X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/cours-maths-dis.git/blobdiff_plain/a0ee5c712b5f4b3e908e2d16da8ff24d414f5f1e..HEAD:/main13.aux diff --git a/main13.aux b/main13.aux index 2be045c..83da33a 100644 --- a/main13.aux +++ b/main13.aux @@ -1,8 +1,9 @@ \relax -\catcode`:\active -\catcode`;\active -\catcode`!\active -\catcode`?\active +\providecommand\hyper@newdestlabel[2]{} +\catcode `:\active +\catcode `;\active +\catcode `!\active +\catcode `?\active \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined \global\let\oldcontentsline\contentsline @@ -18,6 +19,7 @@ \global\let\hyper@last\relax \gdef\HyperFirstAtBeginDocument#1{#1} \providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} \select@language{french} \@writefile{toc}{\select@language{french}} \@writefile{lof}{\select@language{french}} @@ -28,270 +30,322 @@ \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} \@writefile{toc}{\contentsline {section}{\numberline {I}Propri\IeC {\'e}t\IeC {\'e}s g\IeC {\'e}n\IeC {\'e}rales}{4}{section.1.1}} -\@writefile{thm}{\contentsline {Def}{{Définition}{1.{1}}{Algèbre de Boole}}{4}{Def.1}} -\@writefile{thm}{\contentsline {Rem}{{Remarque}{1.{1}}{}}{4}{Rem.1}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{1}}{Somme disjonctive}}{4}{Exo.1}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{2}}{Opérateurs de Sheffer et de Peirce}}{4}{Exo.2}} +\@writefile{thm}{\contentsline {Def}{{Définition}{1.{1}}{Algèbre de Boole}}{4}{Def.1.1}} +\@writefile{thm}{\contentsline {Rem}{{Remarque}{1.{1}}{}}{4}{Rem.1.1}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{1}}{Somme disjonctive}}{4}{Exo.1.1}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{2}}{Opérateurs de Sheffer et de Peirce}}{4}{Exo.1.2}} \@writefile{toc}{\contentsline {section}{\numberline {II}R\IeC {\`e}gles de calcul dans une alg\IeC {\`e}bre de Boole}{5}{section.1.2}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{1.{1}}{Suppression de redondance}}{5}{Th.1}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{1.{1}}{Suppression de redondance}}{5}{Th.1.1}} \global\def\markiThi{\ensuremath {}} -\@writefile{thm}{\contentsline {Proof}{{Preuve}{1.{$\alpha $}}{}}{5}{Proof.1}} +\@writefile{thm}{\contentsline {Proof}{{Preuve}{1.{$\alpha $}}{}}{5}{Proof.1.1}} \global\def\markiProofi{\rule {1ex}{1ex}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{1.{3}}{}}{5}{Exo.3}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{4}}{}}{5}{Exo.4}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{5}}{Calcul booléen élémentaire}}{5}{Exo.5}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{6}}{Calcul booléen}}{5}{Exo.6}} -\@writefile{toc}{\contentsline {section}{\numberline {III}Fonctions bool\IeC {\'e}ennes}{6}{section.1.3}} -\@writefile{thm}{\contentsline {Def}{{Définition}{1.{2}}{Fonction booléenne}}{6}{Def.2}} -\global\def\markiiDefi{\ensuremath {\diamondsuit }} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{1.{7}}{}}{6}{Exo.7}} -\@writefile{thm}{\contentsline {Def}{{Définition}{1.{3}}{Aspect d'une variable}}{6}{Def.3}} -\@writefile{thm}{\contentsline {Def}{{Définition}{1.{4}}{Fonction booléenne nulle}}{6}{Def.4}} -\@writefile{thm}{\contentsline {Def}{{Définition}{1.{5}}{Fonction référentiel}}{6}{Def.5}} -\@writefile{thm}{\contentsline {Def}{{Définition}{1.{6}}{Minterme, maxterme}}{6}{Def.6}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{1.{8}}{Minterme à trois variables}}{6}{Exo.8}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{1.{9}}{Maxterme à trois variables}}{6}{Exo.9}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{10}}{}}{6}{Exo.10}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{11}}{}}{6}{Exo.11}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{1.{2}}{Nombre de mintermes et de maxtermes}}{6}{Th.2}} -\@writefile{toc}{\contentsline {subsection}{\numberline {III.1}Formes canoniques d'une fonction bool\IeC {\'e}enne}{7}{subsection.1.3.1}} -\@writefile{thm}{\contentsline {Def}{{Définition}{1.{7}}{Monômes}}{7}{Def.7}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{12}}{}}{7}{Exo.12}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{1.{3}}{}}{7}{Th.3}} -\@writefile{thm}{\contentsline {Proof}{{Preuve}{1.{$\beta $}}{}}{7}{Proof.2}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{1.{3}}{}}{5}{Exo.1.3}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{4}}{}}{5}{Exo.1.4}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{5}}{Calcul booléen élémentaire}}{5}{Exo.1.5}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{6}}{Calcul booléen}}{5}{Exo.1.6}} +\@writefile{toc}{\contentsline {section}{\numberline {III}Fonctions bool\IeC {\'e}ennes}{5}{section.1.3}} +\@writefile{thm}{\contentsline {Def}{{Définition}{1.{2}}{Fonction booléenne}}{5}{Def.1.2}} +\global\def\markiiDefi{\ensuremath {}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{1.{7}}{}}{6}{Exo.1.7}} +\@writefile{thm}{\contentsline {Def}{{Définition}{1.{3}}{Aspect d'une variable}}{6}{Def.1.3}} +\@writefile{thm}{\contentsline {Def}{{Définition}{1.{4}}{Fonction booléenne nulle}}{6}{Def.1.4}} +\@writefile{thm}{\contentsline {Def}{{Définition}{1.{5}}{Fonction référentiel}}{6}{Def.1.5}} +\@writefile{thm}{\contentsline {Def}{{Définition}{1.{6}}{Minterme, maxterme}}{6}{Def.1.6}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{1.{8}}{Minterme à trois variables}}{6}{Exo.1.8}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{1.{9}}{Maxterme à trois variables}}{6}{Exo.1.9}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{10}}{}}{6}{Exo.1.10}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{11}}{}}{6}{Exo.1.11}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{1.{2}}{Nombre de mintermes et de maxtermes}}{6}{Th.1.2}} +\@writefile{toc}{\contentsline {subsection}{\numberline {III.1}Formes canoniques d'une fonction bool\IeC {\'e}enne}{6}{subsection.1.3.1}} +\@writefile{thm}{\contentsline {Def}{{Définition}{1.