Restriction of Transitive Relation is Transitive

Theorem
Let $S$ be a set.

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

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

Let $\mathcal R \restriction_T \ \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$.

Then $\mathcal R \restriction_T$ is a transitive relation on $T$.

Proof
Suppose $\mathcal R$ is transitive on $S$.

Then by definition:
 * $\left\{{\left({x, y}\right), \left({y, z}\right)}\right\} \subseteq \mathcal R \implies \left({x, z}\right) \in \mathcal R$

So:

Therefore, if $x, y, z \in T$, it follows that:
 * $\left\{{\left({x, y}\right), \left({y, z}\right)}\right\} \subseteq \mathcal R \restriction_T \implies \left({x, z}\right) \in \mathcal R \restriction_T$

and so by definition $\mathcal R \restriction_T$ is a transitive relation on $T$.

Also see

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