# Restriction of Symmetric Relation is Symmetric

## 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 $T$.

$\blacksquare$