# Equivalence of Definitions of Symmetric Relation

## Theorem

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

### Definition 1

$\RR$ is symmetric if and only if:

$\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR$

### Definition 2

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

$\RR^{-1} = \RR$

### Definition 3

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

$\RR \subseteq \RR^{-1}$

## Proof

### Definition 1 implies Definition 3

Let $\RR$ be a relation which fulfils the condition:

$\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR$

Then:

 $\ds$  $\ds \tuple {x, y} \in \RR$ $\ds$ $\leadsto$ $\ds \tuple {y, x} \in \RR$ by hypothesis $\ds$ $\leadsto$ $\ds \tuple {x, y} \in \RR^{-1}$ Definition of Inverse Relation $\ds$ $\leadsto$ $\ds \RR \subseteq \RR^{-1}$ Definition of Subset

Hence $\RR$ is symmetric by definition 3.

$\Box$

### Definition 3 implies Definition 2

Let $\RR$ be a relation which fulfils the condition:

$\RR \subseteq \RR^{-1}$
$\RR = \RR^{-1}$

Hence $\RR$ is symmetric by definition 2.

$\Box$

### Definition 2 implies Definition 1

Let $\RR$ be a relation which fulfils the condition:

$\RR^{-1} = \RR$

Then:

 $\ds$  $\ds \tuple {x, y} \in \RR$ $\ds$ $\leadsto$ $\ds \tuple {x, y} \in \RR^{-1}$ as $\RR^{-1} = \RR$ $\ds$ $\leadsto$ $\ds \tuple {y, x} \in \RR$ Definition of Inverse Relation

Hence $\RR$ is symmetric by definition 1.

$\blacksquare$