Restriction of Connected Relation is Connected

Theorem
Let $S$ be a set.

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

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

That is:
 * $\forall a, b \in S: a \ne b \implies \tuple {a, b} \in \RR \lor \tuple {b, a} \in \RR$

So:

and so $\RR {\restriction_T}$ is connected on $T$.

Also see

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