Can't prove generalized De Morgan's Law

[john562]

Homework Statement

Let B be a non-empty set, and supose that {S_a : a\inB} 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.