Restriction of Antireflexive Relation is Antireflexive

Theorem

Let $S$ be a set.

Let $\RR \subseteq S \times S$ be an antireflexive 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 an antireflexive relation on $T$.

Proof

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

Then:

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

So:

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

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

$\blacksquare$

Also see

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