{7}}{Monômes}}{6}{Def.1.7}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{12}}{}}{6}{Exo.1.12}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{1.{3}}{}}{6}{Th.1.3}} +\@writefile{thm}{\contentsline {Proof}{{Preuve}{1.{$\beta $}}{}}{6}{Proof.1.2}} \global\def\markiiProofi{\rule {1ex}{1ex}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{1.{4}}{}}{7}{Th.4}} -\@writefile{thm}{\contentsline {Proof}{{Preuve}{1.{$\gamma $}}{}}{7}{Proof.3}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{1.{5}}{Forme canonique disjonctive}}{7}{Th.5}} -\@writefile{thm}{\contentsline {Rem}{{Remarque}{1.{2}}{}}{7}{Rem.2}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{1.{6}}{Forme canonique conjonctive}}{8}{Th.6}} -\@writefile{toc}{\contentsline {subsection}{\numberline {III.2}Obtention des formes canoniques}{8}{subsection.1.3.2}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{1.{13}}{}}{8}{Exo.13}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{1.{14}}{}}{8}{Exo.14}} -\@writefile{thm}{\contentsline {Rem}{{Remarque}{1.{3}}{}}{8}{Rem.3}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{15}}{}}{8}{Exo.15}} -\@writefile{toc}{\contentsline {section}{\numberline {IV}Diagrammes de Karnaugh}{8}{section.1.4}} -\@writefile{thm}{\contentsline {Exoc}{{Exercice (corrigé)}{1.{16}}{}}{9}{Exo.16}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{17}}{Fonctions booléennes}}{10}{Exo.17}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{18}}{}}{10}{Exo.18}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{19}}{Application de la méthode de Karnaugh}}{10}{Exo.19}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{20}}{Composition de la méthode de Karnaugh}}{10}{Exo.20}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{21}}{BTS-2009}}{10}{Exo.21}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{22}}{BTS-2002}}{10}{Exo.22}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{1.{4}}{}}{6}{Th.1.4}} +\@writefile{thm}{\contentsline {Proof}{{Preuve}{1.{$\gamma $}}{}}{7}{Proof.1.3}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{1.{5}}{Forme canonique disjonctive}}{7}{Th.1.5}} +\@writefile{thm}{\contentsline {Rem}{{Remarque}{1.{2}}{}}{7}{Rem.1.2}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{1.{6}}{Forme canonique conjonctive}}{7}{Th.1.6}} +\@writefile{toc}{\contentsline {subsection}{\numberline {III.2}Obtention des formes canoniques}{7}{subsection.1.3.2}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{1.{13}}{}}{7}{Exo.1.13}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{1.{14}}{}}{7}{Exo.1.14}} +\@writefile{thm}{\contentsline {Rem}{{Remarque}{1.{3}}{}}{7}{Rem.1.3}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{15}}{}}{7}{Exo.1.15}} +\@writefile{toc}{\contentsline {section}{\numberline {IV}Diagrammes de Karnaugh}{7}{section.1.4}} +\@writefile{thm}{\contentsline {Exoc}{{Exercice (corrigé)}{1.{16}}{}}{8}{Exo.1.16}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{17}}{Fonctions booléennes}}{9}{Exo.1.17}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{18}}{}}{9}{Exo.1.18}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{19}}{Application de la méthode de Karnaugh}}{9}{Exo.1.19}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{20}}{Composition de la méthode de Karnaugh}}{9}{Exo.1.20}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{21}}{BTS-2009}}{9}{Exo.1.21}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{1.{22}}{BTS-2002}}{9}{Exo.1.22}} \citation{Dowek07} -\@writefile{toc}{\contentsline {chapter}{\numberline {2}Logique des pr\IeC {\'e}dicats}{12}{chapter.2}} +\@writefile{toc}{\contentsline {chapter}{\numberline {2}Logique des pr\IeC {\'e}dicats}{11}{chapter.2}} \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} -\@writefile{toc}{\contentsline {section}{\numberline {I}Les propositions}{12}{section.2.1}} -\newlabel{sub:prop:prop}{{I}{12}{Les propositions\relax }{section.2.1}{}} -\@writefile{thm}{\contentsline {Def}{{Définition}{2.{1}}{Proposition}}{12}{Def.1}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{1}}{}}{12}{Exo.1}} -\@writefile{toc}{\contentsline {section}{\numberline {II}Les connecteurs logiques}{12}{section.2.2}} -\newlabel{prop:sub:cnx}{{II}{12}{Les connecteurs logiques\relax }{section.2.2}{}} -\@writefile{toc}{\contentsline {subsection}{\numberline {II.1}Tables de v\IeC {\'e}rit\IeC {\'e} des connecteurs logiques}{13}{subsection.2.2.1}} -\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{1}}{}}{13}{Rem.1}} +\@writefile{toc}{\contentsline {section}{\numberline {I}Les propositions}{11}{section.2.1}} +\newlabel{sub:prop:prop}{{I}{11}{Les propositions}{section.2.1}{}} +\@writefile{thm}{\contentsline {Def}{{Définition}{2.{1}}{Proposition}}{11}{Def.2.1}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{1}}{}}{11}{Exo.2.1}} +\@writefile{toc}{\contentsline {section}{\numberline {II}Les connecteurs logiques}{11}{section.2.2}} +\newlabel{prop:sub:cnx}{{II}{11}{Les connecteurs logiques}{section.2.2}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {II.1}Tables de v\IeC {\'e}rit\IeC {\'e} des connecteurs logiques}{12}{subsection.2.2.1}} +\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{1}}{}}{12}{Rem.2.1}} \global\def\markivRemii{\ensuremath {}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{2}}{}}{14}{Exo.2}} -\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{2}}{}}{14}{Rem.2}} -\@writefile{thm}{\contentsline {Exoc}{{Exercice (corrigé)}{2.{3}}{}}{14}{Exo.3}} -\newlabel{it:x1}{{1}{14}{Tables de vérité des connecteurs logiques\relax }{Item.48}{}} -\newlabel{it:x2}{{2}{14}{Tables de vérité des connecteurs logiques\relax }{Item.49}{}} -\newlabel{it:x3}{{3}{14}{Tables de vérité des connecteurs logiques\relax }{Item.50}{}} -\newlabel{it:x4}{{4}{14}{Tables de vérité des connecteurs logiques\relax }{Item.51}{}} -\newlabel{it:x5}{{5}{14}{Tables de vérité des connecteurs logiques\relax }{Item.52}{}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{4}}{}}{14}{Exo.4}} -\newlabel{ex2:1}{{1}{14}{Tables de vérité des connecteurs logiques\relax }{Item.53}{}} -\newlabel{ex2:2}{{2}{14}{Tables de vérité des connecteurs logiques\relax }{Item.54}{}} -\newlabel{ex2:3}{{3}{14}{Tables de vérité des connecteurs logiques\relax }{Item.55}{}} -\newlabel{ex2:4}{{4}{14}{Tables de vérité des connecteurs logiques\relax }{Item.56}{}} -\newlabel{ex2:5}{{5}{14}{Tables de vérité des connecteurs logiques\relax }{Item.57}{}} -\newlabel{ex2:6}{{6}{14}{Tables de vérité des connecteurs logiques\relax }{Item.58}{}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{5}}{}}{14}{Exo.5}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{6}}{}}{15}{Exo.6}} -\@writefile{toc}{\contentsline {subsection}{\numberline {II.2}Variables et formules propositionnelles}{15}{subsection.2.2.2}} -\newlabel{prop:sub:vars}{{II.2}{15}{Variables et formules propositionnelles\relax }{subsection.2.2.2}{}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{1}}{}}{15}{Th.1}} -\global\def\markviiTh{\ensuremath {}} -\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{3}}{}}{15}{Rem.3}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{7}}{}}{15}{Exo.7}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{8}}{}}{16}{Exo.8}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{9}}{}}{16}{Exo.9}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{2}}{Règles de priorité des connecteurs logiques}}{16}{Th.2}} -\global\def\markviiiThi{\ensuremath {}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{10}}{}}{16}{Exo.10}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{3}}{Associativité des opérateurs $\ou $ et $\et $}}{16}{Th.3}} -\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{4}}{}}{16}{Rem.4}} -\@writefile{thm}{\contentsline {Exoc}{{Exercice (corrigé)}{2.{11}}{}}{16}{Exo.11}} -\@writefile{thm}{\contentsline {Exoc}{{Exercice (corrigé)}{2.{12}}{}}{17}{Exo.12}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{13}}{}}{17}{Exo.13}} -\@writefile{toc}{\contentsline {section}{\numberline {III}S\IeC {\'e}mantique du calcul propositionnel}{17}{section.2.3}} -\@writefile{toc}{\contentsline {subsection}{\numberline {III.1}Fonctions de v\IeC {\'e}rit\IeC {\'e}}{18}{subsection.2.3.1}} -\@writefile{thm}{\contentsline {Def}{{Définition}{2.{2}}{Fonction de vérité de $F$}}{18}{Def.2}} -\newlabel{item:eqv}{{6}{18}{Fonctions de vérité\relax }{Item.106}{}} -\global\def\markixDef{\ensuremath {\diamondsuit }} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{14}}{}}{18}{Exo.14}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{2}}{}}{12}{Exo.2.2}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{3}}{}}{12}{Exo.2.3}} +\newlabel{it:x1}{{1}{12}{Tables de vérité des connecteurs logiques}{Item.44}{}} +\newlabel{it:x2}{{2}{12}{Tables de vérité des connecteurs logiques}{Item.45}{}} +\newlabel{it:x3}{{3}{12}{Tables de vérité des connecteurs logiques}{Item.46}{}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{4}}{}}{12}{Exo.2.4}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{1}}{}}{12}{Th.2.1}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{5}}{}}{12}{Exo.2.5}} +\@writefile{toc}{\contentsline {subsection}{\numberline {II.2}Variables et formules propositionnelles}{13}{subsection.2.2.2}} +\newlabel{prop:sub:vars}{{II.2}{13}{Variables et formules propositionnelles}{subsection.2.2.2}{}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{2}}{}}{13}{Th.2.2}} +\global\def\markviiiTh{\ensuremath {}} +\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{2}}{}}{13}{Rem.2.2}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{6}}{}}{13}{Exo.2.6}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{7}}{}}{13}{Exo.2.7}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{8}}{}}{13}{Exo.2.8}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{3}}{Règles de priorité des connecteurs logiques}}{13}{Th.2.3}} +\global\def\markixThi{\ensuremath {}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{9}}{}}{14}{Exo.2.9}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{4}}{Associativité des opérateurs $\ou $ et $\et $}}{14}{Th.2.4}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{10}}{}}{14}{Exo.2.10}} +\@writefile{toc}{\contentsline {section}{\numberline {III}S\IeC {\'e}mantique du calcul propositionnel}{14}{section.2.3}} +\@writefile{toc}{\contentsline {subsection}{\numberline {III.1}Fonctions de v\IeC {\'e}rit\IeC {\'e}}{14}{subsection.2.3.1}} +\@writefile{thm}{\contentsline {Def}{{Définition}{2.{2}}{Fonction de vérité de $F$}}{14}{Def.2.2}} +\newlabel{item:eqv}{{6}{14}{Fonctions de vérité}{Item.79}{}} +\global\def\markixDef{\ensuremath {}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{11}}{}}{14}{Exo.2.11}} \global\def\markixEx{\ensuremath {}} -\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{5}}{}}{18}{Rem.5}} -\@writefile{toc}{\contentsline {subsection}{\numberline {III.2}Formules propositionnelles particuli\IeC {\`e}res}{18}{subsection.2.3.2}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {III.2.1}Tautologies}{18}{subsubsection.2.3.2.1}} -\@writefile{thm}{\contentsline {Def}{{Définition}{2.{3}}{Tautologie}}{18}{Def.3}} -\@writefile{thm}{\contentsline {Notation}{{Notation}{2.{1}}{}}{18}{Notation.1}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{15}}{}}{18}{Exo.15}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{16}}{}}{18}{Exo.16}} +\@writefile{toc}{\contentsline {subsection}{\numberline {III.2}Formules propositionnelles particuli\IeC {\`e}res}{14}{subsection.2.3.2}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {III.2.1}Tautologies}{15}{subsubsection.2.3.2.1}} +\@writefile{thm}{\contentsline {Def}{{Définition}{2.{3}}{Tautologie}}{15}{Def.2.3}} +\@writefile{thm}{\contentsline {Notation}{{Notation}{2.