1.3 ALGEBRAIC PROPERTIES OF SETS 29
(b)
Figure 1.6
(d) How many regions (i.e., nonoverlapping subsets of U having all of U as
their union) should there be in a Venn diagram based on four sets A, B, C, and D?
Can you draw a diagram with four circles in which the expected number of regions
actually occurs?
Algebraic Properties of Sets
WHY AN ALGEBRA OF SETS?
Like the rules of high school algebra [such as (a + b)(a - b) = a' - b2 for
any real numbers a and b], which are indispensable for solving applied prob-
lems such as "story problems," the rules of set theory are extraordinarily
useful in dealing with any problem expressible in the language of sets. They
are especially helpful in proving theorems and arise in proofs involving
virtually every area of mathematics.
Many of the laws of set theory have a structure that enables us to simplify
considerably the form of a set. Suppose, for instance, that a problem in
advanced high school algebra (with U = R) has a solution set of the form
(A n B) u (A' nBB) u (A n B') u (A'n B'), where A = (-oo,4) u (7, oo)
and B = [- 2, 111. Calculating this set would be laborious if the problem is
approached directly. But an identity of set theory tells us that any set of the
preceding form equals U, the universal set, so that the solution set for the
original problem equals R. Another instance of the usefulness of the algebra
of sets is suggested in the paragraph following Example 6, Article 1.2, where
we asked whether A - B' could be computed from the information given.
The most accurate answer at that stage should have been "no," since a
universal set must be specified if B', and thus A - B', is to be computed.
