Symmetric Difference of Unions is Subset of Union of Symmetric Differences

Theorem
Let $S_i, T_i$ be sets for $i \in \N$.

Then:


 * $\displaystyle \forall n \in \N_{>0}: \bigcup_{i \mathop = 1}^n S_i * \bigcup_{i \mathop = 1}^n T_i \subseteq \bigcup_{i \mathop = 1}^n \left({S_i * T_i}\right)$

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

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


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


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

where $\setminus$ denotes set difference.

Thus we have: