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How to Prove It: A Structured Approach

Daniel J. Velleman
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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Here are the reasons I’m not using this book in my course on how to do proofs in the Fall semester of this year:

  1. I’ve already ordered another book and can’t begin to imagine what trouble changing horses even at the edge of the stream would cause for our book-store (and, more importantly, for our students);
  2. the book is, if anything, too good: it does exactly what I do in the indicated course but is so thorough and so expansive that I’d want to spend a whole year on the syllabus, not just a semester.

Perhaps, as time goes by, and it’s again my turn to teach this course on “baby proofs,” I’ll just give in and do the sensible thing and opt for Velleman’s How To Prove It. (Even its title evinces good taste: how many of us don’t have fond memories of Polya’s classic by a similar name?).

Some details, then: Velleman’s book is divided into seven chapters (Sentential Logic, Quantificational Logic, Proofs, Relations, Functions, Mathematical Induction, and Infinite Sets), covering over 300 pages. The prose is clear and cogent (and even suited to self-study for the rare lone wolf), the exercises are plentiful and are pitched at the right level (and many of them have solutions in the back of the book), and the paperback version of the book comes in at a penny shy of $30.00 Incredible!

I recommend this book very highly!

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.

 1. Sentential logic: 2. Quantificational logic; 3. Proofs; 4. Relations; 5. Functions; 6. Mathematical induction 7. Infinite sets.