Surjection

Let  be a function defined on a set  and taking values in a set . Then  is said to be a surjection (or surjective map) if, for any , there exists an  for which . A surjection is sometimes referred to as being "onto."

Let the function be an operator which maps points in the domain to every point in the range and let  be a vector space with . Then a transformation  defined on  is a surjection if there is an  such that  for all .

In the categories of sets, groups, modules, etc., an epimorphism is the same as a surjection, and is used synonymously with "surjection" outside of category theory.