# Definition:Involution (Mapping)

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## Definition

Let $A$ be a set.

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

### Definition 1

$f$ is an **involution** if and only if:

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

That is:

- $f \circ f = I_A$

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

### Definition 2

$f$ is an **involution** if and only if:

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

### Definition 3

$f$ 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**.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**involution**