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:


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

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


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

Thus:

$$ $$ $$