--- /dev/null
+\section{Représentation des nombres entiers}
+
+
+
+\begin{Def}[Principe de la numération de position]
+\index{Principe de la numérotation de position}
+Il consiste à choisir une base $b$ de numération, et $b$ symboles qui constitueront les chiffres dans la représentation d'un entier positif en base $b$.
+Celle-ci s'écrira alors
+$$n=n_{p}b^p+n_{p-1}b^{p-1}+\cdots+n_{1}b^1+n_{0}$$
+
+Cette écriture est abrégée en ${\left(\overline{n_{p}n_{p-1}\ldots n_{0}}\right)}_{b}$.
+\end{Def}
+
+En informatique, on utilise couramment les bases 2, 8 et 16.
+
+
+
+
+
+L'algorithme pour obtenir la représentation en base $b$ d'un entier est :
+
+\begin{enumerate}
+ \item Effectuer la division euclidienne de cet entier par $b$, division qui donne un premier quotient et un premier reste.
+ \item Le quotient est à sont tour divisé par $b$ pour donner un second quotient et un second reste, et ainsi de suite jusqu'à obtenir un quotient nul.
+\item Les restes successifs (tous strictement inférieurs à $b$), et en commençant par le dernier, constituent la représentation en base $b$ de l'entier donné.
+\end{enumerate}
+
+\begin{Exo}
+Donner la représentation de 23 en base 2.
+\end{Exo}
+
+
+
+\begin{Exo}[Numération, changements de base]
+\begin{enumerate}
+\item Chercher les entiers dont le carré a, en représentation décimale,
+le même chiffre pour les dizaines et les unités.
+\item On pose $a=2p-1$, $b=2p+1$, $c=2p+3$; trouver l'entier $p$ de manière que $a^2+b^2+c^2$ soit de la forme $\sur{xxxx}_{10}$.
+\item L'entier $n$ s'écrit $\sur{341}_{10}$ et $\sur{2331}_a$. Trouver $a$.
+\item Montrer que, dans toute base $b$ supérieure ou égale à 3, l'entier qui s'écrit $\sur{11211}_b$ n'est pas premier.
+\item Soit $n\geqslant 7$. Donner l'écriture de $(n+1)^4$ en base $n$.
+\end{enumerate}
+\end{Exo}
+
+
+
+\begin{Exo}[Développement décimal]
+On considère le nombre réel $x$ dont le dé\-ve\-lop\-pe\-ment décimal s'écrit $x=0,012\ 345\ 679\ 012\ 345\ 679\ \ldots\ \ldots\ \ldots$ (la séquence $012\ 345\ 679$ est reproduite indéfiniment). Ce développement décimal est périodique, de période 9.
+\begin{enumerate}
+\item Montrer que $x$
+vérifie une équation de la forme $10^kx=n+x$, où $k$ et $n$ sont
+des entiers à déterminer. En résolvant cette équation,
+montrer que $x$ est un nombre rationnel, et le mettre sous la forme
+$x= \fr pq$ , où $p$ et $q$ sont premiers entre eux.
+\item Appliquer
+la même méthode au ``nombre" $y$ dont le développement
+décimal est $y= 0,999\ 999\ 999\ 999\ \ldots$ (périodique de période
+1). Quelle conclusion peut-on en tirer?
+\item Démontrer que tout nombre réel dont le développement
+décimal est fini ou périodique à partir d'un certain rang
+est un nombre rationnel.
+\item Réciproquement, on se propose de démontrer que le
+développement décimal de tout nombre rationnel est fini ou
+périodique à partir d'un certain rang. Pour cela, on
+considère un rationnel $x=\fr pq$ , avec $q>0$, $p\in
+\Z$, $p$ et $q$ premiers entre eux, et on étudiera successivement
+les cas suivants:
+\begin{itemize}
+\item $x$ est entier (c'est à dire $q=1$).
+\item $x$ est rationnel non entier, et $q$ est premier avec 10 (On
+pourra montrer que, si $q$ est premier avec 10, il existe un entier
+$k$, non nul, tel que $10^k\equiv 1\ [q]$).
+\item $x$ est rationnel non entier, mais $q$ n'est pas premier avec 10.
+\end{itemize}
+\end{enumerate}
+
+\end{Exo}
+
+
+
+\section{Arithmétique en informatique}
+
+
+La plupart des langages de programmation utilisés en informatique disposent d'un type de données pour représenter ce que les informaticiens appellent les entiers signés (les entiers relatifs) et possèdent des opérateurs pour effectuer les calculs classiques sur ces nombres.
+
+\subsection{Division entière}
+
+En C ou java, par exemple, le symbole $/$ représente le quotient dans la \og division entière\fg{} et le symbole $\%$ représente ce que les informaticiens appellent improprement le modulo (le reste dans leur \og division entière\fg{} ).
+
+
+Pour des raisons pratiques de réalisation des micro-circuits des processeurs qui réalisent ces opérations, la \og division entière\fg{} ne donne pas exactement le même résultat que la division euclidienne.
+
+
+
+Considérons par exemple les 4 cas possibles de division euclidienne de $a$ par $b$ lorsque $|a|=29$ et $|b|=7$ (en n'oubliant pas que le reste d'une division euclidienne ne peut être que positif)
+
+
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|}
+\hline
+$a$ & $b$ & division euclidienne & $q$ & $r$ & $a/b$ & $a\%b$ \\ \hline
+$29$ &$7$ & $29=4\times 7+1$ & $4$ & $1$ & $4$ & $1$ \\ \hline
+$29$ &$-7$ & $29=(-4)\times (-7)+1$ & $-4$ & $1$ & $-4$ & $1$ \\ \hline
+$-29$ &$7$ & $-29=(-5)\times 7+6$ & $-5$ & $6$ & $-4$ & $-1$ \\ \hline
+$-29$ &$-7$ & $-29=5\times (-7)+6$ & $5$ & $6$ & $4$ & $-1$ \\ \hline
+\end{tabular}
+\end{center}
+
+
+
+Autrement dit, mathématiquement, le quotient est positif lorsque les deux nombres ont le même signe et le reste est toujours positif, et, pour que le reste soit toujours positif, le quotient peut ne pas être le quotient des valeurs absolues.
+
+
+Informatiquement, le \og quotient\fg{} est positif lorsque les nombres ont le même signe, le \og reste\fg{} a le signe du dividende, et la valeur absolue du \og quotient\fg{} est toujours le quotient des valeurs absolues.
+
+
+Dans les applications de calcul arithmétique, par exemple un calcul de PGCD, ce n'est pas gênant parce que les restes \og informatiques\fg{} sont congrus aux restes mathématiques modulo la valeur absolue du
+diviseur, et qu'il ne s'agit alors que du choix d'un représentant de la classe concernée (addition et multiplication étant compatibles avec la congruence modulo $n$).
+
+Mais il faut quand même savoir que l'on peut obtenir un \og reste\fg{} négatif et prendre ses dispositions le cas échéant...
+
+
+\subsection{Arithmétique modulo $2^n$}
+
+
+
+Les calculs sur les entiers, dans un ordinateur, se font dans $\Z/2^n\Z$, où $n$ est le nombre de bits utilisés dans la représentation de ces nombres.
+
+
+Dans la plupart des microprocesseurs, les entiers sont représentés sur 64 bits, les calculs se font donc dans $\Z/2^{64}\Z$.
+
+
+Disposer d'entiers signés ou d'entiers non signés est uniquement une question de choix du représentant dans les classes d'équivalence, mais
+la représentation physique est la même.
+
+
+Comme il nous est difficile de représenter ici la liste compléte de tous ces entiers, nous allons illustrer ce propos en supposant que les entiers sont représentés sur 4 bits.
+Pour des mots de 4 bits, il y a alors 16 entiers représentables : (a.s.= arithmétique signée, a.n.s. = arithmétique non signée)
+
+\begin{center}\begin{tabular}{|c|c|c|c|} \hline
+code binaire & & a.s. & a.n.s. \\ \hline
+0000 & interprété par & 0 & 0 \\ \hline
+0001 & interprété par & 1 & 1 \\ \hline
+0010 & interprété par & 2 & 2 \\ \hline
+0011 & interprété par & 3 & 3 \\ \hline
+0100 & interprété par & 4 & 4 \\ \hline
+0101 & interprété par & 5 & 5 \\ \hline
+0110 & interprété par & 6 & 6 \\ \hline
+0111 & interprété par & 7 & 7 \\ \hline
+1000 & interprété par & 8 & -8 \\ \hline
+1001 & interprété par & 9 & -7 \\ \hline
+1010 & interprété par & 10 & -6 \\ \hline
+1011 & interprété par & 11 & -5 \\ \hline
+1100 & interprété par & 12 & -4 \\ \hline
+1101 & interprété par & 13 & -3 \\ \hline
+1110 & interprété par & 14 & -2 \\ \hline
+1111 & interprété par & 15 & -1 \\ \hline
+\end{tabular}\end{center}\vskip 10pt
+
+
+Pourquoi ce choix ? Pourquoi ne pas avoir, en a.s., représenté les entiers dans l'ordre croissant de 0000 (-8) à 1111 (7)?
+
+\begin{itemize}
+ \item Tout simplement pour des raisons d'efficacité : 0 doit toujours être représenté par le code \og nul\fg{} 0000.
+\item Ensuite, il faut pouvoir comparer efficacement ces codes entre eux, ce qui explique que 0 doit être suivi de 1, arithmétique signée ou pas.
+\end{itemize}
+
+\bigskip
+
+Ces principes ont ainsi conduit à placer les codes interprétés comme entiers négatifs après ceux qui représentent les entiers positifs.
+
+
+Par ailleurs, on s'aperçoit que, de cette manière, les codes des entiers
+négatifs commencent tous par 1.
+On parle improprement de \og bit de signe\fg{}\index{bit de signe}: s'il s'agissait d'un véritable bit de signe, le code 1001 devrait être celui de -1, or c'est celui de -7.
+Mais il n'en reste pas moins que tous les entiers négatifs commencent par 1).
+
+
+Ainsi, il est facile de déduire la comparaison signée de la comparaison non signée : les codes qui commencent par 1 sont \og plus petits\fg{} que ceux qui commencent par 0, et, s'ils commencent par le même bit, c'est la comparaison non signée qui peut être utilisée.
+
+
+
+
+
+Pour l'addition et la soustraction, les opérations et les tests de validité des résultats sont les mêmes en arithmétique signée et non signée.
+Pour la multiplication, l'instruction n'est pas la même (le dépassement de capacité doit être ignoré en a.s. dans le dernier exemple).
+
+\begin{Ex}
+
+Premiers résultats, corrects :
+
+ \begin{center}
+\begin{tabular}{r | r | r}
+Opération binaire & Entiers non signés & Entiers signés \\
+\hline
+0010 & 2 & 2 \\
+\underline{+ 1001} & \underline{+ 9} & \underline{+(-7)} \\
+1011 & 11 & (-5) \\
+\end{tabular}
+ \end{center}
+\end{Ex}
+
+
+\begin{Ex}
+ Un résultat correct en arithmétique non \break signée, et négatif en arithmétique signée, mais correct modulo 16 (-6 et 10 sont dans la même classe, mais cette classe est représentée par 10 en a.n.s. et par -6 en a.s.) :
+\begin{center}
+\begin{tabular}{r | r | r}
+Opération binaire & Entiers non signés & Entiers signés \\
+\hline
+0100 & 4 & 4 \\
+\underline{+ 0110} & \underline{+ 6} & \underline{+ 6} \\
+1010 & 10 & (-6) \\
+\end{tabular}
+\end{center}
+\end{Ex}
+
+
+\begin{Ex}
+Un dépassement de capacité dans les deux cas, mais le résultat est correct modulo 16 : les classes de 21, de -11 et de 5 sont les mêmes :
+ \begin{center}
+\begin{tabular}{r | r | r}
+Opération binaire & Entiers non signés & Entiers signés \\
+\hline
+1100 & 12 & (-4) \\
+\underline{+ 1001} & \underline{+ 9} & \underline{+(-7)} \\
+(1)0101 & 5 & 5 \\
+\end{tabular}
+ \end{center}
+Le résultat (correct modulo 16) est disponible dans tous les cas, les \og dépassement de capacité\fg{} et \og résultat négatif\fg{} sont signalés par le positionnement d'un bit dans un registre spécial.
+\end{Ex}
+
+
+
+
+\begin{Ex}
+Un résultat correct en a.n.s., résultat négatif en a.s., mais correct modulo 16 :
+ \begin{center}
+\begin{tabular}{r | r | r}
+Opération binaire & Entiers non signés & Entiers signés \\
+\hline
+0101 & 5 & 5 \\
+\underline{$\times$ 0010} & \underline{$\times$ 2} &
+\underline{$\times$ 2} \\ 1010 & 10 & (-6) \\
+\end{tabular}
+ \end{center}
+\end{Ex}
+
+
+\begin{Ex}
+Dépassement de capacité dans les deux cas, résultat négatif en a.s., mais résultat correct modulo 16, compte tenu du choix des représentants dans les deux arithmétiques:
+ \begin{center}
+\begin{tabular}{r | r | r}
+Opération binaire & Entiers non signés & Entiers signés \\
+\hline
+0101 & 5 & 5 \\
+\sou{$\times$ 0110} & \sou{$\times$ 6} & \sou{$\times$ 6} \\
+(1)1110 & 14 & (-2) \\
+\end{tabular}
+ \end{center}
+\end{Ex}
+
+
+
+
+\begin{Ex}
+Dépassement de capacité dans les deux cas, résultat correct en a.s., correct modulo 16 en a.n.s.
