Fill in the blanks in the proof of the following theorem.

 

THEOREM: Let A and B be subsets of a universal set U. Then

According to our definition of equality of sets, we must show that

A helpful way to visualize the set operations of union, intersection, and complementation is by use of Venn diagrams, as in Figures 1 and 2. In Figure 1 the shaded area represents the union of A and B, and in Figure 2 the shaded area is the intersection of A and B. In each case the large rectangle represents the universal set U. While Venn diagrams (and other diagrams as well) are useful in seeing the relationship between sets and may be helpful in getting ideas for developing a proof, they should not be viewed as proofs themselves. A diagram necessarily represents only one case, and it may not be obvious whether this is a general case that always applies or whether there may be other cases as well.