Identity function: Difference between revisions
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{{distinguish|Null function|Empty function}} |
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[[image:Function-x.svg|thumb|Graph of the identity function on the real numbers]] |
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In [[mathematics]], an '''identity function''', also called an '''identity relation''' or '''identity map''' or '''identity transformation''', is a [[function (mathematics)|function]] that always returns the same value that was used as its argument. In [[equation]]s, the function is given by {{math|1=''f''(''x'') = ''x''}}. |
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==Definition== |
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Formally, if {{math|''M''}} is a [[Set (mathematics)|set]], the identity function {{math|''f''}} on {{math|''M''}} is defined to be that function with [[domain of a function|domain]] and [[codomain]] {{math|''M''}} which satisfies |
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:{{math|1=''f''(''x'') = ''x''}} for all elements {{math|''x''}} in {{math|''M''}}.<ref>{{Citation |last1=Knapp |first1=Anthony W. |last2= |first2= |title=Basic algebra |url= |edition= |volume= |year=2006 |publisher=Springer |isbn=978-0-8176-3248-9 }}</ref> |
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In other words, the function value {{math|''f''(''x'')}} in {{math|''M''}} (that is, the codomain) is always the same input element {{math|''x''}} of {{math|''M''}} (now considered as the domain). The identity function on {{mvar|M}} is clearly an [[injective function]] as well as a [[surjective function]], so it is also [[bijection|bijective]].<ref>{{cite book |last=Mapa |first=Sadhan Kumar |date= |title=Higher Algebra Abstract and Linear |url= |edition=11th |publisher=Sarat Book House |page=36 |isbn=978-93-80663-24-1}}</ref> |
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The identity function {{math|''f''}} on {{math|''M''}} is often denoted by {{math|id<sub>''M''</sub>}}. |
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In [[set theory]], where a function is defined as a particular kind of [[binary relation]], the identity function is given by the [[identity relation]], or ''diagonal'' of {{math|''M''}}. |
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==Algebraic property== |
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If {{math|''f'' : ''M'' → ''N''}} is any function, then we have {{math|1=''f'' ∘ id<sub>''M''</sub> = ''f'' = id<sub>''N''</sub> ∘ ''f''}} (where "∘" denotes [[function composition]]). In particular, {{math|id<sub>''M''</sub>}} is the [[identity element]] of the [[monoid]] of all functions from {{math|''M''}} to {{math|''M''}}. |
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Since the identity element of a monoid is [[unique (mathematics)|unique]], one can alternately define the identity function on {{math|''M''}} to be this identity element. Such a definition generalizes to the concept of an [[identity morphism]] in [[category theory]], where the [[endomorphism]]s of {{math|''M''}} need not be functions. |
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==Properties== |
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*The identity function is a [[linear map|linear operator]], when applied to [[vector space]]s.<ref>{{Citation|last=Anton|first=Howard|year=2005|title=Elementary Linear Algebra (Applications Version)|publisher=Wiley International|edition=9th}}</ref> |
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*The identity function on the positive [[integer]]s is a [[completely multiplicative function]] (essentially multiplication by 1), considered in [[number theory]].<ref>{{cite book|title=Number Theory through Inquiry|author1=D. Marshall |author2=E. Odell |author3=M. Starbird |year=2007|publisher=Mathematical Assn of Amer|isbn=978-0883857519|series=Mathematical Association of America Textbooks}}</ref> |
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*In an {{math|''n''}}-dimensional [[vector space]] the identity function is represented by the [[identity matrix]] {{math|I<sub>''n''</sub>}}, regardless of the [[Basis (linear algebra)|basis]].<ref>{{cite book|title=Applied Linear Algebra and Matrix Analysis|author=T. S. Shores|year=2007|publisher=Springer|isbn=038-733-195-6|series=Undergraduate Texts in Mathematics|url=https://books.google.com/books?id=8qwTb9P-iW8C&printsec=frontcover&dq=Matrix+Analysis&hl=en&sa=X&ei=SCd1UryWD_LG7Aag_4HwBg&ved=0CGQQ6AEwCA#v=onepage&q=Matrix%20Analysis&f=false}}</ref> |
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*In a [[metric space]] the identity is trivially an [[isometry]]. An object without any [[symmetry]] has as [[symmetry group]] the trivial group only containing this isometry (symmetry type {{math|''C''<sub>1</sub>}}).<ref>{{aut|James W. Anderson}}, ''Hyperbolic Geometry'', Springer 2005, {{isbn|1-85233-934-9}}</ref> |
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*In a [[topological space]], the identity function is always continuous. |
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==See also== |
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*[[Inclusion map]] |
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==References== |
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{{reflist|30em}} |
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{{DEFAULTSORT:Identity Function}} |
{{DEFAULTSORT:Identity Function}} |
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