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Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text 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 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. The author 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. 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.
- Sales Rank: #57863 in eBooks
- Published on: 2006-01-16
- Released on: 2006-01-16
- Format: Kindle eBook
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
About the Author
Daniel J. Velleman received his BA at Dartmouth College in 1976 summa cum laude, the highest distinction in mathematics. He received his PhD from the University of Wisconsin, Madison, in 1980 and was an instructor at the University of Texas, Austin, from 1980 to 1983. His other books include Which Way Did the Bicycle Go? (with Stan Wagon and Joe Konhauser, 1996) and Philosophies of Mathematics (with Alexander George, 2002). Among his awards and distinctions are the Lester R. Ford Award for the paper 'Versatile Coins' (with Istvan Szalkai, 1994), and the Carl B. Allendoerfer Award for the paper 'Permutations and Combination Locks' (with Greg Call, 1996). He has been a member of the editorial board for American Mathematical Monthly since 1997 and was Editor of Dolciani Mathematical Expositions from 1999 to 2004. He published papers in the Journal of Symbolic Logic, Annals of Pure and Applied Logic, Transactions of the American Mathematical Society, Proceedings of the American Mathematical Society, American Mathematical Monthly, the Mathematics Magazine, the Mathematical Intelligencer, the Philosophical Review, and the American Journal of Physics.
Most helpful customer reviews
114 of 115 people found the following review helpful.
This Book Taught Me How to "Get" Math... Please Read On..
By Baze
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.
39 of 41 people found the following review helpful.
Completely changed my view of proofs.
By L. Burton
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!
24 of 25 people found the following review helpful.
Solutions to: Does it work on Kindle? & Are there Solutions to the Exercises?
By DrTrips
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.
1st:
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.
2nd:
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:
http://www.math.ucsb.edu/~dai/813wang.html
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. Mainly because there is no universal method of proofing. All one can then hope to do in order to understand higher level of mathematics, or what I would describe better as the underlying foundation of mathematics, is to understand the basic language of proofs, their structure, and their organization enough for the reader to go forth and gain their own experience.
And this book does precisely that
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