In Mathematics , a Function ''f'' is said to be if its values span its whole Codomain ; that is, for every ''y'' in the codomain, there is at least one ''x'' in the Domain such that ''f''(''x'') = ''y''.
Said another way, a function ''f'': ''X'' → ''Y'' is surjective if and only if its range ''f''(''X'') is equal to its codomain ''Y''. A surjective function is called a , and also said to be '''onto'''.
- For any set ''X'', the identity function id''X'' on ''X'' is surjective.
- The function ''f'': → defined by ''f''(''x'') = 2''x'' + 1 is surjective, because for every real number ''y'' we have ''f''(''x'') = ''y'' where ''x'' is (''y'' - 1)/2.
- The Natural Logarithm function ln: (0,+∞) → is surjective.
- The function ''f'': → '''{0,1,2,3}''' defined by ''f''(''x'') = ''x'' ''' Mod ''' 4 is surjective.
- The function ''g'': → defined by ''g''(''x'') = ''x''&2 is ''not'' surjective, because (for example) there is no real number ''x'' such that ''x''&2 = −1. However, if the codomain is defined as [0,+∞), then ''g'' is surjective.
Every function with a there exists a function ''g'': ''Y'' → ''X'' such that, for every
: (''g'' can be undone by ''f'')
that is a function ''g'' such that ''f'' o ''g'' equals the Identity Function on ''Y'' (cf. with definition of Inverse Function ).
Note that ''g'' may not be a complete Inverse of ''f'' because the composition in the other order, ''g'' o ''f'', may not be the identity on ''X''. In other words, f can undo or "''reverse''" ''g'', but not necessarily can be reversed by it. Surjections are not always Invertible ( Bijective ).
- If ''f'' and ''g'' are both surjective, then ''f'' o ''g'' is surjective.
- If ''f'' o ''g'' is surjective, then ''f'' is surjective (but ''g'' may not be).
- ''f'': ''X'' → ''Y'' is surjective if and only if, given any functions ''g'',''h'':''Y'' → ''Z'', whenever ''g'' o ''f'' = ''h'' o ''f'', then ''g'' = ''h''. In other words, surjective functions are precisely the Epimorphism s in the Category of sets.
- If ''f'': ''X'' → ''Y'' is surjective and ''B'' is a Subset of ''Y'', then ''f''(''f'' −1(''B'')) = ''B''. Thus, ''B'' can be recovered from its Preimage ''f'' −1(''B'').
- Every function ''h'': ''X'' → ''Z'' can be decomposed as ''h'' = ''g'' o ''f'' for a suitable surjection ''f'' and Injective Function ''g''. This decomposition is unique Up To Isomorphism , and ''f'' may be thought of as a function with the same values as ''h'' but with its codomain restricted to the range ''h''(''W'') of ''h'', which is only a subset of the codomain ''Z'' of ''h''.
- By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain. More precisely, every surjection ''f'' : ''A'' → ''B'' can be factored as a projection followed by a bijection as follows. Let ''A''/~ be the equivalence classes 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 and let ''f''''P'' : ''A''/~ → ''B'' be the well-defined function given by ''f''''P''([''x'' ~) = ''f''(''x''). Then ''f'' = ''f''''P'' o ''P''(~).
- If ''f'': ''X'' → ''Y'' is a surjective function, then ''X'' has at least as many elements as ''Y'', in the sense of Cardinal Number s.
- If both ''X'' and ''Y'' are .
In the language of Category Theory , surjective functions are precisely the Epimorphism s in the Category Of Sets .
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