# 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 $T$.

## Proof

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

Then:

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

So:

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

Thus $\RR {\restriction_T}$ is reflexive on $T$.

$\blacksquare$