Function (map)

Let $A$ and $B$ be sets . A function (or map) $f$ from $A$ to $B$, written $f\colon A\to B$, can be defined as a relation $f \subseteq A\times B$ (with $A\times B$ the Cartesian product of $A$ and $B$) such that

where $(a,b)$ is an ordered pair and “$\exists!$” means “there exists a unique.”

The domain of $f$ is the input set $A$ and the codomain is the target set $B$; the actual outputs form the image $f(A)\subseteq B$. Functions may be injective or surjective depending on how they hit the codomain.

Examples:

• $f\colon\mathbb{Z}\to\mathbb{Z}$ given by $f(n)=2n$.
• The inclusion map $\iota\colon \{1,2\}\to\{1,2,3\}$ with $\iota(x)=x$.
• A constant function $c\colon A\to B$ defined by $c(a)=b_0$ for a fixed $b_0\in B$.