Intersection is Commutative/Family of Sets/Proof 1

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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

We have that both $\ds \bigcap_{j \mathop \in J} S_j$ and $\ds \bigcap_{k \mathop \in \relcomp I J} S_k$ are sets.

Hence by Intersection is Commutative we have:

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


It remains to be demonstrated that $\ds \bigcap_{i \mathop \in I} S_i = \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k$.

So:

\(\ds x\) \(\in\) \(\ds \bigcap_{i \mathop \in I} S_i\)
\(\ds \leadstoandfrom \ \ \) \(\ds \forall i \in I: \, \) \(\ds x\) \(\in\) \(\ds S_i\) Definition of Intersection of Family
\(\ds \leadstoandfrom \ \ \) \(\ds \forall j \in J: \, \) \(\ds x\) \(\in\) \(\ds S_j\) Definition of Relative Complement
\(\ds \forall k \in \relcomp I J: \, \) \(\, \ds \land \, \) \(\ds x\) \(\in\) \(\ds S_k\)
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(\in\) \(\ds \bigcap_{j \mathop \in J} S_j\) Definition of Intersection of Family
\(\, \ds \land \, \) \(\ds x\) \(\in\) \(\ds \bigcap_{k \mathop \in \relcomp I J} S_k\) Definition of Intersection of Family
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(\in\) \(\ds \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k\) Definition of Set Intersection

That is:

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

The result follows by definition of set equality.

$\blacksquare$


Sources

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