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Did I assume too much in this proof?

  1. Sep 19, 2011 #1
    1. The problem statement, all variables and given/known data
    Suppose F is a nonempty family of sets and B is a set. Prove that B U (∩F) = ∩AЄF(B U A).

    2. Relevant equations

    3. 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).
  2. jcsd
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