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''}} | {{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> | ||
Latest revision as of 22:39, 26 November 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 f(x) = xScript error: No such module "Check for unknown parameters". is true for all values of Template:Mvar to which Template:Mvar can be applied.
Definition
Formally, if XScript error: No such module "Check for unknown parameters". is a set, the identity function fScript error: No such module "Check for unknown parameters". on XScript error: No such module "Check for unknown parameters". is defined to be a function with XScript error: No such module "Check for unknown parameters". as its domain and codomain, satisfying Template:Bi
In other words, the function value f(x)Script error: No such module "Check for unknown parameters". in the codomain XScript error: No such module "Check for unknown parameters". is always the same as the input element xScript error: No such module "Check for unknown parameters". in the domain XScript error: No such module "Check for unknown parameters".. 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 fScript error: No such module "Check for unknown parameters". on XScript error: No such module "Check for unknown parameters". is often denoted by idXScript error: No such module "Check for unknown parameters"..
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 XScript error: No such module "Check for unknown parameters"..[2]
Algebraic properties
If f : X → YScript error: No such module "Check for unknown parameters". is any function, then f ∘ idX = f = idY ∘ fScript error: No such module "Check for unknown parameters"., where "∘" denotes function composition.[3] In particular, idXScript error: No such module "Check for unknown parameters". is the identity element of the monoid of all functions from XScript error: No such module "Check for unknown parameters". to XScript error: No such module "Check for unknown parameters". (under function composition).
Since the identity element of a monoid is unique,[4] one can alternately define the identity function on MScript error: No such module "Check for unknown parameters". to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of MScript error: No such module "Check for unknown parameters". 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 InScript error: No such module "Check for unknown parameters"., 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 C1Script error: No such module "Check for unknown parameters".).[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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