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

Theorem
Let $S$ be a set.

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

Proof
Proof by Counterexample:

Let $S = \set {a, b}$.

Let $\RR = \set {\tuple {b, b} }$.

$\RR$ is a non-reflexive relation, as can be seen by definition:
 * $\tuple {a, a} \notin \RR$
 * $\tuple {b, b} \in \RR$

Now let $T = \set a$.

Then $\RR {\restriction_T} = \O$.

So:
 * $\forall x \in T: \tuple {x, x} \notin \RR {\restriction_T}$

That is, $\RR {\restriction_T}$ 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.