viii PREFACE
countably infinite collections of sets. The main emphasis here is on stan-
dard approaches to proving set inclusion (e.g., the "choose" method) and
set equality (e.g., mutual inclusion), but we manage also, through the many
solved examples, to anticipate additional techniques of proof that are stud-
ied in detail later. The chapter concludes by digressing to an optional, and
perhaps somewhat offbeat, second look at the limit concept, directed to-
ward an understanding of the epsilon-delta definition.
Chapters 5 and 6 provide the text's most concentrated treatment of proof
writing per se. The general organization of these chapters is in order of
increasing complexity, with special emphasis on the logical structure of
the conclusion of the proposition to be proved. In Articles 5.1, 5.2, and
6.1, we progress from conclusions with the simplest logical structure [i.e.,
(Vx)(p(x))], to conclusions with a more complex form [i.e., (b'x)(p(x) +
q(x))], and then to the most complex case [V followed by 31. Additional
techniques, including induction, indirect proof, specialization, division into
cases, and counterexample, are also studied. Solved examples and exercises
calling for the writing of proofs are selected from set theory, intermediate
algebra, trigonometry, elementary calculus, matrix algebra, and elementary
analysis. Of course, instructors must gear the assignment of exercises to
the students' background. Solved examples, toget her with starred exercises
(whose solutions appear in the back of the book) provide numerous models
of proofs, after which students may pattern their own attempts. An ad-
ditional source of correctly written proofs (as well as some that were de-
liberately written incorrectly) is a "Critique and Complete" category of
exercise that occurs in Article 4.1 and throughout Chapter 5.
Chapters 7 and 8 deal with the most common kinds of relations on sets,
equivalence relations, partial orderings, and functions. Chapter 8 includes
an introduction to cardinality of sets and a brief discussion of arbitrary
collections of sets. Chapters 9 and 10 study the standard number systems
encountered in undergraduate mathematics. Chapter 9 emphasizes the
properties that distinguish the real numbers from other familiar number
systems. Chapter 10 provides an outline of an actual construction of the
real numbers, which would perhaps be most appropriately used in a class
of seniors or as an independent study project for a well-motivated and
relatively advanced student. In addition to treating material that is of con-
siderable value in its own right, Chapters 7 through 10 provide ample
opportunity for students to put into practice proof-writing skills acquired
in earlier chapters. In keeping with the advancing abilities of students,
proofs are deliberately written in an increasingly terse fashion (with less
detailed explanation and less psychological support) in the later chapters.
This may provide a smooth transition from this text to the "real world" of
typical texts for standard junior-senior courses.
Flexibility. Bridging courses in mathematics are by no means an estab-
lished or standardized part of the undergraduate curriculum. Indeed,