Symmetric Difference of Unions is Subset of Union of Symmetric Differences

Theorem

 * $\displaystyle \forall n \in \N^*: \bigcup_{i=1}^n S_i * \bigcup_{i=1}^n T_i \subseteq \bigcup_{i=1}^n \left({S_i * T_i}\right)$

where $S * T$ is the symmetric difference between $S$ and $T$.

Proof
From Difference of Unions Subset of Union of Differences, we have:


 * $\displaystyle \bigcup_{i=1}^n S_i - \bigcup_{i=1}^n T_i \subseteq \bigcup_{i=1}^n \left({S_i - T_i}\right)$


 * $\displaystyle \bigcup_{i=1}^n T_i - \bigcup_{i=1}^n S_i \subseteq \bigcup_{i=1}^n \left({T_i - S_i}\right)$

Thus we have: