Restriction of Non-Reflexive Relation is Not Necessarily Non-Reflexive
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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
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.
$\blacksquare$
Also see
- Properties of Relation Not Preserved by Restriction‎ for other similar results.