+ \begin{center}
+\begin{tabular}{r | r | r}
+Opération binaire & Entiers non signés & Entiers signés \\
+\hline
+1101 & 13 & (-3) \\
+\sou{$\times$ 1110} & \sou{$\times$ 14} & \sou{$\times$
+(-2)} \\ (1011)0110 & 6 & 6 \\
+\end{tabular}
+\end{center}
+\end{Ex}
+\centerline{\x{Fin du Chapitre}}
\ No newline at end of file
\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
\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}}
\@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}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{2}}{}}{12}{Exo.2.2}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{2}}{}}{13}{Rem.2.2}}
+\@writefile{thm}{\contentsline {Exoc}{{Exercice (corrigé)}{2.{3}}{}}{13}{Exo.2.3}}
+\newlabel{it:x1}{{1}{13}{Tables de vérité des connecteurs logiques}{Item.48}{}}
+\newlabel{it:x2}{{2}{13}{Tables de vérité des connecteurs logiques}{Item.49}{}}
+\newlabel{it:x3}{{3}{13}{Tables de vérité des connecteurs logiques}{Item.50}{}}
+\newlabel{it:x4}{{4}{13}{Tables de vérité des connecteurs logiques}{Item.51}{}}
+\newlabel{it:x5}{{5}{13}{Tables de vérité des connecteurs logiques}{Item.52}{}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{4}}{}}{13}{Exo.2.4}}
+\newlabel{ex2:1}{{1}{13}{Tables de vérité des connecteurs logiques}{Item.53}{}}
+\newlabel{ex2:2}{{2}{13}{Tables de vérité des connecteurs logiques}{Item.54}{}}
+\newlabel{ex2:3}{{3}{13}{Tables de vérité des connecteurs logiques}{Item.55}{}}
+\newlabel{ex2:4}{{4}{13}{Tables de vérité des connecteurs logiques}{Item.56}{}}
+\newlabel{ex2:5}{{5}{13}{Tables de vérité des connecteurs logiques}{Item.57}{}}
+\newlabel{ex2:6}{{6}{13}{Tables de vérité des connecteurs logiques}{Item.58}{}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{5}}{}}{13}{Exo.2.5}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{6}}{}}{13}{Exo.2.6}}
+\@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.{1}}{}}{13}{Th.2.1}}
\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}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{3}}{}}{14}{Rem.2.3}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{7}}{}}{14}{Exo.2.7}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{8}}{}}{14}{Exo.2.8}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{9}}{}}{14}{Exo.2.9}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{2}}{Règles de priorité des connecteurs logiques}}{14}{Th.2.2}}
\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 {Ex}{{Exemple}{2.{10}}{}}{14}{Exo.2.10}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{3}}{Associativité des opérateurs $\ou $ et $\et $}}{14}{Th.2.3}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{4}}{}}{15}{Rem.2.4}}
+\@writefile{thm}{\contentsline {Exoc}{{Exercice (corrigé)}{2.{11}}{}}{15}{Exo.2.11}}
+\@writefile{thm}{\contentsline {Exoc}{{Exercice (corrigé)}{2.{12}}{}}{15}{Exo.2.12}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{13}}{}}{15}{Exo.2.13}}
+\@writefile{toc}{\contentsline {section}{\numberline {III}S\IeC {\'e}mantique du calcul propositionnel}{15}{section.2.3}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {III.1}Fonctions de v\IeC {\'e}rit\IeC {\'e}}{16}{subsection.2.3.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{2.{2}}{Fonction de vérité de $F$}}{16}{Def.2.2}}
+\newlabel{item:eqv}{{6}{16}{Fonctions de vérité}{Item.106}{}}
+\global\def\markixDef{\ensuremath {}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{14}}{}}{16}{Exo.2.14}}
\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{thm}{\contentsline {Rem}{{Remarque}{2.{5}}{}}{16}{Rem.2.5}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {III.2}Formules propositionnelles particuli\IeC {\`e}res}{16}{subsection.2.3.2}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {III.2.1}Tautologies}{16}{subsubsection.2.3.2.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{2.{3}}{Tautologie}}{16}{Def.2.3}}
+\@writefile{thm}{\contentsline {Notation}{{Notation}{2.{1}}{}}{16}{Notation.2.1}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{15}}{}}{16}{Exo.2.15}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{16}}{}}{16}{Exo.2.16}}
\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.{17}}{}}{16}{Exo.2.17}}
+\newlabel{item:taut:5}{{1}{16}{Tautologies}{Item.107}{}}
+\newlabel{item:taut:7}{{3}{17}{Tautologies}{Item.109}{}}
+\newlabel{item:taut:8}{{4}{17}{Tautologies}{Item.110}{}}
+\newlabel{item:taut:6}{{6}{17}{Tautologies}{Item.112}{}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {III.2.2}Antilogies}{17}{subsubsection.2.3.2.2}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{2.{4}}{Antilogie}}{17}{Def.2.4}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{18}}{}}{17}{Exo.2.18}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{19}}{}}{17}{Exo.2.19}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {III.3}Cons\IeC {\'e}quences logiques}{17}{subsection.2.3.3}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{2.{5}}{Conséquence logique}}{17}{Def.2.5}}
+\@writefile{thm}{\contentsline {Notation}{{Notation}{2.{2}}{}}{17}{Notation.2.2}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{20}}{}}{17}{Exo.2.20}}
\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 $}}{}}{21}{Proof.1}}
-\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{27}}{}}{21}{Exo.27}}
-\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{28}}{}}{22}{Exo.28}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {III.5.2}Th\IeC {\'e}or\IeC {\`e}me de la validit\IeC {\'e}}{22}{subsubsection.2.3.5.2}}
-\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{5}}{Théorème de la validité}}{22}{Th.5}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{21}}{}}{17}{Exo.2.21}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{22}}{}}{17}{Exo.2.22}}
+\newlabel{item:cons:1}{{1}{17}{Conséquences logiques}{Item.118}{}}
+\newlabel{item:cons:2}{{2}{17}{Conséquences logiques}{Item.119}{}}
+\newlabel{item:cons:3}{{3}{17}{Conséquences logiques}{Item.120}{}}
+\newlabel{item:cons:4}{{4}{17}{Conséquences logiques}{Item.121}{}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{23}}{}}{17}{Exo.2.23}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {III.4}Formules \IeC {\'e}quivalentes}{18}{subsection.2.3.4}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{2.{6}}{Formules équivalentes}}{18}{Def.2.6}}
+\global\def\markxiiiDef{\ensuremath {}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{6}}{}}{18}{Rem.2.6}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{24}}{}}{18}{Exo.2.24}}
+\@writefile{thm}{\contentsline {Exoc}{{Exercice (corrigé)}{2.{25}}{}}{18}{Exo.2.25}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{26}}{}}{18}{Exo.2.26}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {III.5}Simplification du calcul des fonctions de v\IeC {\'e}rit\IeC {\'e}}{19}{subsection.2.3.5}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {III.5.1}Th\IeC {\'e}or\IeC {\`e}me de substitution}{19}{subsubsection.2.3.5.1}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{4}}{Théorème de substitution}}{19}{Th.2.4}}
+\@writefile{thm}{\contentsline {Proof}{{Preuve}{2.{$\alpha $}}{}}{19}{Proof.2.1}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{27}}{}}{19}{Exo.2.27}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{28}}{}}{19}{Exo.2.28}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {III.5.2}Th\IeC {\'e}or\IeC {\`e}me de la validit\IeC {\'e}}{19}{subsubsection.2.3.5.2}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{2.{5}}{Théorème de la validité}}{19}{Th.2.5}}
\global\def\markxiTh{\ensuremath {}}
-\@writefile{thm}{\contentsline {Proof}{{Preuve}{2.{$\beta $}}{}}{22}{Proof.2}}
+\@writefile{thm}{\contentsline {Proof}{{Preuve}{2.{$\beta $}}{}}{19}{Proof.2.2}}
\global\def\markvProofii{\rule {1ex}{1ex}}
-\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{29}}{Exemple d'application}}{22}{Exo.29}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{2.{29}}{Exemple d'application}}{20}{Exo.2.29}}
\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 {Exo}{{Exercice}{2.{30}}{}}{20}{Exo.2.30}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{31}}{}}{20}{Exo.2.31}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{32}}{}}{20}{Exo.2.32}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{33}}{}}{20}{Exo.2.33}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{34}}{}}{21}{Exo.2.34}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {III.6}Conclusion}{21}{subsection.2.3.6}}
+\@writefile{toc}{\contentsline {part}{II\hspace {1em}Th\IeC {\'e}orie des ensembles}{22}{part.2}}
+\@writefile{toc}{\contentsline {chapter}{\numberline {3}Introduction \IeC {\`a} la th\IeC {\'e}orie des ensembles}{23}{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}Rappels de th\IeC {\'e}orie des ensembles}{23}{section.3.1}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {I.1}Notion premi\IeC {\`e}re d'ensemble}{23}{subsection.3.1.1}}
+\@writefile{thm}{\contentsline {Notation}{{Notation}{3.{1}}{}}{23}{Notation.3.1}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{1}}{}}{23}{Exo.3.1}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {I.2}R\IeC {\`e}gles de fonctionnement}{23}{subsection.3.1.2}}
+\@writefile{toc}{\contentsline {paragraph}{Relation d'appartenance.}{23}{section*.2}}
+\@writefile{toc}{\contentsline {paragraph}{Objets distincts.}{23}{section*.3}}
+\@writefile{toc}{\contentsline {paragraph}{Ensemble vide.}{23}{section*.4}}
+\@writefile{toc}{\contentsline {paragraph}{Derni\IeC {\`e}re r\IeC {\`e}gle de fonctionnement des ensembles.}{23}{section*.5}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {I.3}Sous-ensembles, ensemble des parties}{23}{subsection.3.1.3}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{3.{1}}{}}{23}{Def.3.1}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{1}}{}}{23}{Th.3.1}}
+\@writefile{thm}{\contentsline {Proof}{{Preuve}{3.{$\alpha $}}{}}{24}{Proof.3.1}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{2}}{}}{24}{Th.3.2}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{3.{2}}{}}{24}{Def.3.2}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{3}}{}}{24}{Th.3.3}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{3.{2}}{}}{24}{Exo.3.2}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{3}}{}}{24}{Exo.3.3}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{4}}{}}{24}{Exo.3.4}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{5}}{}}{24}{Exo.3.5}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{6}}{}}{24}{Exo.3.6}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{7}}{}}{24}{Exo.3.7}}
+\@writefile{toc}{\contentsline {section}{\numberline {II}Op\IeC {\'e}rations sur les ensembles}{24}{section.3.2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {II.1}\'Egalite de deux ensembles}{24}{subsection.3.2.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{3.{3}}{}}{24}{Def.3.3}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{8}}{}}{24}{Exo.3.8}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {II.2}R\IeC {\'e}union, intersection}{24}{subsection.3.2.2}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{3.{4}}{Reunion}}{24}{Def.3.4}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{3.{9}}{}}{24}{Exo.3.9}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{3.{5}}{Intersection}}{24}{Def.3.5}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{4}}{Propriétés de la réunion et de l'intersection}}{25}{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}}{}}{25}{Exo.3.10}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{11}}{}}{25}{Exo.3.11}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{5}}{Distributivités de $\cup $ et $\cap $}}{25}{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}}{}}{25}{Exo.3.12}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {II.3}Compl\IeC {\'e}mentation}{25}{subsection.3.2.3}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{3.{6}}{Complémentation}}{25}{Def.3.6}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{3.{6}}{}}{25}{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}}{}}{25}{Exo.3.13}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {II.4}Produit cart\IeC {\'e}sien}{25}{subsection.3.2.4}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{14}}{}}{25}{Exo.3.14}}
+\@writefile{toc}{\contentsline {section}{\numberline {III}Exercices suppl\IeC {\'e}mentaires}{26}{section.3.3}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{15}}{}}{26}{Exo.3.15}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{16}}{}}{26}{Exo.3.16}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{3.{17}}{Fonction caractéristique des parties d'un ensemble}}{26}{Exo.3.17}}
+\@writefile{toc}{\contentsline {chapter}{\numberline {4}Relations binaires entre ensembles}{27}{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{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}}{30}{Def.4}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{2}}{}}{30}{Exo.2}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{3}}{}}{31}{Exo.3}}
-\@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.{4}}{}}{31}{Exo.4}}
-\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{5}}{Relation de divisibilité}}{31}{Exo.5}}
+\@writefile{toc}{\contentsline {section}{\numberline {I}Relations}{27}{section.4.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{4.{1}}{Relation binaire}}{27}{Def.4.1}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{1}}{}}{27}{Exo.4.1}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{4.{1}}{}}{27}{Rem.4.1}}
+\@writefile{toc}{\contentsline {section}{\numberline {II}Relations d'ordre}{27}{section.4.2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {II.1}R\IeC {\'e}flexivit\IeC {\'e}, antisym\IeC {\'e}trie, transitivit\IeC {\'e}}{27}{subsection.4.2.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{4.{2}}{Réflexivité}}{27}{Def.4.2}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{4.{3}}{Antisymétrie}}{27}{Def.4.3}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{4.{4}}{Transitivité}}{27}{Def.4.4}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{2}}{}}{27}{Exo.4.2}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{3}}{}}{27}{Exo.4.3}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {II.2}Relation d'ordre}{28}{subsection.4.2.2}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{4.{5}}{Relation d'ordre}}{28}{Def.4.5}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{4}}{}}{28}{Exo.4.4}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{5}}{Relation de divisibilité}}{28}{Exo.4.5}}
\global\def\markxxExi{\ensuremath {}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{6}}{}}{31}{Exo.6}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{7}}{Diagrammes de transitivité}}{31}{Exo.7}}
-\@writefile{toc}{\contentsline {section}{\numberline {III}Relations d'\IeC {\'e}quivalence}{31}{section.4.3}}
-\@writefile{thm}{\contentsline {Def}{{Définition}{4.{6}}{Relation symétrique}}{31}{Def.6}}
-\@writefile{thm}{\contentsline {Def}{{Définition}{4.{7}}{Relation d'équivalence}}{31}{Def.7}}
-\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{8}}{}}{31}{Exo.8}}
-\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{9}}{Relation de congruence modulo $n$ dans $\Z $}}{32}{Exo.9}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{6}}{}}{28}{Exo.4.6}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{7}}{Diagrammes de transitivité}}{28}{Exo.4.7}}
+\@writefile{toc}{\contentsline {section}{\numberline {III}Relations d'\IeC {\'e}quivalence}{28}{section.4.3}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{4.{6}}{Relation symétrique}}{28}{Def.4.6}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{4.{7}}{Relation d'équivalence}}{28}{Def.4.7}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{8}}{}}{28}{Exo.4.8}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{9}}{Relation de congruence modulo $n$ dans $\Z $}}{28}{Exo.4.9}}
\global\def\markxxiiExi{\ensuremath {}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{10}}{}}{32}{Exo.10}}
-\@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.{11}}{}}{32}{Exo.11}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{12}}{}}{32}{Exo.12}}
-\@writefile{thm}{\contentsline {Th}{{Propriété}{4.{1}}{}}{32}{Th.1}}
-\@writefile{thm}{\contentsline {Pre}{{Preuve}{1}{}}{32}{Pre.1}}
-\@writefile{thm}{\contentsline {Def}{{Définition}{4.{9}}{Partition d'un ensemble}}{32}{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.{13}}{}}{33}{Exo.13}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{10}}{}}{28}{Exo.4.10}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {III.1}Classes d'\IeC {\'e}quivalence}{29}{subsection.4.3.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{4.{8}}{Classe d'équivalence}}{29}{Def.4.8}}
+\@writefile{thm}{\contentsline {Notation}{{Notation}{4.{1}}{}}{29}{Notation.4.1}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{11}}{}}{29}{Exo.4.11}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{12}}{}}{29}{Exo.4.12}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{4.{1}}{}}{29}{Th.4.1}}
+\@writefile{thm}{\contentsline {Pre}{{Preuve}{1}{}}{29}{Pre.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{4.{9}}{Partition d'un ensemble}}{29}{Def.4.9}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{4.{2}}{}}{29}{Th.4.2}}
+\@writefile{thm}{\contentsline {Pre}{{Preuve}{2}{}}{29}{Pre.2}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{13}}{}}{29}{Exo.4.13}}
\global\def\markxxiiiEx{\ensuremath {}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{14}}{}}{33}{Exo.14}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{15}}{}}{33}{Exo.15}}
-\@writefile{toc}{\contentsline {part}{III\hspace {1em}Annexes}{34}{part.3}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{14}}{}}{29}{Exo.4.14}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{15}}{}}{29}{Exo.4.15}}
+\@writefile{toc}{\contentsline {part}{III\hspace {1em}Arithm\IeC {\'e}tique}{30}{part.3}}
+\@writefile{toc}{\contentsline {chapter}{\numberline {5}Ensembles de nombres entiers}{31}{chapter.5}}
+\@writefile{lof}{\addvspace {10\p@ }}
+\@writefile{lot}{\addvspace {10\p@ }}
+\@writefile{toc}{\contentsline {section}{\numberline {I}Principe de r\IeC {\'e}currence }{31}{section.5.1}}
+\newlabel{itm:2}{{2}{31}{Principe de récurrence}{Item.174}{}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{1}}{}}{31}{Exo.5.1}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{2}}{}}{31}{Exo.5.2}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{3}}{}}{31}{Exo.5.3}}
+\@writefile{toc}{\contentsline {section}{\numberline {II}Nombres premiers}{31}{section.5.2}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{1}}{Multiple, diviseur}}{31}{Def.5.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{2}}{Nombre premier}}{31}{Def.5.2}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{1}}{}}{31}{Rem.5.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{3}}{Décomposition en facteurs premiers}}{31}{Def.5.3}}
+\global\def\markxxxiDefi{\ensuremath {}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{1}}{}}{31}{Th.5.1}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{4}}{}}{32}{Exo.5.4}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{5}}{}}{32}{Exo.5.5}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{2}}{}}{32}{Th.5.2}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{6}}{Nombres premiers en quantité infinie}}{32}{Exo.5.6}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{4}}{PGCD, PPCM}}{32}{Def.5.4}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{5}}{Nombres premiers entre eux}}{32}{Def.5.5}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{7}}{Nombres de Fermat}}{32}{Exo.5.7}}
