Can't prove generalized De Morgan's Law

[john562]

Homework Statement

Let B be a non-empty set, and supose that {S_a : a$$\in$$B} is an B- indexed family of subsets of a set S. Then we have,
($$\cup$$ _{a\in B} S_a)^c = $$\bigcap_{a\in B}$$ S_a^c.

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.

 

[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.