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| {{redirect|Injective|injective modules|Injective module}}
| | I'm Maddison (19) from Alsemberg, Belgium. <br>Ι'm learning Japanese literature аt ɑ local college and І'm jսst аbout to graduate.<br>Ӏ haνe a ρart time job in ɑ college.<br><br>Here is my web page :: [http://www.mechatrician.com/I_did_not_stumble_upon_the_Romanian_deadlift_until_2007._Reason_why_it_rocks! romanian deadlift to deadlift ratio] |
| {{Redirect|1-to-1|other uses of the term|One-to-one (disambiguation){{!}}One-to-one}}
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| [[Image:Injection.svg|thumb|200px|An injective non-surjective function (not a [[bijection]])]]
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| [[Image:Bijection.svg|thumb|200px|An injective surjective function ([[bijection]])]]
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| [[Image:Surjection.svg|thumb|200px|A non-injective surjective function ([[surjection]])]]
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| In [[mathematics]], an '''injective function''' or '''injection''' or '''one-to-one function''' is a [[function (mathematics)|function]] that preserves [[distinct]]ness: it never maps distinct elements of its [[Domain of a function|domain]] to the same element of its [[codomain]]. In other words, every element of the function's codomain is the [[image (mathematics)|image]] of ''at most'' one element of its domain. The term ''one-to-one function'' must not to be confused with ''one-to-one correspondence'' (aka [[surjective]] injection or [[bijective function]]), which uniquely maps all elements in both domain and codomain to each other, (see figures).
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| Occasionally, an injective function from ''X'' to ''Y'' is denoted {{nowrap|''f'': ''X'' ↣ ''Y''}}, using an arrow with a barbed tail ({{unichar|21A3|RIGHTWARDS ARROW WITH TAIL|ulink=Unicode}}).<ref>{{cite web| title = Unicode| url = http://www.unicode.org/charts/PDF/U2190.pdf| accessdate = 2013-05-11}}</ref> The [[set (mathematics)|set]] of injective functions from ''X'' to ''Y'' may be denoted ''Y''<sup><u>''X''</u></sup> using a notation derived from that used for [[falling factorial power]]s, since if ''X'' and ''Y'' are finite sets with respectively ''m'' and ''n'' elements, the number of injections from ''X'' to ''Y'' is ''n''<sup><u>''m''</u></sup> (see the [[Twelvefold way#case i|twelvefold way]]).
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| A function ''f'' that is not injective is sometimes called many-to-one. However, this terminology is also sometimes used to mean "single-valued", i.e., each argument is mapped to at most one value.
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| A [[monomorphism]] is a generalization of an injective function in [[category theory]].
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| == Definition ==
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| Let ''f'' be a [[Function (mathematics)|function]] whose [[Domain of a function|domain]] is a set ''A''. The function ''f'' is '''injective''' if and only if for all ''a'' and ''b'' in ''A'', if ''f''(''a'') = ''f''(''b''), then ''a'' = ''b''; that is, ''f''(''a'') = ''f''(''b'') implies ''a'' = ''b''. Equivalently, if ''a'' ≠ ''b'', then ''f''(''a'') ≠ ''f''(''b'').
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| Symbolically,
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| : <math>\forall a,b \in A, \;\; f(a)=f(b) \Rightarrow a=b</math>
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| which is logically equivalent to the [[contrapositive]],
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| : <math>\forall a,b \in A, \;\; a \neq b \Rightarrow f(a) \neq f(b)</math>
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| == Examples ==
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| *For any set ''X'' and any subset ''S'' of ''X'' the [[inclusion map]] {{nowrap|''S'' → ''X''}} (which sends any element ''s'' of ''S'' to itself) is injective. In particular the [[identity function]] {{nowrap|''X'' → ''X''}} is always injective (and in fact bijective).
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| *If the domain ''X'' = ∅ or ''X'' has only one element, the function {{nowrap|''X'' → ''Y''}} is always injective.
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| *The function ''f'' : '''R''' → '''R''' defined by ''f''(''x'') = 2''x'' + 1 is injective.
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| *The function ''g'' : '''R''' → '''R''' defined by ''g''(''x'') = ''x''<sup>2</sup> is ''not'' injective, because (for example) ''g''(1) = 1 = ''g''(−1). However, if ''g'' is redefined so that its domain is the non-negative real numbers <nowiki>[0,+∞)</nowiki>, then ''g'' is injective.
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| *The [[exponential function]] exp : '''R''' → '''R''' defined by exp(''x'') = ''e''<sup>''x''</sup> is injective (but not [[surjective]] as no real value maps to a negative number).
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| *The [[natural logarithm]] function ln : (0, ∞) → '''R''' defined by ''x'' ↦ ln ''x'' is injective.
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| *The function ''g'' : '''R''' → '''R''' defined by ''g''(''x'') = ''x''<sup>''n''</sup> − ''x'' is not injective, since, for example, ''g''(0) = ''g''(1).