+\@writefile{toc}{\contentsline {section}{\numberline {III}Division euclidienne dans ${\mathbb Z}$ et applications}{32}{section.5.3}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{3}}{}}{32}{Th.5.3}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{6}}{Division euclidienne}}{32}{Def.5.6}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{8}}{}}{32}{Exo.5.8}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{9}}{}}{32}{Exo.5.9}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{10}}{}}{32}{Exo.5.10}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{11}}{Numéro Sécurité Sociale (\url {www.bibmath.net})}}{33}{Exo.5.11}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{12}}{}}{33}{Exo.5.12}}
+\@writefile{toc}{\contentsline {section}{\numberline {IV}Algorithmes d'Euclide}{33}{section.5.4}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {IV.1}L'algorithme initial}{33}{subsection.5.4.1}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{2}}{}}{34}{Rem.5.2}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{13}}{}}{34}{Exo.5.13}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{14}}{}}{34}{Exo.5.14}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{15}}{}}{34}{Exo.5.15}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{4}}{Théorème de Bézout}}{34}{Th.5.4}}
+\@writefile{thm}{\contentsline {Proof}{{Preuve}{5.{$\alpha $}}{}}{34}{Proof.5.1}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{3}}{}}{34}{Rem.5.3}}
+\@writefile{thm}{\contentsline {Proof}{{Preuve}{5.{$\beta $}}{}}{34}{Proof.5.2}}
+\global\def\markxiiiRemi{\ensuremath {}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{16}}{Application de l'algorithme d'Euclide et de Bézout}}{34}{Exo.5.16}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {IV.2}Algorithme d'Euclide g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}}{35}{subsection.5.4.2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {IV.3}L'algorithme.}{35}{subsection.5.4.3}}
+\@writefile{thm}{\contentsline {Pre}{{Preuve}{3}{}}{35}{Pre.3}}
+\global\def\markiiiPrei{$\dagger $}
+\@writefile{toc}{\contentsline {subsection}{\numberline {IV.4}Exemple.}{35}{subsection.5.4.4}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{17}}{}}{35}{Exo.5.17}}
+\global\def\markxxviEx{\ensuremath {}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{4}}{}}{35}{Rem.5.4}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{18}}{}}{35}{Exo.5.18}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{5}}{Théorème de Gauss}}{35}{Th.5.5}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{19}}{}}{36}{Exo.5.19}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{20}}{}}{36}{Exo.5.20}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{21}}{}}{36}{Exo.5.21}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{22}}{}}{36}{Exo.5.22}}
+\@writefile{toc}{\contentsline {section}{\numberline {V}Arithm\IeC {\'e}tique modulo $n$}{36}{section.5.5}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{7}}{Congruence modulo $n$}}{36}{Def.5.7}}
+\global\def\markxxxvDef{\ensuremath {}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{23}}{}}{36}{Exo.5.23}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{6}}{}}{36}{Th.5.6}}
+\@writefile{thm}{\contentsline {Proof}{{Preuve}{5.{$\gamma $}}{}}{36}{Proof.5.3}}
+\global\def\markixProofi{\rule {1ex}{1ex}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{24}}{}}{37}{Exo.5.24}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{7}}{}}{37}{Th.5.7}}
+\@writefile{thm}{\contentsline {Notation}{{Notation}{5.{1}}{}}{37}{Notation.5.1}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{25}}{}}{37}{Exo.5.25}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{8}}{}}{37}{Def.5.8}}
+\global\def\markxxxviDefi{\ensuremath {}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{8}}{}}{37}{Th.5.8}}
+\@writefile{thm}{\contentsline {Proof}{{Preuve}{5.{$\delta $}}{}}{37}{Proof.5.4}}
+\global\def\markxProofii{\rule {1ex}{1ex}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{9}}{}}{37}{Def.5.9}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{26}}{}}{37}{Exo.5.26}}
+\global\def\markxxixExi{\ensuremath {}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{5}}{}}{38}{Rem.5.5}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{27}}{}}{38}{Exo.5.27}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{28}}{Systèmes de congruences}}{38}{Exo.5.28}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{29}}{}}{38}{Exo.5.29}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{30}}{}}{38}{Exo.5.30}}
+\@writefile{toc}{\contentsline {part}{IV\hspace {1em}Annexes}{39}{part.4}}
\@input{PPN.aux}
\bibstyle{alpha}
\bibdata{biblio}
\bibcite{Dowek07}{Dow07}
-\@writefile{toc}{\contentsline {chapter}{Index}{36}{chapter.5}}
+\@writefile{toc}{\contentsline {chapter}{Index}{41}{chapter.6}}
\@input{Bibliographie.aux}
\indexentry{alg\IeC {\`e}bre de Boole|hyperpage}{4}
-\indexentry{fonction!bool\IeC {\'e}enne|hyperpage}{6}
+\indexentry{fonction!bool\IeC {\'e}enne|hyperpage}{5}
\indexentry{fonction!bool\IeC {\'e}enne!nulle|hyperpage}{6}
\indexentry{fonction!r\IeC {\'e}f\IeC {\'e}rentiel|hyperpage}{6}
\indexentry{minterme|hyperpage}{6}
\indexentry{maxterme|hyperpage}{6}
-\indexentry{mon\IeC {\^o}me|hyperpage}{7}
+\indexentry{mon\IeC {\^o}me|hyperpage}{6}
\indexentry{forme canonique!disjonctive|hyperpage}{7}
-\indexentry{forme canonique!conjonctive|hyperpage}{8}
-\indexentry{proposition|hyperpage}{12}
-\indexentry{principe!de non-contradiction|hyperpage}{12}
-\indexentry{principe!du tiers-exclu|hyperpage}{12}
-\indexentry{variable propositionnelle|hyperpage}{15}
-\indexentry{tautologie|hyperpage}{18}
-\indexentry{antilogie|hyperpage}{19}
-\indexentry{cons\IeC {\'e}quence logique|hyperpage}{19}
-\indexentry{formules \IeC {\'e}quivalentes|hyperpage}{20}
-\indexentry{th\IeC {\'e}or\IeC {\`e}me!de substitution|hyperpage}{21}
-\indexentry{ensemble|hyperpage}{26}
-\indexentry{appartenance|hyperpage}{26}
-\indexentry{ensemble!vide|hyperpage}{26}
-\indexentry{inclusion|hyperpage}{26}
-\indexentry{r\IeC {\'e}union|hyperpage}{28}
-\indexentry{r\IeC {\'e}union|hyperpage}{28}
-\indexentry{ensemble!compl\IeC {\'e}mentaire|hyperpage}{28}
-\indexentry{compl\IeC {\'e}mentation|hyperpage}{28}
-\indexentry{involution|hyperpage}{28}
-\indexentry{loi de De Morgan|hyperpage}{28}
-\indexentry{relation binaire|hyperpage}{30}
-\indexentry{relation!r\IeC {\'e}flexive|hyperpage}{30}
-\indexentry{relation!antisym\IeC {\'e}trique|hyperpage}{30}
-\indexentry{relation!transitive|hyperpage}{30}
-\indexentry{relation!d'ordre|hyperpage}{31}
-\indexentry{relation!sym\IeC {\'e}trique|hyperpage}{31}
-\indexentry{classe d'\IeC {\'e}quivalence|hyperpage}{32}
-\indexentry{partition|hyperpage}{32}
+\indexentry{forme canonique!conjonctive|hyperpage}{7}
+\indexentry{proposition|hyperpage}{11}
+\indexentry{principe!de non-contradiction|hyperpage}{11}
+\indexentry{principe!du tiers-exclu|hyperpage}{11}
+\indexentry{variable propositionnelle|hyperpage}{13}
+\indexentry{tautologie|hyperpage}{16}
+\indexentry{antilogie|hyperpage}{17}
+\indexentry{cons\IeC {\'e}quence logique|hyperpage}{17}
+\indexentry{formules \IeC {\'e}quivalentes|hyperpage}{18}
+\indexentry{th\IeC {\'e}or\IeC {\`e}me!de substitution|hyperpage}{19}
+\indexentry{ensemble|hyperpage}{23}
+\indexentry{appartenance|hyperpage}{23}
+\indexentry{ensemble!vide|hyperpage}{23}
+\indexentry{inclusion|hyperpage}{23}
+\indexentry{r\IeC {\'e}union|hyperpage}{24}
+\indexentry{r\IeC {\'e}union|hyperpage}{24}
+\indexentry{ensemble!compl\IeC {\'e}mentaire|hyperpage}{25}
+\indexentry{compl\IeC {\'e}mentation|hyperpage}{25}
+\indexentry{involution|hyperpage}{25}
+\indexentry{loi de De Morgan|hyperpage}{25}
+\indexentry{relation binaire|hyperpage}{27}
+\indexentry{relation!r\IeC {\'e}flexive|hyperpage}{27}
+\indexentry{relation!antisym\IeC {\'e}trique|hyperpage}{27}
+\indexentry{relation!transitive|hyperpage}{27}
+\indexentry{relation!d'ordre|hyperpage}{28}
+\indexentry{relation!sym\IeC {\'e}trique|hyperpage}{28}
+\indexentry{classe d'\IeC {\'e}quivalence|hyperpage}{29}
+\indexentry{partition|hyperpage}{29}
+\indexentry{r\IeC {\'e}currence!restreinte|hyperpage}{31}
+\indexentry{multiple|hyperpage}{31}
+\indexentry{diviseur|hyperpage}{31}
+\indexentry{nombre!premier|hyperpage}{31}
+\indexentry{d\IeC {\'e}composition en facteurs premiers|hyperpage}{31}
+\indexentry{plus grand commun diviseur|hyperpage}{32}
+\indexentry{PGCD|hyperpage}{32}
+\indexentry{PPCM|hyperpage}{32}
+\indexentry{plus petit commun multiple|hyperpage}{32}
+\indexentry{division euclidienne|hyperpage}{32}
+\indexentry{quotient|hyperpage}{32}
+\indexentry{reste|hyperpage}{32}
+\indexentry{algorithme!d'Euclide|hyperpage}{33}
+\indexentry{th\IeC {\'e}or\IeC {\`e}me!de B\IeC {\'e}zout|hyperpage}{34}
+\indexentry{algorithme!d'Euclide!g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}|hyperpage}{35}
+\indexentry{congru|hyperpage}{36}
+\indexentry{modulo|hyperpage}{36}
+\indexentry{pseudo-premier|hyperpage}{38}
-This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.4.28) 17 OCT 2013 21:26
+This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2013.11.13) 15 NOV 2013 08:37
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
**main13.tex
(./main13.tex
LaTeX2e <2011/06/27>
-Babel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo
-aded.
+Babel <3.9f> and hyphenation patterns for 4 languages loaded.
(/usr/share/texlive/texmf-dist/tex/latex/base/report.cls
Document Class: report 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size11.clo
(Font) OT1/cmtt/m/n --> OT1/lmtt/m/n on input line 38.
)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
-Package: amsmath 2000/07/18 v2.13 AMS math features
+Package: amsmath 2013/01/14 v2.14 AMS math features
\@mathmargin=\skip43
For additional information on amsmath, use the `?' option.
Package: amsopn 1999/12/14 v2.01 operator names
)
\inf@bad=\count90
-LaTeX Info: Redefining \frac on input line 211.
+LaTeX Info: Redefining \frac on input line 210.
\uproot@=\count91
\leftroot@=\count92
-LaTeX Info: Redefining \overline on input line 307.
+LaTeX Info: Redefining \overline on input line 306.
\classnum@=\count93
\DOTSCASE@=\count94
-LaTeX Info: Redefining \ldots on input line 379.
-LaTeX Info: Redefining \dots on input line 382.
-LaTeX Info: Redefining \cdots on input line 467.
+LaTeX Info: Redefining \ldots on input line 378.
+LaTeX Info: Redefining \dots on input line 381.
+LaTeX Info: Redefining \cdots on input line 466.
\Mathstrutbox@=\box26
\strutbox@=\box27
\big@size=\dimen105
-LaTeX Font Info: Redeclaring font encoding OML on input line 567.
-LaTeX Font Info: Redeclaring font encoding OMS on input line 568.
+LaTeX Font Info: Redeclaring font encoding OML on input line 566.
+LaTeX Font Info: Redeclaring font encoding OMS on input line 567.
\macc@depth=\count95
\c@MaxMatrixCols=\count96
\dotsspace@=\muskip10
\multlinegap=\skip44
\multlinetaggap=\skip45
\mathdisplay@stack=\toks22
-LaTeX Info: Redefining \[ on input line 2666.
-LaTeX Info: Redefining \] on input line 2667.
+LaTeX Info: Redefining \[ on input line 2665.
+LaTeX Info: Redefining \] on input line 2666.
)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty
-Package: amsfonts 2009/06/22 v3.00 Basic AMSFonts support
+Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support
\symAMSa=\mathgroup4
\symAMSb=\mathgroup5
LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
-(Font) U/euf/m/n --> U/euf/b/n on input line 96.
+(Font) U/euf/m/n --> U/euf/b/n on input line 106.
)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
-Package: amssymb 2009/06/22 v3.00
+Package: amssymb 2013/01/14 v3.01 AMS font symbols
)
(/usr/share/texlive/texmf-dist/tex/latex/optional/optional.sty
Package: optional 2005/01/26 ver 2.2b; Optional inclusion/omission
\FrameSep=\dimen115
)
(/usr/share/texlive/texmf-dist/tex/latex/ntheorem/ntheorem.sty
-Style `ntheorem', Version 1.31 <2011/02/16>
-Package: ntheorem 2011/02/16 1.31
+Style `ntheorem', Version 1.33 <2011/08/15>
+Package: ntheorem 2011/08/15 1.33
(/usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty
Package: ifthen 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC)
)
\theorempostskipamount=\skip48
\theoremframepreskipamount=\skip49
\theoremframepostskipamount=\skip50
+\theoreminframepreskipamount=\skip51
+\theoreminframepostskipamount=\skip52
+\theorem@preskip=\toks27
+\theorem@postskip=\toks28
+\theorem@framepreskip=\toks29
+\theorem@framepostskip=\toks30
+\theorem@inframepreskip=\toks31
+\theorem@inframepostskip=\toks32
\theoremindent=\dimen116
\theorem@indent=\dimen117
-\theoremheaderfont=\toks27
-\theoremseparator=\toks28
-\theoremprework=\toks29
-\theorempostwork=\toks30
-\theoremsymbol=\toks31
-\qedsymbol=\toks32
-\theoremkeyword=\toks33
-\qedsymbol=\toks34
-\thm@topsepadd=\skip51
-Package ntheorem Info: Option `thmmarks' loaded on input line 1066.
+\theoremrightindent=\dimen118
+\theorem@rightindent=\dimen119
+\theoremheaderfont=\toks33
+\theoremseparator=\toks34
+\theoremprework=\toks35
+\theorempostwork=\toks36
+\theoremsymbol=\toks37
+\qedsymbol=\toks38
+\theoremkeyword=\toks39
+\qedsymbol=\toks40
+\thm@topsep=\skip53
+\thm@topsepadd=\skip54
+Package ntheorem Info: Option `thmmarks' loaded on input line 1175.
\c@endNonectr=\count102
\c@currNonectr=\count103
-\mysavskip=\skip52
-Package ntheorem Info: Option `thref' loaded on input line 1066.
-\mysavskip=\skip53
+\mysavskip=\skip55
+Package ntheorem Info: Option `thref' loaded on input line 1175.
+\mysavskip=\skip56
Package ntheorem Info: option `amsmath' handling for `thref' loaded on input li
-ne 1066.
+ne 1175.
Package ntheorem Info: option `amsmath' handling for `thmmarks' loaded on input
- line 1066.
-\thm@amstmpdepth=\dimen118
-\tagwidth=\dimen119
-\shadecolor=\toks35
+ line 1175.
+\thm@amstmpdepth=\dimen120
+\tagwidth=\dimen121
+\shadecolor=\toks41
)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty
Package: graphics 2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty
Package: keyval 1999/03/16 v1.13 key=value parser (DPC)
-\KV@toks@=\toks36
+\KV@toks@=\toks42
)
-\Gin@req@height=\dimen120
-\Gin@req@width=\dimen121
+\Gin@req@height=\dimen122
+\Gin@req@width=\dimen123
)
-\epsfxsize=\dimen122
-\epsfysize=\dimen123
+\epsfxsize=\dimen124
+\epsfysize=\dimen125
)
(/usr/share/texlive/texmf-dist/tex/latex/psfrag/psfrag.sty
Package: psfrag 1998/04/11 v3.04 PSfrag (MCG)
)
(/usr/share/texlive/texmf-dist/tex/latex/subfigure/subfigure.sty
Package: subfigure 2002/03/15 v2.1.5 subfigure package
-\subfigtopskip=\skip54
-\subfigcapskip=\skip55
-\subfigcaptopadj=\dimen124
-\subfigbottomskip=\skip56
-\subfigcapmargin=\dimen125
-\subfiglabelskip=\skip57
+\subfigtopskip=\skip57
+\subfigcapskip=\skip58
+\subfigcaptopadj=\dimen126
+\subfigbottomskip=\skip59
+\subfigcapmargin=\dimen127
+\subfiglabelskip=\skip60
\c@subfigure=\count106
\c@lofdepth=\count107
\c@subtable=\count108
* Local config file subfigure.cfg used *
****************************************
(/usr/share/texlive/texmf-dist/tex/latex/subfigure/subfigure.cfg)
-\subfig@top=\skip58
-\subfig@bottom=\skip59
+\subfig@top=\skip61
+\subfig@bottom=\skip62
)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/color.sty
Package: color 2005/11/14 v1.0j Standard LaTeX Color (DPC)
Package: calc 2007/08/22 v4.3 Infix arithmetic (KKT,FJ)
\calc@Acount=\count111
\calc@Bcount=\count112
-\calc@Adimen=\dimen126
-\calc@Bdimen=\dimen127
-\calc@Askip=\skip60
-\calc@Bskip=\skip61
+\calc@Adimen=\dimen128
+\calc@Bdimen=\dimen129
+\calc@Askip=\skip63
+\calc@Bskip=\skip64
LaTeX Info: Redefining \setlength on input line 76.
LaTeX Info: Redefining \addtolength on input line 77.
\calc@Ccount=\count113
-\calc@Cskip=\skip62
+\calc@Cskip=\skip65
)
(/usr/share/texlive/texmf-dist/tex/latex/listings/listings.sty
\lst@mode=\count114
\lst@gtempboxa=\box28
-\lst@token=\toks37
+\lst@token=\toks43
\lst@length=\count115
-\lst@currlwidth=\dimen128
+\lst@currlwidth=\dimen130
\lst@column=\count116
\lst@pos=\count117
-\lst@lostspace=\dimen129
-\lst@width=\dimen130
+\lst@lostspace=\dimen131
+\lst@width=\dimen132
\lst@newlines=\count118
\lst@lineno=\count119
-\lst@maxwidth=\dimen131
+\lst@maxwidth=\dimen133
(/usr/share/texlive/texmf-dist/tex/latex/listings/lstmisc.sty
-File: lstmisc.sty 2007/02/22 1.4 (Carsten Heinz)
+File: lstmisc.sty 2013/07/10 1.5 (Carsten Heinz)
\c@lstnumber=\count120
\lst@skipnumbers=\count121
\lst@framebox=\box29
)
(/usr/share/texlive/texmf-dist/tex/latex/listings/listings.cfg
-File: listings.cfg 2007/02/22 1.4 listings configuration
+File: listings.cfg 2013/06/27 1.5pre1 listings configuration
))
-Package: listings 2007/02/22 1.4 (Carsten Heinz)
+Package: listings 2013/07/10 1.5 (Carsten Heinz)
(/usr/share/texlive/texmf-dist/tex/latex/url/url.sty
\Urlmuskip=\muskip11
)
(/usr/share/texlive/texmf-dist/tex/latex/tools/longtable.sty
Package: longtable 2004/02/01 v4.11 Multi-page Table package (DPC)
-\LTleft=\skip63
-\LTright=\skip64
-\LTpre=\skip65
-\LTpost=\skip66
+\LTleft=\skip66
+\LTright=\skip67
+\LTpre=\skip68
+\LTpost=\skip69
\LTchunksize=\count122
-\LTcapwidth=\dimen132
+\LTcapwidth=\dimen134
\LT@head=\box30
\LT@firsthead=\box31
\LT@foot=\box32
\LT@rows=\count124
\c@LT@tables=\count125
\c@LT@chunks=\count126
-\LT@p@ftn=\toks38
+\LT@p@ftn=\toks44
)
(/usr/share/texlive/texmf-dist/tex/latex/tabls/tabls.sty
-\tablinesep=\dimen133
-\arraylinesep=\dimen134
-\extrarulesep=\dimen135
-\@arstdepth=\dimen136
-\@arstheight=\dimen137
-\@skip@bove=\dimen138
+\tablinesep=\dimen135
+\arraylinesep=\dimen136
+\extrarulesep=\dimen137
+\@arstdepth=\dimen138
+\@arstheight=\dimen139
+\@skip@bove=\dimen140
)
(/usr/share/texlive/texmf-dist/tex/latex/base/textcomp.sty
Package: textcomp 2005/09/27 v1.99g Standard LaTeX package
\@slashboxc=\box36
\@slashboxwd=\count127
\@slashboxht=\count128
-\@slashsepl=\dimen139
-\@slashsepr=\dimen140
+\@slashsepl=\dimen141
+\@slashsepr=\dimen142
) (/usr/share/texlive/texmf-dist/tex/latex/psnfss/times.sty
Package: times 2005/04/12 PSNFSS-v9.2a (SPQR)
)
\c@cnt@a=\count129
\c@cnt@b=\count130
\c@cnt@c=\count131
-\dim@x=\dimen141
-\dim@y=\dimen142
+\dim@x=\dimen143
+\dim@y=\dimen144
\temp@box=\box37
-\d@my@unit=\dimen143
+\d@my@unit=\dimen145
\c@cnt@@a=\count132
\c@cnt@@b=\count133
\c@cnt@@c=\count134
)
(/usr/share/texlive/texmf-dist/tex/latex/multirow/multirow.sty
-\bigstrutjot=\dimen144
+\bigstrutjot=\dimen146
)
(/usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty
-Package: babel 2008/07/08 v3.8m The Babel package
+Package: babel 2013/05/16 v3.9f The Babel package
-(/usr/share/texlive/texmf-dist/tex/generic/babel/frenchb.ldf
-Language: frenchb 2010/08/21 v2.5a French support from the babel system
+(/usr/share/texlive/texmf-dist/tex/generic/babel-french/frenchb.ldf
+Language: frenchb 2013/07/06 v2.6e French support from the babel system
(/usr/share/texlive/texmf-dist/tex/generic/babel/babel.def
-File: babel.def 2008/07/08 v3.8m Babel common definitions
+File: babel.def 2013/05/16 v3.9f Babel common definitions
\babel@savecnt=\count135
-\U@D=\dimen145
+\U@D=\dimen147
LaTeX Warning: Command \@testdef has changed.
