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Summary: Probably the best book out there but not perfect
Review: 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.)
Rating: 
Summary: A very tasty pudding
Review: A mathematically inclined student can expect to reap a bountiful harvest from D.J. Velleman's 'How to Prove It.' You needn't be a computer type to benefit. In fact, the book avoids computer gobbledygook and, in a highly disciplined manner, hones in on the essentials of proof techniques. Though Dr. Velleman's overt aim is to familiarize the student -- no advanced math necessary -- with the reading and writing of mathematical proofs, he also succeeds admirably in teaching basic logic and set theory as a useful mathematical tool, rather than as a mere corpus of interesting ideas. Velleman writes in a spare, lucid style and his exercises are well chosen to illustrate his lessons, though for some reason the book omits the customary answers to alternate exercises, which is useful for someone, such as myself, engaged in self study. Even so, other writers could take pointers from Velleman. I had very little trouble using the book for self-instruction, which is more than I can say for the Schaum's guides and numerous other math textbooks. I found no significant errors in the text or exercises, though Velleman and I did have a bit of an email dustup over 'vacuous truth' .... A quibble: Velleman omits mention of the foundational problems of set theory, other than to stick Russell's paradox in as an exercise. The final (and very good) chapter gives us Cantor's theorem without mentioning Cantor's paradox. Though beginners may shrink from foundational subtleties, a few more words would have been useful. Yet, all in all, this is an elegant, succint and enormously useful text.
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Summary: I wish I had such a book before taking advanced calculus
Review: Believe it or not, I graduated with a BS in math without being able to write proofs all that well. I got an "A" in advanced calculus and abstract algebra due mostly to the fact that the majority of the students in the class couldn't write proofs. Over a decade later, I was browsing through the math books at my local book store and found this book. After working through some of the problems and studying some of the material, I wished that I had this book a year or so before taking advanced calculus (introductory real analysis). Actually, this book can be handled by a person just finishing high school. My advice to all math majors who don't have a solid foundation in mathematical proofs is to get this book as soon as you can, study it and work many of the problems. This way when you have to take advanced calculus, topology or abstract algebra you will not be struggling to learn how to write proofs. I can not guarrantee that you will breeze through these courses after studying this book, but you will be spending more time on learning concepts and little or no time on the methods and techniques of proofs. Set Theory is the foundation on which mathematical proofs are based. This book emphasizes set theory.
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Summary: Lack of answers
Review: Good book but the greatest fault with the book is its lack of anwsers to the end of chapter questions. If it did have anwsers this book will definetly be worth a five star rating
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Summary: A superb Introduction to Proof Writing and Set Theory
Review: I am a philosophy student with no real mathematics training, though I am very familiar with first order predicate logic. My view is that this book is perfect for anyone who has mastered first-order predicate logic and wants to expand their powers. One of the most instructive and enjoyable books I have read in a long time.
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Summary: Starts off good, and then goes off on a tangent.
Review: 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.
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Summary: Correcting two submitted reviews
Review: I did not notice your ban on URL addresses when I reviewed 'How to Prove It' and 'The Advent of the Algorithm.' Please delete the URL addresses so that the reviews can be posted. Thanks.--P.R. Conant
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Summary: A good start on writing proofs, but falls short!
Review: I found that this book utilized a little too much set theory for beginning students. If the author could have given more concrete examples, perhaps from group theory or simpler ones from analysis or number theory, it would have been much better. For students wanting a more lucid exposition of proof techniques, I highly recommend, "100% Mathematical Proof" by Rowan Garnier and someone else,whos name escapes me at the moment. "100% Mathematical Proof" is far superior to this book, and it has answers to the exercises which is crucial to the beginning student learning on his/her own. Velleman needs to bring the abstract nearer to the concrete for the beginning student.
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Summary: Great book on proofs
Review: I really enjoy this book because it presents all you have to know about the technics of proof. On the first reading I found the book confusing because of it presentation but on a second reading the book appears to me very clear and complete. The author is very clear in is explanation and he present a lot of example that are very pertinent. The book is moslty based on strategic method of proof. The book cover proofs on sets, functions, relations, number theory and induction. The subjects are classical and well know. This book miss some area of interesting mathematics such that topology, analysis, abstract algegra, ...If this book would have this then I definitely buy it again. I recommend this book to those who want an introduction to proofs
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Summary: Breakthrough and Original ......
Review: 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!
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