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.

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