Surjection
In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x) = y. It is not required that x is unique; the function f may map one or more elements of X to the same element of Y.
The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who under this pseudonym wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French prefix means over or above and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain.
Any function induces a surjection by restricting its codomain to its range. Every surjective function has a right inverse, and every function with a right inverse is necessarily a surjection. The composite of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.
A surjective function is a function whose image is equal to its codomain. Equivalently, a function f with domain X and codomain Y is surjective if for every y in Y there exists at least one x in X with . Surjections are sometimes denoted by a two-headed rightwards arrow (U+21A0 ↠ RIGHTWARDS TWO HEADED ARROW), as in f : X ↠ Y.
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