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How to Prove It: A Structured Approach

23 customer reviews
ISBN-13: 978-0521446631
ISBN-10: 0521446635
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Editorial Reviews


'... we can warmly advise this excellent book for those who need to get acquainted with or must teach course on formalism and proof techniques.' Acta Scientiarum Mathematicarum

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Product Details

  • Paperback: 309 pages
  • Publisher: Cambridge University Press (1994)
  • Language: English
  • ISBN-10: 0521446635
  • ISBN-13: 978-0521446631
  • Product Dimensions: 6 x 0.8 x 9 inches
  • Shipping Weight: 13 ounces
  • Average Customer Review: 4.7 out of 5 stars  See all reviews (23 customer reviews)
  • Amazon Best Sellers Rank: #283,030 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

132 of 136 people found the following review helpful By Haseeb on August 3, 2001
Format: Paperback
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.
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73 of 74 people found the following review helpful By A Customer on October 23, 2002
Format: Paperback
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!
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48 of 50 people found the following review helpful By Eskychesser on July 26, 2005
Format: Paperback Verified Purchase
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!
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35 of 41 people found the following review helpful By Amazon Customer on October 12, 2004
Format: Paperback
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
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19 of 21 people found the following review helpful By UNPINGCO on June 12, 2000
Format: Paperback
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.
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