A description/proof of that intersection of products of sets is product of intersections of sets
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of product of infinite number of sets.
Target Context
- The reader will have a description and a proof of the proposition that the intersection of the same-indices-set products of possibly uncountable number of sets is the product of the intersections of the sets.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any possibly uncountable indices sets, \(A, B\), and any sets, \(S_{\alpha \in A, \beta \in B}\), \(\cap_{\alpha \in A} \times_{\beta \in B} S_{\alpha, \beta} = \times_{\beta \in B} \cap_{\alpha \in A} S_{\alpha, \beta}\).
2: Proof
For any \(p \in \cap_{\alpha \in A} \times_{\beta \in B} S_{\alpha, \beta}\), \(p \in \times_{\beta \in B} S_{\alpha, \beta}\) for each \(\alpha\). \(p (\beta) \in S_{\alpha, \beta}\) for each \(\alpha, \beta\). \(p (\beta) \in \cap_{\alpha \in A} S_{\alpha, \beta}\), \(p \in \times_{\beta \in B} \cap_{\alpha \in A} S_{\alpha, \beta}\). For any \(p \in \times_{\beta \in B} \cap_{\alpha \in A} S_{\alpha, \beta}\), \(p (\beta) \in \cap_{\alpha \in A} S_{\alpha, \beta}\) for each \(\beta\). \(p (\beta) \in S_{\alpha, \beta}\) for each \(\beta, \alpha\). \(p \in \times_{\beta} S_{\alpha, \beta}\) for each \(\alpha\). \(p \in \cap_{\alpha \in A} \times_{\beta \in B} S_{\alpha, \beta}\).
3: Note
When \(B\) is a finite indices set, the proposition states that \(\cap_{\alpha \in A} (S_{\alpha, 1} \times S_{\alpha, 2} \times . . . \times S_{\alpha, n}) = (\cap_{\alpha \in A} S_{\alpha, 1}) \times (\cap_{\alpha \in A} S_{\alpha, 2}) \times . . . \times (\cap_{\alpha \in A} S_{\alpha, n})\).
