Symmetric Difference of Unions is Subset of Union of Symmetric Differences

Theorem

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

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

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


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


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

Thus we have:

$$ $$ $$ $$