Check if current package is valid.
)
-
-Package babel Warning: No hyphenation patterns were loaded for
-(babel) the language `French'
-(babel) I will use the patterns loaded for \language=0 instead.
-
-\l@french = a dialect from \language0
-Package babel Info: Making : an active character on input line 234.
-Package babel Info: Making ; an active character on input line 235.
-Package babel Info: Making ! an active character on input line 236.
-Package babel Info: Making ? an active character on input line 237.
-\FB@Mht=\dimen146
-\std@mcc=\count136
-\dec@mcc=\count137
-\parindentFFN=\dimen147
-*************************************
-* Local config file frenchb.cfg used
-*
-(/usr/share/texlive/texmf-dist/tex/generic/babel/frenchb.cfg))
-(/usr/share/texlive/texmf-dist/tex/generic/babel/frenchb.ldf
-Language: frenchb 2010/08/21 v2.5a French support from the babel system
+\FB@interchartokenstateORI=\count136
+Package babel Info: Making : an active character on input line 240.
+Package babel Info: Making ; an active character on input line 241.
+Package babel Info: Making ! an active character on input line 242.
+Package babel Info: Making ? an active character on input line 243.
+\FB@Mht=\dimen148
+\std@mcc=\count137
+\dec@mcc=\count138
+\listindentFB=\skip70
+\labelwidthFB=\skip71
+\leftmarginFB=\skip72
+\parindentFFN=\dimen149
+) (/usr/share/texlive/texmf-dist/tex/generic/babel-french/frenchb.ldf
+Language: frenchb 2013/07/06 v2.6e French support from the babel system
))
(/usr/share/texlive/texmf-dist/tex/latex/carlisle/scalefnt.sty)
(/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty
(/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty
Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional
)
-\Gm@cnth=\count138
-\Gm@cntv=\count139
-\c@Gm@tempcnt=\count140
-\Gm@bindingoffset=\dimen148
-\Gm@wd@mp=\dimen149
-\Gm@odd@mp=\dimen150
-\Gm@even@mp=\dimen151
-\Gm@layoutwidth=\dimen152
-\Gm@layoutheight=\dimen153
-\Gm@layouthoffset=\dimen154
-\Gm@layoutvoffset=\dimen155
-\Gm@dimlist=\toks39
+\Gm@cnth=\count139
+\Gm@cntv=\count140
+\c@Gm@tempcnt=\count141
+\Gm@bindingoffset=\dimen150
+\Gm@wd@mp=\dimen151
+\Gm@odd@mp=\dimen152
+\Gm@even@mp=\dimen153
+\Gm@layoutwidth=\dimen154
+\Gm@layoutheight=\dimen155
+\Gm@layouthoffset=\dimen156
+\Gm@layoutvoffset=\dimen157
+\Gm@dimlist=\toks45
)
(./symboles.sty)
-\c@currExoctr=\count141
-\c@endExoctr=\count142
-\c@Exo=\count143
-\c@currExocctr=\count144
-\c@endExocctr=\count145
-\c@currExctr=\count146
-\c@endExctr=\count147
-\c@currRemctr=\count148
-\c@endRemctr=\count149
-\c@Rem=\count150
-\c@currNotationctr=\count151
-\c@endNotationctr=\count152
-\c@Notation=\count153
-\c@currDefctr=\count154
-\c@endDefctr=\count155
-\c@Def=\count156
-\c@currThctr=\count157
-\c@endThctr=\count158
-\c@Th=\count159
-\c@currCorollaryctr=\count160
-\c@endCorollaryctr=\count161
-\c@currPrectr=\count162
-\c@endPrectr=\count163
-\c@Pre=\count164
-\c@currLemmactr=\count165
-\c@endLemmactr=\count166
-\c@Lemma=\count167
-\c@currProofctr=\count168
-\c@endProofctr=\count169
-\c@Proof=\count170
-\c@currQqctr=\count171
-\c@endQqctr=\count172
-\c@Qq=\count173
+\c@currExoctr=\count142
+\c@endExoctr=\count143
+\c@Exo=\count144
+\c@currExocctr=\count145
+\c@endExocctr=\count146
+\c@currExctr=\count147
+\c@endExctr=\count148
+\c@currRemctr=\count149
+\c@endRemctr=\count150
+\c@Rem=\count151
+\c@currNotationctr=\count152
+\c@endNotationctr=\count153
+\c@Notation=\count154
+\c@currDefctr=\count155
+\c@endDefctr=\count156
+\c@Def=\count157
+\c@currThctr=\count158
+\c@endThctr=\count159
+\c@Th=\count160
+\c@currCorollaryctr=\count161
+\c@endCorollaryctr=\count162
+\c@currPrectr=\count163
+\c@endPrectr=\count164
+\c@Pre=\count165
+\c@currLemmactr=\count166
+\c@endLemmactr=\count167
+\c@Lemma=\count168
+\c@currProofctr=\count169
+\c@endProofctr=\count170
+\c@Proof=\count171
+\c@currQqctr=\count172
+\c@endQqctr=\count173
+\c@Qq=\count174
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty
-Package: hyperref 2012/05/13 v6.82q Hypertext links for LaTeX
+Package: hyperref 2012/11/06 v6.83m Hypertext links for LaTeX
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty
Package: hobsub-hyperref 2012/05/28 v1.13 Bundle oberdiek, subset hyperref (HO)
Package: refcount 2011/10/16 v3.4 Data extraction from label references (HO)
Package: hycolor 2011/01/30 v1.7 Color options for hyperref/bookmark (HO)
)
+(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/auxhook.sty
+Package: auxhook 2011/03/04 v1.3 Hooks for auxiliary files (HO)
+)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty
Package: kvoptions 2011/06/30 v3.11 Key value format for package options (HO)
)
-\@linkdim=\dimen156
-\Hy@linkcounter=\count174
-\Hy@pagecounter=\count175
+\@linkdim=\dimen158
+\Hy@linkcounter=\count175
+\Hy@pagecounter=\count176
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def
-File: pd1enc.def 2012/05/13 v6.82q Hyperref: PDFDocEncoding definition (HO)
+File: pd1enc.def 2012/11/06 v6.83m Hyperref: PDFDocEncoding definition (HO)
Now handling font encoding PD1 ...
... no UTF-8 mapping file for font encoding PD1
)
-\Hy@SavedSpaceFactor=\count176
+\Hy@SavedSpaceFactor=\count177
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg
File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive
)
-Package hyperref Info: Hyper figures OFF on input line 4062.
-Package hyperref Info: Link nesting OFF on input line 4067.
-Package hyperref Info: Hyper index ON on input line 4070.
-Package hyperref Info: Plain pages OFF on input line 4077.
-Package hyperref Info: Backreferencing OFF on input line 4082.
+Package hyperref Info: Hyper figures OFF on input line 4443.
+Package hyperref Info: Link nesting OFF on input line 4448.
+Package hyperref Info: Hyper index ON on input line 4451.
+Package hyperref Info: Plain pages OFF on input line 4458.
+Package hyperref Info: Backreferencing OFF on input line 4463.
Package hyperref Info: Implicit mode ON; LaTeX internals redefined.
-Package hyperref Info: Bookmarks ON on input line 4300.
-\c@Hy@tempcnt=\count177
-LaTeX Info: Redefining \url on input line 4653.
-\Fld@menulength=\count178
-\Field@Width=\dimen157
-\Fld@charsize=\dimen158
-Package hyperref Info: Hyper figures OFF on input line 5773.
-Package hyperref Info: Link nesting OFF on input line 5778.
-Package hyperref Info: Hyper index ON on input line 5781.
-Package hyperref Info: backreferencing OFF on input line 5788.
-Package hyperref Info: Link coloring OFF on input line 5793.
-Package hyperref Info: Link coloring with OCG OFF on input line 5798.
-Package hyperref Info: PDF/A mode OFF on input line 5803.
-LaTeX Info: Redefining \ref on input line 5843.
-LaTeX Info: Redefining \pageref on input line 5847.
-\Hy@abspage=\count179
-\c@Item=\count180
-\c@Hfootnote=\count181
+Package hyperref Info: Bookmarks ON on input line 4688.
+\c@Hy@tempcnt=\count178
+LaTeX Info: Redefining \url on input line 5041.
+\XeTeXLinkMargin=\dimen159
+\Fld@menulength=\count179
+\Field@Width=\dimen160
+\Fld@charsize=\dimen161
+Package hyperref Info: Hyper figures OFF on input line 6295.
+Package hyperref Info: Link nesting OFF on input line 6300.
+Package hyperref Info: Hyper index ON on input line 6303.
+Package hyperref Info: backreferencing OFF on input line 6310.
+Package hyperref Info: Link coloring OFF on input line 6315.
+Package hyperref Info: Link coloring with OCG OFF on input line 6320.
+Package hyperref Info: PDF/A mode OFF on input line 6325.
+LaTeX Info: Redefining \ref on input line 6365.
+LaTeX Info: Redefining \pageref on input line 6369.
+\Hy@abspage=\count180
+\c@Item=\count181
+\c@Hfootnote=\count182
)
Package hyperref Message: Driver (autodetected): hpdftex.
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def
-File: hpdftex.def 2012/05/13 v6.82q Hyperref driver for pdfTeX
-\Fld@listcount=\count182
-\c@bookmark@seq@number=\count183
+File: hpdftex.def 2012/11/06 v6.83m Hyperref driver for pdfTeX
+\Fld@listcount=\count183
+\c@bookmark@seq@number=\count184
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty
Package: rerunfilecheck 2011/04/15 v1.7 Rerun checks for auxiliary files (HO)
Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2
82.
)
-\Hy@SectionHShift=\skip67
+\Hy@SectionHShift=\skip73
)
-Package hyperref Info: Option `colorlinks' set `true' on input line 190.
-Package hyperref Info: Option `breaklinks' set `true' on input line 190.
-Package hyperref Info: Option `bookmarksopen' set `true' on input line 190.
+Package hyperref Info: Option `colorlinks' set `true' on input line 184.
+Package hyperref Info: Option `breaklinks' set `true' on input line 184.
+Package hyperref Info: Option `bookmarksopen' set `true' on input line 184.
\@indexfile=\write4
\openout4 = `main13.idx'.
(./main13.aux (./Remerciements.aux) (./PPN.aux) (./Bibliographie.aux))
\openout1 = `main13.aux'.
-LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 217.
-LaTeX Font Info: ... okay on input line 217.
-LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 217.
-LaTeX Font Info: ... okay on input line 217.
-LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 217.
-LaTeX Font Info: ... okay on input line 217.
-LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 217.
-LaTeX Font Info: ... okay on input line 217.
-LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 217.
-LaTeX Font Info: ... okay on input line 217.
-LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 217.
-LaTeX Font Info: ... okay on input line 217.
-LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 217.
-LaTeX Font Info: Try loading font information for TS1+cmr on input line 217.
+LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 211.
+LaTeX Font Info: ... okay on input line 211.
+LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 211.
+LaTeX Font Info: ... okay on input line 211.
+LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 211.
+LaTeX Font Info: ... okay on input line 211.
+LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 211.
+LaTeX Font Info: ... okay on input line 211.
+LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 211.
+LaTeX Font Info: ... okay on input line 211.
+LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 211.
+LaTeX Font Info: ... okay on input line 211.
+LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 211.
+LaTeX Font Info: Try loading font information for TS1+cmr on input line 211.
(/usr/share/texlive/texmf-dist/tex/latex/base/ts1cmr.fd
File: ts1cmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions
)
-LaTeX Font Info: ... okay on input line 217.
-LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 217.
-LaTeX Font Info: ... okay on input line 217.
-LaTeX Font Info: Try loading font information for T1+ptm on input line 217.
+LaTeX Font Info: ... okay on input line 211.
+LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 211.
+LaTeX Font Info: ... okay on input line 211.
+LaTeX Font Info: Try loading font information for T1+ptm on input line 211.
(/usr/share/texlive/texmf-dist/tex/latex/psnfss/t1ptm.fd
File: t1ptm.fd 2001/06/04 font definitions for T1/ptm.
\UseOption=
Package optional Warning: No options were selected, so all optional text will b
-e printed on input line 217.
+e printed on input line 211.
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/epstopdf-base.sty
Package: epstopdf-base 2010/02/09 v2.5 Base part for package epstopdf
))
(/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).]
-\scratchcounter=\count184
-\scratchdimen=\dimen159
+\scratchcounter=\count185
+\scratchdimen=\dimen162
\scratchbox=\box38
-\nofMPsegments=\count185
-\nofMParguments=\count186
-\everyMPshowfont=\toks40
-\MPscratchCnt=\count187
-\MPscratchDim=\dimen160
-\MPnumerator=\count188
-\makeMPintoPDFobject=\count189
-\everyMPtoPDFconversion=\toks41
+\nofMPsegments=\count186
+\nofMParguments=\count187
+\everyMPshowfont=\toks46
+\MPscratchCnt=\count188
+\MPscratchDim=\dimen163
+\MPnumerator=\count189
+\makeMPintoPDFobject=\count190
+\everyMPtoPDFconversion=\toks47
)
-\c@lstlisting=\count190
-LaTeX Info: Redefining \degres on input line 217.
-LaTeX Info: Redefining \dots on input line 217.
-LaTeX Info: Redefining \up on input line 217.
+\c@lstlisting=\count191
+LaTeX Info: Redefining \degres on input line 211.
+LaTeX Info: Redefining \dots on input line 211.
+LaTeX Info: Redefining \up on input line 211.
*geometry* driver: auto-detecting
*geometry* detected driver: pdftex
* layout: <same size as paper>
* layoutoffset:(h,v)=(0.0pt,0.0pt)
* modes:
-* h-part:(L,W,R)=(71.13188pt, 455.24411pt, 71.13188pt)
-* v-part:(T,H,B)=(71.13188pt, 702.78308pt, 71.13188pt)
+* h-part:(L,W,R)=(42.67912pt, 512.14963pt, 42.67912pt)
+* v-part:(T,H,B)=(28.45274pt, 788.14136pt, 28.45274pt)
* \paperwidth=597.50787pt
* \paperheight=845.04684pt
-* \textwidth=455.24411pt
-* \textheight=702.78308pt
-* \oddsidemargin=-1.1381pt
-* \evensidemargin=-1.1381pt
-* \topmargin=-38.1381pt
+* \textwidth=512.14963pt
+* \textheight=788.14136pt
+* \oddsidemargin=-29.59087pt
+* \evensidemargin=-29.59087pt
+* \topmargin=-80.81725pt
* \headheight=12.0pt
* \headsep=25.0pt
* \topskip=11.0pt
* (1in=72.27pt=25.4mm, 1cm=28.453pt)
\AtBeginShipoutBox=\box39
-Package hyperref Info: Link coloring ON on input line 217.
+Package hyperref Info: Link coloring ON on input line 211.
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty
-Package: nameref 2010/04/30 v2.40 Cross-referencing by name of section
+Package: nameref 2012/10/27 v2.43 Cross-referencing by name of section
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty
Package: gettitlestring 2010/12/03 v1.4 Cleanup title references (HO)
)
-\c@section@level=\count191
+\c@section@level=\count192
)
-LaTeX Info: Redefining \ref on input line 217.
-LaTeX Info: Redefining \pageref on input line 217.
-LaTeX Info: Redefining \nameref on input line 217.
+LaTeX Info: Redefining \ref on input line 211.
+LaTeX Info: Redefining \pageref on input line 211.
+LaTeX Info: Redefining \nameref on input line 211.
(./main13.out) (./main13.out)
\@outlinefile=\write5
\openout5 = `main13.out'.
-LaTeX Font Info: Try loading font information for OT1+lmr on input line 221.
+LaTeX Font Info: Try loading font information for OT1+lmr on input line 215.
(/usr/share/texmf/tex/latex/lm/ot1lmr.fd
File: ot1lmr.fd 2009/10/30 v1.6 Font defs for Latin Modern
)
-LaTeX Font Info: Try loading font information for OML+lmm on input line 221.
+LaTeX Font Info: Try loading font information for OML+lmm on input line 215.
(/usr/share/texmf/tex/latex/lm/omllmm.fd
File: omllmm.fd 2009/10/30 v1.6 Font defs for Latin Modern
)
-LaTeX Font Info: Try loading font information for OMS+lmsy on input line 221
+LaTeX Font Info: Try loading font information for OMS+lmsy on input line 215
.
