1.41 THEOREM [DEMORGAN'S LAWS]. Let X be a set and {Ea.}a.EA be a collection of subsets of X. If for each E ~ X the symbol EC represents the set X \ E, then

(15) (u Ear n E~

a.EA a.EA and

(16) (n Ear U E~.

a.EA a.EA PROOF. Suppose that x belongs to the left side of (15); i.e., x E X and x 1-

Ua.EA Ea.. By definition, x E X and x 1- Ea. for all 0: E A. Hence, x E E~ for all 0: E A; i.e., x belongs to the right side of (15). These steps are reversible. This verifies (15). A similar argument verifies (16) .