Editing Surjective function (section)

==Properties==
A function is [[bijective function|bijective]] if and only if it is both surjective and [[injective function|injective]].

If (as is often done) a function is identified with its [[graph of a function|graph]], then surjectivity is not a property of the function itself, but rather a property of the [[Map (mathematics)|mapping]].<ref>{{cite book|author=T. M. Apostol|title=Mathematical Analysis|year=1981|publisher=Addison-Wesley|page=35}}</ref> This is, the function together with its codomain. Unlike injectivity, surjectivity cannot be read off of the graph of the function alone.

===Surjections as right invertible functions===
The function {{Nowrap|''g'' : ''Y'' → ''X''}} is said to be a [[Inverse function#Left and right inverses|right inverse]] of the function {{Nowrap|''f'' : ''X'' → ''Y''}} if {{Nowrap begin}}''f''(''g''(''y'')) = ''y''{{Nowrap end}} for every ''y'' in ''Y'' (''g'' can be undone by ''f'').  In other words, ''g'' is a right inverse of ''f'' if the [[function composition|composition]] {{Nowrap|''f'' <small>o</small> ''g''}} of ''g'' and ''f'' in that order is the [[identity function]] on the domain ''Y'' of ''g''. The function ''g'' need not be a complete [[inverse function|inverse]] of ''f'' because the composition in the other order, {{Nowrap|''g'' <small>o</small> ''f''}}, may not be the identity function on the domain ''X'' of ''f''. In other words, ''f'' can undo or "''reverse''" ''g'', but cannot necessarily be reversed by it.

Every function with a right inverse is necessarily a surjection. The proposition that every surjective function has a right inverse is equivalent to the [[axiom of choice]].

If {{Nowrap|''f'' : ''X'' → ''Y''}} is surjective and ''B'' is a [[subset]] of ''Y'', then {{Nowrap begin}}''f''(''f''<sup> −1</sup>(''B'')) = ''B''{{Nowrap end}}. Thus, ''B'' can be recovered from its [[preimage]] {{Nowrap|''f''<sup> −1</sup>(''B'')}}.

For example, in the first illustration in the [[#Gallery|gallery]], there is some function ''g'' such that ''g''(''C'') = 4. There is also some function ''f'' such that ''f''(4) = ''C''.  It doesn't matter that ''g'' is not unique (it would also work if ''g''(''C'') equals 3); it only matters that ''f''  "reverses" ''g''.

===Surjections as epimorphisms===
A function {{Nowrap|''f'' : ''X'' → ''Y''}} is surjective if and only if it is [[right-cancellative]]:<ref>{{Cite book
|first=Robert
|last=Goldblatt
|title=Topoi, the Categorial Analysis of Logic
|url=http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3|access-date=2009-11-25
|edition=Revised
|year=2006
|orig-year=1984
|publisher=[[Dover Publications]]
|isbn=978-0-486-45026-1}}</ref> given any functions {{Nowrap|''g'',''h'' : ''Y'' → ''Z''}}, whenever {{Nowrap begin}}''g'' <small>o</small> ''f'' = ''h'' <small>o</small> ''f''{{Nowrap end}}, then {{Nowrap begin}}''g'' = ''h''{{Nowrap end}}. This property is formulated in terms of functions and their [[function composition|composition]] and can be generalized to the more general notion of the [[morphism]]s of a [[category (mathematics)|category]] and their composition. Right-cancellative morphisms are called [[epimorphism]]s. Specifically, surjective functions are precisely the epimorphisms in the [[category of sets]]. The prefix ''epi'' is derived from the Greek preposition ''ἐπί'' meaning ''over'', ''above'', ''on''.

Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse ''g'' of a morphism ''f'' is called a [[section (category theory)|section]] of ''f''. A morphism with a right inverse is called a [[split epimorphism]].

===Surjections as binary relations===
Any function with domain ''X'' and codomain ''Y'' can be seen as a [[left-total relation|left-total]] and [[right-unique relation|right-unique]] binary relation between ''X'' and ''Y'' by identifying it with its [[function graph]]. A surjective function with domain ''X'' and codomain ''Y'' is then a binary relation between ''X'' and ''Y'' that is right-unique and both left-total and [[right-total relation|right-total]].

===Cardinality of the domain of a surjection===
The [[cardinality]] of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If {{Nowrap|''f'' : ''X'' → ''Y''}} is a surjective function, then ''X'' has at least as many elements as ''Y'', in the sense of [[cardinal number]]s.   (The proof appeals to the [[axiom of choice]] to show that a function
{{Nowrap|''g'' : ''Y'' → ''X''}} satisfying ''f''(''g''(''y'')) = ''y'' for all ''y'' in ''Y'' exists. ''g'' is easily seen to be injective, thus the [[Cardinal number#Formal definition|formal definition]] of |''Y''| ≤ |''X''| is satisfied.)

Specifically, if both ''X'' and ''Y'' are [[finite set|finite]] with the same number of elements, then {{Nowrap|''f'' : ''X'' → ''Y''}} is surjective if and only if ''f'' is [[injective]].

Given two sets ''X'' and ''Y'', the notation {{nowrap|''X'' ≤<sup>*</sup> ''Y''}} is used to say that either ''X'' is empty or that there is a surjection from ''Y'' onto ''X''. Using the axiom of choice one can show that {{nowrap|''X'' ≤<sup>*</sup> ''Y''}} and {{nowrap|''Y'' ≤<sup>*</sup> ''X''}} together imply that {{nowrap begin}}|''Y''| = |''X''|,{{nowrap end}} a variant of the [[Schröder–Bernstein theorem]].

===Composition and decomposition===
The [[function composition|composition]] of surjective functions is always surjective: If ''f'' and ''g'' are both surjective, and the codomain of ''g'' is equal to the domain of ''f'', then {{Nowrap|''f'' <small>o</small> ''g''}} is surjective. Conversely, if {{Nowrap|''f'' <small>o</small> ''g''}} is surjective, then ''f'' is surjective (but ''g'', the function applied first, need not be). These properties generalize from surjections in the [[category of sets]] to any [[epimorphism]]s in any [[category (mathematics)|category]].

Any function can be decomposed into a surjection and an [[injective function|injection]]: For any function {{Nowrap|''h'' : ''X'' → ''Z''}} there exist a surjection {{Nowrap|''f'' : ''X'' → ''Y''}} and an injection {{Nowrap|''g'' : ''Y'' → ''Z''}} such that {{Nowrap begin}}''h'' = ''g'' <small>o</small> ''f''{{Nowrap end}}. To see this, define ''Y'' to be the set of [[preimage]]s {{Nowrap|''h''<sup>−1</sup>(''z'')}} where ''z'' is in {{Nowrap|''h''(''X'')}}. These preimages are [[disjoint sets|disjoint]] and [[partition of a set|partition]] ''X''. Then ''f'' carries each ''x'' to the element of ''Y'' which contains it, and ''g'' carries each element of ''Y'' to the point in ''Z'' to which ''h'' sends its points. Then ''f'' is surjective since it is a projection map, and ''g'' is injective by definition.

===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''(~).