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How to Prove It: A Structured Approach [Paperback]

Daniel J. Velleman
5.0 out of 5 stars  See all reviews (4 customer reviews)

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How to Prove It: A Structured Approach How to Prove It: A Structured Approach 4.5 out of 5 stars (4)
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Book Description

25 Nov 1994 0521446635 978-0521446631
Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This textbook will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarise students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratchwork' sections to expose the machinery of proofs about the natural numbers, relations, functions and infinite sets. Numerous exercises give students the opportunity to construct their own proofs. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.

Product details

  • Paperback: 319 pages
  • Publisher: Cambridge University Press (25 Nov 1994)
  • Language: English
  • ISBN-10: 0521446635
  • ISBN-13: 978-0521446631
  • Product Dimensions: 22.6 x 15 x 2 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (4 customer reviews)
  • Amazon Bestsellers Rank: 823,111 in Books (See Top 100 in Books)

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Product Description


'… 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

From the Author

Related software available
Macintosh software that helps you learn to write proofs using the methods explained in this book can be downloaded from my homepage at:

Inside This Book (Learn More)
First Sentence
As we saw in the introduction, proofs play a central role in mathematics, and deductive reasoning is the foundation on which proofs are based. Read the first page
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Front Cover | Copyright | Excerpt | Index | Back Cover
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Most Helpful Customer Reviews
15 of 15 people found the following review helpful
5.0 out of 5 stars Wonderful into to rigorous mathematics 29 Dec 1998
By A Customer
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.
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24 of 26 people found the following review helpful
5.0 out of 5 stars MUST OWN!!! 20 July 1999
By A Customer
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!!
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12 of 13 people found the following review helpful
5.0 out of 5 stars Engaging book on producing mathematical proofs 18 Sep 1997
By A Customer
The emphasis of Vellemans book on the difference between manufacturing a proof and the proof's final presentation speaks directly to the confusion of the uninitiated to proofs. It meets the (perhaps frequent) naive expectation of an invariable and immediate recognition of a polished proofs rhyme and reason. It consequently points to the often necessary autonomous efforts of the student to independantly unravel the proof of a theorem or definition.

The book moves rapidly from the necessary setential logic and truth tabels (a Wittgensteinian invention) to the chapters on proof writing and follows with chapters on functions, relations, closures, and more.
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5.0 out of 5 stars Very good purchase 30 Jun 2013
Format:Paperback|Verified Purchase
The book is like new, but I have yet to read it. It's a book for who really loves Math
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Amazon.com: 4.7 out of 5 stars  23 reviews
131 of 134 people found the following review helpful
5.0 out of 5 stars I wish I had such a book before taking advanced calculus 3 Aug 2001
By Haseeb - Published on Amazon.com
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.
72 of 73 people found the following review helpful
5.0 out of 5 stars Breakthrough and Original ...... 24 Oct 2002
By "self-study" - Published on Amazon.com
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!
47 of 48 people found the following review helpful
5.0 out of 5 stars The best PROOF book I've ever seen. 26 July 2005
By Eskychesser - Published on Amazon.com
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! 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.
34 of 40 people found the following review helpful
3.0 out of 5 stars Lack of answers 12 Oct 2004
By Amazon Customer - Published on Amazon.com
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
18 of 20 people found the following review helpful
5.0 out of 5 stars develop an algorithmic structure for proofs 12 Jun 2000
By UNPINGCO - Published on Amazon.com
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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