Union of Symmetric Relations is Symmetric
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Theorem
The union of two symmetric relations is also a symmetric relation.
Proof
Let $\RR_1$ and $\RR_2$ be symmetric relations on a set $S$.
Let $\RR_3 = \RR_1 \cup \RR_2$.
Then:
\(\ds \tuple {x, y}\) | \(\in\) | \(\ds \RR_3\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tuple {x, y}\) | \(\in\) | \(\ds \RR_1\) | Definition of Set Union | ||||||||||
\(\, \ds \lor \, \) | \(\ds \tuple {x, y}\) | \(\in\) | \(\ds \RR_2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tuple {y, x}\) | \(\in\) | \(\ds \RR_1\) | Definition of Symmetric Relation | ||||||||||
\(\, \ds \lor \, \) | \(\ds \tuple {y, x}\) | \(\in\) | \(\ds \RR_2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tuple {y, x}\) | \(\in\) | \(\ds \RR_3\) | Definition of Set Union |
$\blacksquare$