# Definition:Involution (Mapping)

## Definition

An involution is a mapping which is its own inverse.

### Definition 1

$f: A \to A$ is an involution precisely when:

$\forall x \in A: \map f {\map f x} = x$

That is:

$f \circ f = I_A$

where $I_A$ denotes the identity mapping on $A$.

### Definition 2

$f: A \to A$ is an involution precisely when:

$\forall x, y \in A: \map f x = y \implies \map f y = x$

### Definition 3

Let $f: A \to A$ be a mapping on $A$.

Then $f$ is an involution if and only if $f$ is both a bijection and a symmetric relation.

That is, if and only if $f$ is a bijection such that $f = f^{-1}$.

## Also known as

An involution is also known as an involutive mapping or an involutive function.

An involutive mapping can also be found described as self-inverse.

## Also see

• Results about involutions can be found here.