbf x$

$2^{101}$

$\prec$

$\langle \mathbb R, \leq \rangle$

$\leq$

$0 < 1$

$0$

$\mathbb R$

$x \in \mathbb R$

$x > 0$

$y \in \mathbb R$

$0 < y < x$

$\mathbb R$

$f(x)=1$

$1$

$\{1\}$

$V:s \mapsto V(s)$

$s$

$n$

${^-2}$

${^-6}$

$\log_{10}(10^{-6}) - \log_{10}(10^{-2})$

${^-4}$

${^-13.3}$

$\log_{10}(\frac{0.10}{0.90}) - \log_{10}(\frac{0.11}{0.89}) \approx {^-0.954}-{^-0.907} \approx {^-0.046}$

${^-0.153}$

$x^{-1}$

$\rho_{x^{-1}}$

$\rho_x$

$\rho_{x^{-1}}$

$\rho_\epsilon$

$\log_{10}(500)$

$p \prec q$

$q$

$P$

$p$

$p \prec q$

$p$

$q$

$p$

$q$

$q$

$p$

$x,$

$\lceil x \rceil$

$\operatorname{ceil}(x),$

$n \ge x.$

$\lceil 3.72 \rceil = 4, \lceil 4 \rceil = 4,$

$\lceil -3.72 \rceil = -3.$

$[n]$

$k$

$[n]$

$k$

$\pi_5$

$V$

$V$

$V$

$(x : y)$

$\alpha$

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

$x$

$y$

$\frac{x}{y}.$

$\frac{x}{y}$

$(x : y),$

$\left(\frac{x}{y} : 1\right).$

$x/y$

$\ \mathbb P(a_x \ \square \! \! \rightarrow o_i).$

$\mathbb P(X \wedge Y) = \mathbb P(Y) \cdot \mathbb P(X|Y).$

$1 : 4$

$3 : 1$

$(1 \cdot 3) : (4 \cdot 1) = 3 : 4$

$H$

$\frac{1}{8}$

$\lnot H$

$\frac{1}{4}$

$\lnot H$

$H,$

$\mathbb P(e \mid H)$

$\mathbb P(e \mid \lnot H)$

$\left(\frac{1}{8} : \frac{1}{4}\right)$

$=$

$(1 : 2),$

$H.$

$\mathbb{C}$

$S$

$S$

$S$

$S$

$n\in\mathbb{N}$

$f(b)=1\Rightarrow f(b^q)=q$

$q\in\mathbb{Q}$

$f$

$bayes_rule_details,$

$\mathbb P(\mathbf x | \operatorname{do}(X_j=x_j))$

$\mathbb P(X \mid Y)$

$X$

$Y$

$\mathbb{N} \setminus \{1, 2, 3, 4, 5\}$

${5}$

$1/8$

$1$

$a$

$b$

$b$

$a$

$a$

$c$

$a$

$b$

$b$

$c$

$a$

$b$

$c$

$A$

$B$

$A$

$1$

$B$

$0$

$A$

$B$

$A$

$1$

$B$

$A$

$B$

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

$\frac{a}{m}$

$a$

$\frac{1}{m}$

$\frac{1}{m}$

$n$

$a$

$\frac{1}{m}$

$n$

$n$

$\frac{1}{m}$

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

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

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

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

$V_i$

$v_i.$

$v_i$

$v_i^*$

$V_i$

$X = \{ a, b \}$

$\log_b(316) \approx \frac{5\log_b(10)}{2}$

$n$

$\sqrt{n}$

$n$

$x$

$x$

$x \cdot x$

$n$

$n$

$\sqrt{n}$

$\log$

$\log_2(3)$

$1$

$\log_2(6),$

$\log_2(9)$

$\log_2(3^{10}),$

$\log_2(3^9)$

$\log_2(3^{10}).$

$\log_2(3)$

$x$

$x$

$n$

$x$

$n$

$(5 : 3 : 2) \cdot (2 : 1 : 5) \cdot (12 : 10 : 1) = (120 : 30 : 10) \cong (12/16 : 3/16 : 1/16)$

$D$

$A$

$A$

$\{0,1\}^*$

$C_2$

$2$

$1$

$-1$

$1$

$-1$

$f(x)$

$f(-x)$

$f(x)$

$(-1) \times (-1) = 1$

$f(-(-x)) = f(x)$

$P$

$\leq$

${+1}$

${^+1}$

$0.01$

$0.000001$

$0.11$

$0.100001.$

$\mathbb P(f\mid e\!=\!\textbf {THT}) = \dfrac{\mathcal L(e\!=\!\textbf{THT}\mid f) \cdot \mathbb P(f)}{\mathbb P(e\!=\!\textbf {THT})} = **\dfrac{(1 - x) \cdot x \cdot (1 - x) \cdot 1}{\int_0^1 (1 - x) \cdot x \cdot (1 - x) \cdot 1 \** \operatorname{d}\!f} = 12 \cdot f(1 - f)^2$

$\emptyset$

$57$

$\mathrm{sin}$

$f : X \times X \to X$

$~$x, y, z$~$

$X$

$f(x, f(y, z)) = f(f(x, y), z)$

$+$

$(x + y) + z = x + (y + z)$

$x, y,$

$z$

$0 + 2\mathbb Z$

$1 + 2\mathbb Z$

$+$

$\text{even}$

$\text{odd}$

$\text{even}+ \text{even} = \text{even}$

$\text{even} + \text{odd} = \text{odd}$

$\text{odd} + \text{odd} = \text{odd}$

$x$

$x$

$n$

$c$

$n$

$c.$

$\mathbb P(X \wedge Y) = \mathbb P(Y) \cdot \mathbb P(X|Y).$

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

$g$

$g$

$h$

$h(x+y)=h(x)+h(y)$

$$h(g(x\cdot y))=h(g(x))+h(g(y))$$

$h$

$h(x)=ch(x)$

$c$

$\mathbb{R}$

$\mathbb{Q}$

$\mathbb{R}$

$f$

$f$

$f$

$\mathsf{Fairbot}$

$\mathsf {Fairbot}$

$\mathsf {Fairbot}$

$\mathsf {Fairbot}$

$\mathsf {Fairbot}$

$\mathsf {CooperateBot},$

$\mathsf {Fairbot}$

$\mathsf {CooperateBot},$

$\mathsf {Fairbot}$

$\mathbb P(X \wedge Y) = \mathbb P(Y) \cdot \mathbb P(X|Y).$

$0.999\dotsc$

$1$

$1+2+4+8+\dotsc=-1$

$0.999\dotsc$

$0.999\dots\neq1$

$0.999\dots$

$1$

$0.999\dots$

$9$

$\sum_{k=1}^\infty 9 \cdot 10^{-k}$

$(\sum_{k=1}^n 9 \cdot 10^{-k})_{n\in\mathbb N}$

$a_n$

$n$

$1$

$\varepsilon>0$

$N\in\mathbb N$

$n>N$

$|1-a_n|<\varepsilon$

$1-a_n=10^{-n}$

$a_0$

$0$

$a_0=0$

$1-a_0=1=10^0$

$1-a_i=10^{-i}$

$1-a_n=10^{-n}$

$n$

$10^{-n}$

$10^{-n}$

$0.999\dotsc=1$

$0.999\dotsc=1$

$0.999\dotsc$

$1$

$0.999\dotsc$

$0.$

$0$

$1-0.999\dotsc=0.000\dotsc001\neq0$

$0.000\dotsc001$

$1$

$0$

$0.000\dotsc001$

$0$

$0.999\dotsc$

$0.9, 0.99, 0.999, \dotsc$

$1$

$1$

$1$

$1$

$1$

$1$

$0.999\dotsc$

$9.999\dotsc$

$9$

$9.99-0.999=8.991$

$9.999\dotsc-0.999\dotsc=8.999\dotsc991$

$9$

$0.999\dotsc$

$8.999\dotsc991$

$1$

$10^{100},$

$10^{10^{100}}$

$10^{googol}$

$10^{10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}.$

$1 to be effectively +$

$E = -mc^2,$

$\theta$

$\theta$

$0$

$1.$

$M$

$N$

$$\frac{M + 1}{M + N + 2} : \frac{N + 1}{M + N + 2}$$

$$HTHTHTHTHTHTHTHT…$$

$H.$

$HTTHTTHTTHTT$

$\mathbb{E}$

$\mathbb{E}$

$\Sigma_1$

$\Sigma_2$

$tl$

$l$

$t$

$\text{AIXI}^{tl}$

$l$

$t$

$tl$

$G$

$(X, \bullet)$

$X$

$\bullet$

$x, y$

$X$

$x \bullet y$

$X$

$x \bullet y$

$xy$

$x(yz) = (xy)z$

$x, y, z$

$X$

$e$

$x$

$X$

$xe=ex=x$

$x$

$X$

$x^{-1}$

$X$

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

$x, y$

$X$

$xy=yx$

$G=(X, \bullet)$

$\bullet$

$x, y$

$X$

$x \bullet y$

$X$

$x \bullet y$

$xy$

$x(yz) = (xy)z$

$x, y, z$

$X$

$e$

$x$

$X$

$xe=ex=x$

$x$

$X$

$x^{-1}$

$X$

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

$x, y$

$X$

$xy=yx$

$\{1, a, a^{-1}, b, b^{-1}, c, c^{-1}, d\}$

$aba^{-1}db^{-1}=d^{-1}$

$aa^{-1}bb^{-1}d=d^{-1}$

$d=d^{-1}$

$aba^{-1}$

$aa^{-1}b$

$(\exists v: \forall w > v: \forall x>0, y>0, z>0: x^w + y^w \neq z^w) \rightarrow ((1 = 0) \vee (1 + 0 = 0 + 1))$

$p$

$p$

$p$

$1 - p$

$p$

$1 - p$

$p^2$

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

$4 -6p$

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

$\$0\cdot\frac{1}{3} + \$4\cdot\frac{2}{3}\frac{1}{3} + \$1\cdot\frac{2}{3}\frac{2}{3} = \$\frac{4}{3} \approx \$1.33.$

$p$

$q.$

$1 : q,$

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

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

$p,$

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

$p$

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

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

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

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

$q$

$p$

$q,$

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

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

$q$

$p$

$q$

$q,$

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

$p,$

$q$

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

$p$

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

$q,$

$p$

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

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

$p.$

$q$

$q$

$p$

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

$p$

$q$

$p$

$p$

$A^\complement$

$A$

$A$

$U$

$A^\complement = U \setminus A$

$A^\complement$

$U$

$A$

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

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

$A ^ B$

$A \uparrow B$

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

$\uparrow^n$

$n$

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

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

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

$A(6)$

$A(1)=1$

$A(2)=4$

$A(3)$

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

$5$

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

$a+b$

$a$

$b$

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

$\frac{n}{n}$

$n$

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

$\frac{1}{3}$

$\frac{1}{3}$

$5+8=13$

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

$\frac{1}{3}$

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

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

$\frac{1}{3}$

$\frac{1}{4}$

$\frac{1}{12}$

$\frac{1}{3}$

$\frac{1}{4}$

$\frac{1}{12}$

$\frac{1}{4}$

$\frac{1}{12}$

$\frac{1}{3}$

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

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

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

$\frac{1}{3}$

$\frac{1}{12}$

$\frac{1}{3}$

$\frac{1}{12}$

$\frac{1}{12}$

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

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

$\frac{1}{4}$

$\frac{1}{12}$

$5 \times 3 = 15$

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

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

$\frac{35}{12}$

$\frac{1}{12}$

$\frac{1}{3}$

$\frac{1}{4}$

$\frac{1}{2}$

$\frac{1}{5}$

$\frac{1}{2}$

$\frac{1}{5}$

$2 \times 5 = 10$

$\frac{1}{10}$

$\frac{1}{2}$

$\frac{1}{10}$

$\frac{1}{5}$

$\frac{1}{m}$

$\frac{1}{n}$

$m$

$2$

$n$

$5$

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

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

$\frac{1}{n}$

$m$

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

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

$\frac{1}{m}$

$n$

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

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

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

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

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

$a$

$b$

$\frac{1}{m}$

$\frac{1}{n}$

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

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

$\frac{1}{4}$

$\frac{1}{3}$

$\frac{1}{12}$

$3 \times 4 = 12$

$\frac{1}{12}$

$\frac{5}{4}$

$\frac{15}{12}$

$\frac{1}{4}$

$\frac{1}{12}$

$5 \times 3$

$\frac{1}{12}$

$\frac{5}{3}$

$\frac{20}{12}$

$\frac{1}{3}$

$\frac{1}{12}$

$5 \times 4 = 20$

$\frac{1}{12}$

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

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

$a \times n + b \times m$

$a \times n$

$b \times m$

$a, b, m, n$

$m$

$n$

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

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

$\frac{3}{10}$

$\frac{3}{10}$

$\frac{1}{10}$

$15$

$\frac{1}{50}$

$\frac{1}{10}$

$\frac{1}{5}$

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

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

$\frac{3}{10}$

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

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

$\frac{1}{30}$

$\frac{1}{10}$

$\frac{1}{15}$

