# Surjective function

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 $$a+5 = 2$$.

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