Logic and set theory - proofs of theorems

Question

I am not sure if I am allowed to post here questions like this one, so write if I ma not.

So I have to learn to prove each of these things:

1. Tautologies (e.g. which implication is true, give a counterexample to the false one)
2. Theorem on superposition of functions, one-to-one functions.
3. Properties of images and inverse-images of functions.
4. Theorem about equivalence relation defined by partition.
5. Properties of equivalence classes.
6. Cantor's Theorem on the powerset.

Does anyone have a good source to learn it?

Answer 1

In my opinion, there is one book that fits perfectly your request (I mean the conjunction of all your requests 1. to 6.):

Daniel J. Velleman: How to Prove It: A Structured Approach (2nd Edition), Cambridge University Press, 2006.

This book, beautifully self-contained and with a lot of examples and exercises, is conceived to preparing students to make the transition from solving problems (the mathematics learned at school) to proving theorems, and teaches them the techniques needed to read and write proofs, exploring the subjects in your request (and even more) from scratch.

Disclaimer: I am not Velleman! It is just that I have borrowed many examples and exercises from this book in my introductory courses in mathematical logic and elementary set theory: they seem to be perfect to help students to think and deeply understand not trivial concepts in these disciplines.

Answer 2

As an alternative to Taroccoesbrocco's suggestion, I recommend the following:

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand, Polimeni, Zhang

In my opinion, this is the clearest textbook on proofs, moreso than Velleman. It contains everything besides number 6 on your list. The material is tediously explained, proofs are fully worked out, and there's a wealth of exercises of varying difficulty for your edification.