# Definition:Involution (Mapping)/Definition 1

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## Definition

Let $A$ be a set.

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

$f$ 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 $A$.

## 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

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Exercise $11$: $(2)$