Intersection Distributes over Union/Family of Sets

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Theorem

Let $I$ be an indexing set.

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

Let $B \subseteq S$.


Then:

$\displaystyle \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cap B} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$

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


Corollary

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

\(\displaystyle x\) \(\in\) \(\displaystyle \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap B}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \exists \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha \cap B\) Definition of Union of Family
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha\) Definition of Set Intersection
\(\, \displaystyle \land \, \) \(\displaystyle x\) \(\in\) \(\displaystyle B\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha}\) Set is Subset of Union
\(\, \displaystyle \land \, \) \(\displaystyle x\) \(\in\) \(\displaystyle B\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B\) Definition of Set Intersection

By definition of subset:

$\displaystyle \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap B} \subseteq \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$

$\Box$


\(\displaystyle x\) \(\in\) \(\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha}\) Definition of Set Intersection
\(\, \displaystyle \land \, \) \(\displaystyle x\) \(\in\) \(\displaystyle B\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \exists \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha\) Definition of Union of Family
\(\, \displaystyle \land \, \) \(\displaystyle x\) \(\in\) \(\displaystyle B\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \exists \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha \cap B\) Definition of Set Intersection
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap B}\) Set is Subset of Union

By definition of subset:

$\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B \subseteq \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cap B}$

$\Box$


By definition of set equality:

$\displaystyle \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cap B} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$

$\blacksquare$


Also see


Sources