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How to Prove It: A Structured Approach
Binding: Paperback
Author: Daniel J. Velleman
Manufacturer: Cambridge University Press
Availability: Usually ships in 24 hours
Features:
Average Rating: 4.5
Total Customer Reviews: 29
List Price: $29.99
Our Price: $21.59
Sales Rank: 19701
Product Description
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.
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Users Product Reviews: |
Product Review Summary: good for self study or to supplement an advanced maths course Velleman presents a really basic yet thorough treatment of the various structure for proof-writing. It's important to understand quantifier negation and existence proofs, and most books gloss over those structures to focus on just quantifiers generally, set formation, and induction. It's an excellent book for people of all math backgrounds (computer science, physics, etc), although it requires a bit of maturity for a high school student to patiently reread and digest the proofs, it doesn't require advanced math in the sense of having taken lots of classes. I recommended this book to some friends in the economics PhD programs and got positive feedback.
He really breaks down the structure and intuition of both simple and famous proofs. The book reads more like a logic or puzzle book, since he isn't intending to present a structured study of proofs applied to one area of math. Also, he translates plain English sentences that lack the normal "if...then" structure into mathematical statements with the quantifiers.
The answers included are great for hints and extra exercises. It gives a good basis for relations, partial orders, equivalence classes, functions, cardinality, although perhaps too heavy on the partial and total orders than is necessary for analysis or linear algebra courses.
If you buy this book, use the techniques to dissect other proofs in other courses and to have an understanding of set theoretic definitions of concepts you will use in advanced courses. The idea is to gain an intuition when reading proofs.
Product Review Summary: Completely changed my view of proofs. 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!
Product Review Summary: Outstanding! This is an incredible book -- one of the best mathematics books I have ever read. I have not yet found any other books that cover the material of sets, logic, proof nearly as effectively than this book does.
The most unique and helpful part for me was the concept of "structured proofs" and the proof strategies. I have never encountered this information elsewhere, and it helped me progess very much with my mathematical thinking.
I just wish I had found this book earlier on in my studies...
Product Review Summary: How To Prove It: A Structured Approach 2nd Edition How To Prove It: A Secured Approach 2nd Edition by Daniel J. Velleman
I highly recommend this book for any student who is trying to improve his or her understanding of mathematical or computational proofs. I am an undergraduate student in the Northeastern United States and this book will undoubtly help me with my mathematics course that involve formal proofs. Some people try to understand proofs "naturally" in advanced math courses. This can be a time consuming and unfruitful venture if you don't get the proofs right away or have a bad TA. This book's step by step approach to mathematical logic will give you excellent strategies for tackling any type of proof.
I recommend only buying this book if you have a lot of time to invest. If you are looking for light reading or a quick review this is the wrong book. It took me about 2-4 hours to fully digest each chapter.
Product Review Summary: Good book, but it requires a lot of work to be useful Reading this book will not help you write proofs in the least. How to Prove It, to help you at all, will require lots of time and effort after reading each chapter, actually doing proofs.
Anyone who has done proofs, or attempted to do proofs, should not be surprised that there is no magic key. Working through this book is required for improvement, just a quick read will get you no where.
That said, the time is well worth it. When you think back to how hard proofs used to be after working through the book, you will feel like a super genius with your new found skill.
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