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

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

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

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

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

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

$\frac{1}{6}$

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

$5$

$\frac{1}{5}$

$0$

$2$

$2$

$\frac{2}{5}$

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

$\frac{1}{7}$

$\frac{1}{5}$

$\frac{2}{5}$

$\frac{1}{7}$

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

$\frac{2}{5}$

$\frac{1}{5}$

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

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

$\frac{7}{8}$

$\frac{13}{8}$

$\frac{a}{b}$

$a$

$b$

$\frac{1}{8}$

$\frac{1}{8}$

$7$

$13$

$6$

$\frac{6}{8}$

$\frac{3}{4}$

$\frac{7}{8}$

$\frac{13}{7}$

$\frac{a}{b}$

$a$

$b$

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

$\frac{1}{8}$

$\frac{1}{7}$

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

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

$49$

$104$

$55$

$\frac{1}{56}$

$\frac{55}{56}$

$\mathbb P(Y|X)$

$X$

$Y$

$Y,$

$X.$

$X$

$Y.$

$\pi_0$

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

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

$(R, +, \times)$

$R$

$1$

$0$

$r \in R$

$x \in R$

$xr = rx = 1$

$0 \not = 1$

$X$

$*$

$\circ$

$*$

$\circ$

$\circ$

$*$

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

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

$*$

$\circ$

$*$

$*$

$*$

$\circ$

$\circ$

$*$

$\circ$

$\circ$

$*$

$\circ$

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

$*$

$\circ$

$\wedge$

$\vee$

$3\uparrow\uparrow\uparrow3$

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

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

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

$\rightarrow$

$\rightarrow$

$\rightarrow$

$\rightarrow$

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

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

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

${a_2}x+{b_2}y={c_2}$

$\frac{a_1}{a_2}\neq \frac{b_1}{b_2}$

$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq \frac{c_1}{c_2}$

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

$\big(x-h)^2+\big(y-k)^2=r^2$

$\big(h,k)$

$r=\sqrt{r^2}$

${x^2}+{y^2}+{ax}+{by}+c=0$

$\big(\frac{-a}{2},\frac{-b}{2})$

$\sqrt{\big(\frac{a}{2})^2+(\frac{b}{2})^2-c}$

$\big({x_1},{y_1})$

$\big(x_1-h)^2+\big(y_1-k)^2<r^2$

$\big(x_1-h)^2+\big(y_1-k)^2=r^2$

$\big(x_1-h)^2+\big(y_1-k)^2>r^2$

$A_n$

$S_n$

$A_n$

$S_n$

$S_n$

$(132)$

$(13)(23)$

$(1354)$

$(54)(34)(14)$

$A_4$

$(12)(34)$

$(13)(24)$

$(14)(23)$

$(123)$

$(124)$

$(134)$

$(234)$

$(132)$

$(143)$

$(142)$

$(243)$

$A_n$

$2$

$S_n$

$A_n$

$S_n$

$A_n$

$A_n$

$3$

$A_n$

$A_n$

$A_n$

$3$

$A_n$

$3$

$3$

$(ij)(kl) = (ijk)(jkl)$

$(ij)(jk) = (ijk)$

$(ij)(ij) = e$

$A_n$

$3$

$3$

$A_n$

$(ijk) = (ij)(jk)$

$$\textbf{ Expected Value of Project } = \textbf{Decision Relevant Info} + \textbf{Rare Signals} + \textbf{Cross-Domain Skills}$$

$$\textbf{ Expected Value of Project } = \textbf{Decision Relevant Info} + \textbf{Rare Signals} + \textbf{Cross-Domain Skills}$$

$\phi$

$\phi$

$=, \wedge, \implies$

$0$

$n+1$

$n$

$\forall n. 0 \not = n+1$

$\forall$

$A\implies B$

$A$

$B$

$A$

$w$

$A$

$w$

$w$

$A$

$PA$

$\phi$

$PA$

$\phi$

$PA$

$PA\vdash \phi$

$PA$

$\phi$

$1$

$=$

$1$

$a$

$0$

$n$

$2^{a_1}3^{a_2}5^{a_3}\cdots p(n)^{a_n}$

$n$

$Axiom(x)$

$IsEqualTo42(x)$

$x = 42$

$PA\vdash IsEqualTo42(42)$

$PA\vdash \exists x IsEqualTo42(x)$

$PA\not\vdash IsEqualTo42(7)$

$PA\vdash Axiom(\textbf{n})$

$n$

$PA$

$n$

$n+1$

$Rule(p_1, p_2,…, p_n, r)$

$PA$

$p_1, …., p_n$

$r$

$Proof(x,y)$

$PA$

$x$

$y$

$\exists x. Proof(x,y)$

$\square_{PA}(y)$

$\exists$

$\square_{PA}(x)$

$PA$

$x$

$\ulcorner 1+1=2 \urcorner$

$1+1=2$

$PA\vdash \square_{PA}(\ulcorner 1+1=2 \urcorner)$

$PA$

$PA$

$1+1=2$

$PA$

$Proof(x,y)$

$PA$

$PA$

$PA$

$PA$

$PA$

$\square_{PA}$

$PA\vdash A$

$PA\vdash \square_{PA}(\ulcorner A\urcorner)$

$PA\vdash \square_{PA}(\ulcorner A\rightarrow B\urcorner) \rightarrow [\square_{PA}(\ulcorner A \urcorner)\rightarrow \square_{PA}(\ulcorner B \urcorner)]$

$PA\vdash \square_{PA}(\ulcorner A\urcorner) \rightarrow \square_{PA} \square_{PA} (\ulcorner A\urcorner)$

