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| [[Image:Surjection.svg|thumb|A surjective function from [[domain of a function|domain]] ''X'' to [[codomain]] ''Y''. The function is surjective because every point in the codomain is the value of ''f''(''x'') for at least one point ''x'' in the domain.]]
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| In [[mathematics]], a function ''f'' from a [[set (mathematics)|set]] ''X'' to a set ''Y'' is '''surjective''' (or '''onto'''), or a '''surjection''', if every [[Element (mathematics)|element]] ''y'' in ''Y'' has a corresponding element ''x'' in ''X'' such that ''f''(''x'') = ''y''. The function ''f'' may map more than one element of ''X'' to the same element of ''Y''.
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| 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}}.</ref> a group of mainly [[France|French]] 20th-century [[mathematician]]s who wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French [[prefix]] ''[[wiktionary:sur#French|sur]]'' means ''over'' or ''above'' and relates to the fact that the [[image (mathematics)|image]] of the [[domain of a function|domain]] of a surjective function completely covers the function's [[codomain]].
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| ==Definition==
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| A '''surjective function''' is a [[function(mathematics)|function]] whose [[image (mathematics)|image]] is equal to its [[codomain]]. Equivalently, a function ''f'' with [[domain of a function|domain]] ''X'' and codomain ''Y'' is surjective if for every ''y'' in ''Y'' there exists at least one ''x'' in ''X'' with <math>f(x)=y</math>. Surjections are sometimes denoted by a two-headed rightwards arrow ({{unichar|21A0|RIGHTWARDS TWO HEADED ARROW|ulink=Unicode}}),<ref name="Unicode Arrows">{{cite web| title = Arrows – Unicode| url = http://www.unicode.org/charts/PDF/U2190.pdf| accessdate = 2013-05-11}}</ref> as in ''f'' : ''X'' ↠ ''Y''.
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| Symbolically,
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| :If <math>f\colon X \rightarrow Y</math>, then <math>f</math> is said to be surjective if
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| :<math>\forall y \in Y, \, \exists x \in X, \;\; f(x)=y</math>.
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| ==Examples==
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| [[Image:Codomain2.SVG|right|thumb|250px|'''A non-surjective function''' from [[domain of a function|domain]] ''X'' to [[codomain]] ''Y''. The smaller oval inside ''Y'' is the [[Image (mathematics)|image]] (also called [[range (mathematics)|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'' is surjective only if there are no blue points.]]
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| For any set ''X'', the [[identity function]] id<sub>''X''</sub> on ''X'' is surjective.
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| The function {{Nowrap|''f'' : '''Z''' → '''{0,1}'''}} defined by ''f''(''n'') = ''n'' '''[[Modular arithmetic|mod]]''' 2 (that is, [[even number|even]] [[integer]]s are mapped to 0 and [[odd number|odd]] integers to 1) is surjective.
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| The function {{Nowrap|''f'' : '''R''' → '''R'''}} defined by ''f''(''x'') = 2''x'' + 1 is surjective (and even [[bijective function|bijective]]), because for every [[real number]] ''y'' we have an ''x'' such that ''f''(''x'') = ''y'': an appropriate ''x'' is (''y'' − 1)/2.
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| The function {{Nowrap|''f'' : '''R''' → '''R'''}} defined by ''f''(''x'') = ''x''<sup>3</sup>-3''x'' is surjective, because the pre-image of any [[real number]] ''y'' is the solution set of the cubic polynomial equation ''x''<sup>3</sup>-3''x''-''y''=0 and every cubic polynomial with real coefficients has at least one real root.<ref>[[Cubic function#Roots of a cubic function]]</ref> However, this function is not [[injective function|injective]] (and hence not [[bijective function|bijective]]) since e.g. the pre-image of ''y''=2 is {''x''=-1,''x''=2}. (In fact, the pre-image of this function for every ''y'', -2≤''y''≤2 has more than one element.)
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| The function {{Nowrap|''g'' : '''R''' → '''R'''}} defined by {{Nowrap begin}}''g''(''x'') = ''x''<sup>2</sup>{{Nowrap end}} is ''not'' surjective, because there is no real number ''x'' such that {{Nowrap begin}}''x''<sup>2</sup> = −1{{Nowrap end}}. However, the function {{Nowrap|''g'' : '''R''' → '''R<sup>nn</sup>'''}} defined by {{Nowrap begin}}''g''(''x'') = ''x''<sup>2</sup>{{Nowrap end}} (with restricted codomain) ''is'' surjective because for every ''y'' in the nonnegative real codomain ''Y'' there is at least one ''x'' in the real domain ''X'' such that {{Nowrap begin}}''x''<sup>2</sup> = ''y''{{Nowrap end}}.
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| The [[natural logarithm]] function {{Nowrap|ln : <nowiki>(0,+∞)</nowiki> → '''R'''}} is a surjective and even bijective mapping from the set of positive real numbers to the set of all real numbers. Its inverse, the [[exponential function]], is not surjective as its range is the set of positive real numbers and its domain is usually defined to be the set of all real numbers. The [[matrix exponential]] is not surjective when seen as a map from the space of all ''n''×''n'' [[matrix (mathematics)|matrices]] to itself. It is, however, usually defined as a map from the space of all ''n''×''n'' matrices to the [[general linear group]] of degree ''n'', i.e. the [[group (mathematics)|group]] of all ''n''×''n'' [[invertible matrix|invertible matrices]]. Under this definition the matrix exponential is surjective for complex matrices, although still not surjective for real matrices.
