# Inverse of Transitive Relation is Transitive/Proof 1

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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.

Then:

$\tuple {x, y}, \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$

Thus:

$\tuple {y, x}, \tuple {z, y} \in \RR^{-1} \implies \tuple {z, x} \in \RR^{-1}$

and so $\RR^{-1}$ is transitive.

$\blacksquare$