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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