This text is directed toward the sophomore through senior levels of university 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 juniorsenior 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 exposition. 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 maturity" 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 techniques.
The first three chapters of the text deal with material that is important 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.
