Mapping/Mother of/Example

Let ${}H$ denote the set of all (living or dead) people. We study the mapping

$\varphi \colon H\longrightarrow H,$

which assigns to every person his or her (biological) mother. This is well-defined, as every person has a uniquely determined mother. This mapping is not injective, since there exists different people (brothers and sisters) with the same mother. It is also not surjective, since not every person is a mother of somebody.