# Equivalence of Definitions of Self-Inverse

## Theorem

Let $\struct {S, \circ}$ 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 $x \circ x = e_S$.

### 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

Let $x \in S$.

 $\ds x \circ x$ $=$ $\ds e_S$ by hypothesis $\ds \leadstoandfrom \ \$ $\ds \paren {x \circ x} \circ x^{-1}$ $=$ $\ds e_S \circ x^{-1}$ Monoid Axiom $\text S 0$: Closure $\ds \leadstoandfrom \ \$ $\ds x \circ \paren {x \circ x^{-1} }$ $=$ $\ds e_S \circ x^{-1}$ Monoid Axiom $\text S 1$: Associativity $\ds \leadstoandfrom \ \$ $\ds x \circ \paren {x \circ x^{-1} }$ $=$ $\ds x^{-1}$ Monoid Axiom $\text S 2$: Identity $\ds \leadstoandfrom \ \$ $\ds x \circ e_S$ $=$ $\ds x^{-1}$ Definition of Inverse Element $\ds \leadstoandfrom \ \$ $\ds x$ $=$ $\ds x^{-1}$ Monoid Axiom $\text S 2$: Identity

$\blacksquare$