Editing Identity function (section)

==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
{{bi|left=1.6|{{math|1=''f''(''x'') = ''x''}} &nbsp;&nbsp;for all elements {{math|''x''}} in {{math|''X''}}.<ref>{{Citation |last1=Knapp |first1=Anthony W. |title=Basic algebra |year=2006 |publisher=Springer |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>

The identity function {{math|''f''}} on {{math|''X''}} is often denoted by {{math|id<sub>''X''</sub>}}.

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''}}.<ref>{{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...}}</ref>