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| More generally, when ''X'' and ''Y'' are both the [[real line]] '''R''', then an injective function ''f'' : '''R''' → '''R''' is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the ''[[horizontal line test]]''.
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| [[File:Injective function.svg|310px|"310px"|left|thumb|Injective functions. Diagramatic interpretation in the [[Cartesian plane]], defined by the [[Map (mathematics)|mapping]] ''f'' : ''X'' → ''Y'', where ''y'' = ''f''(''x''), ''X'' = [[Domain of a function|domain of function]], ''Y'' = [[range (mathematics)|range of function]], and im(''f'') denotes [[Image (mathematics)|image]] of ''f''. Every one ''x'' in ''X'' maps to exactly one unique ''y'' in ''Y''. The circled parts of the axes represent domain and range sets – in accordance with the standard diagrams above.]]
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| [[File:Non-injective function1.svg|400px|"400px"|right|thumb|Not an injective function. Here ''X''<sub>1</sub> and ''X''<sub>2</sub> are subsets of ''X'', ''Y''<sub>1</sub> and ''Y''<sub>2</sub> are subsets of ''Y'': for two regions where the function is not injective because more than one domain [[Element (mathematics)|element]] can map to a single range element. That is, it is possible for ''more than one'' ''x'' in ''X'' to map to the ''same'' ''y'' in ''Y''.]]
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| [[File:Non-injective function2.svg|550px|"550px"|right|thumb|Making functions injective. The previous function ''f'' : ''X'' → ''Y'' can be reduced to one or more injective functions (say) ''f'' : ''X''<sub>1</sub> → ''Y''<sub>1</sub> and ''f'' : ''X''<sub>2</sub> → ''Y''<sub>2</sub>, shown by solid curves (long-dash parts of initial curve are not mapped to anymore). Notice how the rule ''f'' has not changed – only the domain and range. ''X''<sub>1</sub> and ''X''<sub>2</sub> are subsets of ''X'', ''Y''<sub>1</sub> and ''Y''<sub>2</sub> are subsets of ''R'': for two regions where the initial function can be made injective so that one domain element can map to a single range element. That is, only one ''x'' in ''X'' maps to one ''y'' in ''Y''.]]
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| {{-}}
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| == Injections can be undone ==
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| Functions with [[Inverse_function#Left_and_right_inverses|left inverses]] are always injections. That is, given ''f'' : ''X'' → ''Y'', if there is a function ''g'' : ''Y'' → ''X'' such that, for every ''x'' ∈ ''X''
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| :''g''(''f''(''x'')) = ''x'' (''f'' can be undone by ''g'')
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|
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| then ''f'' is injective. In this case, ''f'' is called a [[Retract (category theory)|section]] of ''g'' and ''g'' is called a [[Retract (category theory)|retraction]] of ''f''.
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| Conversely, every injection ''f'' with non-empty domain has a left inverse ''g'' (in conventional mathematics<ref>This principle is valid in conventional mathematics, but may fail in [[constructive mathematics]]. For instance, a left inverse of the inclusion {0,1} → '''R''' of the two-element set in the reals violates [[indecomposability]] by giving a [[Retract (category theory)|retraction]] of the real line to the set {0,1}.</ref>). Note that ''g'' may not be a complete [[inverse function|inverse]] of ''f'' because the composition in the other order, {{Nowrap|''f'' <small>o</small> ''g''}}, may not be the identity on ''Y''. In other words, a function that can be undone or "''reversed''", such as ''f'', is not necessarily [[inverse function|invertible]] ([[bijective]]). Injections are "''reversible''" but not always invertible.
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| Although it is impossible to reverse a non-injective (and therefore information-losing) function, one can at least obtain a "quasi-inverse" of it, that is a [[multivalued function|multiple-valued]] function.
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| == Injections may be made invertible ==
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| In fact, to turn an injective function ''f'' : ''X'' → ''Y'' into a [[bijective function|bijective]] (hence [[Inverse function|invertible]]) function, it suffices to replace its codomain ''Y'' by its actual range ''J'' = ''f''(''X''). That is, let ''g'' : ''X'' → ''J'' such that ''g''(''x'') = ''f''(''x'') for all ''x'' in ''X''; then ''g'' is bijective. Indeed, ''f'' can be factored as {{Nowrap| incl<sub>''J'',''Y''</sub> <small>o</small> ''g''}}, where incl<sub>''J'',''Y''</sub> is the [[inclusion function]] from ''J'' into ''Y''.
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| == Other properties ==
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| * If ''f'' and ''g'' are both injective, then {{Nowrap|''f'' <small>o</small> ''g''}} is injective.
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| [[Image:Injective composition2.svg|thumb|300px|The composition of two injective functions is injective.]]
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| * If {{Nowrap|''g'' <small>o</small> ''f''}} is injective, then ''f'' is injective (but ''g'' need not be).
