# Did I assume too much in this proof?

1. Sep 19, 2011

### IntroAnalysis

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 $\forall$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$\forall$A(AЄF$\rightarrow$xЄ(BUA)}. Assume AЄF, then xЄ(BUA). Thus, this is equivalent to BU(∩F).
Therefore, BU(∩F) = ∩AЄF(BUA).