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What (who) is surjective - definition
![range]]) of ''f''. This function is '''not''' surjective, because the image does not fill the whole codomain. In other words, ''Y'' is colored in a two-step process: First, for every ''x'' in ''X'', the point ''f''(''x'') is colored yellow; Second, all the rest of the points in ''Y'', that are not yellow, are colored blue. The function ''f'' would be surjective only if there were no blue points.](https://commons.wikimedia.org/wiki/Special:FilePath/Codomain2.SVG?width=200 "range]]) of ''f''. This function is '''not''' surjective, because the image does not fill the whole codomain. In other words, ''Y'' is colored in a two-step process: First, for every ''x'' in ''X'', the point ''f''(''x'') is colored yellow; Second, all the rest of the points in ''Y'', that are not yellow, are colored blue. The function ''f'' would be surjective only if there were no blue points.")
Wikipedia
In mathematics, a surjective function (also known as surjection, or onto function ) is a function f such that every element y can be mapped from element x so that f(x) = y. In other words, every element of the function's codomain is the image of at least one element of its domain. It is not required that x be 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 word sur 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 the image of its domain. Every surjective function has a right inverse assuming the axiom of choice, and every function with a right inverse is necessarily a surjection. The composition of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.
