How to Prove It: A Structured Approach: Amazon.ca: Daniel J. Velleman: Books

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'... we can warmly advise this excellent book for those who need to get acquainted with or must teach course on formalism and proof techniques.' Acta Scientiarum Mathematicarum

Book Description

Many mathematics students have trouble the first time they take a course, such as linear algebra, abstract algebra, introductory analysis, or discrete mathematics, in which they are asked to prove various theorems. This textbook 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 "scratchwork" sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. Numerous exercises give students the opportunity to construct their own proofs. 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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First Sentence
As we saw in the introduction, proofs play a central role in mathematics, and deductive reasoning is the foundation on which proofs are based. Read the first page

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Front Cover | Copyright | Excerpt | Index | Back Cover

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17 Reviews

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4.8 out of 5 stars (17 customer reviews)

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3 of 3 people found the following review helpful

5.0 out of 5 stars An excellent book, Feb 29 2004

By

Kent S. Kapitan (Kansas, USA) - See all my reviews
(REAL NAME)

This review is from: How to Prove It: A Structured Approach (Hardcover)

This is an excellent book for the early undergraduate student. It is actually two books in one. The first half is a careful review of Logic and the essentials of Set Theory with an emphasis on precise language. Thereafter a structured development of proof techniques is clearly presented using these tools. The second half of the book is a detailed presentation of introductory material about functions, relations, and a few aspects of more advanced set theory. These chapters serve as a wonderful introduction and show applications of the proof techniques developed earlier.
I have referred back to this book often in my own study of analysis and number theory. I recommend it highly. It will be very useful to any undergraduate proceeding through a mathematics curriculum. I recommend studying it early in the first semester, and re-reading it as time goes on.

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2 of 2 people found the following review helpful

5.0 out of 5 stars Almost perfect, Sep 14 2008

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Khadija "Khadija" (Toronto) - See all my reviews

This review is from: How to Prove It: A Structured Approach (Paperback)

I just bought this book as a supplement to help me understand Spivak's Calculus, a proof based treatment of calculus, which is the text of my analysis 1 class.

Velleman elucidates complex concepts in math extremely clearly, by carefully and thoroughly building on concepts one at a time. His explanation of basic set theory had me looking at the definitions I had already known in an entirely new, fresher and clearier perspective, like I was seeing them again for the first time. Flipping through the book at the beginning, without any knowledge of the symbols, it feels like it's written in code. Two chapters in and you'll find that voila! it is you who are writing that code, fluently.

I am eagerly anticipating finishing the book. The later chapters from my current level of understanding seem sort of incomprehensible. All the greater the sense of accomplishment however, when I reach the end and find I can read it as well as if I've learned another language.

I've been studiously working through the problems Velleman includes solved at the back of the book. This book would be *perfect* if all the problems included solutions, then I would attempt all of them, since there would be no fear of reinforcing misunderstanding by believing incorrect answers to be correct.

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1 of 1 people found the following review helpful

5.0 out of 5 stars I wish I had such a book before taking advanced calculus, Aug 3 2001

By

Haseeb - See all my reviews

This review is from: How to Prove It: A Structured Approach (Paperback)

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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Most recent customer reviews

4.0 out of 5 stars Starts off good, and then goes off on a tangent.
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... Read more

Published on July 25 2003 by D. Goldman

5.0 out of 5 stars Breakthrough and Original ......
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. Read more