(/usr/share/texmf/tex/latex/lm/omslmsy.fd
File: omslmsy.fd 2009/10/30 v1.6 Font defs for Latin Modern
)
-LaTeX Font Info: Try loading font information for OMX+lmex on input line 221
+LaTeX Font Info: Try loading font information for OMX+lmex on input line 215
.
(/usr/share/texmf/tex/latex/lm/omxlmex.fd
File: omxlmex.fd 2009/10/30 v1.6 Font defs for Latin Modern
)
LaTeX Font Info: External font `lmex10' loaded for size
-(Font) <12> on input line 221.
+(Font) <12> on input line 215.
LaTeX Font Info: External font `lmex10' loaded for size
-(Font) <8> on input line 221.
+(Font) <8> on input line 215.
LaTeX Font Info: External font `lmex10' loaded for size
-(Font) <6> on input line 221.
-LaTeX Font Info: Try loading font information for U+msa on input line 221.
+(Font) <6> on input line 215.
+LaTeX Font Info: Try loading font information for U+msa on input line 215.
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd
-File: umsa.fd 2009/06/22 v3.00 AMS symbols A
+File: umsa.fd 2013/01/14 v3.01 AMS symbols A
)
-LaTeX Font Info: Try loading font information for U+msb on input line 221.
+LaTeX Font Info: Try loading font information for U+msb on input line 215.
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd
-File: umsb.fd 2009/06/22 v3.00 AMS symbols B
+File: umsb.fd 2013/01/14 v3.01 AMS symbols B
)
-LaTeX Font Info: Try loading font information for T1+pcr on input line 221.
+LaTeX Font Info: Try loading font information for T1+pcr on input line 215.
(/usr/share/texlive/texmf-dist/tex/latex/psnfss/t1pcr.fd
File: t1pcr.fd 2001/06/04 font definitions for T1/pcr.
Non-PDF special ignored!{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]
LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <24.88> not available
-(Font) Font shape `T1/ptm/b/n' tried instead on input line 221.
+(Font) Font shape `T1/ptm/b/n' tried instead on input line 215.
(./main13.toc
LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <12> not available
been already used, duplicate ignored
<to be read again>
\relax
-l.48 ...IeC {\'e}quivalence}{32}{subsection.4.3.1}
+l.50 ...sembles de nombres entiers}{31}{chapter.5}
[1
])
No file Remerciements.tex.
LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <20.74> not available
-(Font) Font shape `T1/ptm/b/n' tried instead on input line 229.
+(Font) Font shape `T1/ptm/b/n' tried instead on input line 223.
[3
(Font) Font shape `T1/ptm/b/n' tried instead on input line 2.
Package hyperref Info: bookmark level for unknown Def defaults to 0 on input li
ne 6.
-
-Overfull \hbox (8.67114pt too wide) in paragraph at lines 6--11
-[]\T1/ptm/m/sl/10.95 On ap-pelle \T1/ptm/m/n/10.95 al-gèbre de Boole[] \T1/ptm/
-m/sl/10.95 la struc-ture al-gébrique $\OT1/lmr/m/n/10.95 (\OMS/lmsy/m/n/10.95 A
-\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 +\OML/lmm/m/it/10.95 ; :; []\OT1/lmr/m
-/n/10.95 )$
- []
-
LaTeX Font Info: External font `lmex10' loaded for size
(Font) <10> on input line 36.
LaTeX Font Info: External font `lmex10' loaded for size
ne 68.
Package hyperref Info: bookmark level for unknown Exo defaults to 0 on input li
ne 74.
-[4
+ [4
]
Package hyperref Info: bookmark level for unknown Th defaults to 0 on input lin
Package hyperref Info: bookmark level for unknown Ex defaults to 0 on input lin
e 172.
[5]
-Overfull \hbox (0.52283pt too wide) in paragraph at lines 327--329
-[]\T1/ptm/m/sl/10.95 Un \T1/ptm/m/n/10.95 monôme[] \T1/ptm/m/sl/10.95 est une f
-onc-tion booléenne pro-duit de vari-ables booléennes
+Underfull \hbox (badness 2762) in paragraph at lines 309--312
+[]\T1/ptm/m/n/10.95 Les min-termes et max-termes, pour un
[]
-[6] [7] [8]
+[6] [7]
Package hyperref Info: bookmark level for unknown Exoc defaults to 0 on input l
ine 500.
- [9]
-Overfull \hbox (0.39145pt too wide) in paragraph at lines 626--627
-[]\T1/ptm/m/it/10.95 Donner la forme canon-ique dis-jonc-tive de la fonc-tion b
-ooléeene
+ [8]
+Overfull \hbox (10.98196pt too wide) in paragraph at lines 653--655
+[]\T1/ptm/m/it/10.95 On consi-dère deux fonc-tions boo-léennes $\OML/lmm/m/it/1
+0.95 u$ \T1/ptm/m/it/10.95 et $\OML/lmm/m/it/10.95 v$ \T1/ptm/m/it/10.95 des qu
+atres
[]
-[10]) [11]
+[9]) [10]
Chapitre 2.
-(./logique/Propositions13.texpdfTeX warning (ext4): destination with the same i
-dentifier (name{Def.1}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.34 \begin{Def}
- [Proposition]pdfTeX warning (ext4): destination with the same i
-dentifier (name{Exo.1}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.43 \begin{Ex}
- [12
+(./logique/Propositions13.tex [11
-]pdfTeX warning (ext4): destination with the same identifier (name{Rem.1}) has
-been already used, duplicate ignored
-<to be read again>
- \relax
-l.175 \begin{Rem}
- [13]pdfTeX warning (ext4): destination with the same identifi
-er (name{Exo.2}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.197 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Rem.2}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.219 \begin{Rem}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.3}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.225 \begin{Exoc}
- pdfTeX warning (ext4): destination with the same identifier (
-name{Exo.4}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.256 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.5}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.289 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.6}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.300 \begin{Exo}
- [14]
-LaTeX Font Info: Try loading font information for TS1+ptm on input line 309.
+]
+Overfull \hbox (0.7556pt too wide) in paragraph at lines 220--222
+[]\T1/ptm/m/n/10.95 Même re-marque que pour l'im-pli-ca-tion lo-gique : l'équi-
+va-lence lo-gique de deux pro-po-si-tions fausses
+ []
+[12]
+LaTeX Font Info: Try loading font information for TS1+ptm on input line 309.
-(/usr/share/texlive/texmf-dist/tex/latex/psnfss/ts1ptm.fd
+ (/usr/share/texlive/texmf-dist/tex/latex/psnfss/ts1ptm.fd
File: ts1ptm.fd 2001/06/04 font definitions for TS1/ptm.
-)pdfTeX warning (ext4): destination with the same identifier (name{Th.1}) has b
-een already used, duplicate ignored
-<to be read again>
- \relax
-l.400 \begin{Th}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Rem.3}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.420 \begin{Rem}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.7}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.431 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.8}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.452 \begin{Exo}
- [15]pdfTeX warning (ext4): destination with the same identifi
-er (name{Exo.9}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.474 \begin{Exo}
-
-Overfull \hbox (2.21603pt too wide) in paragraph at lines 482--487
-[]\T1/ptm/m/n/10.95 Lorsqu'on rem-place, dans une for-mule propo-si-tion-nelle,
- les vari-ables propo-si-tion-nelles par des propo-
+) [13]
+Underfull \hbox (badness 2717) in paragraph at lines 489--491
+[]\T1/ptm/m/n/10.95 Les conven-tions de prio-rité des
[]
-pdfTeX warning (ext4): destination with the same identifier (name{Th.2}) has be
-en already used, duplicate ignored
-<to be read again>
- \relax
-l.488 \begin{Th}
- [Règles de priorité des connecteurs logiques]
-Underfull \hbox (badness 10000) in paragraph at lines 489--491
-[]\T1/ptm/m/n/10.95 Les con-ven-tions de
- []
-
-pdfTeX warning (ext4): destination with the same identifier (name{Exo.10}) has
-been already used, duplicate ignored
-<to be read again>
- \relax
-l.499 \begin{Ex}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Th.3}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.507 \begin{Th}
- [Associativité des opérateurs $\ou$ et $\et$]pdfTeX warning (
-ext4): destination with the same identifier (name{Exo.11}) has been already use
-d, duplicate ignored
-<to be read again>
- \relax
-l.528 \begin{Exoc}
- [16]pdfTeX warning (ext4): destination with the same identif
-ier (name{Exo.12}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.573 \begin{Exoc}
- pdfTeX warning (ext4): destination with the same identifier (
-name{Exo.13}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.602 \begin{Exo}
- [17]pdfTeX warning (ext4): destination with the same identifi
-er (name{Def.2}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.750 \begin{Def}
- [Fonction de vérité de $F$]pdfTeX warning (ext4): destinatio
-n with the same identifier (name{Exo.14}) has been already used, duplicate igno
-red
-<to be read again>
- \relax
-l.803 \begin{Ex}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Def.3}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.842 \begin{Def}
- [Tautologie]
+[14] [15]
Package hyperref Info: bookmark level for unknown Notation defaults to 0 on inp
ut line 857.
-pdfTeX warning (ext4): destination with the same identifier (name{Exo.15}) has
-been already used, duplicate ignored
-<to be read again>
- \relax
-l.861 \begin{Ex}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Exo.16}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.867 \begin{Ex}
- [18]pdfTeX warning (ext4): destination with the same identifie
-r (name{Exo.17}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.923 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Def.4}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.950 \begin{Def}
- [Antilogie]pdfTeX warning (ext4): destination with the same id
-entifier (name{Exo.18}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.958 \begin{Ex}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Exo.19}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.969 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Def.5}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.1005 \begin{Def}
- [Conséquence logique]
-Overfull \hbox (9.10168pt too wide) in paragraph at lines 1015--1017
-[]\T1/ptm/m/n/10.95 On note ce ré-sul-tat : $\OMS/lmsy/m/n/10.95 f\OML/lmm/m/it
-/10.95 F[]; [] ; F[]\OMS/lmsy/m/n/10.95 g j[]\OT1/lmr/m/n/10.95 = \OML/lmm/m/it
-/10.95 A$ \T1/ptm/m/n/10.95 (se lit : $\OML/lmm/m/it/10.95 A$ \T1/ptm/m/n/10.95
- est con-séquence logique de $\OMS/lmsy/m/n/10.95 f\OML/lmm/m/it/10.95 F[]; []
-; F[]\OMS/lmsy/m/n/10.95 g$\T1/ptm/m/n/10.95 ). $$
+ [16] [17] [18]
+Overfull \hbox (20.57855pt too wide) in paragraph at lines 1352--1361
+\OMS/lmsy/m/n/10.95 f\OML/lmm/m/it/10.95 G[]; G[]; [] ; G[]\OMS/lmsy/m/n/10.95
+g$ \T1/ptm/m/n/10.95 sont vraies, $\OML/lmm/m/it/10.95 G[] \OMS/lmsy/m/n/10.95
+) \OML/lmm/m/it/10.95 H$ \T1/ptm/m/n/10.95 est vraie. Re-gar-dons si $\OML/lmm/
+m/it/10.95 H$ \T1/ptm/m/n/10.95 est une consé-quence lo-gique de $\OMS/lmsy/m/n
+/10.95 f\OML/lmm/m/it/10.95 G[]; G[]; [] ; G[]\OMS/lmsy/m/n/10.95 g$
[]
-pdfTeX warning (ext4): destination with the same identifier (name{Exo.20}) has
-been already used, duplicate ignored
-<to be read again>
- \relax
-l.1023 \begin{Ex}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.21}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.1065 \begin{Exo}
- [19]pdfTeX warning (ext4): destination with the same identif
-ier (name{Exo.22}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.1086 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (
-name{Def.6}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.1119 \begin{Def}
- [Formules équivalentes]
-Overfull \hbox (12.18623pt too wide) in paragraph at lines 1120--1124
-[]\T1/ptm/m/sl/10.95 Si la for-mule propo-si-tion-nelle $\OML/lmm/m/it/10.95 G$
- \T1/ptm/m/sl/10.95 est con-séquence logique
- []
+[19] [20]) [21] [22
-[20]pdfTeX warning (ext4): destination with the same identifier (name{Th.4}) ha
-s been already used, duplicate ignored
-<to be read again>
- \relax
-l.1270 \begin{Th}
- [Théorème de substitution]
-Underfull \hbox (badness 1552) in paragraph at lines 1272--1277
-[]\T1/ptm/m/n/10.95 Soit $\OML/lmm/m/it/10.95 F$ \T1/ptm/m/n/10.95 une for-mule
- propo-si-tion-nelle dans
+]
+Chapitre 3.
+(./ensembles/IntroAuxEnsembles13.tex [23
+
+] [24]
+Overfull \hbox (37.774pt too wide) in paragraph at lines 186--187
+[]\T1/ptm/m/it/10.95 Faire la réunion des en-sembles $\OML/lmm/m/it/10.95 A$ \T
+1/ptm/m/it/10.95 et $\OML/lmm/m/it/10.95 B$\T1/ptm/m/it/10.95 , quand $\OML/lmm
+/m/it/10.95 A \OT1/lmr/m/n/10.95 = \OMS/lmsy/m/n/10.95 f\OML/lmm/m/it/10.95 x \
+OMS/lmsy/m/n/10.95 2 \U/msb/m/n/10.95 N\OMS/lmsy/m/n/10.95 j\OML/lmm/m/it/10.95
+ x[]\OMS/lmsy/m/n/10.95 g$\T1/ptm/m/it/10.95 , et $\OML/lmm/m/it/10.95 B \OT1/l
+mr/m/n/10.95 = \OMS/lmsy/m/n/10.95 f\OML/lmm/m/it/10.95 x \OMS/lmsy/m/n/10.95 2
+ \U/msb/m/n/10.95 N\OMS/lmsy/m/n/10.95 j\OML/lmm/m/it/10.95 x[]\OMS/lmsy/m/n/10
+.95 g$\T1/ptm/m/it/10.95 .
[]
-pdfTeX warning (ext4): destination with the same identifier (name{Proof.1}) has
- been already used, duplicate ignored
-<to be read again>
- \relax
-l.1284 \begin{Proof}
- [21]pdfTeX warning (ext4): destination with the same ident
-ifier (name{Th.5}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.1326 \begin{Th}
- [Théorème de la validité]pdfTeX warning (ext4): destination
- with the same identifier (name{Proof.2}) has been already used, duplicate igno
-red
-<to be read again>
- \relax
-l.1338 \begin{Proof}
- [22]
-Overfull \hbox (1.48752pt too wide) in paragraph at lines 1490--1492
-[]\T1/ptm/m/it/10.95 Après une in-tense réflex-ion, l'in-specteur fait sor-tir
-l'un des deux de l'hôpi-tal. Lequel (et pourquoi?)
+[25]
+Overfull \hbox (2.81169pt too wide) in paragraph at lines 301--302
+[]\T1/ptm/m/it/10.95 On ap-pelle fonc-tion ca-rac-té-ris-tique de la par-
[]
-[23]) [24] [25
+) [26]
+Chapitre 4.
+(./ensembles/relbin13.tex [27
-]
-Chapitre 3.
-(./ensembles/IntroAuxEnsembles13.texpdfTeX warning (ext4): destination with the
- same identifier (name{Notation.1}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.16 \begin{Notation}
- pdfTeX warning (ext4): destination with the same identifie
-r (name{Exo.1}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.20 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Def.1}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.55 \begin{Def}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Th.1}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.60 \begin{Th}
- [26
+] [28]
+Package hyperref Info: bookmark level for unknown Pre defaults to 0 on input li
+ne 301.
-]pdfTeX warning (ext4): destination with the same identifier (name{Proof.1}) ha
-s been already used, duplicate ignored
-<to be read again>
- \relax
-l.64 \begin{Proof}
- pdfTeX warning (ext4): destination with the same identifier (
-name{Th.2}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.72 \begin{Th}
- pdfTeX warning (ext4): destination with the same identifier (nam
-e{Def.2}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.77 \begin{Def}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Th.3}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.83 \begin{Th}
- pdfTeX warning (ext4): destination with the same identifier (nam
-e{Exo.2}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.88 \begin{Ex}
- pdfTeX warning (ext4): destination with the same identifier (nam
-e{Exo.3}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.93 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Exo.4}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.98 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Exo.5}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.110 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.6}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.117 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.7}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.122 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Def.3}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.131 \begin{Def}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.8}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.139 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Def.4}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.151 \begin{Def}
- [Reunion]\index{réunion} [27]pdfTeX warning (ext4): destinati
-on with the same identifier (name{Exo.9}) has been already used, duplicate igno
-red
-<to be read again>
- \relax
-l.155 \begin{Ex}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Def.5}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.160 \begin{Def}
- [Intersection]\index{réunion}
-Overfull \hbox (1.73936pt too wide) in paragraph at lines 161--163
-[]\T1/ptm/m/sl/10.95 L'\T1/ptm/m/n/10.95 intersection \T1/ptm/m/sl/10.95 des de
-ux en-sem-bles $\OML/lmm/m/it/10.95 A$ \T1/ptm/m/sl/10.95 et $\OML/lmm/m/it/10.
-95 B$\T1/ptm/m/sl/10.95 , notée $\OML/lmm/m/it/10.95 A \OMS/lmsy/m/n/10.95 \ \O
-ML/lmm/m/it/10.95 B$\T1/ptm/m/sl/10.95 , est l'ensem-
+Overfull \hbox (350.24652pt too wide) in paragraph at lines 344--350
+[]\T1/ptm/m/it/10.95 Soit $\OMS/lmsy/m/n/10.95 R$ \T1/ptm/m/it/10.95 la re-la-t
+ion d'équi-va-lence sui-vante dans l'en-semble $\OML/lmm/m/it/10.95 A \OT1/lmr/
+m/n/10.95 = \OMS/lmsy/m/n/10.95 f\OT1/lmr/m/n/10.95 1\OML/lmm/m/it/10.95 ; \OT1
+/lmr/m/n/10.95 2\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 3\OML/lmm/m/it/10.95 ;
+ \OT1/lmr/m/n/10.95 4\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 5\OML/lmm/m/it/10
+.95 ; \OT1/lmr/m/n/10.95 6\OMS/lmsy/m/n/10.95 g$ \T1/ptm/m/it/10.95 : $\OMS/lms
+y/m/n/10.95 R \OT1/lmr/m/n/10.95 = \OMS/lmsy/m/n/10.95 f\OT1/lmr/m/n/10.95 (1\O
+ML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 1)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.
