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Homework Help: Can't prove generalized De Morgan's Law

  1. Sep 23, 2010 #1
    1. The problem statement, all variables and given/known data
    Let B be a non-empty set, and supose that {Sa : a[tex]\in[/tex]B} is an B- indexed family of subsets of a set S. Then we have,
    ([tex]\cup[/tex] a\in B Sa)c = [tex]\bigcapa\in B[/tex] Sac.


    2. Relevant equations



    3. The attempt at a solution
    I tried to show that the two were both subsets of each other, but I'm not sure how to do that.
     
  2. jcsd
  3. Sep 24, 2010 #2
    Suppose [tex]a\in \bigcap S_\alpha^c[/tex]. Then a is in all the sets [tex]S_\alpha ^c[/tex], and so it is not contained in any of [tex]S_\alpha[/tex]. Therefore, it is not contained in their union (by definition). It is therefore contained in union's complement.

    Suppose a is in union's complement. No [tex]S_\alpha[/tex] contains a, so all the [tex]S_\alpha^c[/tex]'s contain a. Therefore, a is in their intersection.
     
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