# 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.