{1}}{}}{15}{Notation.2.1}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{12}}{}}{15}{Exo.2.12}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{13}}{}}{15}{Exo.2.13}} \global\def\markxiEx{\ensuremath {}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{17}}{}}{19}{Exo.17}} -\newlabel{item:taut:5}{{1}{19}{Tautologies\relax }{Item.107}{}} -\newlabel{item:taut:7}{{3}{19}{Tautologies\relax }{Item.109}{}} -\newlabel{item:taut:8}{{4}{19}{Tautologies\relax }{Item.110}{}} -\newlabel{item:taut:6}{{6}{19}{Tautologies\relax }{Item.112}{}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {III.2.2}Antilogies}{19}{subsubsection.2.3.2.2}} -\@writefile{thm}{\contentsline {Def}{{Définition}{2.{4}}{Antilogie}}{19}{Def.4}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{18}}{}}{19}{Exo.18}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{19}}{}}{19}{Exo.19}} -\@writefile{toc}{\contentsline {subsection}{\numberline {III.3}Cons\IeC {\'e}quences logiques}{19}{subsection.2.3.3}} -\@writefile{thm}{\contentsline {Def}{{Définition}{2.{5}}{Conséquence logique}}{19}{Def.5}} -\@writefile{thm}{\contentsline {Notation}{{Notation}{2.{2}}{}}{19}{Notation.2}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{20}}{}}{19}{Exo.20}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{14}}{}}{15}{Exo.2.14}} +\newlabel{item:taut:5}{{1}{15}{Tautologies}{Item.80}{}} +\newlabel{item:taut:7}{{3}{15}{Tautologies}{Item.82}{}} +\newlabel{item:taut:8}{{4}{15}{Tautologies}{Item.83}{}} +\newlabel{item:taut:6}{{6}{15}{Tautologies}{Item.85}{}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {III.2.2}Antilogies}{15}{subsubsection.2.3.2.2}} +\@writefile{thm}{\contentsline {Def}{{Définition}{2.{4}}{Antilogie}}{15}{Def.2.4}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{15}}{}}{15}{Exo.2.15}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{16}}{}}{15}{Exo.2.16}} +\@writefile{toc}{\contentsline {subsection}{\numberline {III.3}Cons\IeC {\'e}quences logiques}{15}{subsection.2.3.3}} +\@writefile{thm}{\contentsline {Def}{{Définition}{2.{5}}{Conséquence logique}}{15}{Def.2.5}} +\@writefile{thm}{\contentsline {Notation}{{Notation}{2.{2}}{}}{15}{Notation.2.2}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{17}}{}}{16}{Exo.2.17}} \global\def\markxiiiExi{\ensuremath {}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{21}}{}}{20}{Exo.21}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{22}}{}}{20}{Exo.22}} -\newlabel{item:cons:1}{{1}{20}{Conséquences logiques\relax }{Item.118}{}} -\newlabel{item:cons:2}{{2}{20}{Conséquences logiques\relax }{Item.119}{}} -\newlabel{item:cons:3}{{3}{20}{Conséquences logiques\relax }{Item.120}{}} -\newlabel{item:cons:4}{{4}{20}{Conséquences logiques\relax }{Item.121}{}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{23}}{}}{20}{Exo.23}} -\@writefile{toc}{\contentsline {subsection}{\numberline {III.4}Formules \IeC {\'e}quivalentes}{20}{subsection.2.3.4}} -\@writefile{thm}{\contentsline {Def}{{Définition}{2.{6}}{Formules équivalentes}}{20}{Def.6}} -\global\def\markxiiiDef{\ensuremath {\diamondsuit }} -\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{6}}{}}{20}{Rem.6}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{24}}{}}{20}{Exo.24}} -\@writefile{thm}{\contentsline {Exoc}{{Exercice (corrigé)}{2.{25}}{}}{21}{Exo.25}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{26}}{}}{21}{Exo.26}} -\@writefile{toc}{\contentsline {subsection}{\numberline {III.5}Simplification du calcul des fonctions de v\IeC {\'e}rit\IeC {\'e}}{21}{subsection.2.3.5}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {III.5.1}Th\IeC {\'e}or\IeC {\`e}me de substitution}{21}{subsubsection.2.3.5.1}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{4}}{Théorème de substitution}}{21}{Th.4}} -\@writefile{thm}{\contentsline {Proof}{{Preuve}{2.{$\alpha $}}{}}{21}{Proof.1}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{27}}{}}{21}{Exo.27}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{28}}{}}{22}{Exo.28}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {III.5.2}Th\IeC {\'e}or\IeC {\`e}me de la validit\IeC {\'e}}{22}{subsubsection.2.3.5.2}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{5}}{Théorème de la validité}}{22}{Th.5}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{18}}{}}{16}{Exo.2.18}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{19}}{}}{16}{Exo.2.19}} +\newlabel{item:cons:1}{{1}{16}{Conséquences logiques}{Item.91}{}} +\newlabel{item:cons:2}{{2}{16}{Conséquences logiques}{Item.92}{}} +\newlabel{item:cons:3}{{3}{16}{Conséquences logiques}{Item.93}{}} +\newlabel{item:cons:4}{{4}{16}{Conséquences logiques}{Item.94}{}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{20}}{}}{16}{Exo.2.20}} +\@writefile{toc}{\contentsline {subsection}{\numberline {III.4}Formules \IeC {\'e}quivalentes}{16}{subsection.2.3.4}} +\@writefile{thm}{\contentsline {Def}{{Définition}{2.{6}}{Formules équivalentes}}{16}{Def.2.6}} +\global\def\markxiiiDef{\ensuremath {}} +\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{3}}{}}{16}{Rem.2.3}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{21}}{}}{16}{Exo.2.21}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{22}}{}}{17}{Exo.2.22}} +\@writefile{toc}{\contentsline {subsection}{\numberline {III.5}Simplification du calcul des fonctions de v\IeC {\'e}rit\IeC {\'e}}{17}{subsection.2.3.5}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{5}}{Théorème de la validité}}{17}{Th.2.5}} \global\def\markxiTh{\ensuremath {}} -\@writefile{thm}{\contentsline {Proof}{{Preuve}{2.{$\beta $}}{}}{22}{Proof.2}} -\global\def\markvProofii{\rule {1ex}{1ex}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{29}}{Exemple d'application}}{22}{Exo.29}} -\global\def\markxviEx{\ensuremath {}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{30}}{}}{23}{Exo.30}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{31}}{}}{23}{Exo.31}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{32}}{}}{23}{Exo.32}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{33}}{}}{23}{Exo.33}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{34}}{}}{23}{Exo.34}} -\@writefile{toc}{\contentsline {subsection}{\numberline {III.6}Conclusion}{24}{subsection.2.3.6}} -\@writefile{toc}{\contentsline {part}{II\hspace {1em}Th\IeC {\'e}orie des ensembles}{25}{part.2}} -\@writefile{toc}{\contentsline {chapter}{\numberline {3}Introduction \IeC {\`a} la th\IeC {\'e}orie des ensembles}{26}{chapter.3}} +\@writefile{thm}{\contentsline {Proof}{{Preuve}{2.