Equivalence of Definitions of Involutive Mapping
Theorem
The following definitions of the concept of Involutive Mapping are equivalent:
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:
- $f = f^{-1}$
Proof
$(1)$ implies $(2)$
Let $f: A \to A$ be an involutive mapping by definition 1.
Then by definition:
- $(1): \quad \forall x \in A: \map f {\map f x} = x$
Let $\map f x = y$.
Then by substituting $y$ for $\map f x$ into $(1)$:
- $\forall x \in A: \map f y = x$
Thus $f: A \to A$ is an involutive mapping by definition 2.
$\Box$
$(2)$ implies $(1)$
Let $f: A \to A$ be an involutive mapping by definition 2.
Then by definition:
- $\forall x, y \in A: \map f x = y \implies \map f y = x$
and so substituting $\map f x$ for $y$ we have:
- $\map f {\map f x} = x$
Thus $f: A \to A$ is an involutive mapping by definition 1.
$\Box$
$(1)$ iff $(3)$
This is demonstrated in Mapping is Involution iff Bijective and Symmetric.
$\blacksquare$