Mathematical Proofs: A Transition to Advanced Mathematics [Hardcover]

Gary Chartrand (Author), Albert D. Polimeni (Author), Ping Zhang (Author)

Book Description

ISBN-10: 0201710900 | ISBN-13: 978-0201710908 | Publication Date: June 7, 2002

Mathematical Proofs is designed to prepare students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise providing solid introductions to relations, functions, and cardinalities of sets.

Product Details

• Hardcover: 384 pages
• Publisher: Addison Wesley (June 7, 2002)
• Language: English
• ISBN-10: 0201710900
• ISBN-13: 978-0201710908
• Product Dimensions: 9.3 x 7.6 x 0.8 inches
• Shipping Weight: 1.5 pounds
• Average Customer Review: 4.7 out of 5 stars  See all reviews (19 customer reviews)
• Amazon Best Sellers Rank: #392,036 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

121 of 123 people found the following review helpful:

5.0 out of 5 stars Very Readable Textbook, January 24, 2003

By 

Penelope W. Yaqub "Fawzi M. Yaqub" (Fredonia, NY, U.S.A.) - See all my reviews
(REAL NAME)   

This review is from: Mathematical Proofs: A Transition to Advanced Mathematics (Hardcover)

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

24 of 24 people found the following review helpful:

5.0 out of 5 stars A veil has been lifted..., March 29, 2008

By 

A Student (USA) - See all my reviews

This review is from: Mathematical Proofs: A Transition to Advanced Mathematics (Hardcover)

This book pretty much changed my life. My only regret is I didn't find it earlier. What different choices would I have made if I were comfortable with mathematical proofs in high school or early college? I recommend this book to anyone who has an interest in understanding mathematics but for whatever reason never learned or were never taught it by their teachers in school - I'm certain I was never taught these ideas. This book is great for self study and the material should be accessible to most high school students.

The book starts off with basic ideas about sets and some logic. The real "Aha!" moment for me was the explanation of implication and biconditional in chapter 2 and how they're used in direct proofs in chapter 3. The examples about even/odd numbers are perfect for someone feeling their way thru new ideas.

After reading this book I went back to college as an adult and obtained a BS in mathematics. It's foolish to dwell on what might have been, but one can only imagine.

16 of 16 people found the following review helpful:

5.0 out of 5 stars Learning Proofs, on your own, March 14, 2007

By 

Stephan Hartwell "Stephan Hartwell" (Aurora, CO) - See all my reviews

Amazon Verified Purchase

This review is from: Mathematical Proofs: A Transition to Advanced Mathematics (Hardcover)

I bought Chartrand's book to teach myself how to
understand and to do proofs. I worked every exercise
in the text. Now taking some upper level proof based
courses, after being out of school for 20 years, I am
finding that I am more comfortable with proofs than
most of the people in my classes. The main thing that
helped me was the clear communication of the methods
and the ample opportunities to test out my knowledge.
The only thing I that would have helped me more is that
most problems at the end of the chapters do not provide
explanation. I had to trust my knowledge, which is not
always a good idea. Still, the authors do a good
job of conveying the concepts and I do very much like
chapter zero. I am a school teacher and I show that
chapter to my secondary students. Oh,that chapter
explains "good" mathematical writing style.