A function is surjective if every $b \in B$ has some $a \in A$ such that $f (a) = b$ . 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 $N \to {6}$ (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 $a + 5 = 2$ .