Identity function: Difference between revisions
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==Definition== | ==Definition== | ||
Formally, if {{math|''X''}} is a [[Set (mathematics)|set]], the identity function {{math|''f''}} on {{math|''X''}} is defined to be a function with {{math|''X''}} as its [[domain of a function|domain]] and [[codomain]], satisfying | Formally, if {{math|''X''}} is a [[Set (mathematics)|set]], the identity function {{math|''f''}} on {{math|''X''}} is defined to be a function with {{math|''X''}} as its [[domain of a function|domain]] and [[codomain]], satisfying | ||
{{bi|left=1.6|{{math|1=''f''(''x'') = ''x''}} for all elements {{math|''x''}} in {{math|''X''}}.<ref>{{ | {{bi|left=1.6|{{math|1=''f''(''x'') = ''x''}} for all elements {{math|''x''}} in {{math|''X''}}.<ref>{{Cite book |last=Knapp |first=Anthony W. |title=Basic algebra |publisher=Springer |year=2006 |isbn=978-0-8176-3248-9}}</ref>}} | ||
In other words, the function value {{math|''f''(''x'')}} in the codomain {{math|''X''}} is always the same as the input element {{math|''x''}} in the domain {{math|''X''}}. The identity function on {{mvar|X}} is clearly an [[injective function]] as well as a [[surjective function]] (its codomain is also its [[range (function)|range]]), so it is [[bijection|bijective]].<ref>{{cite book |last=Mapa |first=Sadhan Kumar |date= 7 April 2014|title=Higher Algebra Abstract and Linear |edition=11th |publisher=Sarat Book House |page=36 |isbn=978-93-80663-24-1}}</ref> | In other words, the function value {{math|''f''(''x'')}} in the codomain {{math|''X''}} is always the same as the input element {{math|''x''}} in the domain {{math|''X''}}. The identity function on {{mvar|X}} is clearly an [[injective function]] as well as a [[surjective function]] (its codomain is also its [[range (function)|range]]), so it is [[bijection|bijective]].<ref>{{cite book |last=Mapa |first=Sadhan Kumar |date= 7 April 2014|title=Higher Algebra Abstract and Linear |edition=11th |publisher=Sarat Book House |page=36 |isbn=978-93-80663-24-1}}</ref> | ||
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*In an {{mvar|n}}-[[dimension (vector space)|dimensional]] [[vector space]] the identity function is represented by the [[identity matrix]] {{math|''I''<sub>''n''</sub>}}, regardless of the [[basis (linear algebra)|basis]] chosen for the space.<ref>{{cite book|title=Applied Linear Algebra and Matrix Analysis|author=T. S. Shores|year=2007|publisher=Springer|isbn=978-038-733-195-9|series=Undergraduate Texts in Mathematics|url=https://books.google.com/books?id=8qwTb9P-iW8C&q=Matrix+Analysis}}</ref> | *In an {{mvar|n}}-[[dimension (vector space)|dimensional]] [[vector space]] the identity function is represented by the [[identity matrix]] {{math|''I''<sub>''n''</sub>}}, regardless of the [[basis (linear algebra)|basis]] chosen for the space.<ref>{{cite book|title=Applied Linear Algebra and Matrix Analysis|author=T. S. Shores|year=2007|publisher=Springer|isbn=978-038-733-195-9|series=Undergraduate Texts in Mathematics|url=https://books.google.com/books?id=8qwTb9P-iW8C&q=Matrix+Analysis}}</ref> | ||
*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> | *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> | ||
*In a [[metric space]] the identity function is trivially an [[isometry]]. An object without any [[symmetry]] has as its [[symmetry group]] the [[trivial group]] containing only this isometry (symmetry type {{math|C<sub>1</sub>}}).<ref>{{ | *In a [[metric space]] the identity function is trivially an [[isometry]]. An object without any [[symmetry]] has as its [[symmetry group]] the [[trivial group]] containing only this isometry (symmetry type {{math|C<sub>1</sub>}}).<ref>{{Cite book |last=Anderson |first=James W. |title=Hyperbolic geometry |date=2007 |publisher=Springer |isbn=978-1-85233-934-0 |edition=2. ed., corr. print |series=Springer undergraduate mathematics series |location=London}}</ref> | ||
*In a [[topological space]], the identity function is always [[Continuous_function#Continuous_functions_between_topological_spaces|continuous]].<ref>{{Cite book|last=Conover|first=Robert A.|url=https://books.google.com/books?id=KCziAgAAQBAJ&q=identity+function+is+always+continuous&pg=PA65|title=A First Course in Topology: An Introduction to Mathematical Thinking|date=2014-05-21|publisher=Courier Corporation|isbn=978-0-486-78001-6|pages=65|language=en}}</ref> | *In a [[topological space]], the identity function is always [[Continuous_function#Continuous_functions_between_topological_spaces|continuous]].<ref>{{Cite book|last=Conover|first=Robert A.|url=https://books.google.com/books?id=KCziAgAAQBAJ&q=identity+function+is+always+continuous&pg=PA65|title=A First Course in Topology: An Introduction to Mathematical Thinking|date=2014-05-21|publisher=Courier Corporation|isbn=978-0-486-78001-6|pages=65|language=en}}</ref> | ||
*The identity function is [[Idempotence|idempotent]].<ref>{{Cite book|last=Conferences|first=University of Michigan Engineering Summer|url=https://books.google.com/books?id=AvAfAAAAMAAJ&q=The+identity+function+is+idempotent.|title=Foundations of Information Systems Engineering|date=1968|language=en|quote=we see that an identity element of a semigroup is idempotent.}}</ref> | *The identity function is [[Idempotence|idempotent]].<ref>{{Cite book|last=Conferences|first=University of Michigan Engineering Summer|url=https://books.google.com/books?id=AvAfAAAAMAAJ&q=The+identity+function+is+idempotent.|title=Foundations of Information Systems Engineering|date=1968|language=en|quote=we see that an identity element of a semigroup is idempotent.}}</ref> | ||
Latest revision as of 16:15, 2 July 2025
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In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when Template:Mvar is the identity function, the equality Template:Math is true for all values of Template:Mvar to which Template:Mvar can be applied.
Definition
Formally, if Template:Math is a set, the identity function Template:Math on Template:Math is defined to be a function with Template:Math as its domain and codomain, satisfying Template:Bi
In other words, the function value Template:Math in the codomain Template:Math is always the same as the input element Template:Math in the domain Template:Math. The identity function on Template:Mvar is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective.[1]
The identity function Template:Math on Template:Math is often denoted by Template:Math.
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 Template:Math.[2]
Algebraic properties
If Template:Math is any function, then Template:Math, where "∘" denotes function composition.[3] In particular, Template:Math is the identity element of the monoid of all functions from Template:Math to Template:Math (under function composition).
Since the identity element of a monoid is unique,[4] one can alternately define the identity function on Template:Math to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of Template:Math need not be functions.
Properties
- The identity function is a linear operator when applied to vector spaces.[5]
- In an Template:Mvar-dimensional vector space the identity function is represented by the identity matrix Template:Math, regardless of the basis chosen for the space.[6]
- The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.[7]
- In a metric space the identity function is trivially an isometry. An object without any symmetry has as its symmetry group the trivial group containing only this isometry (symmetry type Template:Math).[8]
- In a topological space, the identity function is always continuous.[9]
- The identity function is idempotent.[10]
See also
References
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