\documentclass[letterpaper]{article}

\pagestyle{empty}

\usepackage{amssymb}
\usepackage{amsmath}

% USER DEFINED

\def\gap{\vspace{10pt}\noindent}
\def\rr{\mathbb{R}}
\newcommand{\norm}[1]{\| #1 \|}
\newcommand{\innprod}[2]{\langle #1, #2 \rangle}
 
\begin{document}

\noindent
The congruence symbol $\cong$ does not require an extra package, but 
not-congruence symbol $\ncong$ would require the package `amssymb'.

\gap
It is not so hard to create a matrix:
$$ \begin{pmatrix}
1 & 0 & 1 \\
0 & 1 & 0
\end{pmatrix} $$

\gap
For every vector $x \in \rr^3$, we define the norm of $x$
by $\norm{x}^2 = \innprod{x}{x}$, i.e. 
$$ \norm{x} = \sqrt{\innprod{x}{x}}. $$

\gap
Normal parentheses would not adjust their size to fit the
expression, but we have a solution for that, too:
$$ (\sum_{n=1}^{\infty} \frac{1}{n^s})^2 \,\,\, 
\mbox{versus} \,\,\, \left(\sum_{n=1}^{\infty} \frac{1}{n^s}\right)^2 $$

\gap
Also,
$$ A = \{ \frac{m}{n} : m, n \in \rr \} \,\,\, 
\mbox{versus} \,\,\, A = \left\{ \frac{m}{n} : m, n \in \rr \right\} $$

\gap
We can also define a function implicitly by
$$ f(x) = \left\{
\begin{array}{cl}
x - 1 & \mbox{if } x < 0 \\
x^2 & \mbox{if } 0 \le x < 1 \\
\sin x & \mbox{otherwise}
\end{array}\right. $$

\gap
We can also do the following:
\begin{eqnarray}
(x + y)^3 & = & (x + y)(x + y)(x + y) \\
					& = & (x^2 + 2xy + y^2)(x + y) \\
					& = & x^3 + 3x^2y + 3xy^2 + y^3
\end{eqnarray}

\gap
We could also do without the numberings:
\begin{eqnarray*}
(x + y)^3 & = & (x + y)(x + y)(x + y) \\
					& \le & (x^2 + 2xy + y^2)(x + y) \\
					& < & x^3 + 3x^2y + 3xy^2 + y^3 + 1
\end{eqnarray*}

\end{document}