Inverse of Inverse Relation

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Theorem

The inverse of an inverse relation is the relation itself:

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


Proof

\(\displaystyle \tuple {s, t}\) \(\in\) \(\displaystyle \mathcal R\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \tuple {t, s}\) \(\in\) \(\displaystyle \mathcal R^{-1}\) Definition of Inverse Relation
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \tuple {s, t}\) \(\in\) \(\displaystyle \paren {\mathcal R^{-1} }^{-1}\) Definition of Inverse Relation

$\blacksquare$


Sources