# Involution

<https://en.wikipedia.org/wiki/Involution_(mathematics>)

An **involution**, or an involutory function, is a function `f` that is its own inverse, `f(f(x)) = f(x)` (for all `x` in the domain of f). Equivalently, applying f twice produces the original value.

> Any involution is a bijection.

Example of an involuntary function: `f(x) = -1 * x`

In set theory, involution or *double complement law* states:\
(Aᶜ)ᶜ = A

The term *\*anti-involution* refers to involutions based on *antihomomorphisms* (see Quaternion algebra, groups, semigroups)\
`f(xy) = f(x)f(y)`\
such that\
`xy = f(f(xy)) = f( f(x) f(y) ) = f(f(x)) f(f(y)) = xy`

Examples of involution:

* identity map is a trivial example of an involution
* multiplication by −1 in arithmetic
* taking of reciprocals
* complementation in set theory
* complex conjugation
* circle inversion
* rotation by a half-turn
* reciprocal ciphers such as the ROT13 transformation