+95 (1\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 5)\OML/lmm/m/it/10.95 ; \OT1/lmr/
+m/n/10.95 (2\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 2)\OML/lmm/m/it/10.95 ; \O
+T1/lmr/m/n/10.95 (2\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 3)\OML/lmm/m/it/10.
+95 ; \OT1/lmr/m/n/10.95 (2\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 6)\OML/lmm/m
+/it/10.95 ; \OT1/lmr/m/n/10.95 (3\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 2)\OM
+L/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 (3\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.9
+5 3)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 (3\OML/lmm/m/it/10.95 ; \OT1/lmr/m
+/n/10.95 6)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 (4\OML/lmm/m/it/10.95 ; \OT
+1/lmr/m/n/10.95 4)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 (5\OML/lmm/m/it/10.9
+5 ; \OT1/lmr/m/n/10.95 1)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 (5\OML/lmm/m/
+it/10.95 ; \OT1/lmr/m/n/10.95 5)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 (6\OML
+/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 2)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95
+ (6\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 3)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/
+n/10.95 (6\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 6)\OMS/lmsy/m/n/10.95 g\OML/
+lmm/m/it/10.95 :$
[]
-pdfTeX warning (ext4): destination with the same identifier (name{Th.4}) has be
-en already used, duplicate ignored
-<to be read again>
- \relax
-l.171 \begin{Th}
- [Propriétés de la réunion et de l'intersection]
-Underfull \hbox (badness 1496) in paragraph at lines 172--173
-[]\T1/ptm/m/n/10.95 La réu-nion de deux
+) [29] [30
+
+]
+Chapitre 5.
+(./arithmetique/entiersNaturels13.tex
+Overfull \hbox (11.42195pt too wide) in paragraph at lines 134--134
+[]\T1/ptm/m/n/10.95 L'écriture d'un en-tier $\OML/lmm/m/it/10.95 n$ \T1/ptm/m/n
+/10.95 sous la forme $\OML/lmm/m/it/10.95 n \OT1/lmr/m/n/10.95 = \OML/lmm/m/it/
+10.95 a[]b[]c[] []$\T1/ptm/m/n/10.95 ,
[]
-pdfTeX warning (ext4): destination with the same identifier (name{Exo.10}) has
-been already used, duplicate ignored
-<to be read again>
- \relax
-l.181 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.11}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.185 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Th.5}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.190 \begin{Th}
- [Distributivités de $\cup$ et $\cap$]pdfTeX warning (ext4): de
-stination with the same identifier (name{Exo.12}) has been already used, duplic
-ate ignored
-<to be read again>
- \relax
-l.198 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Def.6}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.205 \begin{Def}
- [Complémentation]pdfTeX warning (ext4): destination with the
-same identifier (name{Th.6}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.211 \begin{Th}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Exo.13}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.221 \begin{Exo}
- [28]pdfTeX warning (ext4): destination with the same identifi
-er (name{Exo.14}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.247 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.15}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.257 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.16}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.277 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.17}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.300 \begin{Exo}
- [Fonction caractéristique des parties d'un ensemble])
-[29]
-Chapitre 4.
-(./ensembles/relbin13.texpdfTeX warning (ext4): destination with the same ident
-ifier (name{Def.1}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.6 \begin{Def}
- [Relation binaire]pdfTeX warning (ext4): destination with the sa
-me identifier (name{Exo.1}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.15 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Rem.1}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.27 \begin{Rem}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Def.2}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.54 \begin{Def}
- [Réflexivité]pdfTeX warning (ext4): destination with the same
- identifier (name{Def.3}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.61 \begin{Def}
- [Antisymétrie]pdfTeX warning (ext4): destination with the same
- identifier (name{Def.4}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.71 \begin{Def}
- [Transitivité]pdfTeX warning (ext4): destination with the same
- identifier (name{Exo.2}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.81 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Exo.3}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.93 \begin{Exo}
- [30
+[31
+
+]
+LaTeX Font Info: External font `lmex10' loaded for size
+(Font) <14.4> on input line 255.
+
+
+Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
+(hyperref) removing `math shift' on input line 255.
+
+
+Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
+(hyperref) removing `math shift' on input line 255.
+
+LaTeX Font Info: Font shape `T1/pcr/bx/n' in size <10.95> not available
+(Font) Font shape `T1/pcr/b/n' tried instead on input line 293.
+[32] [33] [34] [35]
+
+Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
+(hyperref) removing `math shift' on input line 710.
+
+
+Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
+(hyperref) removing `math shift' on input line 710.
+
+[36]
+Underfull \hbox (badness 10000) in paragraph at lines 842--843
-]pdfTeX warning (ext4): destination with the same identifier (name{Def.5}) has
-been already used, duplicate ignored
-<to be read again>
- \relax
-l.115 \begin{Def}
- [Relation d'ordre]pdfTeX warning (ext4): destination with the
-same identifier (name{Exo.4}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.120 \begin{Ex}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Exo.5}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.124 \begin{Ex}
- [Relation de divisibilité]pdfTeX warning (ext4): destination w
-ith the same identifier (name{Exo.6}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.145 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.7}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.156 \begin{Exo}
- [Diagrammes de transitivité]pdfTeX warning (ext4): destinatio
-n with the same identifier (name{Def.6}) has been already used, duplicate ignor
-ed
-<to be read again>
- \relax
-l.185 \begin{Def}
- [Relation symétrique]pdfTeX warning (ext4): destination with
-the same identifier (name{Def.7}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.193 \begin{Def}
- [Relation d'équivalence]
-Overfull \hbox (1.20601pt too wide) in paragraph at lines 194--196
-[]$\OMS/lmsy/m/n/10.95 R$ \T1/ptm/m/sl/10.95 est une re-la-tion d'équiv-a-lence
- lorsqu'elle est réflex-
[]
-pdfTeX warning (ext4): destination with the same identifier (name{Exo.8}) has b
-een already used, duplicate ignored
-<to be read again>
- \relax
-l.199 \begin{Ex}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Exo.9}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.205 \begin{Ex}
- [Relation de congruence modulo $n$ dans $\Z$] [31]pdfTeX warnin
-g (ext4): destination with the same identifier (name{Exo.10}) has been already
-used, duplicate ignored
-<to be read again>
- \relax
-l.225 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Notation.1}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.250 \begin{Notation}
- pdfTeX warning (ext4): destination with the same identifi
-er (name{Exo.11}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.255 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.12}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.264 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Th.1}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.294 \begin{Th}
-
-Package hyperref Info: bookmark level for unknown Pre defaults to 0 on input li
-ne 301.
-pdfTeX warning (ext4): destination with the same identifier (name{Th.2}) has be
-en already used, duplicate ignored
-<to be read again>
- \relax
-l.318 \begin{Th}
- [32]pdfTeX warning (ext4): destination with the same identifie
-r (name{Exo.13}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.328 \begin{Ex}
- pdfTeX warning (ext4): destination with the same identifier (na
-me{Exo.14}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.343 \begin{Exo}
- pdfTeX warning (ext4): destination with the same identifier (n
-ame{Exo.15}) has been already used, duplicate ignored
-<to be read again>
- \relax
-l.355 \begin{Exo}
- ) [33] [34
+[37]) [38] [39
]
\openout2 = `PPN.aux'.
(./PPN.tex
-Chapitre 5.
-) [35
+Chapitre 6.
+) [40
]
No file main13.ind.
-(./main13.bbl) [36
+(./main13.bbl) [41
]
(./Bibliographie.tex
LaTeX Font Info: Font shape `T1/ptm/bx/it' in size <10.95> not available
(Font) Font shape `T1/ptm/b/it' tried instead on input line 5.
-
-Overfull \hbox (1.34502pt too wide) in paragraph at lines 5--6
-[]\T1/ptm/m/n/10.95 ] : les thèmes abor-dés sont
- []
-
-
-Overfull \hbox (11.59552pt too wide) in paragraph at lines 13--14
-[]\T1/ptm/m/n/10.95 ] : Pour un pub-lic aver-tis, souhai-
- []
-
-) [37
+) [42
]
Package atveryend Info: Empty hook `BeforeClearDocument' on input line 336.
Package atveryend Info: Empty hook `AfterLastShipout' on input line 336.
- (./main13.aux (./Remerciements.aux) (./PPN.aux) (./Bibliographie.aux))
+ (./main13.aux
+(./Remerciements.aux) (./PPN.aux) (./Bibliographie.aux))
Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 336.
Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 336.
Package rerunfilecheck Info: File `main13.out' has not changed.
-(rerunfilecheck) Checksum: B510A10B4C2809C1121D155E1D0BF426;3178.
+(rerunfilecheck) Checksum: 10B0C3771DF7ECC4CEE9D23226FE986C;3951.
Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 336.
)
Here is how much of TeX's memory you used:
- 11789 strings out of 495059
- 159454 string characters out of 3182030
- 286421 words of memory out of 3000000
- 14355 multiletter control sequences out of 15000+200000
- 97094 words of font info for 97 fonts, out of 3000000 for 9000
+ 12395 strings out of 495002
+ 167172 string characters out of 6180261
+ 283229 words of memory out of 5000000
+ 14738 multiletter control sequences out of 15000+600000
+ 109186 words of font info for 111 fonts, out of 8000000 for 9000
14 hyphenation exceptions out of 8191
- 30i,13n,32p,469b,618s stack positions out of 5000i,500n,10000p,200000b,50000s
+ 30i,13n,32p,479b,676s stack positions out of 5000i,500n,10000p,200000b,80000s
{/usr/share/texlive/texmf-dist/fonts/enc/dvips/base/8r.enc}{/usr/share/texmf/
-fonts/enc/dvips/lm/lm-mathex.enc}{/usr/share/texmf/fonts/enc/dvips/lm/lm-mathsy
-.enc}{/usr/share/texmf/fonts/enc/dvips/lm/lm-mathit.enc}{/usr/share/texmf/fonts
+fonts/enc/dvips/lm/lm-mathit.enc}{/usr/share/texmf/fonts/enc/dvips/lm/lm-mathex
+.enc}{/usr/share/texmf/fonts/enc/dvips/lm/lm-mathsy.enc}{/usr/share/texmf/fonts
/enc/dvips/lm/lm-rm.enc}</usr/share/texlive/texmf-dist/fonts/type1/public/amsfo
nts/latxfont/line10.pfb></usr/share/texmf/fonts/type1/public/lm/lmex10.pfb></us
r/share/texmf/fonts/type1/public/lm/lmmi10.pfb></usr/share/texmf/fonts/type1/pu
-blic/lm/lmmi6.pfb></usr/share/texmf/fonts/type1/public/lm/lmmi8.pfb></usr/share
-/texmf/fonts/type1/public/lm/lmr10.pfb></usr/share/texmf/fonts/type1/public/lm/
-lmr6.pfb></usr/share/texmf/fonts/type1/public/lm/lmr8.pfb></usr/share/texmf/fon
-ts/type1/public/lm/lmsy10.pfb></usr/share/texmf/fonts/type1/public/lm/lmsy6.pfb
-></usr/share/texmf/fonts/type1/public/lm/lmsy8.pfb></usr/share/texlive/texmf-di
-st/fonts/type1/public/amsfonts/symbols/msam10.pfb></usr/share/texlive/texmf-dis
-t/fonts/type1/public/amsfonts/symbols/msbm10.pfb></usr/share/texlive/texmf-dist
-/fonts/type1/urw/courier/ucrr8a.pfb></usr/share/texlive/texmf-dist/fonts/type1/
-urw/times/utmb8a.pfb></usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmbi
-8a.pfb></usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmr8a.pfb></usr/sh
-are/texlive/texmf-dist/fonts/type1/urw/times/utmr8a.pfb></usr/share/texlive/tex
-mf-dist/fonts/type1/urw/times/utmri8a.pfb>
-Output written on main13.pdf (38 pages, 320746 bytes).
+blic/lm/lmmi12.pfb></usr/share/texmf/fonts/type1/public/lm/lmmi6.pfb></usr/shar
+e/texmf/fonts/type1/public/lm/lmmi8.pfb></usr/share/texmf/fonts/type1/public/lm
+/lmr10.pfb></usr/share/texmf/fonts/type1/public/lm/lmr6.pfb></usr/share/texmf/f
+onts/type1/public/lm/lmr8.pfb></usr/share/texmf/fonts/type1/public/lm/lmsy10.pf
+b></usr/share/texmf/fonts/type1/public/lm/lmsy6.pfb></usr/share/texmf/fonts/typ
+e1/public/lm/lmsy8.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfon
+ts/symbols/msam10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfont
+s/symbols/msbm10.pfb></usr/share/texlive/texmf-dist/fonts/type1/urw/courier/ucr
+b8a.pfb></usr/share/texlive/texmf-dist/fonts/type1/urw/courier/ucrr8a.pfb></usr
+/share/texlive/texmf-dist/fonts/type1/urw/times/utmb8a.pfb></usr/share/texlive/
+texmf-dist/fonts/type1/urw/times/utmbi8a.pfb></usr/share/texlive/texmf-dist/fon
+ts/type1/urw/times/utmr8a.pfb></usr/share/texlive/texmf-dist/fonts/type1/urw/ti
+mes/utmr8a.pfb></usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmri8a.pfb
+>
+Output written on main13.pdf (43 pages, 369126 bytes).
