Editing Surjective function (section)

===Induced surjection and induced bijection===
Any function induces a surjection by restricting its codomain to its range. Any surjective function induces a bijection defined on a [[quotient set|quotient]] of its domain by collapsing all arguments mapping to a given fixed image. More precisely, every surjection {{Nowrap|''f'' : ''A'' → ''B''}} can be factored as a projection followed by a bijection as follows. Let ''A''/~ be the [[equivalence class]]es of ''A'' under the following [[equivalence relation]]: ''x'' ~ ''y'' if and only if ''f''(''x'') = ''f''(''y''). Equivalently, ''A''/~ is the set of all preimages under ''f''. Let ''P''(~) : ''A'' → ''A''/~ be the [[projection map]] which sends each ''x'' in ''A'' to its equivalence class [''x'']<sub>~</sub>, and let ''f''<sub>''P''</sub> : ''A''/~ → ''B'' be the well-defined function given by ''f''<sub>''P''</sub>([''x'']<sub>~</sub>) = ''f''(''x''). Then ''f'' = ''f''<sub>''P''</sub> o ''P''(~).