$PA$

$A$

$A$

$A$

$B$

$A$

$B$

$A$

$A\rightarrow B$

$B$

$\square_{PA}(\ulcorner A \urcorner)$

$\phi(x)$

$\psi$

$PA\vdash \psi \leftrightarrow \phi(\ulcorner \psi \urcorner)$

$PA\vdash \square_{PA}(\ulcorner A\urcorner) \rightarrow A$

$PA\vdash A$

$PA\not\vdash A$

$PA\not\vdash \square_{PA}(\ulcorner A\urcorner) \rightarrow A$

$A$

$A$

$A$

$PA$

$A$

$n$

$PA\vdash Proof(\textbf n, \ulcorner A\urcorner)$

$A$

$PA$

$PA$

$n$

$Proof(\textbf n,\ulcorner A\urcorner)$

$PA$

$n$

$PA$

$P\wedge \neg P$

$P\wedge \neg P$

$P$

$\bot$

$PA$

$PA\not \vdash \neg \square_{PA}(\bot)$

$PA$

$PA$

$\square_{PA}$

$PA$

$PA$

$A$

$\square_{PA}(\ulcorner A\urcorner$

$A$

$PA$

$R$

$(aRb ∧ bRa) → a = b$

$a ≠ b → (¬aRb ∨ ¬bRa)$

$aRa$

$\{(0,0), (1,1), (2,2)…\}$

$\{(0,1), (1,2), (2,3), (3,4)…\}$

$\{…(9,3),(10,5),(10,2),(14,7),(14,2)…)\}$

$ax^2 + bx + c = 0$

$ax^2 + bx + c = 0$

$$\lim_{N \to \infty} \sum_{k=1}^N f(t_k) \Delta t$$

$$\lim_{N \to \infty} \sum_{k=1}^N f(t_k) \Delta t$$

$n^\text{th}$

$\Pi_1$

$\Sigma_1$

$\Pi_n$

$\Sigma_{n+1}$

$\Sigma_n$

$\Pi_{n+1}$

$\Pi_n$

$\Sigma_n$

$\Delta_n$

$\Pi_1$

$\Sigma_1$

$\Delta_0$

$\Pi_0$

$\Sigma_0$

$\forall x < 10: \exists y < x: x + y < 10$

$x, y, z…$

$\phi(x, y, z…)$

$\Sigma_n,$

$\forall x: \forall y: \forall z: … \phi(x, y, z…)$

$\Pi_{n+1}$

$\Pi_n$

$\Sigma_{n+1}$

$\Pi_n$

$\Sigma_n$

$\Delta_n$

$\Pi_1$

$\Sigma_1$

$\forall x$

$\exists y$

$\phi(x, y) \leftrightarrow [(x + y) = (y + x)],$

$x$

$y$

$\Delta_0 = \Pi_0 = \Sigma_0.$

$+$

$=$

$\Delta_0$

$c$

$d$

$c + d = d + c$

$\forall x_1: \forall x_2: …$

$\Sigma_n$

$x_i$

$\Pi_{n+1}.$

$\forall x: (x + 3) = (3 + x)$

$\Pi_1.$

$\exists x_1: \exists x_2: …$

$\Pi_n$

$\Sigma_{n+1}.$

$\exists y: \forall x: (x + y) = (y + x)$

$\Sigma_2$

$\exists y: \exists x: (x + y) = (y + x)$

$\Sigma_1.$

$\Pi_n$

$\Sigma_n$

$\Delta_n.$

$\Delta_0$

$\forall x: \exists y < x: (x + y) = (y + x)$

$\Pi_1$

$\Pi_2$

$c,$

$\forall x < c: \phi(x)$

$\phi(0) \wedge \phi(1) … \wedge \phi(c)$

$\exists x < c: \phi(x)$

$\phi(0) \vee \phi(1) \vee …$

$z = 2^x \cdot 3^y$

$\Delta_{n+1}$

$\Pi_n$

$\Sigma_n$

$\Pi_{n}$

$\Pi_{n+1}$

$\exists$

$\Sigma_{n+1}$

$\Pi_{n}$

$\forall$

$\phi \in \Pi_n$