PDF statistics:
- 792 PDF objects out of 1000 (max. 8388607)
- 725 compressed objects within 8 object streams
- 327 named destinations out of 1000 (max. 500000)
- 337 words of extra memory for PDF output out of 10000 (max. 10000000)
+ 1150 PDF objects out of 1200 (max. 8388607)
+ 1073 compressed objects within 11 object streams
+ 566 named destinations out of 1000 (max. 500000)
+ 425 words of extra memory for PDF output out of 10000 (max. 10000000)
\BOOKMARK [2][]{subsection.4.2.2}{Relation d'ordre}{section.4.2}% 37
\BOOKMARK [1][]{section.4.3}{Relations d'\351quivalence}{chapter.4}% 38
\BOOKMARK [2][]{subsection.4.3.1}{Classes d'\351quivalence}{section.4.3}% 39
-\BOOKMARK [-1][]{part.3}{III Annexes}{}% 40
-\BOOKMARK [0][]{chapter.5}{Programme P\351dagogique National 2005 \(PPN\)}{part.3}% 41
-\BOOKMARK [0][]{chapter.5}{Index}{part.3}% 42
+\BOOKMARK [-1][]{part.3}{III Arithm\351tique}{}% 40
+\BOOKMARK [0][]{chapter.5}{Ensembles de nombres entiers}{part.3}% 41
+\BOOKMARK [1][]{section.5.1}{Principe de r\351currence }{chapter.5}% 42
+\BOOKMARK [1][]{section.5.2}{Nombres premiers}{chapter.5}% 43
+\BOOKMARK [1][]{section.5.3}{Division euclidienne dans Z et applications}{chapter.5}% 44
+\BOOKMARK [1][]{section.5.4}{Algorithmes d'Euclide}{chapter.5}% 45
+\BOOKMARK [2][]{subsection.5.4.1}{L'algorithme initial}{section.5.4}% 46
+\BOOKMARK [2][]{subsection.5.4.2}{Algorithme d'Euclide g\351n\351ralis\351}{section.5.4}% 47
+\BOOKMARK [2][]{subsection.5.4.3}{L'algorithme.}{section.5.4}% 48
+\BOOKMARK [2][]{subsection.5.4.4}{Exemple.}{section.5.4}% 49
+\BOOKMARK [1][]{section.5.5}{Arithm\351tique modulo n}{chapter.5}% 50
+\BOOKMARK [-1][]{part.4}{IV Annexes}{}% 51
+\BOOKMARK [0][]{chapter.6}{Programme P\351dagogique National 2005 \(PPN\)}{part.4}% 52
+\BOOKMARK [0][]{chapter.6}{Index}{part.4}% 53
\input{symboles.sty}
-\geometry{hmargin=2.5cm, vmargin=2.5cm }
+\geometry{hmargin=1.5cm, vmargin=1cm }
\theoremstyle{plain}
\theoremstyle{plain}
\theoremheaderfont{\normalfont\bfseries\sc}
-\theorembodyfont{\slshape}
-\theoremsymbol{\ensuremath{\diamondsuit}}
-\theoremprework{\bigskip}
+\theorembodyfont{\normalfont}
+%\theoremsymbol{\ensuremath{\diamondsuit}}
+%\theoremprework{\bigskip}
\theoremseparator{.}
\newtheorem{Def}{Définition}[chapter]
%\theoremstyle{break}
\theoremstyle{plain}
-%\theoremheaderfont{\normalfont\bfseries\sc}
\theoremheaderfont{\normalfont\sc}
\theorembodyfont{\normalfont}
-\theoremsymbol{\ensuremath{}}
-%\theoremsymbol{\rule{1ex}{1ex}}
\theoremseparator{ :}
-%\theoremnumbering{roman}
-%\theoremprework{\begin{block}}
-%\theorempostwork{\end{block}}
\newframedtheorem{Th}{Propriété}[chapter]
-\theoremindent0cm
-\theoremsymbol{\ensuremath{\spadesuit}}
+%\theoremindent0cm
+%\theoremsymbol{\ensuremath{\spadesuit}}
\theoremnumbering{arabic}
-\theoremprework{\begin{block}}
-\theorempostwork{\end{block}}
+%\theoremprework{\begin{block}}
+%\theorempostwork{\end{block}}
\newtheorem{Corollary}[Th]{Corollary}
% \chapter{Relations $n$-aires}
% \input{ensembles/relnaire}
-% \part{Arithmétique}
-% \chapter{Ensembles de nombres entiers}
-% \input{arithmetique/entiersNaturels}
+ \part{Arithmétique}
+ \chapter{Ensembles de nombres entiers}
+ \input{arithmetique/entiersNaturels13}
+
+ % \chapter{Représentation des nombres entiers en machine}
+ % \input{arithmetique/enInfo13}
+
+
+
+ % \chapter{Représentation des nombres réels en machine}
-% \chapter{Représentation des nombres réels en machine}
% \input{arithmetique/representation}
% \chapter{Cryptologie et arithmétique.}
-\contentsline {Def}{{Définition}{1.{1}}{Algèbre de Boole}}{4}{Def.1}
-\contentsline {Rem}{{Remarque}{1.{1}}{}}{4}{Rem.1}
-\contentsline {Exo}{{Exercice}{1.{1}}{Somme disjonctive}}{4}{Exo.1}
-\contentsline {Exo}{{Exercice}{1.{2}}{Opérateurs de Sheffer et de Peirce}}{4}{Exo.2}
-\contentsline {Th}{{Propriété}{1.{1}}{Suppression de redondance}}{5}{Th.1}
-\contentsline {Proof}{{Preuve}{1.{$\alpha $}}{}}{5}{Proof.1}
-\contentsline {Ex}{{Exemple}{1.{3}}{}}{5}{Exo.3}
-\contentsline {Exo}{{Exercice}{1.{4}}{}}{5}{Exo.4}
-\contentsline {Exo}{{Exercice}{1.{5}}{Calcul booléen élémentaire}}{5}{Exo.5}
-\contentsline {Exo}{{Exercice}{1.{6}}{Calcul booléen}}{5}{Exo.6}
-\contentsline {Def}{{Définition}{1.{2}}{Fonction booléenne}}{6}{Def.2}
-\contentsline {Ex}{{Exemple}{1.{7}}{}}{6}{Exo.7}
-\contentsline {Def}{{Définition}{1.{3}}{Aspect d'une variable}}{6}{Def.3}
-\contentsline {Def}{{Définition}{1.{4}}{Fonction booléenne nulle}}{6}{Def.4}
-\contentsline {Def}{{Définition}{1.{5}}{Fonction référentiel}}{6}{Def.5}
-\contentsline {Def}{{Définition}{1.{6}}{Minterme, maxterme}}{6}{Def.6}
-\contentsline {Ex}{{Exemple}{1.{8}}{Minterme à trois variables}}{6}{Exo.8}
-\contentsline {Ex}{{Exemple}{1.{9}}{Maxterme à trois variables}}{6}{Exo.9}
-\contentsline {Exo}{{Exercice}{1.{10}}{}}{6}{Exo.10}
-\contentsline {Exo}{{Exercice}{1.{11}}{}}{6}{Exo.11}
-\contentsline {Th}{{Propriété}{1.{2}}{Nombre de mintermes et de maxtermes}}{6}{Th.2}
-\contentsline {Def}{{Définition}{1.{7}}{Monômes}}{7}{Def.7}
-\contentsline {Exo}{{Exercice}{1.{12}}{}}{7}{Exo.12}
-\contentsline {Th}{{Propriété}{1.{3}}{}}{7}{Th.3}
-\contentsline {Proof}{{Preuve}{1.{$\beta $}}{}}{7}{Proof.2}
-\contentsline {Th}{{Propriété}{1.{4}}{}}{7}{Th.4}
-\contentsline {Proof}{{Preuve}{1.{$\gamma $}}{}}{7}{Proof.3}
-\contentsline {Th}{{Propriété}{1.{5}}{Forme canonique disjonctive}}{7}{Th.5}
-\contentsline {Rem}{{Remarque}{1.{2}}{}}{7}{Rem.2}
-\contentsline {Th}{{Propriété}{1.{6}}{Forme canonique conjonctive}}{8}{Th.6}
-\contentsline {Ex}{{Exemple}{1.{13}}{}}{8}{Exo.13}
-\contentsline {Ex}{{Exemple}{1.{14}}{}}{8}{Exo.14}
-\contentsline {Rem}{{Remarque}{1.{3}}{}}{8}{Rem.3}
-\contentsline {Exo}{{Exercice}{1.{15}}{}}{8}{Exo.15}
-\contentsline {Exoc}{{Exercice (corrigé)}{1.{16}}{}}{9}{Exo.16}
-\contentsline {Exo}{{Exercice}{1.{17}}{Fonctions booléennes}}{10}{Exo.17}
-\contentsline {Exo}{{Exercice}{1.{18}}{}}{10}{Exo.18}
-\contentsline {Exo}{{Exercice}{1.{19}}{Application de la méthode de Karnaugh}}{10}{Exo.19}
-\contentsline {Exo}{{Exercice}{1.{20}}{Composition de la méthode de Karnaugh}}{10}{Exo.20}
-\contentsline {Exo}{{Exercice}{1.{21}}{BTS-2009}}{10}{Exo.21}
-\contentsline {Exo}{{Exercice}{1.{22}}{BTS-2002}}{10}{Exo.22}
-\contentsline {Def}{{Définition}{2.{1}}{Proposition}}{12}{Def.1}
-\contentsline {Ex}{{Exemple}{2.{1}}{}}{12}{Exo.1}
-\contentsline {Rem}{{Remarque}{2.{1}}{}}{13}{Rem.1}
-\contentsline {Exo}{{Exercice}{2.{2}}{}}{14}{Exo.2}
-\contentsline {Rem}{{Remarque}{2.{2}}{}}{14}{Rem.2}
-\contentsline {Exoc}{{Exercice (corrigé)}{2.{3}}{}}{14}{Exo.3}
-\contentsline {Exo}{{Exercice}{2.{4}}{}}{14}{Exo.4}
-\contentsline {Exo}{{Exercice}{2.{5}}{}}{14}{Exo.5}
-\contentsline {Exo}{{Exercice}{2.{6}}{}}{15}{Exo.6}
-\contentsline {Th}{{Propriété}{2.{1}}{}}{15}{Th.1}
-\contentsline {Rem}{{Remarque}{2.{3}}{}}{15}{Rem.3}
-\contentsline {Exo}{{Exercice}{2.{7}}{}}{15}{Exo.7}
-\contentsline {Exo}{{Exercice}{2.{8}}{}}{16}{Exo.8}
-\contentsline {Exo}{{Exercice}{2.{9}}{}}{16}{Exo.9}
-\contentsline {Th}{{Propriété}{2.{2}}{Règles de priorité des connecteurs logiques}}{16}{Th.2}
-\contentsline {Ex}{{Exemple}{2.{10}}{}}{16}{Exo.10}
-\contentsline {Th}{{Propriété}{2.{3}}{Associativité des opérateurs $\ou $ et $\et $}}{16}{Th.3}
-\contentsline {Rem}{{Remarque}{2.{4}}{}}{16}{Rem.4}
-\contentsline {Exoc}{{Exercice (corrigé)}{2.{11}}{}}{16}{Exo.11}
-\contentsline {Exoc}{{Exercice (corrigé)}{2.{12}}{}}{17}{Exo.12}
-\contentsline {Exo}{{Exercice}{2.{13}}{}}{17}{Exo.13}
-\contentsline {Def}{{Définition}{2.{2}}{Fonction de vérité de $F$}}{18}{Def.2}
-\contentsline {Ex}{{Exemple}{2.{14}}{}}{18}{Exo.14}
-\contentsline {Rem}{{Remarque}{2.{5}}{}}{18}{Rem.5}
-\contentsline {Def}{{Définition}{2.{3}}{Tautologie}}{18}{Def.3}
-\contentsline {Notation}{{Notation}{2.{1}}{}}{18}{Notation.1}
-\contentsline {Ex}{{Exemple}{2.{15}}{}}{18}{Exo.15}
-\contentsline {Ex}{{Exemple}{2.{16}}{}}{18}{Exo.16}
-\contentsline {Exo}{{Exercice}{2.{17}}{}}{19}{Exo.17}
-\contentsline {Def}{{Définition}{2.{4}}{Antilogie}}{19}{Def.4}
-\contentsline {Ex}{{Exemple}{2.{18}}{}}{19}{Exo.18}
-\contentsline {Exo}{{Exercice}{2.{19}}{}}{19}{Exo.19}
-\contentsline {Def}{{Définition}{2.{5}}{Conséquence logique}}{19}{Def.5}
-\contentsline {Notation}{{Notation}{2.{2}}{}}{19}{Notation.2}
-\contentsline {Ex}{{Exemple}{2.{20}}{}}{19}{Exo.20}
-\contentsline {Exo}{{Exercice}{2.{21}}{}}{20}{Exo.21}
-\contentsline {Exo}{{Exercice}{2.{22}}{}}{20}{Exo.22}
-\contentsline {Exo}{{Exercice}{2.{23}}{}}{20}{Exo.23}
-\contentsline {Def}{{Définition}{2.{6}}{Formules équivalentes}}{20}{Def.6}
-\contentsline {Rem}{{Remarque}{2.{6}}{}}{20}{Rem.6}
-\contentsline {Exo}{{Exercice}{2.{24}}{}}{20}{Exo.24}
-\contentsline {Exoc}{{Exercice (corrigé)}{2.{25}}{}}{21}{Exo.25}
-\contentsline {Exo}{{Exercice}{2.{26}}{}}{21}{Exo.26}
-\contentsline {Th}{{Propriété}{2.{4}}{Théorème de substitution}}{21}{Th.4}
-\contentsline {Proof}{{Preuve}{2.{$\alpha $}}{}}{21}{Proof.1}
-\contentsline {Ex}{{Exemple}{2.{27}}{}}{21}{Exo.27}
-\contentsline {Ex}{{Exemple}{2.{28}}{}}{22}{Exo.28}
-\contentsline {Th}{{Propriété}{2.{5}}{Théorème de la validité}}{22}{Th.5}
-\contentsline {Proof}{{Preuve}{2.{$\beta $}}{}}{22}{Proof.2}
-\contentsline {Ex}{{Exemple}{2.{29}}{Exemple d'application}}{22}{Exo.29}
-\contentsline {Exo}{{Exercice}{2.{30}}{}}{23}{Exo.30}
-\contentsline {Exo}{{Exercice}{2.{31}}{}}{23}{Exo.31}
-\contentsline {Exo}{{Exercice}{2.{32}}{}}{23}{Exo.32}
-\contentsline {Exo}{{Exercice}{2.{33}}{}}{23}{Exo.33}
-\contentsline {Exo}{{Exercice}{2.{34}}{}}{23}{Exo.34}
-\contentsline {Notation}{{Notation}{3.{1}}{}}{26}{Notation.1}
-\contentsline {Exo}{{Exercice}{3.{1}}{}}{26}{Exo.1}
-\contentsline {Def}{{Définition}{3.{1}}{}}{26}{Def.1}
-\contentsline {Th}{{Propriété}{3.{1}}{}}{27}{Th.1}
-\contentsline {Proof}{{Preuve}{3.{$\alpha $}}{}}{27}{Proof.1}
-\contentsline {Th}{{Propriété}{3.{2}}{}}{27}{Th.2}
-\contentsline {Def}{{Définition}{3.{2}}{}}{27}{Def.2}
-\contentsline {Th}{{Propriété}{3.{3}}{}}{27}{Th.3}
-\contentsline {Ex}{{Exemple}{3.{2}}{}}{27}{Exo.2}
-\contentsline {Exo}{{Exercice}{3.{3}}{}}{27}{Exo.3}
-\contentsline {Exo}{{Exercice}{3.{4}}{}}{27}{Exo.4}
-\contentsline {Exo}{{Exercice}{3.{5}}{}}{27}{Exo.5}
-\contentsline {Exo}{{Exercice}{3.{6}}{}}{27}{Exo.6}
-\contentsline {Exo}{{Exercice}{3.{7}}{}}{27}{Exo.7}
-\contentsline {Def}{{Définition}{3.{3}}{}}{27}{Def.3}
-\contentsline {Exo}{{Exercice}{3.{8}}{}}{27}{Exo.8}
-\contentsline {Def}{{Définition}{3.{4}}{Reunion}}{28}{Def.4}
-\contentsline {Ex}{{Exemple}{3.{9}}{}}{28}{Exo.9}
-\contentsline {Def}{{Définition}{3.{5}}{Intersection}}{28}{Def.5}
-\contentsline {Th}{{Propriété}{3.{4}}{Propriétés de la réunion et de l'intersection}}{28}{Th.4}
-\contentsline {Exo}{{Exercice}{3.{10}}{}}{28}{Exo.10}
-\contentsline {Exo}{{Exercice}{3.{11}}{}}{28}{Exo.11}
-\contentsline {Th}{{Propriété}{3.{5}}{Distributivités de $\cup $ et $\cap $}}{28}{Th.5}
-\contentsline {Exo}{{Exercice}{3.{12}}{}}{28}{Exo.12}
-\contentsline {Def}{{Définition}{3.{6}}{Complémentation}}{28}{Def.6}
-\contentsline {Th}{{Propriété}{3.{6}}{}}{28}{Th.6}
-\contentsline {Exo}{{Exercice}{3.{13}}{}}{29}{Exo.13}
-\contentsline {Exo}{{Exercice}{3.{14}}{}}{29}{Exo.14}
-\contentsline {Exo}{{Exercice}{3.{15}}{}}{29}{Exo.15}
-\contentsline {Exo}{{Exercice}{3.{16}}{}}{29}{Exo.16}
-\contentsline {Exo}{{Exercice}{3.{17}}{Fonction caractéristique des parties d'un ensemble}}{29}{Exo.17}
-\contentsline {Def}{{Définition}{4.{1}}{Relation binaire}}{30}{Def.1}
-\contentsline {Exo}{{Exercice}{4.{1}}{}}{30}{Exo.1}
-\contentsline {Rem}{{Remarque}{4.{1}}{}}{30}{Rem.1}
-\contentsline {Def}{{Définition}{4.{2}}{Réflexivité}}{30}{Def.2}
-\contentsline {Def}{{Définition}{4.{3}}{Antisymétrie}}{30}{Def.3}
-\contentsline {Def}{{Définition}{4.{4}}{Transitivité}}{30}{Def.4}
-\contentsline {Exo}{{Exercice}{4.{2}}{}}{30}{Exo.2}
-\contentsline {Exo}{{Exercice}{4.{3}}{}}{31}{Exo.3}
-\contentsline {Def}{{Définition}{4.{5}}{Relation d'ordre}}{31}{Def.5}
-\contentsline {Ex}{{Exemple}{4.{4}}{}}{31}{Exo.4}
-\contentsline {Ex}{{Exemple}{4.{5}}{Relation de divisibilité}}{31}{Exo.5}
-\contentsline {Exo}{{Exercice}{4.{6}}{}}{31}{Exo.6}
-\contentsline {Exo}{{Exercice}{4.{7}}{Diagrammes de transitivité}}{31}{Exo.7}
-\contentsline {Def}{{Définition}{4.{6}}{Relation symétrique}}{31}{Def.6}
-\contentsline {Def}{{Définition}{4.{7}}{Relation d'équivalence}}{31}{Def.7}
