Intersection Distributes over Intersection/Families of Sets

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Theorem

Let $I$ be an indexing set.

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


Then:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha}$

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


Proof

\(\displaystyle x\) \(\in\) \(\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \forall \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha \cap B_\alpha\) Definition of Intersection of Family
\(\displaystyle \leadsto \ \ \) \(\displaystyle \forall \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha\) Definition of Set Intersection
\(\, \displaystyle \land \, \) \(\displaystyle x\) \(\in\) \(\displaystyle B_\alpha\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha\) Definition of Intersection of Family
\(\, \displaystyle \land \, \) \(\displaystyle x\) \(\in\) \(\displaystyle \bigcap_{\alpha \mathop \in I} B_\alpha\) Definition of Intersection of Family
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha}\) Definition of Set Intersection

Thus by definition of subset:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha} \subseteq \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha}$

$\Box$


\(\displaystyle x\) \(\in\) \(\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha\) Definition of Set Intersection
\(\, \displaystyle \land \, \) \(\displaystyle x\) \(\in\) \(\displaystyle \bigcap_{\alpha \mathop \in I} B_\alpha\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \forall \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha\) Definition of Intersection of Family
\(\, \displaystyle \land \, \) \(\displaystyle x\) \(\in\) \(\displaystyle B_\alpha\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \forall \alpha \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\alpha \cap B_\alpha\) Definition of Set Intersection
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha}\) Definition of Intersection of Family

Thus by definition of subset:

$\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha} \subseteq \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha}$

$\Box$


By definition of set equality:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha}$

$\blacksquare$


Sources