\documentclass{article}

\usepackage[english]{babel}

\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{amsthm}
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\theoremstyle{definition}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{axiom}{Axiom}

\begin{document}

\section{Functions}

% function
\begin{definition}
  Let $A$ and $B$ be sets. A \emph{function} from $A$ into $B$ is a set $f\subseteq A\times B$ such that
  $$
    \forall a\in A \ \exists!b\in B \quad (a,b)\in f.
  $$
  We write $f: A\to B$ to denote that $f$ is a function from $A$ into $B$. 
  
  If $a\in A$, we write $f(a)$ to denote the unique $b\in B$ such that $(a,b)\in f$.
\end{definition}
    
% injective
\begin{definition}
  A function $f:A\to B$ is \emph{injective} if 
  $$
    \forall x,y\in A \quad f(x)=f(y) \implies x=y
  $$
\end{definition}

% surjective
\begin{definition}
  A function $f: A\to B$ is \emph{surjective} if
  $$
    \forall b\in B \ \exists a\in A \quad f(a)=b
  $$
  If $f: A\to B$ is surjective, we say $f$ is a function from $A$ \emph{onto} $B$.
\end{definition}

% bijective
\begin{definition}
  A function is \emph{bijective} if it is both injective and surjective.
\end{definition}

\end{document}