A function $f:A \to B$ 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 $\mathbb{N} \to \{ 6 \}$ (where $\mathbb{N}$ is the set of natural numbers) given by $n \mapsto 6$ is surjective. However, the same function viewed as a function $\mathbb{N} \to \mathbb{N}$ is not surjective, because it does not hit the number $4$, for instance.
• The function $\mathbb{N} \to \mathbb{N}$ given by $n \mapsto n+5$ is not surjective, because it does not hit the number $2$, for instance: there is no $a \in \mathbb{N}$ such that $a+5 = 2$.