Inverse of Inverse Relation

Theorem

The inverse of an inverse relation is the relation itself:

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

$\ds \tuple {s, t}$

$\in$

$\ds \RR$

$\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

$\blacksquare$

1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations: Theorem $5 \ \text{(a)}$
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Problem $\text{AA}$: Relations
1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.11$: Relations