A function $f:A\to B$ is surjective if every $b\in B$ has some $a\in A$ such that $f\left(a\right)=b$. That is, its codomain is equal to its image.

This concept is commonly referred to as being “onto”, as in “The function $f$ is onto.”

Examples

• The function $N\to \left\{6\right\}$ (where $N$ is the set of natural numbers) given by $n\mapsto 6$ is surjective. However, the same function viewed as a function $N\to N$ is not surjective, because it does not hit the number $4$, for instance.

• The function $N\to N$ given by $n\mapsto n+5$ is not surjective, because it does not hit the number $2$, for instance: there is no $a\in N$ such that $a+5=2$.