{$\alpha $}}{}}{17}{Proof.2.1}} +\global\def\markivProofii{\rule {1ex}{1ex}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{23}}{Exemple d'application}}{17}{Exo.2.23}} +\global\def\markxivEx{\ensuremath {}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{24}}{}}{18}{Exo.2.24}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{25}}{}}{18}{Exo.2.25}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{26}}{}}{18}{Exo.2.26}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{27}}{}}{18}{Exo.2.27}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{28}}{}}{18}{Exo.2.28}} +\@writefile{toc}{\contentsline {part}{II\hspace {1em}Th\IeC {\'e}orie des ensembles}{19}{part.2}} +\@writefile{toc}{\contentsline {chapter}{\numberline {3}Introduction \IeC {\`a} la th\IeC {\'e}orie des ensembles}{20}{chapter.3}} \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} -\@writefile{toc}{\contentsline {section}{\numberline {I}Rappels de th\IeC {\'e}orie des ensembles}{26}{section.3.1}} -\@writefile{toc}{\contentsline {subsection}{\numberline {I.1}Notion premi\IeC {\`e}re d'ensemble}{26}{subsection.3.1.1}} -\@writefile{thm}{\contentsline {Notation}{{Notation}{3.{1}}{}}{26}{Notation.1}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{1}}{}}{26}{Exo.1}} -\@writefile{toc}{\contentsline {subsection}{\numberline {I.2}R\IeC {\`e}gles de fonctionnement}{26}{subsection.3.1.2}} -\@writefile{toc}{\contentsline {paragraph}{Relation d'appartenance.}{26}{section*.2}} -\@writefile{toc}{\contentsline {paragraph}{Objets distincts.}{26}{section*.3}} -\@writefile{toc}{\contentsline {paragraph}{Ensemble vide.}{26}{section*.4}} -\@writefile{toc}{\contentsline {paragraph}{Derni\IeC {\`e}re r\IeC {\`e}gle de fonctionnement des ensembles.}{26}{section*.5}} -\@writefile{toc}{\contentsline {subsection}{\numberline {I.3}Sous-ensembles, ensemble des parties}{26}{subsection.3.1.3}} -\@writefile{thm}{\contentsline {Def}{{Définition}{3.{1}}{}}{26}{Def.1}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{1}}{}}{27}{Th.1}} -\@writefile{thm}{\contentsline {Proof}{{Preuve}{3.{$\alpha $}}{}}{27}{Proof.1}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{2}}{}}{27}{Th.2}} -\@writefile{thm}{\contentsline {Def}{{Définition}{3.{2}}{}}{27}{Def.2}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{3}}{}}{27}{Th.3}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{3.{2}}{}}{27}{Exo.2}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{3}}{}}{27}{Exo.3}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{4}}{}}{27}{Exo.4}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{5}}{}}{27}{Exo.5}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{6}}{}}{27}{Exo.6}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{7}}{}}{27}{Exo.7}} -\@writefile{toc}{\contentsline {section}{\numberline {II}Op\IeC {\'e}rations sur les ensembles}{27}{section.3.2}} -\@writefile{toc}{\contentsline {subsection}{\numberline {II.1}\'Egalite de deux ensembles}{27}{subsection.3.2.1}} -\@writefile{thm}{\contentsline {Def}{{Définition}{3.{3}}{}}{27}{Def.3}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{8}}{}}{27}{Exo.8}} -\@writefile{toc}{\contentsline {subsection}{\numberline {II.2}R\IeC {\'e}union, intersection}{28}{subsection.3.2.2}} -\@writefile{thm}{\contentsline {Def}{{Définition}{3.{4}}{Reunion}}{28}{Def.4}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{3.{9}}{}}{28}{Exo.9}} -\@writefile{thm}{\contentsline {Def}{{Définition}{3.{5}}{Intersection}}{28}{Def.5}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{4}}{Propriétés de la réunion et de l'intersection}}{28}{Th.4}} +\@writefile{toc}{\contentsline {section}{\numberline {I}Des D\IeC {\'e}finitions}{20}{section.3.1}} +\@writefile{toc}{\contentsline {subsection}{\numberline {I.1}Notion premi\IeC {\`e}re d'ensemble}{20}{subsection.3.1.1}} +\@writefile{thm}{\contentsline {Notation}{{Notation}{3.{1}}{}}{20}{Notation.3.1}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{1}}{}}{20}{Exo.3.1}} +\@writefile{toc}{\contentsline {subsection}{\numberline {I.2}R\IeC {\`e}gles de fonctionnement}{20}{subsection.3.1.2}} +\@writefile{toc}{\contentsline {paragraph}{Relation d'appartenance.}{20}{section*.2}} +\@writefile{toc}{\contentsline {paragraph}{Objets distincts.}{20}{section*.3}} +\@writefile{toc}{\contentsline {paragraph}{Ensemble vide.}{20}{section*.4}} +\@writefile{toc}{\contentsline {paragraph}{Derni\IeC {\`e}re r\IeC {\`e}gle de fonctionnement des ensembles.}{20}{section*.5}} +\@writefile{toc}{\contentsline {subsection}{\numberline {I.3}Sous-ensembles, ensemble des parties}{20}{subsection.3.1.3}} +\@writefile{thm}{\contentsline {Def}{{Définition}{3.{1}}{}}{20}{Def.3.1}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{1}}{}}{20}{Th.3.1}} +\@writefile{thm}{\contentsline {Proof}{{Preuve}{3.{$\alpha $}}{}}{21}{Proof.3.1}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{2}}{}}{21}{Th.3.2}} +\@writefile{thm}{\contentsline {Def}{{Définition}{3.{2}}{}}{21}{Def.3.2}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{3}}{}}{21}{Th.3.3}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{3.{2}}{}}{21}{Exo.3.2}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{3}}{}}{21}{Exo.3.3}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{4}}{}}{21}{Exo.3.4}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{5}}{}}{21}{Exo.3.5}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{6}}{}}{21}{Exo.3.6}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{7}}{}}{21}{Exo.3.7}} +\@writefile{toc}{\contentsline {section}{\numberline {II}Op\IeC {\'e}rations sur les ensembles}{21}{section.3.2}} +\@writefile{toc}{\contentsline {subsection}{\numberline {II.1}\'Egalite de deux ensembles}{21}{subsection.3.2.1}} +\@writefile{thm}{\contentsline {Def}{{Définition}{3.