PREFACE vllIMPORTANT FEATURESReadability. The author's primary pedagogical goal in writing the text was
to produce a book that students can read. Since many colleges and uni-
versities in the United States do not currently have a "bridging" course in
mathematics, it was a goal to make the book suitable for the individual
student who might want to study it independently. Toward this end, an in-
troduction is provided for each chapter, and for many articles within chap-
ters, to place content in perspective and relate it to other parts of the book,
providing both an overall point of view and specific suggestions for work-
ing through the unit. Solved examples are distributed liberally through-
out the text. Abstract definitions are amplified, whenever appropriate, by
a number of concrete examples. Occasionally, the presentation of material
is interrupted, so the author can "talk to" the reader and explain various
mathematical "facts of life." The numerous exercises at the end of articles
have been carefully selected and placed to illustrate and supplement ma-
terial in the article. In addition, exercises are often used to anticipate results
or concepts in the next article. Of course, most students who use the text
will do so under the direction of an instructor. Both instructor and students
reap the benefit of enhanced opportunity for efficient classroom coverage
of material when students are able to read a text.
Organization. In Chapter 1, we introduce basic terminology and notation
of set theory and provide an informal study of the algebra of sets. Beyond
this, we use set theory as a device to indicate to the student what serious
mathematics is really about, that is, the discovery of general theorems.
Such discovery devices as examples, pictures, analogies, and counterexam-
ples are brought into play. Rhetorical questions are employed often in this
chapter to instill in the student the habit of thinking aggressively, of looking
for questions as well as answers. Also, a case is made at this stage for both
the desirability of a systematic approach to manipulating statements (i.e.,
logic) and the necessity of abstract proof to validate our mathematical
beliefs.
In Chapters 2 and 3, we study logic from a concrete and common-sense
point of view. Strong emphasis is placed on those logical principles that
are most commonly used in everyday mathematics (i.e., tautologies of the
propositional calculus and theorems of the predicate calculus). The goal
of these chapters is to integrate principles of logic into the student's way
of thinking so that they are applied correctly, though most often only
implicitly, to the solving of mathematical problems, including the writing
of proofs.
In Chapter 4, we begin to some mathematics, with an emphasis on
topics whose understanding is enhanced by a knowledge of elementary
logic. Most important, we begin in this chapter to deal with proofs, limiting
ourselves at this stage to theorems of set theory, including properties of