Department of Mathematics

https://www.univ-soukahras.dz/en/dept/math

Lecturer	Amer MESBAHI
Information	Bachelor - Socle Commun Mathématiques et mathématiques appliquées L2 Department of Mathematics Website : https://www.univ-soukahras.dz/en/module/1202 Semester : S3 Unit : UEM312 Credit : 3 Coefficient: 2
Content	Chapter 1: Introduction Mathematical language elements: 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-theoretic definition of cartesian product. Power sets. Set-theoretic definition of relations. Set-theoretic definition of functions. Russell\'s paradox. Other versions of Russell\'s paradox (Liar paradox, Librarian paradox, Cretan liar paradox). Optional: Zermelo-Fraenkel theory. Equivalence relation. Cardinality of sets. Cantor-Bernstein theorem. Countable set, power of the continuum. Continuum hypothesis. Paul Cohen\'s theorem. Axiom of choice. Gödel\'s theorem. Chapter 3: Propositional Calculus and Predicate Calculus Logical proposition, conjunction, disjunction, implication, equivalence, negation. Truth table. Logical formula, tautology, contradiction. Rules of inference or deduction, Modus Ponens rule. Modus Tollens rule. Predicate calculus, Universal and existential quantifiers, Unique existence quantifier. Multiple quantifiers, Negation of a quantifier, Quantifiers and connectors. Note: It is important to address logical implication in the context of classical mathematical definitions. Thus, a good part of students think that the < relation in R is not an antisymmetric relation. Chapter 4: Well-ordering and Proof by Recurrence Reminder proof by recurrence. Theorem of proof by recurrence. Strong recurrence proof. Example of the existence of a prime factorization of a natural number. Optional (Cauchy\'s proof by recurrence. Proof of Cauchy-Schwarz inequality by recurrence). Well-founded order. Proof by the well-ordering principle. Zermelo\'s general well-ordering theorem.
Evaluation	Exam: 60% Directed activities: 40% Evaluation method: Mini-quiz: 12 points Participation: 04 points Discipline: 04 points