Preface
This text is directed toward the sophomore through senior levels of uni-
versity mathematics, with a tilt toward the former. It presumes that the
student has completed at least one semester, and preferably a full year, of
calculus. The text is a product of fourteen years of experience, on the part
of the author, in teaching a not-too-common course to students with a very
common need. The course is taken predominantly by sophomores and
juniors from various fields of concentration who expect to enroll in junior-
senior mathematics courses that include significant abstract content. It
endeavors to provide a pathway, or bridge, to the level of mathematical
sophistication normally desired by instructors in such courses, but generally
not provided by the standard freshman-sophomore program. Toward this
end, the course places strong emphasis on mathematical reasoning and ex-
position. Stated differently, it endeavors to serve as a significant first step
toward the goal of precise thinking and effective communication of one's
thoughts in the language of science.
Of central importance in any overt attempt to instill "mathematical ma-
turity" in students is the writing and comprehension of proofs. Surely, the
requirement that students deal seriously with mathematical proofs is the
single factor that most strongly differentiates upper-division courses from
the calculus sequence and other freshman-sophomore classes. Accordingly,
the centerpiece of this text is a substantial body of material that deals
explicitly and systematically with mathematical proof (Article 4.1, Chapters
\ 5 and 6). A primary feature of this material is a recognition of and reliance
on the student's background in mathematics (e.g., algebra, trigonometry,
calculus, set theory) for a context in which to present proof-writing tech-
niques. The first three chapters of the text deal with material that is impor-
tant in its own right (sets, logic), but their major role is to lay groundwork
for the coverage of proofs. Likewise, the material in Chapters 7 through 10
(relations, number systems) is of independent value to any student going
on in mathematics. It is not inaccurate, however, in the context of this
book, to view it primarily as a vehicle by which students may develop
further the incipient ability to read and write proofs.