Restriction of Reflexive Relation is Reflexive
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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 $T$.
$\blacksquare$
Also see
- Properties of Restriction of Relation‎ for other similar properties of the restriction of a relation.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings