\documentclass[12pt]{article}
\usepackage[english]{babel}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amssymb}

\begin{document}
	\begin{equation}
		(p \vee q) \wedge (p \rightarrow r) \wedge (q \rightarrow r) \rightarrow r
	\end{equation}

	\begin{equation}
		X^n +  Y^n = Z^n
	\end{equation}

	\begin{equation}
		6.7^n - 2.3^n
	\end{equation}

	\begin{equation}
		a^n + 1 = b^n
	\end{equation}

	\begin{equation}
		n!-(n-1)! = (n-1)(n-1)!
	\end{equation}

	\begin{equation}
		7x + 5y = 3 \ (\text{mod} \ 4)
	\end{equation}

	\begin{equation}
		\sum_{k=1}^5 k+1
	\end{equation}

	\begin{equation}
		\sum_{i=1}^n i^k
	\end{equation}

	\begin{equation}
		O(n!) \in O(n^n)
	\end{equation}

	\begin{equation}
		\sum_{i=1}^n x^{i!}
	\end{equation}

	\begin{equation}
		(p \leftrightarrow q)
	\end{equation}

	\begin{equation}
		\neg(p \leftrightarrow q) = \neg q \leftrightarrow p
	\end{equation}

	\begin{equation}
		1.1! + \dots + n . n! = (n+1)!
	\end{equation}

	\begin{equation}
		2+4+ \dots + 2n = n(n+1)
	\end{equation}

	\begin{equation}
		(n-m) \mid (n^k - m^k)
	\end{equation}

	\begin{equation}
		\neg(p \vee q) \vee (\neg p \wedge q) = \neg p
	\end{equation}

	\begin{equation}
		(\neg q \vee \neg q) \leftrightarrow (p \rightarrow \neg q)
	\end{equation}

	\begin{equation}
		\sum_{j=1}^{n}j^k\binom{n}{j}
	\end{equation}

	\begin{equation}
		[A \rightarrow (B \vee C)] \wedge C' \rightarrow (A \rightarrow B)
	\end{equation}

	\begin{equation}
		\sum_{i=1}^\infty P(X > i) < \infty
	\end{equation}

	\begin{equation}
		\sum_{m=1}^n \frac{e^{-am^2}}{m^2}
	\end{equation}

	\begin{equation}
		(a^2 +b^2 +c^2)^2 =2(a^4 +b^4 +c^4)
	\end{equation}

	\begin{equation}
		\int \sin x \cos^2 x \ dx
	\end{equation}

	\begin{equation}
		\frac{0}{0}
	\end{equation}

	\begin{equation}
		10\sin^2(\theta) - 13\sin(\theta)-3 = 0
	\end{equation}

	\begin{equation}
		\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9}...
	\end{equation}

	\begin{equation}
		\int x^3(1-x^2)^{1/2}
	\end{equation}

	\begin{equation}
		\int e^{-x^2} dx
	\end{equation}

	\begin{equation}
		\int \exp(-x^2)dx
	\end{equation}

	\begin{equation}
		p(xy|z)=p(x|z)p(y|xz)
	\end{equation}

	\begin{equation}
		ab=c(\text{mod} \ n)
	\end{equation}

	\begin{equation}
		\tan(x)=5/12
	\end{equation}

	\begin{equation}
		T(n)=T(n-1)+T(n-2)
	\end{equation}

	\begin{equation}
		\int x^2f(x) \ dx
	\end{equation}

	\begin{equation}
		{f_{n+1}}^2 + {f_{n}}^2 = f_{2n+1}
	\end{equation}

	\begin{equation}
		\int_{-\infty}^{\infty} \exp -r^2 \ dr
	\end{equation}

	\begin{equation}
		e^\pi > \pi^e
	\end{equation}

	\begin{equation}
		\frac{d}{dx} \int \sqrt{4-x^2} \ dx
	\end{equation}

	\begin{equation}
		\int_{0}^{2\pi} \frac{\cos(2x)}{\exp(x)} \ dx
	\end{equation}

	\begin{equation}
		(p-1)! = -1(\text{mod} \ p)
	\end{equation}

	\begin{equation}
		X^2 = 10^X
	\end{equation}

	\begin{equation}
		F_n^2 - (F_{n+1})(F_{n-1}) = (-1)^{n-1}
	\end{equation}

	\begin{equation}
		f(x) = (1+\frac{1}{x})^x
	\end{equation}

	\begin{equation}
		\int e^{-\frac{r^2}{k}} \ dr
	\end{equation}

	\begin{equation}
		z \tan z
	\end{equation}

	\begin{equation}
		\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}
	\end{equation}

