Mathematics consists of calculations and proofs. Elementary mathematics consists mainly of calculations, and students often have difficulty when they advance to the point where proofs become important. This book is intended to help students develop the ``mathematical sophistication'' they will need in advanced courses. That sophistication involves concepts and skills from three areas: logic and proofs, sets and functions, and the basic number systems used in mathematics.
The first four chapters of the book are devoted to logic and proofs; the next three to sets and functions; and the last three to number systems. Everything that should be in such a book is included: propositional and predicate logic, proof by induction, Russell's paradox, functions as sets of ordered pairs, the concept of cardinality; examples of rings and fields, the completeness axiom for the real numbers; complex numbers as pairs of real numbers.
Dr. Wolf has brought to this book a lively wit, twenty years of teaching experience with the target audience, and the acumen and scholarship of a highly-trained mathematician and logician. The book thus entertains and educates, without sacrificing accuracy or precision. The twenty years of experience, for example, is highly visible in the section on "Hints for Finding Proofs". The scholarship is visible in the "Suggestions for Further Reading" at the end of each chapter. The wit is visible in the examples. The scholarship and experience are both visible in the selection of exercises.
The subjects of proofs and their logical foundations have challenged the minds of some of the world's deepest thinkers. Both the difficulties of the subject, and its beauties, are extraordinary. This book will help the reader to appreciate the beauties and overcome the difficulties.
