Intersection is Associative/Family of Sets
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Theorem
Let $\family {S_i}_{i \mathop \in I}$ and $\family {I_\lambda}_{\lambda \mathop \in \Lambda}$ be indexed families of sets.
Let $\ds I = \bigcap_{\lambda \mathop \in \Lambda} I_\lambda$.
Then:
- $\ds \bigcap_{i \mathop \in I} S_i = \bigcap_{\lambda \mathop \in \Lambda} \paren {\bigcap_{i \mathop \in I_\lambda} S_i}$
Proof 1
For every $\lambda \in \Lambda$, let $\ds T_\lambda = \bigcap_{i \mathop \in I_\lambda} S_i$.
Then:
\(\ds x\) | \(\in\) | \(\ds \bigcap_{i \mathop \in I} S_i\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \forall i \in I: \, \) | \(\ds x\) | \(\in\) | \(\ds S_i\) | Definition of Intersection of Family | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \forall \lambda \in \Lambda: \forall i \in I_\lambda: \, \) | \(\ds x\) | \(\in\) | \(\ds S_i\) | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \forall \lambda \in \Lambda: \, \) | \(\ds x\) | \(\in\) | \(\ds \bigcap_{i \mathop \in I_\lambda} S_i = T_\lambda\) | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds \bigcap_{\lambda \mathop \in \Lambda} T_\lambda\) |
Thus:
- $\ds \bigcap_{i \mathop \in I} S_i = \bigcap_{\lambda \mathop \in \Lambda} T_\lambda = \bigcap_{\lambda \mathop \in \Lambda} \paren {\bigcap_{i \mathop \in I_\lambda} S_i}$
$\blacksquare$
Proof 2
\(\ds \bigcap_{i \mathop \in I} S_i\) | \(=\) | \(\ds \map \complement {\map \complement {\bigcap_{i \mathop \in I} S_i} }\) | Complement of Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \complement {\bigcup_{i \mathop \in I} \map \complement {S_i} }\) | De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \complement {\bigcup_{\lambda \mathop \in \Lambda} \paren {\bigcup_{i \mathop \in I_\lambda} \map \complement {S_i} } }\) | General Associativity of Set Union | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \complement {\bigcup_{\lambda \mathop \in \Lambda} \paren {\map \complement {\bigcap_{i \mathop \in I_\lambda} S_i} } }\) | De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcap_{\lambda \mathop \in \Lambda} \paren {\map \complement {\map \complement {\bigcap_{i \mathop \in I_\lambda} S_i} } }\) | De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcap_{\lambda \mathop \in \Lambda} \paren {\bigcap_{i \mathop \in I_\lambda} S_i}\) | Complement of Complement |
$\blacksquare$