Definition:Involution (Mapping)
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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.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: involution