Restriction of Non-Reflexive Relation is Not Necessarily Non-Reflexive

Theorem
Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be a non-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 not necessarily a non-reflexive relation on $T$.

Proof
Proof by Counterexample:

Let $S = \left\{{a, b}\right\}$.

Let $\mathcal R = \left\{{\left({b, b}\right)}\right\}$.

$\mathcal R$ is a non-reflexive relation, as can be seen by definition:
 * $\left({a, a}\right) \notin \mathcal R$
 * $\left({b, b}\right) \in \mathcal R$

Now let $T = \left\{{a}\right\}$.

Then $\mathcal R \restriction_T \ = \varnothing$.

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

That is, $\mathcal R \restriction_T$ is an antireflexive relation on $T$.

That is, specifically not a non-reflexive relation.

Also see

 * Properties of Relation Not Preserved by Restriction‎ for other similar results.