{3}}{}}{21}{Def.3.3}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{8}}{}}{21}{Exo.3.8}} +\@writefile{toc}{\contentsline {subsection}{\numberline {II.2}R\IeC {\'e}union, intersection}{21}{subsection.3.2.2}} +\@writefile{thm}{\contentsline {Def}{{Définition}{3.{4}}{Reunion}}{21}{Def.3.4}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{3.{9}}{}}{21}{Exo.3.9}} +\@writefile{thm}{\contentsline {Def}{{Définition}{3.{5}}{Intersection}}{21}{Def.3.5}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{4}}{Propriétés de la réunion et de l'intersection}}{22}{Th.3.4}} \global\def\markxvThi{\ensuremath {}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{10}}{}}{28}{Exo.10}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{11}}{}}{28}{Exo.11}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{5}}{Distributivités de $\cup $ et $\cap $}}{28}{Th.5}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{10}}{}}{22}{Exo.3.10}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{11}}{}}{22}{Exo.3.11}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{5}}{Distributivités de $\cup $ et $\cap $}}{22}{Th.3.5}} \global\def\markxviThi{\ensuremath {}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{12}}{}}{28}{Exo.12}} -\@writefile{toc}{\contentsline {subsection}{\numberline {II.3}Compl\IeC {\'e}mentation}{28}{subsection.3.2.3}} -\@writefile{thm}{\contentsline {Def}{{Définition}{3.{6}}{Complémentation}}{28}{Def.6}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{6}}{}}{28}{Th.6}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{12}}{}}{22}{Exo.3.12}} +\@writefile{toc}{\contentsline {subsection}{\numberline {II.3}Compl\IeC {\'e}mentation}{22}{subsection.3.2.3}} +\@writefile{thm}{\contentsline {Def}{{Définition}{3.{6}}{Complémentation}}{22}{Def.3.6}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{6}}{}}{22}{Th.3.6}} \global\def\markxviiThi{\ensuremath {}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{13}}{}}{29}{Exo.13}} -\@writefile{toc}{\contentsline {subsection}{\numberline {II.4}Produit cart\IeC {\'e}sien}{29}{subsection.3.2.4}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{14}}{}}{29}{Exo.14}} -\@writefile{toc}{\contentsline {section}{\numberline {III}Exercices suppl\IeC {\'e}mentaires}{29}{section.3.3}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{15}}{}}{29}{Exo.15}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{16}}{}}{29}{Exo.16}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{17}}{Fonction caractéristique des parties d'un ensemble}}{29}{Exo.17}} -\@writefile{toc}{\contentsline {chapter}{\numberline {4}Relations binaires entre ensembles}{30}{chapter.4}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{13}}{}}{22}{Exo.3.13}} +\@writefile{toc}{\contentsline {subsection}{\numberline {II.4}Produit cart\IeC {\'e}sien}{22}{subsection.3.2.4}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{14}}{}}{22}{Exo.3.14}} +\@writefile{toc}{\contentsline {section}{\numberline {III}Exercices suppl\IeC {\'e}mentaires}{23}{section.3.3}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{15}}{}}{23}{Exo.3.15}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{16}}{}}{23}{Exo.3.16}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{17}}{Fonction caractéristique des parties d'un ensemble}}{23}{Exo.3.17}} +\@writefile{toc}{\contentsline {chapter}{\numberline {4}Relations binaires entre ensembles}{24}{chapter.4}} \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} -\@writefile{toc}{\contentsline {section}{\numberline {I}Relations}{30}{section.4.1}} -\@writefile{thm}{\contentsline {Def}{{Définition}{4.{1}}{Relation binaire}}{30}{Def.1}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{1}}{}}{30}{Exo.1}} -\@writefile{thm}{\contentsline {Rem}{{Remarque}{4.{1}}{}}{30}{Rem.1}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{2}}{}}{30}{Exo.2}} -\@writefile{toc}{\contentsline {section}{\numberline {II}Relations d'ordre}{30}{section.4.2}} -\@writefile{toc}{\contentsline {subsection}{\numberline {II.1}R\IeC {\'e}flexivit\IeC {\'e}, antisym\IeC {\'e}trie, transitivit\IeC {\'e}}{30}{subsection.4.2.1}} -\@writefile{thm}{\contentsline {Def}{{Définition}{4.{2}}{Réflexivité}}{30}{Def.2}} -\@writefile{thm}{\contentsline {Def}{{Définition}{4.{3}}{Antisymétrie}}{30}{Def.3}} -\@writefile{thm}{\contentsline {Def}{{Définition}{4.{4}}{Transitivité}}{31}{Def.4}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{3}}{}}{31}{Exo.3}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{4}}{}}{31}{Exo.4}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{5}}{}}{31}{Exo.5}} -\@writefile{toc}{\contentsline {subsection}{\numberline {II.2}Relation d'ordre}{31}{subsection.4.2.2}} -\@writefile{thm}{\contentsline {Def}{{Définition}{4.{5}}{Relation d'ordre}}{31}{Def.5}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{6}}{}}{31}{Exo.6}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{7}}{Relation de divisibilité}}{31}{Exo.7}} +\@writefile{toc}{\contentsline {section}{\numberline {I}Relations}{24}{section.4.1}} +\@writefile{thm}{\contentsline {Def}{{Définition}{4.{1}}{Relation binaire}}{24}{Def.4.1}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{1}}{}}{24}{Exo.4.1}} +\@writefile{thm}{\contentsline {Rem}{{Remarque}{4.{1}}{}}{24}{Rem.4.1}} +\@writefile{toc}{\contentsline {section}{\numberline {II}Relations d'ordre}{24}{section.4.2}} +\@writefile{toc}{\contentsline {subsection}{\numberline {II.1}R\IeC {\'e}flexivit\IeC {\'e}, antisym\IeC {\'e}trie, transitivit\IeC {\'e}}{24}{subsection.4.2.1}} +\@writefile{thm}{\contentsline {Def}{{Définition}{4.