Set Intersection is Self-Distributive/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:

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

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


Proof

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

Thus by definition of subset:

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

$\Box$


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

Thus by definition of subset:

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

$\Box$


By definition of set equality:

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

$\blacksquare$


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