Inverse of Inverse Relation

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Theorem

The inverse of an inverse relation is the relation itself:

$\left({\mathcal R^{-1}}\right)^{-1} = \mathcal R$


Proof

\(\displaystyle \left({s, t}\right)\) \(\in\) \(\displaystyle \mathcal R\) $\quad$ $\quad$
\(\displaystyle \iff \ \ \) \(\displaystyle \left({t, s}\right)\) \(\in\) \(\displaystyle \mathcal R^{-1}\) $\quad$ Definition of Inverse Relation $\quad$
\(\displaystyle \iff \ \ \) \(\displaystyle \left({s, t}\right)\) \(\in\) \(\displaystyle \left({\mathcal R^{-1} }\right)^{-1}\) $\quad$ Definition of Inverse Relation $\quad$

$\blacksquare$


Sources