Restriction of Connected Relation is Connected

Theorem
Let $S$ be a set.

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

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

That is:
 * $\forall a, b \in S: a \ne b \implies \left({a, b}\right) \in \mathcal R \lor \left({b, a}\right) \in \mathcal R$

So:

and so $\mathcal R \restriction_T$ is connected on $T$.

Also see

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