Brackets and Parentheses in LaTeX - Complete Guide

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

% Parentheses
$(a + b)$

% Square brackets  
$[x, y]$

% Curly braces (must be escaped)
$\{z : z > 0\}$

% Angle brackets
$\langle u, v \rangle$

% Absolute value
$|x - y|$

% Norm
$\|v\|$

% Floor and ceiling
$\lfloor x \rfloor$ and $\lceil x \rceil$

\end{document}

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

% Without automatic sizing
$(\frac{1}{2})$

% With automatic sizing
$\left(\frac{1}{2}\right)$

% Works with any delimiter type
$\left[\frac{x^2}{y}\right]$

$\left\{\sqrt{\frac{a}{b}}\right\}$

$\left|\sum_{i=1}^n x_i\right|$

% Nested fractions
$\left(\frac{\frac{a}{b}}{\frac{c}{d}}\right)$

\end{document}

% Using \right. for invisible right delimiter
$\left\{\begin{array}{ll}
x + y = 1 \\
x - y = 0
\end{array}\right.$

% Using \left. for invisible left delimiter
$\left.\frac{dy}{dx}\right|_{x=0}$

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

% Normal size
$( \frac{1}{2} )$

% \big
$\big( \frac{1}{2} \big)$

% \Big
$\Big( \frac{1}{2} \Big)$

% \bigg
$\bigg( \frac{1}{2} \bigg)$

% \Bigg
$\Bigg( \frac{1}{2} \Bigg)$

% For opening and closing separately
$\bigl( x \bigr)$  % 'l' for left, 'r' for right

\end{document}

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

% Poor: all brackets same size
$[1 + [2 + [3 + 4]]]$

% Better: graduated sizes
$\Big[1 + \big[2 + [3 + 4]\big]\Big]$

% Best: automatic sizing
$\left[1 + \left[2 + \left[3 + 4\right]\right]\right]$

% Complex nesting
$\left\{x : \left[a + \left(\frac{b}{c}\right)\right] > 0\right\}$

\end{document}

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

% Using \right. and \left. for invisible delimiters
\begin{align}
f(x) = & \left[ x^2 + 2x \right. \\
       & \left. + 1 \right]
\end{align}

% Alternative using \big commands
\begin{align}
g(x) = & \Big[ x^3 + 3x^2 \\
       & \phantom{\Big[} + 3x + 1 \Big]
\end{align}

% For cases/piecewise functions
\begin{equation}
f(x) = \begin{cases}
x^2 & \text{if } x \geq 0 \\
-x^2 & \text{if } x < 0
\end{cases}
\end{equation}

\end{document}

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

% Different matrix environments
$\begin{pmatrix}  % parentheses
a & b \\
c & d
\end{pmatrix}$

$\begin{bmatrix}  % brackets
a & b \\
c & d
\end{bmatrix}$

$\begin{Bmatrix}  % braces
a & b \\
c & d
\end{Bmatrix}$

$\begin{vmatrix}  % vertical bars
a & b \\
c & d
\end{vmatrix}$

$\begin{Vmatrix}  % double bars
a & b \\
c & d
\end{Vmatrix}$

\end{document}

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

% Basic set
$A = \{1, 2, 3, 4, 5\}$

% Set builder notation
$B = \{x \in \mathbb{R} : x^2 < 4\}$

% Set with conditions
$C = \left\{x \in \mathbb{Z} : \begin{array}{l}
x > 0 \\
x \text{ is even}
\end{array}\right\}$

% Empty set
$\emptyset = \{\}$

\end{document}

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

% Open interval
$(a, b)$

% Closed interval
$[a, b]$

% Half-open intervals
$[a, b)$ and $(a, b]$

% Infinite intervals
$(-\infty, a]$ and $[b, \infty)$

% Union of intervals
$[0, 1] \cup [2, 3]$

\end{document}

\documentclass{article}
\usepackage{amsmath}
\usepackage{physics} % provides \bra, \ket, \braket
\begin{document}

% Quantum mechanics bra-ket notation
$\langle \psi | \phi \rangle$

% With physics package
$\bra{\psi}\ket{\phi}$
$\braket{\psi|\phi}$

% Commutator
$[A, B] = AB - BA$

% Anticommutator  
$\{A, B\} = AB + BA$

% Poisson bracket
$\{f, g\} = \sum_i \left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)$

\end{document}

% Good spacing with \bigl and \bigr
$\bigl( x + y \bigr)$

% Poor spacing with just \big
$\big( x + y \big)$

% For better spacing around bars
$\left\lvert x \right\rvert$  % Better than |x|

% Instead of manual brackets
$|| v ||$

% Use semantic commands
$\lVert v \rVert$      % Double bars for norm
$\lvert x \rvert$      % Single bars for absolute value
$\langle u, v \rangle$ % Angle brackets for inner product