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I am a Math fan and a self-learner. I try to look at least once a day at a Linear Algebra or Calculus problem to keep myself in shape and to learn.

I also like Analysis, Abstract Algebra and Discrete Math, but I feel I need to and would like to get proficient at proofs.

I have much harder time finding good online sources of solved problems and step-by-step guides for practicing proofs (induction, contradiction).

I would ideally like to have a list of solved proofs that progress in difficulty.

Something like this but in the area of analysis and abstract algebra:
http://archives.math.utk.edu/visual.calculus/0/domain.1/index.html
http://archives.math.utk.edu/visual.calculus/

Please suggest any resources and thank you in advance.

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3 Answers 3

Daniel Solow has an excellent book on proofs that my undergraduate mathematics program used.

Solow, Daniel. How to Read and Do Proofs: An Introduction to Mathematical Thought Processes. Hoboken, NJ: Wiley, 2009. Print.

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For induction I highly recommend this book: http://www.amazon.com/Handbook-Mathematical-Induction-Applications-Mathematics/dp/1420093649

It has many problems on induction, strong induction, and many variations of induction. Also the answers are in the back of the book.

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I always like to plug How To Prove It by Daniel Velleman when I talk about proof-writing, but it sounds like you're looking for specific topics, rather than a general proofs book. I don't know any websites, but here is my advice:

If you want to practice induction, the best thing to do is pick up a cheap graph theory book. The subject is entry level, so they are good about progressing in difficulty. Furthermore graph theoretic proofs also tend to be quite intuitive, yet notation heavy, so they are great practice for mathematical writing. Luckily pirates have uploaded instructor's solution manuals for most popular mathematics texts onto the internet so that students may more efficiently cheat at their coursework, so if you find a book with problems you like you should be able to get their solutions.

As a subject abstract algebra is full of ways to practice direct proofs, as every step requires you to pick apart the expressions you write and explain them from the definitions. Personally I started on Fraleigh, though most people will recommend Dummit and Foote (which I view as too advanced for a beginner).

I don't know of any subject that is particularly good for contrapositive proofs (other than ANY subject - it's my favorite proof method!). I'd say practice this by writing any direct proof you do "backwards" after you're done, until you're comfortable with the contrapositive.

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