X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/cours-maths-dis.git/blobdiff_plain/a0ee5c712b5f4b3e908e2d16da8ff24d414f5f1e..11feb823ac14af10bbfd2e14b897a46b1b3d8b15:/main13.toc

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 \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}{6}{section.1.3}
-\contentsline {subsection}{\numberline {III.1}Formes canoniques d'une fonction bool\IeC {\'e}enne}{7}{subsection.1.3.1}
-\contentsline {subsection}{\numberline {III.2}Obtention des formes canoniques}{8}{subsection.1.3.2}
-\contentsline {section}{\numberline {IV}Diagrammes de Karnaugh}{8}{section.1.4}
-\contentsline {chapter}{\numberline {2}Logique des pr\IeC {\'e}dicats}{12}{chapter.2}
-\contentsline {section}{\numberline {I}Les propositions}{12}{section.2.1}
-\contentsline {section}{\numberline {II}Les connecteurs logiques}{12}{section.2.2}
-\contentsline {subsection}{\numberline {II.1}Tables de v\IeC {\'e}rit\IeC {\'e} des connecteurs logiques}{13}{subsection.2.2.1}
-\contentsline {subsection}{\numberline {II.2}Variables et formules propositionnelles}{15}{subsection.2.2.2}
-\contentsline {section}{\numberline {III}S\IeC {\'e}mantique du calcul propositionnel}{17}{section.2.3}
-\contentsline {subsection}{\numberline {III.1}Fonctions de v\IeC {\'e}rit\IeC {\'e}}{18}{subsection.2.3.1}
-\contentsline {subsection}{\numberline {III.2}Formules propositionnelles particuli\IeC {\`e}res}{18}{subsection.2.3.2}
-\contentsline {subsubsection}{\numberline {III.2.1}Tautologies}{18}{subsubsection.2.3.2.1}
-\contentsline {subsubsection}{\numberline {III.2.2}Antilogies}{19}{subsubsection.2.3.2.2}
-\contentsline {subsection}{\numberline {III.3}Cons\IeC {\'e}quences logiques}{19}{subsection.2.3.3}
-\contentsline {subsection}{\numberline {III.4}Formules \IeC {\'e}quivalentes}{20}{subsection.2.3.4}
-\contentsline {subsection}{\numberline {III.5}Simplification du calcul des fonctions de v\IeC {\'e}rit\IeC {\'e}}{21}{subsection.2.3.5}
-\contentsline {subsubsection}{\numberline {III.5.1}Th\IeC {\'e}or\IeC {\`e}me de substitution}{21}{subsubsection.2.3.5.1}
-\contentsline {subsubsection}{\numberline {III.5.2}Th\IeC {\'e}or\IeC {\`e}me de la validit\IeC {\'e}}{22}{subsubsection.2.3.5.2}
-\contentsline {subsection}{\numberline {III.6}Conclusion}{24}{subsection.2.3.6}
-\contentsline {part}{II\hspace {1em}Th\IeC {\'e}orie des ensembles}{25}{part.2}
-\contentsline {chapter}{\numberline {3}Introduction \IeC {\`a} la th\IeC {\'e}orie des ensembles}{26}{chapter.3}
-\contentsline {section}{\numberline {I}Rappels de th\IeC {\'e}orie des ensembles}{26}{section.3.1}
-\contentsline {subsection}{\numberline {I.1}Notion premi\IeC {\`e}re d'ensemble}{26}{subsection.3.1.1}
-\contentsline {subsection}{\numberline {I.2}R\IeC {\`e}gles de fonctionnement}{26}{subsection.3.1.2}
-\contentsline {paragraph}{Relation d'appartenance.}{26}{section*.2}
-\contentsline {paragraph}{Objets distincts.}{26}{section*.3}
-\contentsline {paragraph}{Ensemble vide.}{26}{section*.4}
-\contentsline {paragraph}{Derni\IeC {\`e}re r\IeC {\`e}gle de fonctionnement des ensembles.}{26}{section*.5}
-\contentsline {subsection}{\numberline {I.3}Sous-ensembles, ensemble des parties}{26}{subsection.3.1.3}
-\contentsline {section}{\numberline {II}Op\IeC {\'e}rations sur les ensembles}{27}{section.3.2}
-\contentsline {subsection}{\numberline {II.1}\'Egalite de deux ensembles}{27}{subsection.3.2.1}
-\contentsline {subsection}{\numberline {II.2}R\IeC {\'e}union, intersection}{28}{subsection.3.2.2}
-\contentsline {subsection}{\numberline {II.3}Compl\IeC {\'e}mentation}{28}{subsection.3.2.3}
-\contentsline {subsection}{\numberline {II.4}Produit cart\IeC {\'e}sien}{29}{subsection.3.2.4}
-\contentsline {section}{\numberline {III}Exercices suppl\IeC {\'e}mentaires}{29}{section.3.3}
-\contentsline {chapter}{\numberline {4}Relations binaires entre ensembles}{30}{chapter.4}
-\contentsline {section}{\numberline {I}Relations}{30}{section.4.1}
-\contentsline {section}{\numberline {II}Relations d'ordre}{30}{section.4.2}
-\contentsline {subsection}{\numberline {II.1}R\IeC {\'e}flexivit\IeC {\'e}, antisym\IeC {\'e}trie, transitivit\IeC {\'e}}{30}{subsection.4.2.1}
-\contentsline {subsection}{\numberline {II.2}Relation d'ordre}{31}{subsection.4.2.2}
-\contentsline {section}{\numberline {III}Relations d'\IeC {\'e}quivalence}{32}{section.4.3}
-\contentsline {subsection}{\numberline {III.1}Classes d'\IeC {\'e}quivalence}{32}{subsection.4.3.1}
-\contentsline {subsection}{\numberline {III.2}Ensemble-quotient}{34}{subsection.4.3.2}
-\contentsline {part}{III\hspace {1em}Annexes}{35}{part.3}
-\contentsline {chapter}{\numberline {5}Programme P\IeC {\'e}dagogique National 2005 (PPN)}{36}{chapter.5}
-\contentsline {chapter}{Index}{37}{chapter.5}
+\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}{14}{section.2.3}
+\contentsline {subsection}{\numberline {III.1}Fonctions de v\IeC {\'e}rit\IeC {\'e}}{14}{subsection.2.3.1}
+\contentsline {subsection}{\numberline {III.2}Formules propositionnelles particuli\IeC {\`e}res}{14}{subsection.2.3.2}
+\contentsline {subsubsection}{\numberline {III.2.1}Tautologies}{15}{subsubsection.2.3.2.1}
+\contentsline {subsubsection}{\numberline {III.2.2}Antilogies}{15}{subsubsection.2.3.2.2}
+\contentsline {subsection}{\numberline {III.3}Cons\IeC {\'e}quences logiques}{15}{subsection.2.3.3}
+\contentsline {subsection}{\numberline {III.4}Formules \IeC {\'e}quivalentes}{16}{subsection.2.3.4}
+\contentsline {subsection}{\numberline {III.5}Simplification du calcul des fonctions de v\IeC {\'e}rit\IeC {\'e}}{17}{subsection.2.3.5}
+\contentsline {part}{II\hspace {1em}Th\IeC {\'e}orie des ensembles}{19}{part.2}
+\contentsline {chapter}{\numberline {3}Introduction \IeC {\`a} la th\IeC {\'e}orie des ensembles}{20}{chapter.3}
+\contentsline {section}{\numberline {I}Des D\IeC {\'e}finitions}{20}{section.3.1}
+\contentsline {subsection}{\numberline {I.1}Notion premi\IeC {\`e}re d'ensemble}{20}{subsection.3.1.1}
+\contentsline {subsection}{\numberline {I.2}R\IeC {\`e}gles de fonctionnement}{20}{subsection.3.1.2}
+\contentsline {paragraph}{Relation d'appartenance.}{20}{section*.2}
+\contentsline {paragraph}{Objets distincts.}{20}{section*.3}
+\contentsline {paragraph}{Ensemble vide.}{20}{section*.4}
+\contentsline {paragraph}{Derni\IeC {\`e}re r\IeC {\`e}gle de fonctionnement des ensembles.}{20}{section*.5}
+\contentsline {subsection}{\numberline {I.3}Sous-ensembles, ensemble des parties}{20}{subsection.3.1.3}
+\contentsline {section}{\numberline {II}Op\IeC {\'e}rations sur les ensembles}{21}{section.3.2}
+\contentsline {subsection}{\numberline {II.1}\'Egalite de deux ensembles}{21}{subsection.3.2.1}
+\contentsline {subsection}{\numberline {II.2}R\IeC {\'e}union, intersection}{21}{subsection.3.2.2}
+\contentsline {subsection}{\numberline {II.3}Compl\IeC {\'e}mentation}{22}{subsection.3.2.3}
+\contentsline {subsection}{\numberline {II.4}Produit cart\IeC {\'e}sien}{22}{subsection.3.2.4}
+\contentsline {section}{\numberline {III}Exercices suppl\IeC {\'e}mentaires}{23}{section.3.3}
+\contentsline {chapter}{\numberline {4}Relations binaires entre ensembles}{24}{chapter.4}
+\contentsline {section}{\numberline {I}Relations}{24}{section.4.1}
+\contentsline {section}{\numberline {II}Relations d'ordre}{24}{section.4.2}
+\contentsline {subsection}{\numberline {II.1}R\IeC {\'e}flexivit\IeC {\'e}, antisym\IeC {\'e}trie, transitivit\IeC {\'e}}{24}{subsection.4.2.1}
+\contentsline {subsection}{\numberline {II.2}Relation d'ordre}{25}{subsection.4.2.2}
+\contentsline {section}{\numberline {III}Relations d'\IeC {\'e}quivalence}{25}{section.4.3}
+\contentsline {subsection}{\numberline {III.1}Classes d'\IeC {\'e}quivalence}{26}{subsection.4.3.1}
+\contentsline {part}{III\hspace {1em}Arithm\IeC {\'e}tique}{27}{part.3}
+\contentsline {chapter}{\numberline {5}Ensembles de nombres entiers}{28}{chapter.5}
+\contentsline {section}{\numberline {I}Principe de r\IeC {\'e}currence }{28}{section.5.1}
+\contentsline {section}{\numberline {II}Nombres premiers}{28}{section.5.2}
+\contentsline {section}{\numberline {III}Division euclidienne dans ${\mathbb Z}$ et applications}{29}{section.5.3}
+\contentsline {section}{\numberline {IV}Algorithmes d'Euclide}{30}{section.5.4}
+\contentsline {subsection}{\numberline {IV.1}L'algorithme initial}{30}{subsection.5.4.1}
+\contentsline {subsection}{\numberline {IV.2}Algorithme d'Euclide g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}}{32}{subsection.5.4.2}
+\contentsline {subsection}{\numberline {IV.3}L'algorithme.}{32}{subsection.5.4.3}
+\contentsline {subsection}{\numberline {IV.4}Exemple.}{32}{subsection.5.4.4}
+\contentsline {section}{\numberline {V}Arithm\IeC {\'e}tique modulo $n$}{33}{section.5.5}
+\contentsline {part}{IV\hspace {1em}Annexes}{36}{part.4}
+\contentsline {chapter}{\numberline {6}Programme P\IeC {\'e}dagogique National 2005 (PPN)}{37}{chapter.6}
+\contentsline {chapter}{Index}{38}{chapter.6}