Take a look at this Venn diagram:
You can see that the shaded area is exactly the area not in , so this is the Venn diagram for . Now consider the Venn diagrams for and :
You can see from the diagrams that . This is a general and useful fact, one of De Morgan’s laws.
(De Morgan’s laws for sets). Let and let and denote the complement with respect to . Then
, and
.
These follow from De Morgan’s laws in logic. The left hand side of the first of these is the set of all such that
and the right hand side is the set of all such that
Since is logically equivalent to (Theorem 1.6.3), the two sets have the same elements and so are equal. The second equality follows from the other logical De Morgan law. ∎
De Morgan’s laws also work for unions and intersections of more than two sets.
For any sets
, and