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
- 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