# Definition:Symmetric Relation

## Definition

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

### 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

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

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

## Examples

### Brotherhood Relation

Let $P$ be the set of male people.

Let $\sim$ be the relation on $P$ defined as:

$\forall \tuple {x, y} \in P \times P: x \sim y \iff \text {$x$is a brother of$y$}$

Then $\sim$ is a symmetric relation.

This does not hold if $P$ is the set of all people.

Because if $a$ is male and $b$ are brother and sister, then:

$a \sim b$

but:

$b \not \sim a$

## Also see

• Results about symmetry of relations can be found here.