Editing Surjective function (section)

{{short description|Mathematical function such that every output has at least one input}}
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{{Functions}}
In [[mathematics]], a '''surjective function''' (also known as '''surjection''', or '''onto function''' {{IPAc-en|ˈ|ɒ|n|.|t|uː}}) is a [[Function (mathematics)|function]] {{math|''f''}} such that, for every element {{math|''y''}} of the function's [[codomain]], there exists {{em|at least}} one element {{math|''x''}} in the function's [[Domain of a function|domain]] such that {{math|''f''(''x'') {{=}} ''y''}}. In other words, for a function {{math|''f'' : ''X'' → ''Y''}}, the codomain {{math|''Y''}} is the [[Image (mathematics)|image]] of the function's domain {{math|''X''}}.<ref name=":0">{{Cite web|url=https://www.mathsisfun.com/sets/injective-surjective-bijective.html|title=Injective, Surjective and Bijective|website=www.mathsisfun.com|access-date=2019-12-07}}</ref><ref name=":1">{{Cite web|url=https://brilliant.org/wiki/bijection-injection-and-surjection/|title=Bijection, Injection, And Surjection {{!}} Brilliant Math & Science Wiki|website=brilliant.org|language=en-us|access-date=2019-12-07}}</ref> It is not required that {{math|''x''}} be [[unique (mathematics)|unique]]; the function {{math|''f''}} may map one or more elements of {{math|''X''}} to the same element of {{math|''Y''}}.

The term ''surjective'' and the related terms ''[[injective function|injective]]'' and ''[[bijective function|bijective]]'' were introduced by [[Nicolas Bourbaki]],<ref>{{Citation | url = http://jeff560.tripod.com/i.html | title = Earliest Uses of Some of the Words of Mathematics | contribution = Injection, Surjection and Bijection | publisher = Tripod |first=Jeff|last=Miller}}.</ref><ref>{{Cite book|url=https://books.google.com/books?id=-CXn6y_1nJ8C&q=injection+surjection+bijection+bourbaki&pg=PA106|title=Bourbaki|last=Mashaal|first=Maurice|date=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-3967-6|pages=106|language=en}}</ref> a group of mainly [[France|French]] 20th-century [[mathematician]]s who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word ''[[wikt:sur#French|sur]]'' means ''over'' or ''above'', and relates to the fact that the [[image (mathematics)|image]] of the domain of a surjective function completely covers the function's codomain.

Any function induces a surjection by [[restriction of a function|restricting]] its codomain to the image of its domain. Every surjective function has a [[Inverse function#Left and right inverses|right inverse]] assuming the [[axiom of choice]], and every function with a right inverse is necessarily a surjection. The [[function composition|composition]] of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.