	\begin{equation}
		\begin{bmatrix}
		 x & a & b \\
		 b & x & a \\
		 a & b & x
		\end{bmatrix}
	\end{equation}

	\begin{equation}
		(x \oplus y) = (x+7)(x+4)-6k
	\end{equation}

	\begin{equation}
		\binom{n}{k} = 1+\sum_{j=0}^{k-1} \sum_{i=0}^{n-k-1} \binom{i+j}{j}
	\end{equation}

	\begin{equation}
		(a-b)^2 = a^2 + b^2 - 2ab
	\end{equation}

	\begin{equation}
		y' = -x \sin\theta + y \cos \theta
	\end{equation}

	\begin{equation}
		\sin^2 x + \cos^2 x
	\end{equation}

	\begin{equation}
		\int \frac{\sin^x}{x} \ dx
	\end{equation}

	\begin{equation}
		\log(x+3) + \log x
	\end{equation}

	\begin{equation}
		1 + \tan^2 x
	\end{equation}

	\begin{equation}
		\sqrt{a^2 + b^2}
	\end{equation}

	\begin{equation}
		\sqrt{(a^2 +b^2)}
	\end{equation}

	\begin{equation}
		\frac{x}{1 + \sin x}
	\end{equation}

	\begin{equation}
		\sum \frac{1}{i^2}
	\end{equation}

	\begin{equation}
		\sum_{i=1}^{n} \frac{n!}{i!}2^i
	\end{equation}

	\begin{equation}
		\sum_{i=1}^n 2^i
	\end{equation}

	\begin{equation}
		\int x \ dx
	\end{equation}

	\begin{equation}
		\int x \ dx = 1
	\end{equation}

	\begin{equation}
		x^2 - y^{2^6}
	\end{equation}

	\begin{equation}
		\sum_{x=1}^\infty \frac{1}{x}
	\end{equation}

	\begin{equation}
		ax^2 + bx + c
	\end{equation}

	\begin{equation}
		\Phi([V1]^{[V2]})
	\end{equation}

	\begin{equation}
		g(n) = o(m^n)
	\end{equation}

	\begin{equation}
		poly(n)
	\end{equation}

	\begin{equation}
		\exp\sum_t \lambda f(t,Q,t) \frac{1}{z_\lambda (Q)}
	\end{equation}

	\begin{equation}
		\pi = \frac{c}{d}
	\end{equation}

	\begin{equation}
		F-E+V=2
	\end{equation}

	\begin{equation}
		(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}
	\end{equation}

	\begin{equation}
		F(k) = \frac{\varphi^k - (-\frac{1}{\varphi})^k}{\sqrt{5}}
	\end{equation}

	\begin{equation}
		\sum_{k=1}^{\infty}\frac{1}{k^s}=\prod_{p \;\; \mathrm{prime}}\frac{1}{1-p^{-s}}
	\end{equation}

	\begin{equation}
		f(x) = \frac{d}{dx}\int_{a}^{x}f(s)ds
	\end{equation}

	\begin{equation}
		\int _1^{e^x} \frac{dy}{y}= x
	\end{equation}

	\begin{equation}
		e^x = \lim_{y\to\infty} \left(1+\frac{x}{y}\right)^y
	\end{equation}

	\begin{equation}
		e^{i\pi}+1=0
	\end{equation}

	\begin{equation}
		\Gamma(n) = \int_0^\infty e^{-x}x^{n-1}dx
	\end{equation}

	\begin{equation}
		\int_{-\infty}^{\infty} e^{-x^2}dx=\sqrt{\pi}
	\end{equation}

	\begin{equation}
		F(s) = \int_{-\infty}^{\infty}f(x)e^{-s x}dx
	\end{equation}

	\begin{equation}
		P(A|B) = \frac{P(A \cap B)}{P(B)}
	\end{equation}

	\begin{equation}
		P(B) = \sum_k P(A_k \cap B)
	\end{equation}

	\begin{equation}
		P(A|B) = \frac{P(B|A)P(A)}{P(B)}
	\end{equation}

	\begin{equation}
		p(x) = \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}
	\end{equation}

	\begin{equation}
		H(X) = -\sum_{x \in \mathbf{X}} p(x) \log p(x)
	\end{equation}

	\begin{equation}
		f(x) = \sum_{k=0} ^ {\infty} \frac {f^{(k)}(a)}{k!} \, (x-a)^{k}
	\end{equation}

	\begin{equation}
		e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}
	\end{equation}

