How to Prove It: A Structured Approach [Paperback]

Daniel J. Velleman (Author)

Book Description

Publication Date: January 16, 2006 | ISBN-10: 0521675995 | ISBN-13: 978-0521675994 | Edition: 2

Geared to preparing students to make the transition from solving problems to proving theorems, this text teaches them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize 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. To help students construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. Previous Edition Hb (1994) 0-521-44116-1 Previous Edition Pb (1994) 0-521-44663-5

Editorial Reviews

Review

"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.

See all Editorial Reviews

Product Details

• Paperback: 384 pages
• Publisher: Cambridge University Press; 2 edition (January 16, 2006)
• Language: English
• ISBN-10: 0521675995
• ISBN-13: 978-0521675994
• Product Dimensions: 6 x 0.9 x 9 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 4.4 out of 5 stars  See all reviews (33 customer reviews)
• Amazon Best Sellers Rank: #10,338 in Books (See Top 100 in Books)

Most Helpful Customer Reviews

25 of 25 people found the following review helpful

5.0 out of 5 stars This Book Taught Me How to "Get" Math... Please Read On.. July 17, 2012

By Baze

Format:Paperback|Amazon 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. Making the material accessible is the mark of a real "teacher," and if you read through this book yourself, I believe you'd agree that Velleman is a pretty legit teacher.

(2) Examples. There are plenty- plenty that Velleman works out himself. Reading the examples alone- and actually taking the time to understand them- is a task that's up to the reader, obviously, but they do show results almost immediately in understanding discrete math.

(3) Problems (exercises). There's never a shortage of exercises, I found, as I tried to work through the problem set. There are plenty. Fortunately, there are some answers in the back, but just enough so that you can verify to see if you're understanding the material, and not enough so that you find yourself copying every answer in the back (even the best students get tempted to do that). Velleman gives the proper amount of answers in the back and a ton of exercises to do. If you complete them all properly, you'd be far ahead of the curve amongst math majors.

I know my review may have been too wordy, or too optimistic. However, my feelings are very honest and not exaggerated: this book is written so one can learn discrete mathematics, and really helps the reader understand what higher math is all about- and how mathematicians think, write, and communicate. This book deserves an A+, and I've only given that score out to a handful of books.

13 of 13 people found the following review helpful

5.0 out of 5 stars Buy it early in your math major and keep it throughout April 1, 2011

By Charles Ashbacher HALL OF FAMETOP 500 REVIEWERVINE™ VOICE

Format:Paperback

All math teachers at the college level are familiar with students hitting the "struggling with proofs" wall. Students take calculus and do fairly well using the algorithms to differentiate and integrate functions and this continues into the first part of linear algebra. However, when it is time to understand and execute the proofs they experience a great deal of difficulty that many simply cannot overcome.
This book is designed to present a set of techniques used in mathematical proofs and that aspect is well done. Yet, this book is also just as valuable for the thorough treatment of many of the foundational structures of mathematics. Those topics are:

*) The logic of propositions and predicates
*) Set theory
*) Relations and functions
*) Mathematical induction and recursion
*) Infinite sets

The combination of a thorough introduction to these topics as well as demonstrating proof techniques applied to these objects is an excellent way to learn about them, so this book would be a valuable text in the foundations of mathematics.
The more complex or difficult proofs are also presented in a very stepwise deconstruction, begun using a technique called scratch wok, where even the most insignificant details are included. Once the preliminary scratch work is completed, the formal proof is given. While experienced readers will find this tedious, beginners will find the clarity a relief. A large number of exercises are given at the ends of sections and chapters and solutions to many are included in an appendix.
If you have a course in the foundations of mathematics for the early math major, this is a book that would be an excellent text. It would also be valuable as a supplemental reference text for all students taking a math course where understanding of proofs is required. Think of it as a boost over the wall.

16 of 17 people found the following review helpful

5.0 out of 5 stars Completely changed my view of proofs. March 21, 2009

By L. Burton

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.

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