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IntroAnalysis
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Homework Statement
Suppose F is a nonempty family of sets and B is a set. Prove that B U (∩F) = ∩AЄF(B U A).
Homework Equations
The Attempt at a Solution
Let x Є (B U (∩F)). This means (x Є B) ν (x Є ∩F). x Є F = {x l [itex]\forall[/itex]AЄF(xЄA)}.
Assume AЄF, then xЄA. Therefore, xЄB ν xЄA. This is equivalent to ∩AЄF(BUA).
Let xЄ ∩AЄF(BUA). This means xЄ{x l[itex]\forall[/itex]A(AЄF[itex]\rightarrow[/itex]xЄ(BUA)}. Assume AЄF, then xЄ(BUA). Thus, this is equivalent to BU(∩F).
Therefore, BU(∩F) = ∩AЄF(BUA).