Equivalence of Definitions of Self-Inverse

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Theorem

Let $\left({S, \circ}\right)$ be a monoid whose identity is $e_S$.

Let $x \in S$.


The following definitions of the concept of Self-Inverse Element in the context of Abstract Algebra are equivalent:

Definition 1

$x$ is a self-inverse element of $\struct {S, \circ}$ if and only if $x \circ x = e$.

Definition 2

$x$ is a self-inverse element of $\struct {S, \circ}$ if and only if:

$x$ is invertible

and:

$x = x^{-1}$, where $x^{-1}$ is the inverse of $x$.


Proof

\(\displaystyle x \circ x\) \(=\) \(\displaystyle e_S\)
\(\displaystyle \iff \ \ \) \(\displaystyle x \circ x \circ x^{-1}\) \(=\) \(\displaystyle e_S \circ x^{-1}\)
\(\displaystyle \iff \ \ \) \(\displaystyle x \circ e_S\) \(=\) \(\displaystyle x^{-1}\) Definition of Inverse Element
\(\displaystyle \iff \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle x^{-1}\) Definition of Identity Element

$\blacksquare$