How to Prove It: A Structured Approach

Set Theory is the foundation on which mathematical proofs are based. This book emphasizes set theory.


Breakthrough and Original ......   October 24, 2002
38 out of 41 found this review helpful

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!


develop an algorithmic structure for proofs   June 12, 2000
UNPINGCO (Los Angeles, CA)
16 out of 17 found this review helpful

The strength of this book is that it tries to develop an algorithmic structure for the approach of proofs that is very similar to computer programming. This means that the logic is easier to understand because of the way he standardizes his symbols and lays out the logical flow of different prove techniques. Many examples are worked out in detail. I recommend this book to anyone (especially engineering students) without formal training in mathematics (but who can program computers), who need to understand very formal mathematical material. The presentation is strengthened by the author's use of basic set theory to illustrate the proof technique. This means that the results you're trying to prove are often pretty obvious, but this allows you to concentrate on the technique of proof in question. Also check out Polya's book of the same name.


Rigorous but accesible..an engineer's intro to proofs   June 18, 2006
Humberto Mejias (Caracas, Venezuela)
4 out of 7 found this review helpful

I write this review on the context of having done all the math required for mechanical engineer but never havin to do proofs... as B.Rusell once said do and the by faith you will believe... trying to escape that state of mind I got this book some years ago, Im glad for the author follows a structure approach similar to learning a programming language, once you master some elemenatry techniques you can stack up to create refined algorithms.


An excellent book   March 1, 2004
Kent S. Kapitan (Kansas, USA)
12 out of 12 found this review helpful

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.