Inverse of Inverse Relation

From ProofWiki
Jump to navigation Jump to search

Theorem

The inverse of an inverse relation is the relation itself:

$\paren {\RR^{-1} }^{-1} = \RR$


Proof

\(\ds \tuple {s, t}\) \(\in\) \(\ds \RR\)
\(\ds \leadstoandfrom \ \ \) \(\ds \tuple {t, s}\) \(\in\) \(\ds \RR^{-1}\) Definition of Inverse Relation
\(\ds \leadstoandfrom \ \ \) \(\ds \tuple {s, t}\) \(\in\) \(\ds \paren {\RR^{-1} }^{-1}\) Definition of Inverse Relation

$\blacksquare$


Sources