Intersection is Commutative/Family of Sets/Proof 2

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Theorem

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets.

Let $\ds I = \bigcap_{i \mathop \in I} S_i$ denote the intersection of $\family {S_i}_{i \mathop \in I}$.

Let $J \subseteq I$ be a subset of $I$.


Then:

$\ds \bigcap_{i \mathop \in I} S_i = \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k = \bigcap_{k \mathop \in \relcomp I J} S_k \cap \bigcap_{j \mathop \in J} S_j$

where $\relcomp I J$ denotes the complement of $J$ relative to $I$.


Proof

\(\ds \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k\) \(=\) \(\ds \map \complement {\map \complement {\bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k} }\) Complement of Complement
\(\ds \) \(=\) \(\ds \map \complement {\map \complement {\bigcap_{j \mathop \in J} S_j} \cup \map \complement {\bigcap_{k \mathop \in \relcomp I J} S_k} }\) De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \map \complement {\bigcup_{j \mathop \in J} \map \complement {S_j} \cup \bigcup_{k \mathop \in \relcomp I J} \map \complement {S_k} }\) De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection
\(\ds \) \(=\) \(\ds \map \complement {\bigcup_{k \mathop \in \relcomp I J} \map \complement {S_k} \cup \bigcup_{j \mathop \in J} \map \complement {S_j} }\) General Commutativity of Set Union
\(\ds \) \(=\) \(\ds \map \complement {\map \complement {\bigcap_{k \mathop \in \relcomp I J} S_k}\cup \map \complement {\bigcap_{j \mathop \in J} S_j} }\) De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection
\(\ds \) \(=\) \(\ds \map \complement {\map \complement {\bigcap_{k \mathop \in \relcomp I J} \cap S_k \bigcap_{j \mathop \in J} S_j} }\) De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection
\(\ds \) \(=\) \(\ds \bigcap_{k \mathop \in \relcomp I J} S_k \cap \bigcap_{j \mathop \in J} S_j\) Complement of Complement


Also:

\(\ds \bigcap_{i \mathop \in I} S_i\) \(=\) \(\ds \map \complement {\map \complement {\bigcap_{i \mathop \in I} S_i} }\) 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
\(\ds \) \(=\) \(\ds \map \complement {\bigcup_{j \mathop \in J} \map \complement {S_j} \cup \bigcup_{k \mathop \in \relcomp I J} \map \complement {S_k} }\) General Commutativity of Set Union
\(\ds \) \(=\) \(\ds \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k\) from $(1)$ above

$\blacksquare$


Sources

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