An Introduction to Mathematical Reasoning: Numbers, Sets and Functions

Part I. Mathematical Statements and Proofs: 1. The language of mathematics 2. Implications 3. Proofs 4. Proof by contradiction 5. The induction principle Part II. Sets and Functions: 6. The language of set theory 7. Quantifiers 8. Functions 9. Injections, surjections and bijections Part III. Numbers and Counting: 10. Counting 11. Properties of finite sets 12. Counting functions and subsets 13. Number systems 14. Counting infinite sets Part IV. Arithmetic: 15. The division theorem 16. The Euclidean algorithm 17. Consequences of the Euclidean algorithm 18. Linear diophantine equations Part V. Modular Arithmetic: 19. Congruences of integers 20. Linear congruences 21. Congruence classes and the arithmetic of remainders 22. Partitions and equivalence relations Part VI. Prime Numbers: 23. The sequence of prime numbers 24. Congruence modulo a prime Solutions to exercises.