surjektívust

Surjektívust, often translated as surjectivity or onto-ness, is a fundamental property of functions in mathematics. A function f from a set A to a set B is called surjective if every element in the codomain B is mapped to by at least one element in the domain A. In simpler terms, for every element y in B, there exists at least one element x in A such that f(x) = y.

To illustrate, consider a function f mapping students in a class (set A) to their assigned grades (set B). If the function is surjective, it means that every possible grade in set B (e.g., A, B, C, D, F) has been awarded to at least one student. If there's a grade that no student received, the function would not be surjective.

A common notation to express that f is surjective from A to B is f: A → B (surjective) or sometimes f: A ↠ B. Alternatively, one might state that f maps A onto B.

The concept of surjectivity is closely related to injectivity (one-to-one) and bijectivity (both injective and surjective). A bijective function is both injective and surjective, meaning each element in the codomain is mapped to by exactly one element in the domain. Surjectivity alone does not imply injectivity, as multiple elements in the domain can map to the same element in the codomain.