Union of Symmetric Differences

Theorem

 * $$R * S \subseteq \left({R * T}\right) \cup \left({S * T}\right)$$

Proof
From the definition of symmetric difference, we have:


 * $$R * S = \left({R - S}\right) \cup \left({S - R}\right)$$

Then from Set Difference Subset of Union of Differences, we have:
 * $$R - S \subseteq \left({R - T}\right) \cup \left({T - S}\right)$$;


 * $$S - R \subseteq \left({S - T}\right) \cup \left({T - R}\right)$$.

Thus:

$$ $$ $$