Let f be a function defined on a set A and taking values in a set B. Then f is said to be a surjection (or surjective map) if, for any b element B, there exists an a element A for which b = f(a). 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 V be a vector space with A, B element V. Then a transformation T defined on V is a surjection if there is an A element V such that T(A) = B for all B. 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.
bijection | domain | epimorphism | injection | many-to-one | range