Bridge to Abstract Mathematics: Mathematical Proof and Structures

1.2 OPERATIONS ON SETS 17

solution to Example 1 lists numbers in increasing order, this is not neces-
sary. Observe also that the sets that result from the operation of union
tend to be relatively large, whereas those obtained through intersection are
relatively small. You should formulate a more exact statement of this idea,
using one of of the concepts introduced in Article 1.1. Finally, our previous
example introduces the use of parentheses, as in the algebra of numbers,
to set priorities when an expression contains more than one instance of a
set theoretic operation. In view of this can you suggest how to apply the
operation of intersection to three sets? union also? What would you expect
to be the intersection of the preceding three sets A, B, and C? the union?

EXAMPLE 2 Let D = (2,4,6,8, 101, E =(-5,5), F= [3, w),and G= a.

Calculate E n F, E u F, D n E, D u F, and D u G. Also, using the sets

A and C defined in Example 1, calculate C n E and A n D.

Solution E n F = [3, 5), whereas E u F = (-5, a). Graphing along a
number line is perhaps the easiest way of arriving at these answers.
D n E = (2,4), since 2 and 4 are the only elements of D that are between
-5 and 5. D u F can perhaps be best expressed as (2) u [3, a). (The
other elements of D, besides 2, are already accounted for in F). What
about D u G? What is the result of taking the union of a set with the
empty set? The answer is either D u (a = D = (2,4,6,8, 10) or,
D u 0 = {(a, 2,4,6,8, 10). Which do you think is correct? (The answer
is in Article 1.3.) Using a set defined in Example 1, we note that C n E =
( - 3, - 1, 1,3). The numbers - 5 and 5 are not in C n E, because they
are not in E, an open interval. Finally, A n D = a. 0

The intersection of A and D in Example 2 provides another justification
for the existence of an empty set since A and D have no elements in common.
Pairs of sets such as A and D, having no elements in common, are said to
be disjoint. You should perform other calculations involving the sets in
Examples 1 and 2, for instance, B n F. What about the intersection of the
empty set with another set? In particular, what is A n a? Finally, does
our calculation of E n F and E u F suggest any possible theorem about
intervals? (See Article 5.2, Example 6 and Exercise 3.)
One reason that union and intersection are of value in mathematics is
that, like the subject of set theory itself, they provide mathematicians with
a convenient language for expressing solutions to problems.

EXAMPLE 3 (a) Solve the inequalities 12x + 31 2 5 and 2x2 + x - 28 > 0.

(b) Find all real numbers that satisfy both inekpalities in (a) simulta-

neously.

Solution (a) If a E R, then 1x1 2 a is equivalent to "either x < -a or x 2 a.

Hence 12x + 31 2 5 becomes "either 2x + 3 s -5 or 2x + 3 2 5," which