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| The [[projection (set theory)|projection]] from a [[cartesian product]] {{Nowrap|''A'' × ''B''}} to one of its factors is surjective.
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| In a 3D video game vectors are projected onto a 2D flat screen by means of a surjective function.
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| [[File:Surjective function.svg|500px|"500px"|thumb|left|Interpretation for '''surjective functions''' in the Cartesian plane, defined by the mapping ''f'' : ''X'' → ''Y'', where ''y'' = ''f''(''x''), ''X'' = domain of function, ''Y'' = range of function. Every element in the range is mapped onto from an element in the domain, by the rule ''f''. There may be a number of domain elements which map to the same range element. That is, every ''y'' in ''Y'' is mapped from an element ''x'' in ''X'', more than one ''x'' can map to the same ''y''. '''Left:''' Only one domain is shown which makes ''f'' surjective. '''Right:''' two possible domains ''X''<sub>1</sub> and ''X''<sub>2</sub> are shown.]]
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| [[File:Non-surjective function2.svg|500px|"500px"|thumb|right|'''Non-surjective functions''' in the Cartesian plane. Although some parts of the function are surjective, where elements ''y'' in ''Y'' do have a value ''x'' in ''X'' such that ''y'' = ''f''(''x''), some parts are not. '''Left:''' There is ''y''<sub>0</sub> in ''Y'', but there is no ''x''<sub>0</sub> in ''X'' such that ''y''<sub>0</sub> = ''f''(''x''<sub>0</sub>). '''Right:''' There are ''y''<sub>1</sub>, ''y''<sub>2</sub> and ''y''<sub>3</sub> in ''Y'', but there are no ''x''<sub>1</sub>, ''x''<sub>2</sub>, and ''x''<sub>3</sub> in ''X'' such that ''y''<sub>1</sub> = ''f''(''x''<sub>1</sub>), ''y''<sub>2</sub> = ''f''(''x''<sub>2</sub>), and ''y''<sub>3</sub> = ''f''(''x''<sub>3</sub>).]]
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| {{-}}
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| ==Properties==
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| A function is [[bijective function|bijective]] if and only if it is both surjective and [[injective function|injective]].
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| 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 relationship between the function and its codomain. Unlike injectivity, surjectivity cannot be read off of the graph of the function alone.
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| ===Surjections as right invertible functions===
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| 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.
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| 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]].
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| 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'')}}.
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| For example, in the first illustration, 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''(''C'') can also equal 3; it only matters that ''f'' "reverses" ''g''.
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| <gallery> | |
| Image:Bijection.svg|Another surjective function. (This one happens to be a [[Bijective function|bijection]])
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| Image:Injection.svg|A '''non'''-surjective function. (This one happens to be an [[Injective function|injection]])
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| Image:Surjective_composition.svg|Surjective composition: the first function need not be surjective.
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| </gallery> | |
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| ===Surjections as epimorphisms===
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| A function {{Nowrap|''f'' : ''X'' → ''Y''}} is surjective if and only if it is [[right-cancellative]]:<ref>{{Cite book
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| |first=Robert
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| |last=Goldblatt
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| |title=Topoi, the Categorial Analysis of Logic
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| |url=http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3|accessdate=2009-11-25
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| |edition=Revised
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| |year=2006
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| |origyear=1984
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| |publisher=[[Dover Publications]]
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| |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''.
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| 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]].
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| ===Surjections as binary relations===
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| 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]].
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| ===Cardinality of the domain of a surjection===
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| 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
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| {{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.)
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| 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]].
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| ===Composition and decomposition===
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| The [[function composition|composite]] 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]].
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| 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 sets {{Nowrap|''h''<sup> −1</sup>(''z'')}} where ''z'' is in ''Z''. These sets 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.
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| ===Induced surjection and induced bijection===
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| 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''(~).
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| ==See also==
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| {{Wiktionary|surjective|surjection|onto}}
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| *[[Bijection, injection and surjection]]
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| *[[Cover (algebra)]]
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| *[[Covering map]]
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| *[[Enumeration]]
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| *[[Fiber bundle]]
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| *[[Index set]]
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| *[[Section (category theory)]]
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| ==Notes==
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| <references/>
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| ==References==
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| * {{Cite book
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| |title=Theory of Sets
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| |last=Bourbaki
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| |first=Nicolas
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| |authorlink=Nicolas Bourbaki
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| |year=2004
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| |origyear=1968
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| |publisher=Springer
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| |isbn=978-3-540-22525-6
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| |ref=bourbaki}}
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| {{DEFAULTSORT:Surjective Function}}
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| [[Category:Functions and mappings]]
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| [[Category:Basic concepts in set theory]]
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| [[Category:Mathematical relations]]
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| [[Category:Types of functions]]
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