Union Distributes over Intersection/Family of Sets

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Theorem

Let $I$ be an indexing set.

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

Let $B \subseteq S$.


Then:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B$

where $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha$ denotes the intersection 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:

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

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


Proof

\(\displaystyle x\) \(\in\) \(\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \forall \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha \cup B\) Intersection is Subset
\(\displaystyle \leadsto \ \ \) \(\displaystyle \forall \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha\) Definition of Set Union
\(\, \displaystyle \lor \, \) \(\displaystyle x\) \(\in\) \(\displaystyle B\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha}\) Definition of Intersection of Family
\(\, \displaystyle \lor \, \) \(\displaystyle x\) \(\in\) \(\displaystyle B\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B\) Definition of Set Union

By definition of subset:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B} \subseteq \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B$

$\Box$


\(\displaystyle x\) \(\in\) \(\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha}\) Definition of Set Union
\(\, \displaystyle \lor \, \) \(\displaystyle x\) \(\in\) \(\displaystyle B\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \forall \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha\) Intersection is Subset
\(\, \displaystyle \lor \, \) \(\displaystyle x\) \(\in\) \(\displaystyle B\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \forall \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha \cup B\) Definition of Set Union
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B}\) Definition of Intersection of Family

By definition of subset:

$\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B \subseteq \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B}$

$\Box$


By definition of set equality:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B$

$\blacksquare$


Also see


Sources