Restriction of Reflexive Relation is Reflexive

Theorem
Let $S$ be a set.

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

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

Then:
 * $\forall x \in S: \left({x, x}\right) \in \mathcal R$

So:
 * $\forall x \in T: \left({x, x}\right) \in \mathcal R {\restriction_T}$

Thus $\mathcal R {\restriction_T}$ is reflexive on $T$.

Also see

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