Restriction of Reflexive Relation is Reflexive

Theorem

Let $S$ be a set.

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

Suppose $\RR$ is reflexive on $S$.

So:

$\forall x \in S: \tuple {x, x} \in \RR$

We are given $T$ is a subset of $S$, so:

$\forall x \in T: \tuple {x, x} \in \RR$

By definition of restriction:

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

Hence $\RR {\restriction_T}$ is by definition reflexive on $T$.

$\blacksquare$

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

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings