Customer Reviews

Mathematical Proofs: A Transition to Advanced Mathematics

5 star:		(15)
4 star:		(3)
3 star:		(1)
2 star:		(0)
1 star:		(0)

 

 

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

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.

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.

If you look at the "Customers Who Bought This Item Also Bought" section on this page you will see several highly rated books that cover similar topics. They all are priced approximately $100 less than this book because Addison-Wesley (the publisher) has decided to market _Mathematical Proofs_ as a text-book rather than a book-book.

This is tragic. The only people who will encounter this fantastic text are students with no choice but to suffer the rapacious fancies of textbook publishers.

I was once such a student. Everything Penelope W. Yaqub "Fawzi M. Yaqub" says in her review is spot on. I have kept only three books on my personal shelf from my days as an undergraduate in mathematics. This slim volume is one of those.

I recommend this book to anyone -- whether a student or hobbyist -- wanting to better their understanding of what mathematics is really about. It is also great training for thinking logically about all topics, from mathematics to current events

I would have rated this 5 stars if the cover price were not so outrageous.

Being a physist and a computer scientist i find this book very helpful for someone to aquire good skills in the mathematical language and thought.
I consider that these two principles enable the creativity in mathematics for someone less involved with formal mathematical thought

This is one of the most well-written textbooks that I have ever had the pleasure to read. I come from a mechanical engineering background, but have a strong interest in mathematics and regularly tutor undergraduates. I picked up this book not for a course, but for self-study. From this book alone I was about to learn the material, teach it to one of my tutoring students, and get her an 'A'! I can't say that about most textbooks I suffered through in college.

The only downside, as some reviewers have mentioned, is the hefty price tag. C'est la vie. I actually like the presentation of this book so much that I'm now looking into other works that the authors have.

A little more than a year back, I was forced to purchase this text for a mathematics course. At the time I lacked the maturity to see what a gem it really was, but it proved its worth the very next semester. I began a self-study real analysis sequence, and I tackled a mathematically rigorous (theorem-proof style) special relativity course. I found myself returning to "Mathematical Proofs" to slog through the dense material of these more advanced subjects.

The exposition of "Mathematical Proofs" is not too terse. The examples are informative and useful. The problem sets are challenging and require a desire towards mathematical maturity, but they are solvable for anyone who has read and understands the preceding chapter.

I keep this on my home desk for quick reference to lucid definitions and explanations of fundamental concepts. I even work an occasional problem to stay sharp in useful skills. It will definitely accompany me to graduate school next year.

A must have for budding mathematicians!

This is the single best book I have ever seen on the art of proofs. You really don't even need a teacher to guide you with this book. It is expensive but then so are all textbooks now.

It's an all around great book. Some people might complain about it holding the reader's hand too much, but that's what makes the book so great. It assumes you never seen a proof before, and it helps guide you towards writing and analyzing proofs. The examples provided are clear. The section problems for the most part mirror the examples provided, so usually you have a model solution to work with. All in all, it's a great book for self study. If you have to purchase it for a class your instructor made an excellent choice.

If you plan to take any seconed year math courses such as linear algebra, calculus or real analysis I strongly suggest you read this book. I found that some seconed year books assume that you have some knowledge of how proofs are constructed. This book assumes that you know nothing about proofs.It cleared up a lot of confusion for me. I am pleased with book and it was well worth the price.