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Can't prove generalized De Morgan's Law

  • Thread starter john562
  • Start date
  • #1
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Homework Statement


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.


Homework Equations





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.
 

Answers and Replies

  • #2
133
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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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