Difference of Unions is Subset of Union of Differences

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Theorem

Let $I$ be an indexing set.

Let $S_\alpha, T_\alpha$ be sets, for all $\alpha \in I$.


Then:

$\displaystyle \paren {\bigcup_{\alpha \mathop \in I} S_\alpha} \setminus \paren {\bigcup_{\alpha \mathop \in I} T_\alpha} \subseteq \bigcup_{\alpha \mathop \in I} \paren {S_\alpha \setminus T_\alpha}$

where $S_\alpha \setminus T_\alpha$ denotes set difference.


Proof

Let $\displaystyle x \in \paren {\bigcup_{\alpha \mathop \in I} S_\alpha} \setminus \paren {\bigcup_{\alpha \mathop \in I} T_\alpha}$.

Then by definition of set difference:

\(\displaystyle x\) \(\in\) \(\displaystyle \bigcup_{\alpha \mathop \in I} S_\alpha\)
\(\displaystyle x\) \(\notin\) \(\displaystyle \bigcup_{\alpha \mathop \in I} T_\alpha\)


By definition of set union, it follows that:

\(\displaystyle \exists \beta \in I: x\) \(\in\) \(\displaystyle S_\beta\)
\(\displaystyle \neg \exists \beta \in I: x\) \(\in\) \(\displaystyle T_\beta\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \forall \beta \in I: x\) \(\notin\) \(\displaystyle T_\beta\) De Morgan's Laws (Predicate Logic)

and so:

$\exists \beta \in I: x \in S_\beta \setminus T_\beta$


Hence:

$\displaystyle x \in \bigcup_{\alpha \mathop \in I} \paren {S_\alpha \setminus T_\alpha}$

by definition of set union.

The result follows by definition of subset.

$\blacksquare$


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