Inverse of Transitive Relation is Transitive/Proof 2
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Theorem
Let $\RR$ be a relation on a set $S$.
Let $\RR$ be transitive.
Then its inverse $\RR^{-1}$ is also transitive.
Proof
Let $\RR$ be transitive.
Thus by definition:
- $\RR \circ \RR \subseteq \RR$
Thus:
\(\ds \RR^{-1} \circ \RR^{-1}\) | \(=\) | \(\ds \paren {\RR \circ \RR}^{-1}\) | Inverse of Composite Relation | |||||||||||
\(\ds \) | \(\subseteq\) | \(\ds \RR^{-1}\) | Inverse of Subset of Relation is Subset of Inverse |
$\blacksquare$
Hence the result by definition of transitive relation.
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations