The structure of proof - with logic and set theory

There are many approaches that one can take with regard to a course dedicated to teaching proof writing. Some prefer to teach the mathematics and the structure of the proofs simultaneously.
Others choose to teach the methods of proof and then apply those methods to various topics. This is the strategy of this text.

Here we boldly jump into a discussion of logic and examine some details with the belief that if the details are not understood well, then the application of those details will suffer later.

Moreover, it is the understanding of how logic interacts with mathematics that empowers the student to have the courage and confidence to tackle greater problems in courses such as Abstract Algebra or Topology.
How we accomplish this is outlined below.

Contents

1- Logical foundations

• Propositional logic
• Predicates and proofs

2- Main topics

• Set theory
• Mathematical induction
• Number theory
• Relations and functions

3- Coming attractions

• Ring theory
• Topology

4- Appendices