Intersection Distributes over Union/Family of Sets/Corollary

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Theorem

Let $I$ and $J$ be indexing sets.

Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\beta}_{\beta \mathop \in J}$ be indexed families of subsets of a set $S$.


Then:

$\displaystyle \bigcup_{\tuple {\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cap B_\beta} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}$

where $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.


Proof

\(\displaystyle \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap B}\) \(=\) \(\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B\) Intersection Distributes over Union: Family of Sets
\(\displaystyle \leadsto \ \ \) \(\displaystyle \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta} }\) \(=\) \(\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}\) setting $\displaystyle B = \paren {\bigcup_{\beta \mathop \in J} B_\beta}$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \bigcup_{\alpha \mathop \in I} \paren {\bigcup_{\beta \mathop \in J} \paren {A_\alpha \cap B_\beta} }\) \(=\) \(\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}\) Intersection Distributes over Union: Family of Sets
\(\displaystyle \leadsto \ \ \) \(\displaystyle \bigcup_{\paren {\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cap B_\beta}\) \(=\) \(\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}\)

$\blacksquare$


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