Contents

Chapter 1: Introduction

Elements of the mathematical language: Axiom, lemma, theorem, conjecture.
Writing mathematical proofs: Basic principles of writing a mathematical proof.
Expression "Without loss of generality". Constructive proof and existential proof.

Chapter 2: Set theory

Naive set theory. Set definition of the Cartesian product. Sets of parts.
Set definition of relations. Set definition of applications.
Russel's paradox. Other versions of Russel's paradox (Liar paradox, Librarian paradox, Cretan liar paradox). Optional: Zermelo-Fraenkel theory. Equipotency relation. Cardinality of sets. Cantor-Betnestein theorem. Countable set, power of the continuum. Continuum hypothesis. Paul Cohen's theorem. Axiom of choice. Godel's Theorem.

Chapter 3: Propositional Calculus and Predicate Calculus

The logical proposition, conjunction, disjunction, implication, equivalence, negation. The
truth table. The logical formula, tautology, contradiction.
Rules of inference or deduction, Modus Ponens rule. Modus Tollens rule.
Predicate Calculus, Universal and existential quantifier, The quantifier of unique existence.
Multiple quantifiers, Negation of a quantifier, Quantifiers and connectors.
Note: It is important to approach logical implication in the context of classical mathematical
definitions. Thus, a good number of students think that the relation < in R is
not an antisymmetric relation.

Chapter 4: Well-ordering and proof by recurrence

Reminder of proof by recurrence. Theorem of proof by recurrence.
Proof by strong recurrence. Example of the existence of a prime decomposition of a
natural integer. Optional (Proof by Cauchy induction. Proof of the Cauchy-Schwhartz inequality
by induction). Well-founded order. Proof by the principle of well-ordering. Zermelo's general well-ordering theorem.