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How to Prove It: A Structured Approach, 2nd Edition 2nd Edition

4.4 out of 5 stars 87 customer reviews
ISBN-13: 978-0521675994
ISBN-10: 0521675995
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  • How to Prove It: A Structured Approach, 2nd Edition
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Editorial Reviews


"The prose is clear and cogent ... the exercises are plentiful and are pitched at the right level.... I recommend this book very highly!"
MAA Reviews

"The book provides a valuable introduction to the nuts and bolts of mathematical proofs in general."
SIAM Review

"This is a good book, and an exceptionally good mathematics book. Thorough and clear explanations, examples, and (especially) exercised with complete solutions all contribute to make this an excellent choice for teaching yourself, or a class, about writing proofs."
Brent Smith, SIGACT News

Book Description

Beginning with the basic concepts of logic and set theory, this book teaches the language of mathematics and how it is interpreted. The author uses these concepts as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. He shows how complex proofs are built up from these smaller steps, using detailed "scratch work" sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software.
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Product Details

  • Paperback: 384 pages
  • Publisher: Cambridge University Press; 2nd edition (January 16, 2006)
  • Language: English
  • ISBN-10: 0521675995
  • ISBN-13: 978-0521675994
  • Product Dimensions: 6 x 0.9 x 9 inches
  • Shipping Weight: 1.5 pounds (View shipping rates and policies)
  • Average Customer Review: 4.4 out of 5 stars  See all reviews (87 customer reviews)
  • Amazon Best Sellers Rank: #13,732 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

Format: Paperback Verified Purchase
Before buying this book, I struggled in math. I excelled at "calculating" stuff by simply plugging in numbers into some sort of equation our high school teachers would spoil us with, but when I got to college, I had to start thinking abstractly- and it bothered me a lot, because I had no idea how to test or prove the logic of some statement. I was doing very poorly in linear algebra and desperately needed help- lo and behold, my professors weren't helpful (at all). Someone recommended this proof writing book to me, and I am VERY grateful for that referral.

The book takes the average student (it's shocking with how little math background one needs) and introduces him to basic boolean logic. You know, material like "If A is true, and B is false, then A implies B is false." In a discrete mathematics course, one would call this "truth tables." From there, the author takes the reader into set theory, basic proofs, group theory, etc- and into more advanced topics, like the Cantor-Schroeder-Bernstein theorem, countability, etc. So what makes this book stand out?

(1) Readability. Many math professors stop just short of taking pride in how confusing, abstract, or daunting their lectures can be. Velleman, however, goes the extra mile in the text to see that the reader UNDERSTANDS the logical buildup and concepts of mathematical proofs. Sure, set theory can be confusing- but after reading several other texts in discrete math, including "Discrete Math and its Applications" by Kenneth Rosen (if you're reading this, no offense) I've found that Velleman by far writes the most comprehensive and cohesive explanations for understanding set theory.
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Format: Paperback
Now I understand how proofs are being constructed. I can read and write them the right way! After reading this book I went back to my Calculus textbook and started looking at the proofs. I was amazed at how differently I perceived them. I actually enjoyed reading them and understood why they were written that way.

A little info about the book. Basically, it teaches you the same material that you learn in a Discrete Mathematics course - Propositional logic, Sets and Proofs, Relations, Functions, and Mathematical Induction. However, it looks at those subjects from a completely different perspective. There's absolutely no practical information - all you do is prove stuff.

I strongly advise to learn Discrete Math before reading this book, because getting straight to the proofs of the material, that you just have learned and have no previous experience with, can get very tough.

The first two chapters were a bit boring and too easy - but only because I have already learned that stuff. Chapter 3 is where you start to do your own proofs and is where it gets fun.

The exercises are not hard, and shouldn't present any trouble for the reader. However, I did find the exercises in the last 3 chapters to be more challenging. There were some problems on which I was simply staring for an hour, literally, trying to figure out the way to prove it. The theorem made sense to me, but I couldn't find a way to put into strict mathematical proof! But let me tell you, there's nothing like getting a "Eureka!" moment and figuring out the answer all by yourself. I have just spent 1.5 hours doing 1 problem, and after getting the answer I've felt like I have accomplished something.

Get this book, NOW!
Comment 39 people found this helpful. Was this review helpful to you? Yes No Sending feedback...
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Format: Paperback
My goal for this review is to make it as helpful as possible to someone considering to buy this Book !
-I will not go over most of the summaries of the text provided by my fellow reviewers but will provide two important clarifications.
I read in another review that that the Kindle version of the text interprets some logical operators or other notation incorrectly causing confusion with what the text or exercises are referring to....
This Statement is FALSE. I have purchased the kindle version and did not find one inconsistency and all notations are indeed accurate.
I have found that a major complaint about this text is that it does not provide enough solutions to its exercises for one to verify whether they have actually learned the material or not.
This Statement is TRUE. On average, out of 7 question in each section, only 2 solutions are given in the back of the text.

HOWEVER!!! There is another way to circumvent this problem. The Department of Mathematics of the University of California, Santa Barbara has been so kind as to post the Solutions to the unlisted problems on their website.
Please visit this site to view them:

Now with both of these clarifications in place, and after going through a couple of other Mathematical Proof books, and I personally prefer this one. It is direct, and covers the basics needed for understanding and doing proofs.
One must understand that doing proofs is a skill a Mathematician gains through vast experience, practice and long hours of thought. Finding a book that breaks down a universal method of proofing in the same simple way an Algebra text shows how to use a formula will not be possible.
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