Mathematical Symbols Guide

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Standard operators
$a + b - c \times d \div e$

% Alternative multiplication
$a \cdot b$ or $a \ast b$ or $a \star b$

% Plus/minus and minus/plus
$x = a \pm b$, $y = c \mp d$

% Advanced operators
$a \oplus b \ominus c \otimes d \oslash e$

% Fractions and ratios
$\frac{a}{b}$, $a/b$, $a:b$

\end{document}

% Set operations
$A \cup B$ (union)
$A \cap B$ (intersection)
$A \setminus B$ (set difference)
$A \triangle B$ (symmetric difference)

% Logic operations
$p \land q$ or $p \wedge q$ (and)
$p \lor q$ or $p \vee q$ (or)
$\neg p$ or $\lnot p$ (not)
$p \oplus q$ (exclusive or)

% Other binary operators
$a \circ b$ (composition)
$a \bullet b$ (bullet)
$a \diamond b$ (diamond)
$a \Box b$ (box)

% Summation
$\sum_{i=1}^{n} a_i$
$\displaystyle\sum_{i=1}^{n} a_i$ % Larger in inline

% Product
$\prod_{i=1}^{n} a_i$

% Integrals
$\int_a^b f(x)\,dx$
$\iint_D f(x,y)\,dA$
$\iiint_V f(x,y,z)\,dV$
$\oint_C F \cdot dr$

% Unions and intersections
$\bigcup_{i=1}^{n} A_i$
$\bigcap_{i=1}^{n} A_i$

% Other large operators
$\coprod$ (coproduct)
$\bigoplus$ (direct sum)
$\bigotimes$ (tensor product)
$\bigvee$ (join)
$\bigwedge$ (meet)

\documentclass{article}
\usepackage{amssymb}
\begin{document}

% Equality and inequality
$a = b$, $a \neq b$ or $a \ne b$

% Comparisons
$a < b$, $a > b$
$a \leq b$ or $a \le b$
$a \geq b$ or $a \ge b$

% Much less/greater
$a \ll b$, $a \gg b$

% Approximately equal
$a \approx b$ (approximately)
$a \simeq b$ (similar equal)
$a \sim b$ (similar)
$a \cong b$ (congruent)

% Equivalence
$a \equiv b$ (equivalent)
$a \triangleq b$ (defined as)

\end{document}

% Membership
$x \in A$ (element of)
$x \notin A$ (not element of)
$A \ni x$ (contains)

% Subset relations
$A \subset B$ (proper subset)
$A \subseteq B$ (subset or equal)
$A \supset B$ (proper superset)
$A \supseteq B$ (superset or equal)

% Special subsets
$A \sqsubset B$ (square subset)
$A \sqsubseteq B$
$A \subsetneq B$ (subset not equal)

% Parallel and perpendicular
$a \parallel b$
$a \perp b$

% Proportional and asymptotic
$y \propto x$ (proportional to)
$f(x) \asymp g(x)$ (asymptotic)

% Order relations
$a \prec b$ (precedes)
$a \preceq b$ (precedes or equal)
$a \succ b$ (succeeds)
$a \succeq b$ (succeeds or equal)

% Other relations
$a \models b$ (models)
$a \vdash b$ (proves)
$a \dashv b$ (reverse proves)
$a \smile b$ (smile)
$a \frown b$ (frown)

% Single arrows
$\rightarrow$ or $\to$
$\leftarrow$ or $\gets$
$\leftrightarrow$
$\uparrow$, $\downarrow$
$\updownarrow$

% Double arrows
$\Rightarrow$ (implies)
$\Leftarrow$ (implied by)
$\Leftrightarrow$ or $\iff$ (if and only if)
$\Uparrow$, $\Downarrow$
$\Updownarrow$

% Long arrows
$\longrightarrow$
$\longleftarrow$
$\longleftrightarrow$
$\Longrightarrow$
$\Longleftarrow$
$\Longleftrightarrow$

% Maps to
$f: A \to B$ or $f: A \rightarrow B$
$x \mapsto f(x)$
$x \longmapsto f(x)$

% Hooked arrows
$\hookrightarrow$ (injection)
$\hookleftarrow$

% Two-headed arrows
$\twoheadrightarrow$ (surjection)
$\twoheadleftarrow$

% Harpoons
$\rightharpoonup$, $\rightharpoondown$
$\leftharpoonup$, $\leftharpoondown$
$\rightleftharpoons$

% Diagonal arrows
$\nearrow$ (northeast)
$\searrow$ (southeast)
$\swarrow$ (southwest)
$\nwarrow$ (northwest)

\documentclass{article}
\usepackage{amsmath}
\begin{document}

% Basic delimiters
$(a + b)$
$[a + b]$
$\{a + b\}$
$\langle a, b \rangle$

% Floor and ceiling
$\lfloor x \rfloor$ (floor)
$\lceil x \rceil$ (ceiling)

% Absolute value and norm
$|x|$ or $\lvert x \rvert$
$\|x\|$ or $\lVert x \rVert$

