1. Sets, subsets, power-sets. Functions, informally. Cantor's theorem and Russell's paradox.
2. Ordered pairs and cartesian products, Functions, formally. Operations on sets (union, intersection), operations on functions (composition).
3. Relations, equivalence relations, partitions. Quotients and the integers.  Well-defined functions.
4. Modular arithmetic. Boolean operators, truth tables, quantifiers.
5. Proof, informally and formally. How to write them and how to read them.
6. Recursion and induction. The well-ordering principle.
7. Groups - definitions, examples, applications.
8. Group homomorphisms.  The symmetric group, the parity of a permutation.
9. Rings - definitions, examples, applications.
10. Fields - definitions, examples, applications.