2023-09-10

364: Intersection of Products of Sets Is Product of Intersections of Sets

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A description/proof of that intersection of products of sets is product of intersections of sets

Topics


About: set

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Starting Context

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, Sα∈A,β∈B, ∩α∈A×β∈BSα,β=×β∈B∩α∈ASα,β.

2: Proof


For any p∈∩α∈A×β∈BSα,β, p∈×β∈BSα,β for each α. p(β)∈Sα,β for each α,β. p(β)∈∩α∈ASα,β, p∈×β∈B∩α∈ASα,β. For any p∈×β∈B∩α∈ASα,β, p(β)∈∩α∈ASα,β for each β. p(β)∈Sα,β for each β,α. p∈×βSα,β for each α. p∈∩α∈A×β∈BSα,β.

3: Note


When B is a finite indices set, the proposition states that ∩α∈A(Sα,1×Sα,2×...×Sα,n)=(∩α∈ASα,1)×(∩α∈ASα,2)×...×(∩α∈ASα,n).

References
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