10 SETS Chapter 1
Solution By completing the square, we can express 5x2 + 3x + 2 in the
form 5(x + + (g), which is clearly positive for any real x. Thus no
real number satisfies the given inequality; we express this by saying that
the solution set is 0.
RELATIONS BETWEEN SETSEquality. Earlier we observed that the set D = {x lx is the name of a state
in the United States beginning with the letter M} could also be described
by means of the roster method. This observation implied an intimate re-
lationship between D and the set M = (Maine, Maryland, Massachusetts,
Michigan, Minnesota, Missouri, Mississippi, Montana}, a relationship
identical to that existing between the sets T = {x E Rlx2 - 8x + 15 = 0)
and P = (3,5), or between the sets G = {xlx was the first president of the
United States) and W = {George Washington}. The relationship is set
theoretic equality. We will defer a formal definition of equality of sets until
Chapter 4 (Definition l(a), Article 4.1), contenting ourselves at this stage
with an informal description.
REMARK (^2) Let A and B be sets. We will regard the statement A equals B,
denoted A = B, to mean that A and B have precisely the same elements.
Applying the criterion of Remark 2 to the preceding examples, we have
D = M, P = T, and G = W. Equality of sets has such a deceptively simple
appearance that it might be questioned at first why we even bother to discuss
it. One reason is that our informal description of set equality highlights the
basic fact that a set is completely determined by its elements. A second rea-
son is that sets that are indeed equal often appear, or are presented in a form,
quite different from each other, with the burden of proof of equality on the
reader. Many of the proofs that the reader is given or challenged to write
later in the text are, ultimately, pmofs that two particular sets are equal.
Such proofs are usually approached by the following alternative description
of equality of sets:
Sets A and B are equal if and only if every element of A is also an element
of B and every element of B is also an element of A. (1)
In Chapters 2 through 4, on logic and proof, we will discuss why this
characterization of equality carries the same meaning as the criterion from
Remark 2. As examples of properties of set equality to be discussed in detail
later, we note that every set equals itself; given sets A and B, if A = B, then
B = A; and given sets A, B, and C, if A = B and B = C, then A = C. These
are called the rejexive, symmetric, and transitive properties of set theoretic
equality, respectively. Finally, we note that A # B symbolizes the state-
ment that sets A and B are not equal.