- 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: 1.2 pounds (View shipping rates and policies)
- Average Customer Review: 4.4 out of 5 stars See all reviews (112 customer reviews)
- Amazon Best Sellers Rank: #53,221 in Books (See Top 100 in Books)
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How to Prove It: A Structured Approach, 2nd Edition 2nd Edition
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"The prose is clear and cogent ... the exercises are plentiful and are pitched at the right level.... I recommend this book very highly!"
"The book provides a valuable introduction to the nuts and bolts of mathematical proofs in general."
"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
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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Top Customer Reviews
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.
My main issues with this book:
1) Most of the problems/exercises in this book revolve around set theory; and towards the end of the book you will find topics in mathematical induction. It would have been nice if the author included a more diverse set of problems, perhaps from elementary geometry and number theory -- that would have kept me more engaged. I eventually got tired of working sets. I know most advanced mathematics requires working with sets, and perhaps the author thought that a book on learning proofs would be the ideal time to introduce students to working with sets, but the amount of set theory was inundating in my opinion.
2) I recall one instance where a problem required a concept that was never introduced in the chapter/section prior: I was stuck on one exercise from the logic section on De Morgan's law. In order to solve that exercise, you had to distribute the logical quantifiers to prove it. However, the fact that you are able to distribute the logical quantifiers is never mentioned in the particular reading. I had to get help from the internet to figure it out.
3) I stopped reading the book towards the end of chapter 4 ("Ordering Relations"). The discussion about the "smallest elements" was really confusing and I stopped reading there. I did read that part multiple times and I still couldn't understand it. The author ought to have provided more clarity and examples here. Again, I had to go on the internet to really figure it out.
You will learn many techniques on doing proofs from this book, but you might have to supplement some of the reading with other sources like I had to. I would have rated the book higher had if the author was a bit more clear on some subjects.
There is also a companion website with a "Proof Designer." I couldn't get the program to work, but it's not necessary at all to go through the book.
This book is very readable and at the beginner's level. The best part I like about this book is he gives a template on how to approach proofs. He provides good examples and the exercises are at a good level, varying in level of difficulty. Now I look back at my calculus book and could really understand the proofs, and I was able to attempt more exercises in the analysis book. If you need a book to prepare you for upper division mathematics, I would buy this book as soon as possible.
TIP: If you have a weak background in your mathematics, I would go back and review those concepts. This book does assume you are proficient in Pre-calculus and lower levels of mathematics.
Also, if you get stuck, that's normal! Don't feel discouraged. Put down the problem aside and maybe an idea might pop up later. Be determined and move on.