identity function
{{short description|In mathematics, a function that always returns the same value that was used as its argument}}
{{distinguish|Null function|Empty function}}
image:Function-x.svg of the identity function on the real numbers]]
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 {{mvar|f}} is the identity function, the equality {{math|1=f(x) = x}} is true for all values of {{mvar|x}} to which {{mvar|f}} can be applied.
Definition
Formally, if {{math|X}} is a set, the identity function {{math|f}} on {{math|X}} is defined to be a function with {{math|X}} as its domain and codomain, satisfying
{{bi|left=1.6|{{math|1=f(x) = x}} for all elements {{math|x}} in {{math|X}}.{{Citation |last1=Knapp |first1=Anthony W. |title=Basic algebra |year=2006 |publisher=Springer |isbn=978-0-8176-3248-9 }}}}
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), so it is bijective.{{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}}
The identity function {{math|f}} on {{math|X}} is often denoted by {{math|idX}}.
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|X}}.{{Cite book|url=https://books.google.com/books?id=oIFLAQAAIAAJ&q=the+identity+function+is+given+by+the+identity+relation,+or+diagonal|title=Proceedings of Symposia in Pure Mathematics|date=1974|publisher=American Mathematical Society|isbn=978-0-8218-1425-3|pages=92|language=en|quote=...then the diagonal set determined by M is the identity relation...}}
Algebraic properties
If {{math|f : X → Y}} is any function, then {{math|1=f ∘ idX = f = idY ∘ f}}, where "∘" denotes function composition.{{cite book
| last = Nel | first = Louis
| year = 2016
| title = Continuity Theory
| url = https://books.google.com/books?id=_JdPDAAAQBAJ&pg=PA21
| page = 21
| publisher = Springer
| location = Cham
| doi = 10.1007/978-3-319-31159-3
| isbn = 978-3-319-31159-3
}} In particular, {{math|idX}} is the identity element of the monoid of all functions from {{math|X}} to {{math|X}} (under function composition).
Since the identity element of a monoid is unique,{{Cite book|last1=Rosales|first1=J. C.|url=https://books.google.com/books?id=LQsH6m-x8ysC&q=identity+element+of+a+monoid+is+unique&pg=PA1|title=Finitely Generated Commutative Monoids|last2=García-Sánchez|first2=P. A.|date=1999|publisher=Nova Publishers|isbn=978-1-56072-670-8|pages=1|language=en|quote=The element 0 is usually referred to as the identity element and if it exists, it is 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 endomorphisms of {{math|M}} need not be functions.
Properties
- The identity function is a linear operator when applied to vector spaces.{{Citation|last=Anton|first=Howard|year=2005|title=Elementary Linear Algebra (Applications Version)|publisher=Wiley International|edition=9th}}
- In an {{mvar|n}}-dimensional vector space the identity function is represented by the identity matrix {{math|In}}, regardless of the basis chosen for the space.{{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}}
- The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.{{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}}
- 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|C1}}).{{aut|James W. Anderson}}, Hyperbolic Geometry, Springer 2005, {{isbn|1-85233-934-9}}
- In a topological space, the identity function is always continuous.{{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}}
- The identity function is idempotent.{{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.}}
See also
References
{{reflist|30em}}
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Category:Functions and mappings
Category:Elementary mathematics