Restriction of Transitive Relation is Transitive

Theorem
Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a transitive relation on $S$.

Let $T \subseteq S$ be a subset of $S$.

Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$.

Then $\RR {\restriction_T}$ is a transitive relation on $T$.

Proof
Suppose $\RR$ is transitive on $S$.

Then by definition:
 * $\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \in \RR$

So:

Therefore, if $x, y, z \in T$, it follows that:
 * $\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR {\restriction_T} \implies \tuple {x, z} \in \RR {\restriction_T}$

and so by definition $\RR {\restriction_T}$ is a transitive relation on $T$.

Also see

 * Properties of Restriction of Relation‎ for other similar properties of the restriction of a relation.