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

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

$\blacksquare$