Mathematical Proofs : A Transition to Advanced Mathematics

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Summary: Mathematical Proofs: A Transition to Advanced Mathematics
Review: I purchased this book as a preparation to take advanced courses in mathematics. It is well laid out and explains the material clearly. I especially liked the chapter on "proofs found in calculus" which explains delta epsilon proofs in a manner that is easy and understandable. Big thumbs up!

Rating:
Summary: Very Readable Textbook
Review: This book is designed to prepare students for upper division math courses-like abstract algebra and advanced calculus-in which mathematical rigor and proofs are emphasized. The authors have made a serious effort to present the material with clarity and sufficient details to make it accessible to students who have completed two courses in calculus. Much of the material covered is fairly standard for such a textbook. Chapters 1-9 are devoted to basic topics from set theory and logic (including four proof techniques: direct proof, proof by contrapositive, proof by contradiction, and mathematical induction), equivalence relations, and functions, as well as a special chapter under the heading, "Prove or Disprove." Chapters 10-13 cover cardinalities of sets and proof techniques applied to results from number theory, calculus, and group theory. In addition, the authors have a web site which includes three additional chapters (Chapters 14-16) dealing with proofs from ring theory, linear algebra, and topology. Thus instructors using this book will have a wide choice of options in selecting the material they want to include after the basic concepts are covered.

The emphasis throughout the book is on proofs and proof techniques--how to recognize proofs, understand them and, above all, how to create and write them. The presentation is leisurely and thorough. Many examples are given, and discussions are always presented with all the details that students at this level would need to follow the argument. There are ample exercises at the end of each chapter (including those in the web site) that range in difficulty from routine to moderately challenging. The book also contains answers and hints to odd-numbered exercises.

There are two features of this textbook that I believe are helpful to students and that set this book apart from others at its level: the detailed way in which proofs are analyzed, and the inclusion of a chapter on how to write mathematics well. In most cases, before a proof is presented the authors offer a "proof strategy": a discussion pointing out what needs to be proved and how one might go about proving it. Also, many proofs are followed by "proof analyses" in which some of the interesting or unusual points of the proof are commented on. I believe that students would find these discussions very helpful. In particular, these discussions offer students concrete pointers from which they would learn how to cope with abstract mathematical proofs.

The chapter on writing mathematics (Chapter 0) is unique. While some mathematics textbooks encourage good writing and might devote a few paragraphs to the subject, the present volume offers a brief manual on mathematical writing. The authors begin by explaining why writing is important in mathematics and follow that by offering detailed instructions that would help students in improving their writing. From specific advice like, "Never start a sentence with a symbol" to explanations of "common words and phrases that are peculiar to mathematics," there is a wealth of material on writing from which students can learn. I believe that, by its very existence, this chapter on writing would have a positive influence on students writing.

This book can be used either as a textbook for a course such as the one described above or as a reference that students can consult on certain topics.

Fawzi M. Yaqub
Emeritus Professor of Mathematics
SUNY College at Fredonia