Symmetric Difference is Subset of Union of Symmetric Differences

Theorem
Let $R, S, T$ be sets.

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

where $R * S$ denotes the symmetric difference between $R$ and $S$.

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 is 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: