\select@language {french}
\contentsline {part}{I\hspace {1em}Logique}{3}{part.1}
\contentsline {chapter}{\numberline {1}Alg\IeC {\`e}bre de Boole}{4}{chapter.1}
\contentsline {section}{\numberline {I}Propri\IeC {\'e}t\IeC {\'e}s g\IeC {\'e}n\IeC {\'e}rales}{4}{section.1.1}
\contentsline {section}{\numberline {II}R\IeC {\`e}gles de calcul dans une alg\IeC {\`e}bre de Boole}{5}{section.1.2}
\contentsline {section}{\numberline {III}Fonctions bool\IeC {\'e}ennes}{5}{section.1.3}
\contentsline {subsection}{\numberline {III.1}Formes canoniques d'une fonction bool\IeC {\'e}enne}{6}{subsection.1.3.1}
\contentsline {subsection}{\numberline {III.2}Obtention des formes canoniques}{7}{subsection.1.3.2}
\contentsline {section}{\numberline {IV}Diagrammes de Karnaugh}{7}{section.1.4}
\contentsline {chapter}{\numberline {2}Logique des pr\IeC {\'e}dicats}{11}{chapter.2}
\contentsline {section}{\numberline {I}Les propositions}{11}{section.2.1}
\contentsline {section}{\numberline {II}Les connecteurs logiques}{11}{section.2.2}
\contentsline {subsection}{\numberline {II.1}Tables de v\IeC {\'e}rit\IeC {\'e} des connecteurs logiques}{12}{subsection.2.2.1}
\contentsline {subsection}{\numberline {II.2}Variables et formules propositionnelles}{13}{subsection.2.2.2}
\contentsline {section}{\numberline {III}S\IeC {\'e}mantique du calcul propositionnel}{15}{section.2.3}
\contentsline {subsection}{\numberline {III.1}Fonctions de v\IeC {\'e}rit\IeC {\'e}}{16}{subsection.2.3.1}
\contentsline {subsection}{\numberline {III.2}Formules propositionnelles particuli\IeC {\`e}res}{16}{subsection.2.3.2}
\contentsline {subsubsection}{\numberline {III.2.1}Tautologies}{16}{subsubsection.2.3.2.1}
\contentsline {subsubsection}{\numberline {III.2.2}Antilogies}{17}{subsubsection.2.3.2.2}
\contentsline {subsection}{\numberline {III.3}Cons\IeC {\'e}quences logiques}{17}{subsection.2.3.3}
\contentsline {subsection}{\numberline {III.4}Formules \IeC {\'e}quivalentes}{18}{subsection.2.3.4}
\contentsline {subsection}{\numberline {III.5}Simplification du calcul des fonctions de v\IeC {\'e}rit\IeC {\'e}}{19}{subsection.2.3.5}
\contentsline {subsubsection}{\numberline {III.5.1}Th\IeC {\'e}or\IeC {\`e}me de substitution}{19}{subsubsection.2.3.5.1}
\contentsline {subsubsection}{\numberline {III.5.2}Th\IeC {\'e}or\IeC {\`e}me de la validit\IeC {\'e}}{19}{subsubsection.2.3.5.2}
\contentsline {subsection}{\numberline {III.6}Conclusion}{21}{subsection.2.3.6}
\contentsline {part}{II\hspace {1em}Th\IeC {\'e}orie des ensembles}{22}{part.2}
\contentsline {chapter}{\numberline {3}Introduction \IeC {\`a} la th\IeC {\'e}orie des ensembles}{23}{chapter.3}
\contentsline {section}{\numberline {I}Rappels de th\IeC {\'e}orie des ensembles}{23}{section.3.1}
\contentsline {subsection}{\numberline {I.1}Notion premi\IeC {\`e}re d'ensemble}{23}{subsection.3.1.1}
\contentsline {subsection}{\numberline {I.2}R\IeC {\`e}gles de fonctionnement}{23}{subsection.3.1.2}
\contentsline {paragraph}{Relation d'appartenance.}{23}{section*.2}
\contentsline {paragraph}{Objets distincts.}{23}{section*.3}
\contentsline {paragraph}{Ensemble vide.}{23}{section*.4}
\contentsline {paragraph}{Derni\IeC {\`e}re r\IeC {\`e}gle de fonctionnement des ensembles.}{23}{section*.5}
\contentsline {subsection}{\numberline {I.3}Sous-ensembles, ensemble des parties}{23}{subsection.3.1.3}
\contentsline {section}{\numberline {II}Op\IeC {\'e}rations sur les ensembles}{24}{section.3.2}
\contentsline {subsection}{\numberline {II.1}\'Egalite de deux ensembles}{24}{subsection.3.2.1}
\contentsline {subsection}{\numberline {II.2}R\IeC {\'e}union, intersection}{24}{subsection.3.2.2}
\contentsline {subsection}{\numberline {II.3}Compl\IeC {\'e}mentation}{25}{subsection.3.2.3}
\contentsline {subsection}{\numberline {II.4}Produit cart\IeC {\'e}sien}{25}{subsection.3.2.4}
\contentsline {section}{\numberline {III}Exercices suppl\IeC {\'e}mentaires}{26}{section.3.3}
\contentsline {chapter}{\numberline {4}Relations binaires entre ensembles}{27}{chapter.4}
\contentsline {section}{\numberline {I}Relations}{27}{section.4.1}
\contentsline {section}{\numberline {II}Relations d'ordre}{27}{section.4.2}
\contentsline {subsection}{\numberline {II.1}R\IeC {\'e}flexivit\IeC {\'e}, antisym\IeC {\'e}trie, transitivit\IeC {\'e}}{27}{subsection.4.2.1}
\contentsline {subsection}{\numberline {II.2}Relation d'ordre}{28}{subsection.4.2.2}
\contentsline {section}{\numberline {III}Relations d'\IeC {\'e}quivalence}{28}{section.4.3}
\contentsline {subsection}{\numberline {III.1}Classes d'\IeC {\'e}quivalence}{29}{subsection.4.3.1}
\contentsline {part}{III\hspace {1em}Arithm\IeC {\'e}tique}{30}{part.3}
\contentsline {chapter}{\numberline {5}Ensembles de nombres entiers}{31}{chapter.5}
\contentsline {section}{\numberline {I}Principe de r\IeC {\'e}currence }{31}{section.5.1}
\contentsline {section}{\numberline {II}Nombres premiers}{31}{section.5.2}
\contentsline {section}{\numberline {III}Division euclidienne dans ${\mathbb Z}$ et applications}{32}{section.5.3}
\contentsline {section}{\numberline {IV}Algorithmes d'Euclide}{33}{section.5.4}
\contentsline {subsection}{\numberline {IV.1}L'algorithme initial}{33}{subsection.5.4.1}
\contentsline {subsection}{\numberline {IV.2}Algorithme d'Euclide g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}}{35}{subsection.5.4.2}
\contentsline {subsection}{\numberline {IV.3}L'algorithme.}{35}{subsection.5.4.3}
\contentsline {subsection}{\numberline {IV.4}Exemple.}{35}{subsection.5.4.4}
\contentsline {section}{\numberline {V}Arithm\IeC {\'e}tique modulo $n$}{36}{section.5.5}
\contentsline {part}{IV\hspace {1em}Annexes}{39}{part.4}
\contentsline {chapter}{\numberline {6}Programme P\IeC {\'e}dagogique National 2005 (PPN)}{40}{chapter.6}
\contentsline {chapter}{Index}{41}{chapter.6}