Topic outline
- General
- Description of the module
Description of the module
Description of the module:
In this resource, we will know :Basic principles of writing a mathematical proof; Set theory; Propositional calculus and predicate calculus;
Well ordering and proof by recurrence.
Target audience:
Students of 2nd year bachelor maths
Essential prerequisites:
The basic vocabulary of Algebra1.
Objectives:
1/ Acquire the foundations of mathematical reasoning.
2/ Acquire the foundations of set theory.
3/ Acquire the elements of writing mathematical proofs.
Content :
Chapter 1: Introduction
Elements of mathematical language: axiom, lemma, theorem, conjecture.
Writing mathematical proofs: Basic principles of writing a mathematical proof. The expression ‘without loss of generality’. Constructive proof and existential proof.
Chapter 2: Set theory
Naive set theory. Set definition of Cartesian product. Sets of parts. Assemblistic definition of relations. Assemblistic definition of applications.
Russel's paradox. Other versions of Russel's paradox (Liar's paradox, Librarian's paradox, Cretan liar's paradox). Optional: Zermelo-Fraenkel theory.
Equipotence relation. Cardinality of sets. Cantor-Betnestein theorem. Countable set, power of 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 quantifiers, The single existence quantifier. Multiple quantifiers, Negation of a quantifier, Quantifiers and connectors.
Chapter 4: Well ordering principle and proof by recurrence
Recall proof by recurrence. Theorem of proof by recurrence.
Proof by strong recurrence. Example of the existence of a prime decomposition of a natural number. Optional (Proof by Cauchy recurrence. Proof of Cauchy Scwhartz inequality by recurrence).
Well ordering proof. Proof by the well ordering principle. Zermelo's general well ordering theorem.
- Chapter 1: Introduction
- Chapter 2: Set theory
- Chapter 3: Propositional calculus and predicate calculus
- Chapter 4: Well ordering principle and proof by recurrence
- Notes TD