Intersection is Associative/Family of Sets/Proof 1

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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

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:

$\ds \bigcap_{i \mathop \in I} S_i = \bigcap_{\lambda \mathop \in \Lambda} T_\lambda = \bigcap_{\lambda \mathop \in \Lambda} \paren {\bigcap_{i \mathop \in I_\lambda} S_i}$

$\blacksquare$


Sources

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