	\begin{equation}
		\ln(1+x) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}x^k}{k}
	\end{equation}

	\begin{equation}
		F=G\frac{m_1 m_2}{d^2}
	\end{equation}

	\begin{equation}
		\frac{\partial^2u}{\partial t ^2}= c^2\nabla^2 u
	\end{equation}

	\begin{equation}
		\frac{\partial \varnothing}{\partial t} = D \nabla^2 \varnothing
	\end{equation}

	\begin{equation}
		\nabla \cdot E = \frac{\rho}{\varepsilon}
	\end{equation}

	\begin{equation}
		\nabla \times E = -\frac{\partial B}{\partial t}
	\end{equation}

	\begin{equation}
		\nabla \cdot B = 0
	\end{equation}

	\begin{equation}
		\nabla \times B = \mu J + \mu \varepsilon \frac{\partial E}{\partial t}
	\end{equation}

	\begin{equation}
		E = mc^2
	\end{equation}

	\begin{equation}
		\mathcal{G} = - \frac{8\pi G}{c^4} \mathcal{T}
	\end{equation}

	\begin{equation}
		i \hbar \frac{\partial \Psi}{\partial t} = \hat H \Psi
	\end{equation}

	\begin{equation}
		\tilde \rho
	\end{equation}

	\begin{equation}
		\overline {u} = (x, y, z)
	\end{equation}

	\begin{equation}
		1 + \tan ^{2} \theta = \sec^{2} \theta
	\end{equation}

	\begin{equation}
		\cos{(\theta_\mathrm{E})} = e ^ {-TR/T_1 }
	\end{equation}

	\begin{equation}
		a = g \frac{m_1-m_2}{m_1+m_2} 
	\end{equation}

	\begin{equation}
		\int_{a}^{b}{ f\left(x\right) dx = F(b) - F(a).}
	\end{equation}

	\begin{equation}
		(1/6, \sqrt{1/28}, -\sqrt{12/7}, 0,\, 0,\, 0,\, 0,\, 0)
	\end{equation}

	\begin{equation}
		\sum_{i=m}^{n}{a_i = a_m + a_{m+1} + a_{m+2} + \cdots + a_{n-1} + a_n.} 
	\end{equation}

	\begin{equation}
		f\left(x; \mu,c\right) = \sqrt{\frac{c}{2 \pi }} \,\,\,\, \frac{ e^{- \frac{c}{2(x - \mu )}}  } {(x - \mu)^{3/2}}
	\end{equation}

	\begin{equation}
		D4 \sigma  = 4 \sigma = 4 \sqrt{ \frac{ \int_{- \infty }^{\infty} \int_{- \infty }^{\infty} I(x,y) (x - \overline{x} )^{2} dx \, dy} { \int_{- \infty }^{\infty} \int_{- \infty }^{\infty}I(x,y) dx\,dy }}
	\end{equation}

	\begin{equation}
		\frac{d}{dx}(cx^n) = ncx^{n-1}
	\end{equation}

	\begin{equation}
		\int udv = uv - \int vdu
	\end{equation}

	\begin{equation}
		\sin^2 x = 1 - \cos^2 x
	\end{equation}

	\begin{equation}
		\begin{vmatrix}
			{x} && {y} && {1} \\
			{x_0} && {y_0} && {1} \\ {x_1} && {y_1} && {1}
		\end{vmatrix} = 0
	\end{equation}

	\begin{equation}
		\frac{\pi}{4} = 1 + \frac{1^2}{2+\frac{3^2}{2+\frac{5^2}{2+\frac{7^2}{2+\cdots}}}}
	\end{equation}

	\begin{equation}
		\frac{(x - h)^2}{a^2}+\frac{(y-k)^2}{b^2} = 1
	\end{equation}

	\begin{equation}
		\cos \theta  = \frac{(x_1,y_1)\cdot(x_2,y_2)}{l_1 l_2}
	\end{equation}

	\begin{equation}
		S(x) = \sum_{d | x} d = \prod_{i = 1}^n \frac{p_i ^{e_i + 1} - 1}{p_i - 1}
	\end{equation}

	\begin{equation}
		\int \sec ^n x dx = \frac{\tan x \sec^{n-1}x}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2} x dx , n \ne 1
	\end{equation}

	\begin{equation}
		\lim_{h \to 0} \frac{1}{h} \left( \frac{-h}{x(x+h)}\right) = \lim_{h \to 0} \frac{-1}{x(x+h)} = \frac{-1}{x^2}
	\end{equation}
\end{document}