Customer Reviews

How to Prove It: A Structured Approach

5 star:		(31)
4 star:		(7)
3 star:		(3)
2 star:		(0)
1 star:		(1)

 

 

Believe it or not, I graduated with a BS in math without being able to write proofs all that well. I got an "A" in advanced calculus and abstract algebra due mostly to the fact that the majority of the students in the class couldn't write proofs. Over a decade later, I was browsing through the math books at my local book store and found this book. After working through some of the problems and studying some of the material, I wished that I had this book a year or so before taking advanced calculus (introductory real analysis). Actually, this book can be handled by a person just finishing high school. My advice to all math majors who don't have a solid foundation in mathematical proofs is to get this book as soon as you can, study it and work many of the problems. This way when you have to take advanced calculus, topology or abstract algebra you will not be struggling to learn how to write proofs. I can not guarrantee that you will breeze through these courses after studying this book, but you will be spending more time on learning concepts and little or no time on the methods and techniques of proofs.

Set Theory is the foundation on which mathematical proofs are based. This book emphasizes set theory.

I recall it was a few years back when I encountered this little gem at my first analysis class. In fact this book wasn't assigned and instead we used Analysis by Lay. I didn't get essential proof tactics/strategies out of Lay's so I plunged myself into Library and after looking up one after another, I finally found this book. It is about as title says and not about Analysis. The book does not cover as much as one expects from Analysis books. But many of them I've seen seem to fail on teaching "how to prove" to study Analysis.

Velleman uses structured style as a technique. Two columns are prepared. The left column is Givens and right Goal. By restructuring Givens and Goal using relationships and definitions, some parts of Goal statement is moved to Givens, like peeling skins of onion. This process iterates until one finds the proving obvious. The whole process is a "scratch work" and a reader is able to see how the author structures the proof step by step, both from Goal and Givens viewpoints.

In past, there was only a Macintosh proofing program, but now Java version called Proof Designer is out. So Windows and Linux users alike can now enjoy this little program in conjunction with the book. Two disappointments with Proof Designer are that the output is only in the form of a traditional proof style which does not expose "the scratch work" and that the program does not use the two column style used in the book.

There are additional materials such as supplementary exercises, documentation, and a list of proof strategies (which is also available at the end of the book as a good reminder and reference), all available from author's site for free. [search in google like this: velleman "how to prove it" inurl:amherst]

After completion of this book, don't throw it away! Advance to Rudin's Principles of Mathematical Analysis and keep Velleman aside. Now one can work on complete proof of materials in Rudin with rigor and study how he constructs logical structures step by step in your own "structured" words!

This is it folks, the best there is!

However, it could have been better. I bought the book almost 10 years ago. I am a secondary ed. math teacher and when I left college I was quite upset with myself that I had this fancy math degree and couldn't prove anything. I picked up this book and today I'm working on my PhD in mathematics! This book inspired me to that.

First - What's wrong with the book. Not that there really is anything wrong with the book. I have attempted this book 3 times. I admit, the first two times I stalled (1997 - 2001) when I got to page 119. For some reason I couldn't grip those concepts such as intersecting families, etc. The preface of the book says only high school mathematics is required - that is just flat out wrong. This book is more for undergrads and maybe older fossils like me that have delved into mathematics a bit more than average. Also, like all the other reviews, there is too many exercises with no solutions. What really threw me with that is I didn't know if I was setting the written argument up properly. Sure, on the one hand, it's better to NOT have answers so you strive like a mad person to find them. Yet, it's so frustrating to not know if you did something right. The best approach is to do your best I suppose. After the third try (2004 & 2005) I finally completed the book on my own volition and I'm assuming most of my content is correct.

