Inverse of Inverse Relation
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
- 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