Relation equals Inverse iff Symmetric

From ProofWiki

Jump to: navigation, search

Theorem

A relation $\mathcal R$ is symmetric iff it equals its inverse:

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

Suppose $\mathcal R \subseteq S \times S$ is symmetric.

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

Thus $\mathcal R^{-1} \subseteq \mathcal R$ and, from Inverse Relation Equal iff Subset, $\mathcal R^{-1} = \mathcal R$.

Now suppose $\mathcal R^{-1} = \mathcal R$.

Thus:

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

... so $\mathcal R$ is symmetric by the definition of a Symmetric Relation.

$\blacksquare$

Some sources use this definition:

$\left\{{\left({t, s}\right): \left({s, t}\right) \in \mathcal R}\right\} = \mathcal R$

as the definition of a symmetric relation.

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

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