# Codomain

Let $A$ and $B$ be any sets, and let $f:A\to B$ be a function. Then $B$ is said to be the codomain of $f$.

In general, a function given by a fixed rule on a fixed domain may have many different codomains. For instance, consider the function $f$ given by the rule $f(x) = x^2$ whose domain is the integers. The range of this function is the nonnegative integers, but its codomain could be any set which contains the nonnegative integers, such as the integers ( $f:\mathbb{Z}\to\mathbb{Z}$), the rationals ( $f:\mathbb{Z}\to\mathbb{Q}$), the reals ( $f:\mathbb{Z}\to\mathbb{R}$), the complex numbers ( $f:\mathbb{Z}\to\mathbb{C}$), or the set $\mathbb{Z}_{\geq 0} \cup \{\textrm{Groucho, Harpo, Chico}\}$. In this last case, there are exactly three elements of the codomain which are not in the range. Technically, each of these examples is a different function. (Of course, a function given by the same rule could also take a variety of different domains as well.)

A function is surjective exactly when the range is equal to the codomain.