# Equivalence of Definitions of Self-Inverse

## Contents

## 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 |

$\blacksquare$