Restriction of Symmetric Relation is Symmetric

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Theorem

Let $S$ be a set.

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


Proof

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


Then:

\(\ds \tuple {x, y}\) \(\in\) \(\ds \RR {\restriction_T}\)
\(\ds \leadsto \ \ \) \(\ds \tuple {x, y}\) \(\in\) \(\ds \paren {T \times T} \cap \RR\) Definition of Restriction of Relation
\(\ds \leadsto \ \ \) \(\ds \tuple {x, y}\) \(\in\) \(\ds T \times T\)
\(\, \ds \land \, \) \(\ds \tuple {x, y}\) \(\in\) \(\ds \RR\) Definition of Set Intersection
\(\ds \leadsto \ \ \) \(\ds \tuple {y, x}\) \(\in\) \(\ds T \times T\)
\(\, \ds \land \, \) \(\ds \tuple {y, x}\) \(\in\) \(\ds \RR\) $\RR$ is symmetric on $S$
\(\ds \leadsto \ \ \) \(\ds \tuple {y, x}\) \(\in\) \(\ds \paren {T \times T} \cap \RR\) Definition of Set Intersection
\(\ds \leadsto \ \ \) \(\ds \tuple {y, x}\) \(\in\) \(\ds \RR {\restriction_T}\) Definition of Restriction of Relation


and so $\RR {\restriction_T}$ is symmetric on $T$.

$\blacksquare$


Also see


Sources