Union of Symmetric Relations is Symmetric

Theorem
The union of two symmetric relations is also a symmetric relation.

Proof
Let $$\mathcal{R}_1$$ and $$\mathcal{R}_2$$ be symmetric relations on a set $$S$$.

Let $$\mathcal{R}_3 = \mathcal{R}_1 \cup \mathcal{R}_2$$.

Then:

$$ $$ $$ $$ $$ $$