# Books for learning how to write proofs

## Main Question or Discussion Point

What are some classic, good books on learning mathematical language and writing proofs?

(to gain facility with mathematical language and method of conjecture, proof and counter example, with emphasis on proofs. Topics: logic, sets, functions and others.)

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Perhaps this isn't quite what you're looking for, but I've found that the best way to learn to write proofs is to simply write proofs!

There are all sorts of resources to help you in this regard. Check out the past IMO exams. Despite being aimed at high school students, they should provide a good challenge even to the average college math major. Also, look at the Putnam exams. These are of course quite harder than the IMO tests, but the questions are all of the same fundamentally proof-based nature.

In this spirit, the book I recommend is https://www.amazon.com/dp/0387257659/?tag=pfamazon01-20. It's a great collection of very challenging proof-based exercises. It covers the whole range of undergraduate mathematics, and the opening chapter is focused on specific methods of proof together with some (surprisingly hard!) exercises that are meant to be solved using each method.

I should mention that it's not only a collection of problems. It does contain a fair amount of discussion of various techniques and topics that are relevant to writing proofs and solving problems.

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"How to Prove it; A structured approach" by Daniel J. Velleman. I just got it at a library, haven't read it yet. But its suppose to teach you proves.

IMO, learning proofs by doing logic and sets is making it harder then it has to be. Doing something natural like number theory or geometry (which are on problems like zpconn suggests) is more enjoyable. Induction and contradiction, for example, can be picked up from number theory . Although it might be a good idea to pick up a book like Bright Wang's from the library to learn the notation.

I agree. I used this text. Solow does a good job systematizing the subject. In order to clearly illustrate the methods, his examples and problems are usually at the level of high school mathematics (e.g. trigonometry). There are supplementary sections at the end of the book, which comprise about one-half of the pages, that explore examples related to more advanced mathematics (e.g. real analysis).

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thrill3rnit3
Gold Member
I thought Velleman's book was pretty good.