Bridge to Abstract Mathematics: Mathematical Proof and Structures

18 SETS Chapter 1

simplifies to "either x I -4 or x 2 1." The solution set is most con-

veniently expressed (- co, -41 u [I, co). On the other hand, the qua-

dratic inequality 2x2 + x - 28 > 0 is solved by all values of x to either

the left of the smaller root or the right of the larger root of the corre-

sponding quadratic equation, that is, by all values of x either less than

-4 or greater than 3. The solution set is (- co, -4) u (z, a).

(b) We find the simultaneous solutions to the two inequalities by in-

tersecting the two solution sets we got in (a). Graphing along a number

line, we arrive at the set ( - GO, - 4) u (z, co).

In Article 1.3, as we develop the algebra of sets, we will discover theorems

of set theory by which we may obtain the last answer in Example 3 system-

atically, avoiding graphing. Although union and intersection are binary

operations, there is nothing to prevent the two sets to which they are applied

from being the same set, so that an expression like X n X or X u X has

meaning. For Examples 1 and 2, calculate F n F and B u B. Does any

general fact suggest itself?

COMPLEMENT

Our third operation, complement, is unary rather than binary; we obtain

a resultant set from a single given set rather than from two such sets. The

role of the universal set is so important in calculating complements that

we mention it explicitly in the following definition.

DEFINITION 3

Let A be a subset of a universal set U. We define the complement of A, denoted A',

by the rule A' = {x E UIX q! A).

The complement of a set consists of all objects in the universe at hand

that are not in the given set. Clearly the complement of A is very much

dependent on the universal set, as well as on A itself. If A = (I}, then A'

i is one thing if U = N, something quite different if U = R, and something
altogether different again (a singleton set in fact as opposed to an infinite
set in the other two cases) if U = (1,2).

EXAMPLE 4 Letting R be the universal set, calculate the complement of

the sets A = [- 1,1], B = (-3,2], C = (- co, 01, and D = (0, co).

Solution A' consists of all real numbers that are not between -1 and 1

inclusive, that is, all numbers either less than - 1 [i.e., in (- co, - I)]

or greater than 1 [i.e., in (1, a)]. In conclusion, A' = (- oo, - 1) u

(1, a). Similarly, B' =(--a, -31 u (2, oo), C' = (0, oo) = D, and D' =

(-GO,O]=C. 0