Logique Mathématique

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  General

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  Propositional logic is a branch of mathematics that studies the logical relationships between
  propositions (or statements, sentences, assertions) taken as a whole, and connected via logical
  connectives. In this chapter, we will cover propositional logic and related topics in detail.
  Logic is the basis of all mathematical reasoning and all automated reasoning. The rules of logic
  specify the meaning of mathematical statements. These rules help us understand and reason
  with statements.
  The rules of logic give precise meaning to mathematical statements. These rules are used
  to distinguish between valid and invalid mathematical arguments. Apart from its importance
  in understanding mathematical reasoning, logic has numerous applications in Computer Science, varying from the design of digital circuits to the construction of computer programs and
  verication of the correctness of programs. 

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• Select topic Contents

  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.
• Select topic Logic and propositional calculus

  Logic and propositional calculus

  Propositional logic is a branch of mathematics that studies the logical relationships between
  propositions (or statements, sentences, assertions) taken as a whole, and connected via logical
  connectives. In this chapter, we will cover propositional logic and related topics in detail.
  Logic is the basis of all mathematical reasoning and all automated reasoning. The rules of logic
  specify the meaning of mathematical statements. These rules help us understand and reason
  with statements. 

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• Select topic Predicats calulus

  Predicats calulus

  A predicate or propositional function is a description of the property (or properties) a variable or subject may have. A proposition may be created from a propositional function by either assigning a value to the variable or by quantication. 

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• Select topic Types of proof

  Types of proof

  A proof is a sequence of logical statements, one implying another, which gives an explanation
  of why a given statement is true. It is based on a logical consequence of axioms, denitions,
  lemmas, theorems, etc.previously established. Mathematical proof is absolute, which means
  that once a theorem is proved, it is proved for ever. Until proven though, the statement is
  never accepted as a true one. 

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