{2}}{Réflexivité}}{24}{Def.4.2}} +\@writefile{thm}{\contentsline {Def}{{Définition}{4.{3}}{Antisymétrie}}{24}{Def.4.3}} +\@writefile{thm}{\contentsline {Def}{{Définition}{4.{4}}{Transitivité}}{24}{Def.4.4}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{2}}{}}{24}{Exo.4.2}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{3}}{}}{24}{Exo.4.3}} +\@writefile{toc}{\contentsline {subsection}{\numberline {II.2}Relation d'ordre}{25}{subsection.4.2.2}} +\@writefile{thm}{\contentsline {Def}{{Définition}{4.{5}}{Relation d'ordre}}{25}{Def.4.5}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{4}}{}}{25}{Exo.4.4}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{5}}{Relation de divisibilité}}{25}{Exo.4.5}} +\global\def\markxviiiExi{\ensuremath {}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{6}}{}}{25}{Exo.4.6}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{7}}{Diagrammes de transitivité}}{25}{Exo.4.7}} +\@writefile{toc}{\contentsline {section}{\numberline {III}Relations d'\IeC {\'e}quivalence}{25}{section.4.3}} +\@writefile{thm}{\contentsline {Def}{{Définition}{4.{6}}{Relation symétrique}}{25}{Def.4.6}} +\@writefile{thm}{\contentsline {Def}{{Définition}{4.{7}}{Relation d'équivalence}}{25}{Def.4.7}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{8}}{}}{25}{Exo.4.8}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{9}}{Relation de congruence modulo $n$ dans $\Z $}}{25}{Exo.4.9}} \global\def\markxxExi{\ensuremath {}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{8}}{}}{31}{Exo.8}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{9}}{Diagrammes de transitivité}}{31}{Exo.9}} -\@writefile{toc}{\contentsline {section}{\numberline {III}Relations d'\IeC {\'e}quivalence}{32}{section.4.3}} -\@writefile{thm}{\contentsline {Def}{{Définition}{4.{6}}{Relation symétrique}}{32}{Def.6}} -\@writefile{thm}{\contentsline {Def}{{Définition}{4.{7}}{Relation d'équivalence}}{32}{Def.7}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{10}}{}}{32}{Exo.10}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{11}}{Relation de congruence modulo $n$ dans $\Z $}}{32}{Exo.11}} -\global\def\markxxiiExi{\ensuremath {}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{12}}{}}{32}{Exo.12}} -\@writefile{toc}{\contentsline {subsection}{\numberline {III.1}Classes d'\IeC {\'e}quivalence}{32}{subsection.4.3.1}} -\@writefile{thm}{\contentsline {Def}{{Définition}{4.{8}}{Classe d'équivalence}}{32}{Def.8}} -\@writefile{thm}{\contentsline {Notation}{{Notation}{4.{1}}{}}{32}{Notation.1}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{13}}{}}{32}{Exo.13}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{14}}{}}{32}{Exo.14}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{4.{1}}{}}{33}{Th.1}} -\@writefile{thm}{\contentsline {Rem}{{Remarque}{4.{2}}{}}{33}{Rem.2}} -\@writefile{thm}{\contentsline {Pre}{{Preuve}{1}{}}{33}{Pre.1}} -\@writefile{thm}{\contentsline {Def}{{Définition}{4.{9}}{Partition d'un ensemble}}{33}{Def.9}} -\@writefile{thm}{\contentsline {Th}{{Propriété}{4.{2}}{}}{33}{Th.2}} -\@writefile{thm}{\contentsline {Pre}{{Preuve}{2}{}}{33}{Pre.2}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{15}}{}}{33}{Exo.15}} -\global\def\markxxiiiEx{\ensuremath {}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{16}}{}}{33}{Exo.16}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{17}}{Une relation d'équivalence}}{33}{Exo.17}} -\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{18}}{}}{33}{Exo.18}} -\@writefile{toc}{\contentsline {subsection}{\numberline {III.2}Ensemble-quotient}{34}{subsection.4.3.2}} -\@writefile{thm}{\contentsline {Def}{{Définition}{4.{10}}{Ensemble-quotient}}{34}{Def.10}} -\@writefile{thm}{\contentsline {Notation}{{Notation}{4.{2}}{}}{34}{Notation.2}} -\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{19}}{Congruence modulo 4}}{34}{Exo.19}} -\@writefile{toc}{\contentsline {part}{III\hspace {1em}Annexes}{35}{part.3}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{10}}{}}{25}{Exo.4.10}} +\@writefile{toc}{\contentsline {subsection}{\numberline {III.1}Classes d'\IeC {\'e}quivalence}{26}{subsection.4.3.1}} +\@writefile{thm}{\contentsline {Def}{{Définition}{4.{8}}{Classe d'équivalence}}{26}{Def.4.8}} +\@writefile{thm}{\contentsline {Notation}{{Notation}{4.{1}}{}}{26}{Notation.4.1}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{11}}{}}{26}{Exo.4.11}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{12}}{}}{26}{Exo.4.12}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{4.{1}}{}}{26}{Th.4.1}} +\@writefile{thm}{\contentsline {Pre}{{Preuve}{1}{}}{26}{Pre.1}} +\@writefile{thm}{\contentsline {Def}{{Définition}{4.{9}}{Partition d'un ensemble}}{26}{Def.4.9}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{4.{2}}{}}{26}{Th.4.2}} +\@writefile{thm}{\contentsline {Pre}{{Preuve}{2}{}}{26}{Pre.2}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{13}}{}}{26}{Exo.4.13}} +\global\def\markxxiEx{\ensuremath {}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{14}}{}}{26}{Exo.4.14}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{15}}{}}{26}{Exo.4.15}} +\@writefile{toc}{\contentsline {part}{III\hspace {1em}Arithm\IeC {\'e}tique}{27}{part.3}} +\@writefile{toc}{\contentsline {chapter}{\numberline {5}Ensembles de nombres entiers}{28}{chapter.5}} +\@writefile{lof}{\addvspace {10\p@ }} +\@writefile{lot}{\addvspace {10\p@ }} +\@writefile{toc}{\contentsline {section}{\numberline {I}Principe de r\IeC {\'e}currence }{28}{section.5.1}} +\newlabel{itm:2}{{2}{28}{Principe de récurrence}{Item.147}{}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{1}}{}}{28}{Exo.5.1}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{2}}{}}{28}{Exo.5.2}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{3}}{}}{28}{Exo.5.3}} +\@writefile{toc}{\contentsline {section}{\numberline {II}Nombres premiers}{28}{section.5.2}} +\@writefile{thm}{\contentsline {Def}{{Définition}{5.