Inverse of Inverse Relation

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

$\blacksquare$