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Keeping with the free theme, Book Of Proof by Richard Hammack is a nice introduction to proofs and formalism. It's available free from the author as a PDF[1], and also as a physical book on Amazon[2].

An alternative if you're willing to spend a little is How to Prove It by Daniel J. Velleman, also available from Amazon[3] and probably many other retailers. Both books cover roughly the same topics.

[1]: http://www.people.vcu.edu/~rhammack/BookOfProof/

[2]: http://www.amazon.com/Book-Proof-Richard-Hammack/dp/09894721...

[3]: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/05...

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