{1}}{Multiple, diviseur}}{28}{Def.5.1}} +\@writefile{thm}{\contentsline {Def}{{Définition}{5.{2}}{Nombre premier}}{28}{Def.5.2}} +\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{1}}{}}{28}{Rem.5.1}} +\@writefile{thm}{\contentsline {Def}{{Définition}{5.{3}}{Décomposition en facteurs premiers}}{28}{Def.5.3}} +\global\def\markxxxiDefi{\ensuremath {}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{1}}{}}{28}{Th.5.1}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{4}}{}}{29}{Exo.5.4}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{5}}{}}{29}{Exo.5.5}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{2}}{}}{29}{Th.5.2}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{6}}{Nombres premiers en quantité infinie}}{29}{Exo.5.6}} +\@writefile{thm}{\contentsline {Def}{{Définition}{5.{4}}{PGCD, PPCM}}{29}{Def.5.4}} +\@writefile{thm}{\contentsline {Def}{{Définition}{5.{5}}{Nombres premiers entre eux}}{29}{Def.5.5}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{7}}{Nombres de Fermat}}{29}{Exo.5.7}} +\@writefile{toc}{\contentsline {section}{\numberline {III}Division euclidienne dans ${\mathbb Z}$ et applications}{29}{section.5.3}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{3}}{}}{29}{Th.5.3}} +\@writefile{thm}{\contentsline {Def}{{Définition}{5.{6}}{Division euclidienne}}{29}{Def.5.6}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{8}}{}}{29}{Exo.5.8}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{9}}{}}{29}{Exo.5.9}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{10}}{}}{29}{Exo.5.10}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{11}}{Numéro Sécurité Sociale (\url {www.bibmath.net})}}{30}{Exo.5.11}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{12}}{}}{30}{Exo.5.12}} +\@writefile{toc}{\contentsline {section}{\numberline {IV}Algorithmes d'Euclide}{30}{section.5.4}} +\@writefile{toc}{\contentsline {subsection}{\numberline {IV.1}L'algorithme initial}{30}{subsection.5.4.1}} +\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{2}}{}}{31}{Rem.5.2}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{13}}{}}{31}{Exo.5.13}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{14}}{}}{31}{Exo.5.14}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{15}}{}}{31}{Exo.5.15}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{4}}{Théorème de Bézout}}{31}{Th.5.4}} +\@writefile{thm}{\contentsline {Proof}{{Preuve}{5.{$\alpha $}}{}}{31}{Proof.5.1}} +\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{3}}{}}{31}{Rem.5.3}} +\@writefile{thm}{\contentsline {Proof}{{Preuve}{5.{$\beta $}}{}}{31}{Proof.5.2}} +\global\def\markxRemi{\ensuremath {}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{16}}{Application de l'algorithme d'Euclide et de Bézout}}{31}{Exo.5.16}} +\@writefile{toc}{\contentsline {subsection}{\numberline {IV.2}Algorithme d'Euclide g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}}{32}{subsection.5.4.2}} +\@writefile{toc}{\contentsline {subsection}{\numberline {IV.3}L'algorithme.}{32}{subsection.5.4.3}} +\@writefile{thm}{\contentsline {Pre}{{Preuve}{3}{}}{32}{Pre.3}} +\global\def\markiiiPrei{$\dagger $} +\@writefile{toc}{\contentsline {subsection}{\numberline {IV.4}Exemple.}{32}{subsection.5.4.4}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{17}}{}}{32}{Exo.5.17}} +\global\def\markxxivEx{\ensuremath {}} +\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{4}}{}}{32}{Rem.5.4}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{18}}{}}{32}{Exo.5.18}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{5}}{Théorème de Gauss}}{32}{Th.5.5}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{19}}{}}{33}{Exo.5.19}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{20}}{}}{33}{Exo.5.20}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{21}}{}}{33}{Exo.5.21}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{22}}{}}{33}{Exo.5.22}} +\@writefile{toc}{\contentsline {section}{\numberline {V}Arithm\IeC {\'e}tique modulo $n$}{33}{section.5.5}} +\@writefile{thm}{\contentsline {Def}{{Définition}{5.{7}}{Congruence modulo $n$}}{33}{Def.5.7}} +\global\def\markxxxvDef{\ensuremath {}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{23}}{}}{33}{Exo.5.23}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{6}}{}}{33}{Th.5.6}} +\@writefile{thm}{\contentsline {Proof}{{Preuve}{5.{$\gamma $}}{}}{33}{Proof.5.3}} +\global\def\markviiiProofi{\rule {1ex}{1ex}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{24}}{}}{34}{Exo.5.24}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{7}}{}}{34}{Th.5.7}} +\@writefile{thm}{\contentsline {Notation}{{Notation}{5.{1}}{}}{34}{Notation.5.1}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{25}}{}}{34}{Exo.5.25}} +\@writefile{thm}{\contentsline {Def}{{Définition}{5.{8}}{}}{34}{Def.5.8}} +\global\def\markxxxviDefi{\ensuremath {}} +\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{8}}{}}{34}{Th.5.8}} +\@writefile{thm}{\contentsline {Proof}{{Preuve}{5.{$\delta $}}{}}{34}{Proof.5.4}} +\global\def\markixProofii{\rule {1ex}{1ex}} +\@writefile{thm}{\contentsline {Def}{{Définition}{5.{9}}{}}{34}{Def.5.9}} +\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{26}}{}}{34}{Exo.5.26}} +\global\def\markxxviiExi{\ensuremath {}} +\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{5}}{}}{35}{Rem.5.5}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{27}}{}}{35}{Exo.5.27}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{28}}{Systèmes de congruences}}{35}{Exo.5.28}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{29}}{}}{35}{Exo.5.29}} +\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{30}}{}}{35}{Exo.5.30}} +\@writefile{toc}{\contentsline {part}{IV\hspace {1em}Annexes}{36}{part.4}} \@input{PPN.aux} +\@writefile{toc}{\contentsline {chapter}{Index}{38}{chapter.6}} \bibstyle{alpha} \bibdata{biblio} \bibcite{Dowek07}{Dow07} -\@writefile{toc}{\contentsline {chapter}{Index}{37}{chapter.5}} \@input{Bibliographie.aux}