# Equivalence of Definitions of Self-Inverse

## 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

$\blacksquare$