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Which of the two books is suited for a student looking to learn how to write proofs? I have a working knowledge of calculus and linear algebra but I'm not good at writing proofs. My intention is to learn proofs in general, not necessarily for the two.

The reason I ask is that the latter is suggested on a highly voted question here but the former has a more apt name. The reviews aren't helping. Please don't suggest books other than these.

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Perhaps this will be an unpopular answer, but I found the best way to learn how to write proofs was to read them and imitate them. –  Holdsworth88 Jul 4 '12 at 16:23

4 Answers 4

up vote 9 down vote accepted

Velleman's How to Prove it is quite a bit more relevant to your needs. It is organized like a conventional text, and pays a lot of attention to proof writing.

Polya's book focuses on problem-solving. One can view it as a better book, certainly a historically far more important book. But it focuses on how one finds the idea that will crack a concrete problem.

There is quite a bit of material in Velleman that is useful for writing proofs in linear algebra, in particular on how to proceed from definitions. There is none of that in Polya. There is also essentially nothing in Polya on basic analysis. Polya beautifully accomplishes his aims: they just happen to be different from what you said you wanted.

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math.stackexchange.com/questions/7743/getting-better-at-proofs suggests otherwise hence the doubt. –  Inquest Jul 4 '12 at 15:35
    
There will inevitably be differences of opinion. But I am quite convinced that Velleman will be far more useful to you in writing proofs associated with your courses. If you were a young person getting into contest-type mathematics, my advice would be different. –  André Nicolas Jul 4 '12 at 15:43

The Polya is more advanced than I think you are looking for. It is a well-known classic, but assumes the reader already knows how to write proofs.

I haven't seen it, but the Velleman should be good for you. Another one I have seen is by Solow: How to Read and Do Proofs

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Can you elaborate on more advanced? I am a programmer and can work with matrices, scipy, numpy and all. Enough? –  Inquest Jul 4 '12 at 15:20
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Edited. It assumes the reader already knows how to write proofs. –  GEdgar Jul 4 '12 at 15:23
    
Assumes reader knows to write proofs? To what extent? –  Inquest Jul 4 '12 at 15:24

I know it's not one of the two you referred to, but you might want to look at How to Think Like A Mathematician by Kevin Houston.

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Another book that you should check out is Gary Chartrand's Mathematical Proofs: A Transition to Advanced Mathematics. It has an instructor solutions manual.

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