Union of Symmetric Differences

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

Then:
 * $\left({R * S}\right) \cup \left({S * T}\right) = \left({R \cup S \cup T}\right) \setminus \left({R \cap S \cap T}\right)$

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

Proof
From the definition of symmetric difference, we have:


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

Thus, expanding: