1.1 BASIC DEFINITIONS AND NOTATION 9Solution
to x I
(b)
7x - 9 5 16 is equivalent to 7x I 25, which is equivalent
The set of solutions is (x E Rlx I 91 = (- m, y].
a E R, then 1x1 < a is equivalent to -a < x < a. Hence
)2x + 31 < 5 is equivalent to -5 < 2x + 3 < 5, or -8 < 2x < 2, or- 4 < x < 1, which is expressed in interval notation as (- 4, 1).
(c) If a > 0, the quadratic inequality axZ + bx + c < 0 is satisfied by
precisely those numbers between and including the roots of the equation
ax2 + bx + c = 0. We find the latter by factoring 2x2 + x - 28 into
(2x - 7)(x + 4), yielding x = $ and x = -4 as roots. Thus we arrive at
the solution set -4 I x I z, which is expressed in interval notation as
[-4,fJ.
The assumption that U = R in problems like the preceding example is
usually made implicitly, that is, without specific mention. As a final re-
mark on intervals, bearing on notation, we observe that it is necessary to
distinguish carefully between (0, I}, a two-element set, and [0, 11, an in-
finite set. This remark suggests a general caveat for beginning students of
abstract mathematics: A small difference between two notations repre-
senting mathematical objects can understate a vast difference between the
objects themselves. The conclusion to be drawn is that we need always to
read and write mathematics with great care!
THE EMPTY SET
Certain special cases of Definition 3 lead to rather surprising facts. For
instance, if we let a = b in (I), we see that [a, a] = (a), a singleton or
single-element set, is an interval. If we do the same thing in (2), we arrive
at an even less intuitive situation, namely, no real number satisfies the
criterion for membership in the open interval (a, a), since no real number is
simultaneously greater than and less than a. *Thus if this special case of (2)
is to be regarded as a set, much less an interval, we must posit the existence
of a set with no elements. This we do under the title of the empty set or
null set, denoted either @ (a derivative of the Greek letter phi in lower case)
or { ). The empty set is, in several senses we will discuss later in this
chapter, at the opposite end of the spectrum from a universal set. It is an
exception to many theorems in mathematics; that is, the hypotheses of many
theorems must include the proviso that some or all the sets involved should
not be empty (i.e., be nonempty), and has properties that quickly lead to
many brain-teasing questions (e.g., Exercise 10, Article 1.3). Another justi-
fication for the existence of an empty set, explored further in Article 1.2, is
the desirability that the intersection of any two sets be a set. Still another
justification is provided by examples such as Example 2.EXAMPLE 2 Solve the quadratic inequality 5x2 + 3x + 2 < 0, U = R.