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| * ''f'' : ''X'' → ''Y'' is injective if and only if, given any functions ''g'', ''h'' : ''W'' → ''X'', whenever {{Nowrap|''f'' <small>o</small> ''g''}} = {{Nowrap|''f'' <small>o</small> ''h''}}, then ''g'' = ''h''. In other words, injective functions are precisely the [[monomorphism]]s in the [[category theory|category]] '''[[Category of sets|Set]]''' of sets.
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| * If ''f'' : ''X'' → ''Y'' is injective and ''A'' is a [[subset]] of ''X'', then ''f''<sup> −1</sup>(''f''(''A'')) = ''A''. Thus, ''A'' can be recovered from its [[image (function)|image]] ''f''(''A'').
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| * If ''f'' : ''X'' → ''Y'' is injective and ''A'' and ''B'' are both subsets of ''X'', then ''f''(''A'' ∩ ''B'') = ''f''(''A'') ∩ ''f''(''B'').
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| * Every function ''h'' : ''W'' → ''Y'' can be decomposed as ''h'' = {{Nowrap|''f'' <small>o</small> ''g''}} for a suitable injection ''f'' and surjection ''g''. This decomposition is unique [[up to isomorphism]], and ''f'' may be thought of as the [[inclusion function]] of the range ''h''(''W'') of ''h'' as a subset of the codomain ''Y'' of ''h''.
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| * If ''f'' : ''X'' → ''Y'' is an injective function, then ''Y'' has at least as many elements as ''X'', in the sense of [[cardinal number]]s. In particular, if, in addition, there is an injection from <math>Y</math> to <math>X</math>, then <math>X</math> and <math>Y</math> have the same cardinal number. (This is known as the [[Cantor–Bernstein–Schroeder theorem]].)
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| * If both ''X'' and ''Y'' are [[finite set|finite]] with the same number of elements, then ''f'' : ''X'' → ''Y'' is injective if and only if ''f'' is [[surjective]] (in which case ''f'' is [[bijective]]).
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| *An injective function which is a [[homomorphism]] between two algebraic structures is an [[embedding]].
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| *Unlike surjectivity, which is a relation between the graph of a function and its codomain, injectivity is a property of the graph of the function alone; that is, whether a function f is injective can be decided by only considering the graph (and not the codomain) of f.
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| ==Proving that functions are injective==
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| A proof that a function ''ƒ'' is injective depends on how the function is presented and what properties the function holds.
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| For functions that are given by some formula there is a basic idea.
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| We use the contrapositive of the definition of injectivity, namely that if ''ƒ''(''x'') = ''ƒ''(''y''), then ''x'' = ''y''.<ref>{{cite web|last=Williams|first=Peter|title=Proving Functions One-to-One|url=http://www.math.csusb.edu/notes/proofs/bpf/node4.html}}</ref>
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| Here is an example:
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| : ''ƒ'' = 2''x'' + 3
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| Proof: Let ''ƒ'' : ''X'' → ''Y''. Suppose ''ƒ''(''x'') = ''ƒ''(''y''). So 2''x'' + 3 = 2''y'' + 3 => 2''x'' = 2''y'' => ''x'' = ''y''. Therefore it follows from the definition that ''ƒ'' is injective. Q.E.D.
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| There are multiple other methods of proving that a function is injective. For example, in calculus if ''ƒ'' is differentiable, then it is sufficient to show that the derivative is always positive or always negative. In linear algebra, if ''ƒ'' is a linear transformation it is sufficient to show that the kernel of ''ƒ'' contains only the zero vector. If ''ƒ'' is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list.
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| == See also ==
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| *[[Surjective function]]
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| *[[Bijective function]]
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| *[[Injective module]]
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| *[[Bijection, injection and surjection]]
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| *[[Horizontal line test]]
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| *[[Injective metric space]]
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| == Notes ==
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| <references/>
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| == References ==
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| * {{Citation | last1=Bartle | first1=Robert G. | title=The Elements of Real Analysis | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | isbn=978-0-471-05464-1 | year=1976}}, p. 17 ''ff''.
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| * {{Citation | last1=Halmos | first1=Paul R. | author1-link=Paul R. Halmos | title=[[Naive Set Theory (book)|Naive Set Theory]] | isbn=978-0-387-90092-6 | year=1974 | publisher=Springer | location=New York}}, p. 38 ''ff''.
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| ==External links==
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| {{Wiktionary|injective}}
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| *[http://jeff560.tripod.com/i.html Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.]
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| *[http://www.khanacademy.org/math/linear-algebra/v/surjective--onto--and-injective--one-to-one--functions Khan Academy – Surjective (onto) and Injective (one-to-one) functions: Introduction to surjective and injective functions]
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| [[Category:Functions and mappings]]
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| [[Category:Basic concepts in set theory]]
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| [[Category:Types of functions]]
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