Editing Involution (mathematics) (section)

== General properties ==
Any involution is a [[bijection]].

The [[identity function|identity map]] is a trivial example of an involution. Examples of nontrivial involutions include [[Additive inverse|negation]] ({{math|''x'' ↦ −''x''}}), [[multiplicative inverse|reciprocation]] ({{math|''x'' ↦ 1/''x''}}), and [[complex conjugate|complex conjugation]] ({{math|''z'' ↦ {{overline|''z''}}}}) in [[arithmetic]]; [[reflection (mathematics)|reflection]], half-turn [[rotation (mathematics)|rotation]], and [[circle inversion]] in [[geometry]]; [[complement (set theory)|complementation]] in [[set theory]]; and [[reciprocal cipher]]s such as the [[ROT13]] transformation and the [[Beaufort cipher|Beaufort]] [[polyalphabetic cipher]].

The [[Function composition|composition]] {{math|''g'' ∘ ''f''}} of two involutions {{math|''f''}} and {{math|''g''}} is an involution if and only if they [[Commutative property|commute]]: {{math|1=''g'' ∘ ''f'' = ''f'' ∘ ''g''}}.<ref>{{citation|title=The Elements of Operator Theory|first=Carlos S.|last=Kubrusly|publisher=Springer Science & Business Media|year=2011|isbn=9780817649982|at=Problem&nbsp;1.11(a), p.&nbsp;27|url=https://books.google.com/books?id=g-UFYTO8SbMC&pg=PA27}}.</ref>