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$~$.