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Title: How to Prove It : A Structured Approach by Daniel J. Velleman ISBN: 0-521-44663-5 Publisher: Cambridge University Press Pub. Date: 25 November, 1994 Format: Paperback Volumes: 1 List Price(USD): $29.00 |
Average Customer Rating: 4.73 (15 reviews)
Rating: 4
Summary: Starts off good, and then goes off on a tangent.
Comment: I bought this book in the hopes that it would help me improve my proof writing skills. Being only a high school graduate (a month ago), I had practically no knowledge of set theory. The initial proof structures were great, and I enjoyed following the proofs from the premises and, through logical steps, to the desired conclusion. However, then the Set Theory came in. I can understand why a certain amount of set theory was necessary in order to be able to talk about certain types of proofs, but he goes so far into set theory in the book, that by a certain point, instead of following the logical flow of the proofs, I was trying to remember abstruse terminology he had mentioned briefly and trying, successfully for the most part, to understand what the actual proof meant, and why it would make sense that it was correct. Its possible that the reason I feel this way is because when I do proofs, I usually need to understand it intuitively first and then go from there, and it could be the case that this isn't possible with more abstract proofs. Overall, it was a good read, but unfortunately, he went a little too far into the set theory than was necessary. Reading it twice would fix that problem though. Another criticism is that there are no solutions to the exercises.
Rating: 5
Summary: Breakthrough and Original ......
Comment: 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!
Rating: 4
Summary: Probably the best book out there but not perfect
Comment: A good basic introduction to understanding math proofs by understanding logic first. Only lacking in its connection to math proofs that one might actually see, in other words too basic (which is as much complement as a critcism.)
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Title: How to Read and Do Proofs : An Introduction to Mathematical Thought Processes by Daniel Solow ISBN: 0471406473 Publisher: Wiley Text Books Pub. Date: 06 July, 2001 List Price(USD): $41.95 |
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Title: How to Solve It by G. Polya ISBN: 0691023565 Publisher: Princeton Univ Pr Pub. Date: 01 November, 1971 List Price(USD): $18.95 |
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Title: An Introduction to Mathematical Reasoning : Numbers, Sets and Functions by Peter J. Eccles ISBN: 0521597188 Publisher: Cambridge University Press Pub. Date: 11 December, 1997 List Price(USD): $32.00 |
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Title: Proofs and Fundamentals: A First Course in Abstract Mathematics by Ethan D. Bloch ISBN: 0817641114 Publisher: Birkhauser Boston Pub. Date: May, 2000 List Price(USD): $49.95 |
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Title: Concepts of Modern Mathematics by Ian Stewart ISBN: 0486284247 Publisher: Dover Pubns Pub. Date: 01 February, 1995 List Price(USD): $10.95 |
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