$\Pi_n$

$\phi$

$\Pi_n$

$\Pi_n.$

$\Sigma_1$

$\phi \in \Delta_0$

$\exists x: \phi(x)$

$\Pi_1$

$\phi$

$\forall x: \phi(x)$

$\phi$

$\Sigma_1$

$\Pi_1.$

$\Pi_2$

$\Delta_0,$

$\Pi_0,$

$\Sigma_0$

$\Pi_1.$

$y^9 = 9^y.$

$y^9 = 9^y.$

$\Delta_0,$

$\Sigma_1.$

$c$

$c$

$\Sigma_1.$

$c,$

$\Sigma_1$

$c,$

$\Pi_2.$

$(x + y) > 10^9$

$\Sigma_2,$

$\Pi_1$

$x.$

$\Pi_n$

$\Sigma_{n+1}$

$\Sigma_n$

$\Pi_{n+1}$

$\Sigma_n$

$\Pi_n$

$\Delta_n.$

$\Sigma_1$

$y$

$y^9 = 9^y$

$y$

$y^9 = 9^y,$

$y^9 = 9^y$

$y$

$\Pi_1$

$\Delta_1$

$\Pi_2$

$\Sigma_2$

$\Pi_2$

$x,$

$y$

$x^x$

$\Pi_1$

$\Pi_2$

$x^x$

$y$

$x^x$

$x,$

$x = 2,$

$y$

$2^2$

$x,$

$y$

$x^x$

$c,$

$c^c,$

$c=1.$

$z = 2^x \cdot 3^y$

$x^3 + y^3 = z^3$

$w$

$w = 2^x \cdot 3^y \cdot 5^z$

$x^3 + y^3 = z^3.$

$\Pi_1,$

$x^w + y^w = z^w.$

$\Pi_1$

$X \rightarrow Y$

$Y$

$X$

$X$

$Y$

$\Pi_2$

$x$

$y$

$\Pi_1$

$x$

$y = f(x) = 4x+1$

$\Pi_2$

$\Pi_2$

$\Pi_1$

$4x+1$

$\Pi_2$

$f(a, b, c, d) = ac - bd$

$+$

$(\mathrm{People} \times \mathrm{Ages})$

$\bullet : X \times X \to X$

$x, y, z$

$X$

$x \bullet (y \bullet z) = (x \bullet y) \bullet z$

$+$

$(x + y) + z = x + (y + z)$

$x, y,$

$z$

$f$

$x, y,$

$z$

$f$

$f$

$f(f(x, y), z) = f(x, f(y, z)),$

$f$

$x$

$y$

$z$

$f$

$y$

$z$

$x$

$f$

$f$

$f_3 : X \times X \times X \to X,$

$f$

$f$

$f$

$f_4, f_5, \ldots,$

$\bullet$

$2 \cdot 3 \cdot 4 \cdot 5$

$x$

$y,$

$y$

$x.$

$a \cdot (b \cdot (c \cdot d)),$

$((a \cdot b) \cdot c) \cdot d.$

$\cdot$

$3 + 2 + (-7) + 5 + (-2) + (-3) + 7,$

$3 - 3 + 2 - 2 + 7 - 7 + 5 = 5,$

$(x + y) + z = x + (y + z)$

$x, y,$

$z.$

$n$

$n$

$(x \times y) \times z = x \times (y \times z)$

$x, y,$

$z.$

$n$

$n$

$x$

$y$

$z$

$(x \times y) \times z$

$x \times (y \times z).$

$x$

$y$

$z$

$z$

$(5-3)-2=0$

$5-(3-2)=4.$

$\uparrow$

$\uparrow$

$\uparrow\downarrow.$

$\uparrow\downarrow$

$\uparrow,$

$\uparrow\downarrow\downarrow,$

$\uparrow$

$\uparrow\downarrow,$

$\uparrow\downarrow\uparrow,$

$?$

$(red\ ?\ green)\ ?\ blue = blue$

$red\ ?\ (green\ ?\ blue)=red.$

$f : X \times X \to X$

$X$

$3 + 4 + 5 + 6,$

$+$

$[a, b, c, d, \ldots]$

$a$

$b$

$[a, b]$

$c$

$b$

$c$

$[b, c]$

$a$

$[a, b, c]$

$f : X \times X \to Y$