Velleman describes math so well that I honestly admit, I have a full repetoire of tactics to use to solve mathematical proofs. I don't have the confidence to toy with the big boys yet, like correcting a 49 page proof pertaining to the 'Twin Prime Conjecture' ... but it is SO NICE to UNDERSTAND the arguments! When I took Number Theory, I knew induction well, I know the If P Then Q arguments, it was just a blessing to know what the angle that the provers were using to prove mathematical theorems. I absolutely love this book. The cover is falling off and the pages are wearing out. I'm about to buy a new copy and start all over again. Mastery of this book, will certainly lead to a mastery of proof-writing in mathematics. I totally 100% recommend you buy this book if you are interested in mathematical proofs.

Good book but the greatest fault with the book is its lack of anwsers to the end of chapter questions. If it did have anwsers this book will definetly be worth a five star rating

The strength of this book is that it tries to develop an algorithmic structure for the approach of proofs that is very similar to computer programming. This means that the logic is easier to understand because of the way he standardizes his symbols and lays out the logical flow of different prove techniques. Many examples are worked out in detail. I recommend this book to anyone (especially engineering students) without formal training in mathematics (but who can program computers), who need to understand very formal mathematical material. The presentation is strengthened by the author's use of basic set theory to illustrate the proof technique. This means that the results you're trying to prove are often pretty obvious, but this allows you to concentrate on the technique of proof in question. Also check out Polya's book of the same name.

I agree with Usispaul's comments.

I only want to add that this is a wonderful introduction to mathematical thinking. It is completely engaging, and not like other textbooks. [This is a rigorous math book (not a book about math) and covers the material of first course for mathematics majors, logic, sets, relations, functions.] There are exercises (do them!) and examples.

I took math in college, but this book made me want to know MORE about mathematics.

This is an excellent book for the early undergraduate student. It is actually two books in one. The first half is a careful review of Logic and the essentials of Set Theory with an emphasis on precise language. Thereafter a structured development of proof techniques is clearly presented using these tools. The second half of the book is a detailed presentation of introductory material about functions, relations, and a few aspects of more advanced set theory. These chapters serve as a wonderful introduction and show applications of the proof techniques developed earlier.
I have referred back to this book often in my own study of analysis and number theory. I recommend it highly. It will be very useful to any undergraduate proceeding through a mathematics curriculum. I recommend studying it early in the first semester, and re-reading it as time goes on.

I found that this book utilized a little too much set theory for beginning students. If the author could have given more concrete examples, perhaps from group theory or simpler ones from analysis or number theory, it would have been much better. For students wanting a more lucid exposition of proof techniques, I highly recommend, "100% Mathematical Proof" by Rowan Garnier and someone else,whos name escapes me at the moment. "100% Mathematical Proof" is far superior to this book, and it has answers to the exercises which is crucial to the beginning student learning on his/her own. Velleman needs to bring the abstract nearer to the concrete for the beginning student.

I first purchased this book two years ago and became really excited when I found that I had it in me to write proofs. The truth be told, I have NOT STOPPED writing proofs since being inspired by this book. It is a masterpiece and I think ANY student approaching the writing of proofs for the first time couldn't possibly go wrong with an investment in a copy of this book. I also think that professors teaching a first class in abstract mathematics would be doing a service to their students by either requiring the book or giving it serious recommendation. AWESOME!!

If you're like many math students then there comes a time when you must be prepared to formalise your knowledge and your ability to present this knowledge in a logical and consistent fashion. I've found this volume to be most helpful in allowing me to do both.

Velleman is an excellent expositor and manages to present the art of proof writing in a clear and unintimidating way. The book is not overly formal and therefore doesn't treat the subjects of logic and set theory in too much depth (i.e. such depth that they become tedious). In addition to presenting some solid tactics for writing proofs, Velleman covers many fundamental mathematical concepts. Important topics such as Sets, Relations, Functions, and even some number theory are introduced and expanded. These serve both as the material which one is asked to prove and as useful revisions or introductions to these topics. While lacking solutions, this is not really a loss as the exercises follow very directly on the material presented in each section.

All round this book has helped formalise my proof writing skills enormously and therefore I can highly recommend it. It could well be used as a revision of discrete mathematics (or basic analysis) as well. Definitely worth purchasing!