Function

A function is a relation $f\subseteq A\times B$ between sets $A$ and $B$ such that for every $a\in A$ there exists a unique $b\in B$ with $(a,b)\in f$; we write this $b$ as $f(a)$ and denote the function by $f:A\to B$.

The set $A$ is called the domain and the set $B$ the codomain . Functions allow you to form images and preimages of subsets and to build new functions via composition .

Examples:

• The squaring map $f:\mathbb{N}\to\mathbb{N}$ given by $f(n)=n^2$ is a function on the natural numbers .
• If $S$ is a subset of a set $A$, the inclusion map $i:S\to A$ defined by $i(s)=s$ is a function.