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 \(a+5 = 2\).