Books with proving stuff. Any suggestions.

Question

So I've recently started proofs since I'm taking a proofs class in college. We use Velleman's book "How To Prove It". I've been having a lot of trouble with proofs and Velleman's book just doesn't do the job in helping me out. I have pretty good understanding of the first 3 chapters but I've had so much trouble with proofs of the other 4. This includes relations, functions, strong induction (I'm pretty ok with induction) and infinite sets. I'm almost close to the end of the semester and finals are near. I was wondering if anyone has suggestions for any proof books that use the same axioms that also has answers so I can practice and learn. I don't need the book just to pass finals, I need a book that would help me grasp concepts easily, so I wouldn't have trouble in future classes.

Thanks.

Answer 1

To be honest, I've got mixed feelings about such courses.On the one hand, it's nice to have a course that lays out the basic language and methodology of real mathematics for students who have just calculated their way through math before. But such courses inevitably look like service courses that have little or no connection with subsequent courses. I think there's a lot better ways to construct such courses through a single theme that would connect basic proof methods to other areas of mathematics. But this is a full discussion for another day.

I've looked at a lot of books for such courses and most of them look cut from the same mold. My favorite is a book that's sadly out of print: it's Robert S. Wolf's Proof, Logic, and Conjecture: The Mathematician's Toolbox. It's deep, beautifully written and looks to motivate the methods of proof and basic constructions of sets and the number systems through a single unifying concept: basic mathematical logic. Wolfe also emphasizes methods and understanding why these methods work rather then the basic structures of mathematics alone-this supplies much more motivation for why mathematicians prove things the way they do then other books give. Wolf also writes very clearly and informatively. I think you'll like this book a lot and will find it quite helpful to get through exams.