Union of Symmetric Differences

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

Then:
 * $\paren {R * S} \cup \paren {S * T} = \paren {R \cup S \cup T} \setminus \paren {R \cap S \cap T}$

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

Proof
From the definition of symmetric difference, we have:


 * $R * S = \paren {R \setminus S} \cup \paren {S \setminus R}$

Thus, expanding: