# Equivalence of Definitions of Symmetric Relation

## Theorem

The following definitions of the concept of Symmetric Relation are equivalent:

### Definition 1

$\mathcal R$ is symmetric if and only if:

$\tuple {x, y} \in \mathcal R \implies \tuple {y, x} \in \mathcal R$

### Definition 2

$\mathcal R$ is symmetric if and only if it equals its inverse:

$\mathcal R^{-1} = \mathcal R$

### Definition 3

$\mathcal R$ is symmetric if and only if it is a subset of its inverse:

$\mathcal R \subseteq \mathcal R^{-1}$

## Proof

### Definition 1 implies Definition 3

Let $\mathcal R$ be a relation which fulfils the condition:

$\left({x, y}\right) \in \mathcal R \implies \left({y, x}\right) \in \mathcal R$

Then:

 $\displaystyle$  $\displaystyle \left({x, y}\right) \in \mathcal R$ $\displaystyle$ $\implies$ $\displaystyle \left({y, x}\right) \in \mathcal R$ by hypothesis $\displaystyle$ $\implies$ $\displaystyle \left({x, y}\right) \in \mathcal R^{-1}$ Definition of Inverse Relation $\displaystyle$ $\implies$ $\displaystyle \mathcal R \subseteq \mathcal R^{-1}$ Definition of Subset

Hence $\mathcal R$ is symmetric by definition 3.

$\Box$

### Definition 3 implies Definition 2

Let $\mathcal R$ be a relation which fulfils the condition:

$\mathcal R \subseteq \mathcal R^{-1}$
$\mathcal R = \mathcal R^{-1}$

Hence $\mathcal R$ is symmetric by definition 2.

$\Box$

### Definition 2 implies Definition 1

Let $\mathcal R$ be a relation which fulfils the condition:

$\mathcal R^{-1} = \mathcal R$

Then:

 $\displaystyle$  $\displaystyle \left({x, y}\right) \in \mathcal R$ $\displaystyle$ $\implies$ $\displaystyle \left({x, y}\right) \in \mathcal R^{-1}$ as $\mathcal R^{-1} = \mathcal R$ $\displaystyle$ $\implies$ $\displaystyle \left({y, x}\right) \in \mathcal R$ Definition of Inverse Relation

Hence $\mathcal R$ is symmetric by definition 1.

$\blacksquare$