% Automatic sizing
$\left( \frac{a}{b} \right)$
$\left[ \sum_{i=1}^{n} a_i \right]$
$\left\{ x : x > 0 \right\}$

% Manual sizing
$\big( \Big( \bigg( \Bigg($
$\big] \Big] \bigg] \Bigg]$

\end{document}

% Mixed delimiters
$\left( a, b \right]$ (half-open interval)
$\left[ a, b \right)$

% Invisible delimiters
$\left. \frac{df}{dx} \right|_{x=0}$

% Multiple sizes
\begin{align}
&\text{Small: } (x) \\
&\text{big: } \big(x\big) \\
&\text{Big: } \Big(x\Big) \\
&\text{bigg: } \bigg(x\bigg) \\
&\text{Bigg: } \Bigg(x\Bigg)
\end{align}

% Angle brackets for inner products
$\langle x, y \rangle$
$\langle x \mid y \rangle$ % Quantum mechanics

% Standard vs variant forms
$\epsilon$ vs $\varepsilon$
$\theta$ vs $\vartheta$
$\pi$ vs $\varpi$
$\rho$ vs $\varrho$
$\sigma$ vs $\varsigma$
$\phi$ vs $\varphi$

% Bold Greek (requires bm package)
\usepackage{bm}
$\bm{\alpha}$, $\bm{\beta}$, $\bm{\Omega}$

% Upright Greek (requires upgreek)
\usepackage{upgreek}
$\upalpha$, $\upbeta$, $\upgamma$

% Dots
$a_1 + a_2 + \cdots + a_n$ (centered dots)
$a_1, a_2, \ldots, a_n$ (low dots)
$\vdots$ (vertical dots)
$\ddots$ (diagonal dots)

% Over and under dots
$\dot{x}$ (first derivative)
$\ddot{x}$ (second derivative)
$\dddot{x}$ (third derivative)
$\ddddot{x}$ (fourth derivative)

% Accents
$\hat{a}$ (hat)
$\check{a}$ (check)
$\tilde{a}$ (tilde)
$\acute{a}$ (acute)
$\grave{a}$ (grave)
$\bar{a}$ (bar)
$\vec{a}$ (vector)
$\breve{a}$ (breve)

% Wide accents
$\widehat{ABC}$
$\widetilde{xyz}$
$\overline{a + b}$
$\underline{a + b}$

% Overbraces and underbraces
$\overbrace{a + b + c}^{\text{sum}}$
$\underbrace{x \cdot x \cdot x}_{n \text{ times}}$

% Overlining and underlining
$\overline{AB}$ (line segment)
$\underline{important}$

% Stacking
$\overset{?}{=}$ (question over equals)
$\underset{n \to \infty}{\lim}$ (limit notation)
$\overset{def}{=}$ (defined as)

% Blackboard bold (requires amssymb)
$\mathbb{N}$ (natural numbers)
$\mathbb{Z}$ (integers)
$\mathbb{Q}$ (rationals)
$\mathbb{R}$ (reals)
$\mathbb{C}$ (complex)

% Calligraphic
$\mathcal{A}$, $\mathcal{B}$, $\mathcal{L}$

% Fraktur (requires amssymb)
$\mathfrak{a}$, $\mathfrak{g}$, $\mathfrak{H}$

% Script (requires mathrsfs)
\usepackage{mathrsfs}
$\mathscr{A}$, $\mathscr{F}$, $\mathscr{L}$

% Bold
$\mathbf{x}$, $\mathbf{A}$
$\boldsymbol{\alpha}$ (bold Greek)

% Spacing commands
$ab$ (no space)
$a\,b$ (thin space)
$a\:b$ (medium space)
$a\;b$ (thick space)
$a\quad b$ (quad space)
$a\qquad b$ (double quad)

% Negative spacing
$a\!b$ (negative thin space)

% Common uses
$\int f(x)\,dx$ (before dx)
$n\text{-}th$ (hyphen in text)
$5\,\text{cm}$ (before units)

% Text in math
$x \in \mathbb{R} \text{ such that } x > 0$

% Derivatives
$\frac{d}{dx}$, $\frac{\partial}{\partial x}$
$\nabla$ (gradient)
$\nabla \cdot$ (divergence)
$\nabla \times$ (curl)
$\Box$ or $\square$ (d'Alembertian)

% Quantum mechanics
$\hbar$ (reduced Planck constant)
$\langle \psi | \phi \rangle$ (inner product)
$| \psi \rangle$ (ket)
$\langle \phi |$ (bra)

% Units
$^\circ$ (degree)
$\AA$ (Angstrom)

% Probability
$P(A \mid B)$ (conditional)
$\mathbb{E}[X]$ (expectation)
$\text{Var}(X)$ (variance)
$X \sim N(\mu, \sigma^2)$ (distribution)

% Statistics
$\bar{x}$ (mean)
$\hat{\theta}$ (estimator)
$s^2$ (sample variance)
$r$ (correlation)