Symmetric Difference of Unions

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Theorem

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

Proof

\(\displaystyle \left({R \cup T}\right) * \left({S \cup T}\right)\) \(=\) \(\displaystyle \left({\left({R \cup T}\right) \setminus \left({S \cup T}\right)}\right) \cup \left({\left({S \cup T}\right) \setminus \left({R \cup T}\right)}\right)\) Definition of Symmetric Difference
\(\displaystyle \) \(=\) \(\displaystyle \left({\left({R \setminus S}\right) \setminus T}\right) \cup \left({\left({S \setminus R}\right) \setminus T}\right)\) Set Difference with Union
\(\displaystyle \) \(=\) \(\displaystyle \left ({\left({R \setminus S}\right) \cup \left({S \setminus R}\right)}\right) \setminus T\) Set Difference is Right Distributive over Union
\(\displaystyle \) \(=\) \(\displaystyle \left({R * S}\right) \setminus T\) Definition of Symmetric Difference

$\blacksquare$