# Can't prove generalized De Morgan's Law

1. Sep 23, 2010

### john562

1. The problem statement, all variables and given/known data
Let B be a non-empty set, and supose that {Sa : a$$\in$$B} is an B- indexed family of subsets of a set S. Then we have,
($$\cup$$ a\in B Sa)c = $$\bigcapa\in B$$ 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. Sep 24, 2010

### losiu99

Suppose $$a\in \bigcap S_\alpha^c$$. Then a is in all the sets $$S_\alpha ^c$$, and so it is not contained in any of $$S_\alpha$$. 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 $$S_\alpha$$ contains a, so all the $$S_\alpha^c$$'s contain a. Therefore, a is in their intersection.