Codomain

A codomain is the target set of a function: if $f:A\to B$ is a function , then its codomain is the set $B$.

The image of $f$ is always a subset of the codomain. The function $f$ is surjective exactly when its image equals its codomain.

Examples:

• For $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=x^2$, the codomain is $\mathbb{R}$ even though not every real number occurs as a value of $f$.
• Define $p:\mathbb{Z}\to\{0,1\}$ by $p(n)=0$ if $n$ is even and $p(n)=1$ if $n$ is odd; the codomain is the set $\{0,1\}$.