-\contentsline {Ex}{{Exemple}{4.{8}}{}}{31}{Exo.8}
-\contentsline {Ex}{{Exemple}{4.{9}}{Relation de congruence modulo $n$ dans $\Z $}}{32}{Exo.9}
-\contentsline {Exo}{{Exercice}{4.{10}}{}}{32}{Exo.10}
-\contentsline {Def}{{Définition}{4.{8}}{Classe d'équivalence}}{32}{Def.8}
-\contentsline {Notation}{{Notation}{4.{1}}{}}{32}{Notation.1}
-\contentsline {Exo}{{Exercice}{4.{11}}{}}{32}{Exo.11}
-\contentsline {Exo}{{Exercice}{4.{12}}{}}{32}{Exo.12}
-\contentsline {Th}{{Propriété}{4.{1}}{}}{32}{Th.1}
-\contentsline {Pre}{{Preuve}{1}{}}{32}{Pre.1}
-\contentsline {Def}{{Définition}{4.{9}}{Partition d'un ensemble}}{32}{Def.9}
-\contentsline {Th}{{Propriété}{4.{2}}{}}{33}{Th.2}
-\contentsline {Pre}{{Preuve}{2}{}}{33}{Pre.2}
-\contentsline {Ex}{{Exemple}{4.{13}}{}}{33}{Exo.13}
-\contentsline {Exo}{{Exercice}{4.{14}}{}}{33}{Exo.14}
-\contentsline {Exo}{{Exercice}{4.{15}}{}}{33}{Exo.15}
+\contentsline {Def}{{Définition}{1.{1}}{Algèbre de Boole}}{4}{Def.1.1}
+\contentsline {Rem}{{Remarque}{1.{1}}{}}{4}{Rem.1.1}
+\contentsline {Exo}{{Exercice}{1.{1}}{Somme disjonctive}}{4}{Exo.1.1}
+\contentsline {Exo}{{Exercice}{1.{2}}{Opérateurs de Sheffer et de Peirce}}{4}{Exo.1.2}
+\contentsline {Th}{{Propriété}{1.{1}}{Suppression de redondance}}{5}{Th.1.1}
+\contentsline {Proof}{{Preuve}{1.{$\alpha $}}{}}{5}{Proof.1.1}
+\contentsline {Ex}{{Exemple}{1.{3}}{}}{5}{Exo.1.3}
+\contentsline {Exo}{{Exercice}{1.{4}}{}}{5}{Exo.1.4}
+\contentsline {Exo}{{Exercice}{1.{5}}{Calcul booléen élémentaire}}{5}{Exo.1.5}
+\contentsline {Exo}{{Exercice}{1.{6}}{Calcul booléen}}{5}{Exo.1.6}
+\contentsline {Def}{{Définition}{1.{2}}{Fonction booléenne}}{5}{Def.1.2}
+\contentsline {Ex}{{Exemple}{1.{7}}{}}{6}{Exo.1.7}
+\contentsline {Def}{{Définition}{1.{3}}{Aspect d'une variable}}{6}{Def.1.3}
+\contentsline {Def}{{Définition}{1.{4}}{Fonction booléenne nulle}}{6}{Def.1.4}
+\contentsline {Def}{{Définition}{1.{5}}{Fonction référentiel}}{6}{Def.1.5}
+\contentsline {Def}{{Définition}{1.{6}}{Minterme, maxterme}}{6}{Def.1.6}
+\contentsline {Ex}{{Exemple}{1.{8}}{Minterme à trois variables}}{6}{Exo.1.8}
+\contentsline {Ex}{{Exemple}{1.{9}}{Maxterme à trois variables}}{6}{Exo.1.9}
+\contentsline {Exo}{{Exercice}{1.{10}}{}}{6}{Exo.1.10}
+\contentsline {Exo}{{Exercice}{1.{11}}{}}{6}{Exo.1.11}
+\contentsline {Th}{{Propriété}{1.{2}}{Nombre de mintermes et de maxtermes}}{6}{Th.1.2}
+\contentsline {Def}{{Définition}{1.{7}}{Monômes}}{6}{Def.1.7}
+\contentsline {Exo}{{Exercice}{1.{12}}{}}{6}{Exo.1.12}
+\contentsline {Th}{{Propriété}{1.{3}}{}}{6}{Th.1.3}
+\contentsline {Proof}{{Preuve}{1.{$\beta $}}{}}{6}{Proof.1.2}
+\contentsline {Th}{{Propriété}{1.{4}}{}}{6}{Th.1.4}
+\contentsline {Proof}{{Preuve}{1.{$\gamma $}}{}}{7}{Proof.1.3}
+\contentsline {Th}{{Propriété}{1.{5}}{Forme canonique disjonctive}}{7}{Th.1.5}
+\contentsline {Rem}{{Remarque}{1.{2}}{}}{7}{Rem.1.2}
+\contentsline {Th}{{Propriété}{1.{6}}{Forme canonique conjonctive}}{7}{Th.1.6}
+\contentsline {Ex}{{Exemple}{1.{13}}{}}{7}{Exo.1.13}
+\contentsline {Ex}{{Exemple}{1.{14}}{}}{7}{Exo.1.14}
+\contentsline {Rem}{{Remarque}{1.{3}}{}}{7}{Rem.1.3}
+\contentsline {Exo}{{Exercice}{1.{15}}{}}{7}{Exo.1.15}
+\contentsline {Exoc}{{Exercice (corrigé)}{1.{16}}{}}{8}{Exo.1.16}
+\contentsline {Exo}{{Exercice}{1.{17}}{Fonctions booléennes}}{9}{Exo.1.17}
+\contentsline {Exo}{{Exercice}{1.{18}}{}}{9}{Exo.1.18}
+\contentsline {Exo}{{Exercice}{1.{19}}{Application de la méthode de Karnaugh}}{9}{Exo.1.19}
+\contentsline {Exo}{{Exercice}{1.{20}}{Composition de la méthode de Karnaugh}}{9}{Exo.1.20}
+\contentsline {Exo}{{Exercice}{1.{21}}{BTS-2009}}{9}{Exo.1.21}
+\contentsline {Exo}{{Exercice}{1.{22}}{BTS-2002}}{9}{Exo.1.22}
+\contentsline {Def}{{Définition}{2.{1}}{Proposition}}{11}{Def.2.1}
+\contentsline {Ex}{{Exemple}{2.{1}}{}}{11}{Exo.2.1}
+\contentsline {Rem}{{Remarque}{2.{1}}{}}{12}{Rem.2.1}
+\contentsline {Exo}{{Exercice}{2.{2}}{}}{12}{Exo.2.2}
+\contentsline {Rem}{{Remarque}{2.{2}}{}}{13}{Rem.2.2}
+\contentsline {Exoc}{{Exercice (corrigé)}{2.{3}}{}}{13}{Exo.2.3}
+\contentsline {Exo}{{Exercice}{2.{4}}{}}{13}{Exo.2.4}
+\contentsline {Exo}{{Exercice}{2.{5}}{}}{13}{Exo.2.5}
+\contentsline {Exo}{{Exercice}{2.{6}}{}}{13}{Exo.2.6}
+\contentsline {Th}{{Propriété}{2.{1}}{}}{13}{Th.2.1}
+\contentsline {Rem}{{Remarque}{2.{3}}{}}{14}{Rem.2.3}
+\contentsline {Exo}{{Exercice}{2.{7}}{}}{14}{Exo.2.7}
+\contentsline {Exo}{{Exercice}{2.{8}}{}}{14}{Exo.2.8}
+\contentsline {Exo}{{Exercice}{2.{9}}{}}{14}{Exo.2.9}
+\contentsline {Th}{{Propriété}{2.{2}}{Règles de priorité des connecteurs logiques}}{14}{Th.2.2}
+\contentsline {Ex}{{Exemple}{2.{10}}{}}{14}{Exo.2.10}
+\contentsline {Th}{{Propriété}{2.{3}}{Associativité des opérateurs $\ou $ et $\et $}}{14}{Th.2.3}
+\contentsline {Rem}{{Remarque}{2.{4}}{}}{15}{Rem.2.4}
+\contentsline {Exoc}{{Exercice (corrigé)}{2.{11}}{}}{15}{Exo.2.11}
+\contentsline {Exoc}{{Exercice (corrigé)}{2.{12}}{}}{15}{Exo.2.12}
+\contentsline {Exo}{{Exercice}{2.{13}}{}}{15}{Exo.2.13}
+\contentsline {Def}{{Définition}{2.{2}}{Fonction de vérité de $F$}}{16}{Def.2.2}
+\contentsline {Ex}{{Exemple}{2.{14}}{}}{16}{Exo.2.14}
+\contentsline {Rem}{{Remarque}{2.{5}}{}}{16}{Rem.2.5}
+\contentsline {Def}{{Définition}{2.{3}}{Tautologie}}{16}{Def.2.3}
+\contentsline {Notation}{{Notation}{2.{1}}{}}{16}{Notation.2.1}
+\contentsline {Ex}{{Exemple}{2.{15}}{}}{16}{Exo.2.15}
+\contentsline {Ex}{{Exemple}{2.{16}}{}}{16}{Exo.2.16}
+\contentsline {Exo}{{Exercice}{2.{17}}{}}{16}{Exo.2.17}
+\contentsline {Def}{{Définition}{2.{4}}{Antilogie}}{17}{Def.2.4}
+\contentsline {Ex}{{Exemple}{2.{18}}{}}{17}{Exo.2.18}
+\contentsline {Exo}{{Exercice}{2.{19}}{}}{17}{Exo.2.19}
+\contentsline {Def}{{Définition}{2.{5}}{Conséquence logique}}{17}{Def.2.5}
+\contentsline {Notation}{{Notation}{2.{2}}{}}{17}{Notation.2.2}
+\contentsline {Ex}{{Exemple}{2.{20}}{}}{17}{Exo.2.20}
+\contentsline {Exo}{{Exercice}{2.{21}}{}}{17}{Exo.2.21}
+\contentsline {Exo}{{Exercice}{2.{22}}{}}{17}{Exo.2.22}
+\contentsline {Exo}{{Exercice}{2.{23}}{}}{17}{Exo.2.23}
+\contentsline {Def}{{Définition}{2.{6}}{Formules équivalentes}}{18}{Def.2.6}
+\contentsline {Rem}{{Remarque}{2.{6}}{}}{18}{Rem.2.6}
+\contentsline {Exo}{{Exercice}{2.{24}}{}}{18}{Exo.2.24}
+\contentsline {Exoc}{{Exercice (corrigé)}{2.{25}}{}}{18}{Exo.2.25}
+\contentsline {Exo}{{Exercice}{2.{26}}{}}{18}{Exo.2.26}
+\contentsline {Th}{{Propriété}{2.{4}}{Théorème de substitution}}{19}{Th.2.4}
+\contentsline {Proof}{{Preuve}{2.{$\alpha $}}{}}{19}{Proof.2.1}
+\contentsline {Ex}{{Exemple}{2.{27}}{}}{19}{Exo.2.27}
+\contentsline {Ex}{{Exemple}{2.{28}}{}}{19}{Exo.2.28}
+\contentsline {Th}{{Propriété}{2.{5}}{Théorème de la validité}}{19}{Th.2.5}
+\contentsline {Proof}{{Preuve}{2.{$\beta $}}{}}{19}{Proof.2.2}
+\contentsline {Ex}{{Exemple}{2.{29}}{Exemple d'application}}{20}{Exo.2.29}
+\contentsline {Exo}{{Exercice}{2.{30}}{}}{20}{Exo.2.30}
+\contentsline {Exo}{{Exercice}{2.{31}}{}}{20}{Exo.2.31}
+\contentsline {Exo}{{Exercice}{2.{32}}{}}{20}{Exo.2.32}
+\contentsline {Exo}{{Exercice}{2.{33}}{}}{20}{Exo.2.33}
+\contentsline {Exo}{{Exercice}{2.{34}}{}}{21}{Exo.2.34}
+\contentsline {Notation}{{Notation}{3.{1}}{}}{23}{Notation.3.1}
+\contentsline {Exo}{{Exercice}{3.{1}}{}}{23}{Exo.3.1}
+\contentsline {Def}{{Définition}{3.{1}}{}}{23}{Def.3.1}
+\contentsline {Th}{{Propriété}{3.{1}}{}}{23}{Th.3.1}
+\contentsline {Proof}{{Preuve}{3.{$\alpha $}}{}}{24}{Proof.3.1}
+\contentsline {Th}{{Propriété}{3.{2}}{}}{24}{Th.3.2}
+\contentsline {Def}{{Définition}{3.{2}}{}}{24}{Def.3.2}
+\contentsline {Th}{{Propriété}{3.{3}}{}}{24}{Th.3.3}
+\contentsline {Ex}{{Exemple}{3.{2}}{}}{24}{Exo.3.2}
+\contentsline {Exo}{{Exercice}{3.{3}}{}}{24}{Exo.3.3}
+\contentsline {Exo}{{Exercice}{3.{4}}{}}{24}{Exo.3.4}
+\contentsline {Exo}{{Exercice}{3.{5}}{}}{24}{Exo.3.5}
+\contentsline {Exo}{{Exercice}{3.{6}}{}}{24}{Exo.3.6}
+\contentsline {Exo}{{Exercice}{3.{7}}{}}{24}{Exo.3.7}
+\contentsline {Def}{{Définition}{3.{3}}{}}{24}{Def.3.3}
+\contentsline {Exo}{{Exercice}{3.{8}}{}}{24}{Exo.3.8}
+\contentsline {Def}{{Définition}{3.{4}}{Reunion}}{24}{Def.3.4}
+\contentsline {Ex}{{Exemple}{3.{9}}{}}{24}{Exo.3.9}
+\contentsline {Def}{{Définition}{3.{5}}{Intersection}}{24}{Def.3.5}
+\contentsline {Th}{{Propriété}{3.{4}}{Propriétés de la réunion et de l'intersection}}{25}{Th.3.4}
+\contentsline {Exo}{{Exercice}{3.{10}}{}}{25}{Exo.3.10}
+\contentsline {Exo}{{Exercice}{3.{11}}{}}{25}{Exo.3.11}
+\contentsline {Th}{{Propriété}{3.{5}}{Distributivités de $\cup $ et $\cap $}}{25}{Th.3.5}
+\contentsline {Exo}{{Exercice}{3.{12}}{}}{25}{Exo.3.12}
+\contentsline {Def}{{Définition}{3.{6}}{Complémentation}}{25}{Def.3.6}
+\contentsline {Th}{{Propriété}{3.{6}}{}}{25}{Th.3.6}
+\contentsline {Exo}{{Exercice}{3.{13}}{}}{25}{Exo.3.13}
+\contentsline {Exo}{{Exercice}{3.{14}}{}}{25}{Exo.3.14}
+\contentsline {Exo}{{Exercice}{3.{15}}{}}{26}{Exo.3.15}
+\contentsline {Exo}{{Exercice}{3.{16}}{}}{26}{Exo.3.16}
+\contentsline {Exo}{{Exercice}{3.{17}}{Fonction caractéristique des parties d'un ensemble}}{26}{Exo.3.17}
+\contentsline {Def}{{Définition}{4.{1}}{Relation binaire}}{27}{Def.4.1}
+\contentsline {Exo}{{Exercice}{4.{1}}{}}{27}{Exo.4.1}
+\contentsline {Rem}{{Remarque}{4.{1}}{}}{27}{Rem.4.1}
+\contentsline {Def}{{Définition}{4.{2}}{Réflexivité}}{27}{Def.4.2}
+\contentsline {Def}{{Définition}{4.{3}}{Antisymétrie}}{27}{Def.4.3}
+\contentsline {Def}{{Définition}{4.{4}}{Transitivité}}{27}{Def.4.4}
+\contentsline {Exo}{{Exercice}{4.{2}}{}}{27}{Exo.4.2}
+\contentsline {Exo}{{Exercice}{4.{3}}{}}{27}{Exo.4.3}
+\contentsline {Def}{{Définition}{4.{5}}{Relation d'ordre}}{28}{Def.4.5}
+\contentsline {Ex}{{Exemple}{4.{4}}{}}{28}{Exo.4.4}
+\contentsline {Ex}{{Exemple}{4.{5}}{Relation de divisibilité}}{28}{Exo.4.5}
+\contentsline {Exo}{{Exercice}{4.{6}}{}}{28}{Exo.4.6}
+\contentsline {Exo}{{Exercice}{4.{7}}{Diagrammes de transitivité}}{28}{Exo.4.7}
+\contentsline {Def}{{Définition}{4.{6}}{Relation symétrique}}{28}{Def.4.6}
+\contentsline {Def}{{Définition}{4.{7}}{Relation d'équivalence}}{28}{Def.4.7}
+\contentsline {Ex}{{Exemple}{4.{8}}{}}{28}{Exo.4.8}
+\contentsline {Ex}{{Exemple}{4.{9}}{Relation de congruence modulo $n$ dans $\Z $}}{28}{Exo.4.9}
+\contentsline {Exo}{{Exercice}{4.{10}}{}}{28}{Exo.4.10}
+\contentsline {Def}{{Définition}{4.{8}}{Classe d'équivalence}}{29}{Def.4.8}
+\contentsline {Notation}{{Notation}{4.{1}}{}}{29}{Notation.4.1}
+\contentsline {Exo}{{Exercice}{4.{11}}{}}{29}{Exo.4.11}
+\contentsline {Exo}{{Exercice}{4.{12}}{}}{29}{Exo.4.12}
+\contentsline {Th}{{Propriété}{4.{1}}{}}{29}{Th.4.1}
+\contentsline {Pre}{{Preuve}{1}{}}{29}{Pre.1}
+\contentsline {Def}{{Définition}{4.{9}}{Partition d'un ensemble}}{29}{Def.4.9}
+\contentsline {Th}{{Propriété}{4.{2}}{}}{29}{Th.4.2}
+\contentsline {Pre}{{Preuve}{2}{}}{29}{Pre.2}
+\contentsline {Ex}{{Exemple}{4.{13}}{}}{29}{Exo.4.13}
+\contentsline {Exo}{{Exercice}{4.{14}}{}}{29}{Exo.4.14}
+\contentsline {Exo}{{Exercice}{4.{15}}{}}{29}{Exo.4.15}
+\contentsline {Exo}{{Exercice}{5.{1}}{}}{31}{Exo.5.1}
+\contentsline {Exo}{{Exercice}{5.{2}}{}}{31}{Exo.5.2}
+\contentsline {Exo}{{Exercice}{5.{3}}{}}{31}{Exo.5.3}
+\contentsline {Def}{{Définition}{5.{1}}{Multiple, diviseur}}{31}{Def.5.1}
+\contentsline {Def}{{Définition}{5.{2}}{Nombre premier}}{31}{Def.5.2}
+\contentsline {Rem}{{Remarque}{5.{1}}{}}{31}{Rem.5.1}
+\contentsline {Def}{{Définition}{5.{3}}{Décomposition en facteurs premiers}}{31}{Def.5.3}
+\contentsline {Th}{{Propriété}{5.{1}}{}}{31}{Th.5.1}
+\contentsline {Exo}{{Exercice}{5.{4}}{}}{32}{Exo.5.4}
+\contentsline {Exo}{{Exercice}{5.{5}}{}}{32}{Exo.5.5}
+\contentsline {Th}{{Propriété}{5.{2}}{}}{32}{Th.5.2}
+\contentsline {Exo}{{Exercice}{5.{6}}{Nombres premiers en quantité infinie}}{32}{Exo.5.6}
+\contentsline {Def}{{Définition}{5.{4}}{PGCD, PPCM}}{32}{Def.5.4}
+\contentsline {Def}{{Définition}{5.{5}}{Nombres premiers entre eux}}{32}{Def.5.5}
+\contentsline {Exo}{{Exercice}{5.{7}}{Nombres de Fermat}}{32}{Exo.5.7}
+\contentsline {Th}{{Propriété}{5.{3}}{}}{32}{Th.5.3}
+\contentsline {Def}{{Définition}{5.{6}}{Division euclidienne}}{32}{Def.5.6}
+\contentsline {Ex}{{Exemple}{5.{8}}{}}{32}{Exo.5.8}
+\contentsline {Ex}{{Exemple}{5.{9}}{}}{32}{Exo.5.9}
+\contentsline {Exo}{{Exercice}{5.{10}}{}}{32}{Exo.5.10}
+\contentsline {Exo}{{Exercice}{5.{11}}{Numéro Sécurité Sociale (\url {www.bibmath.net})}}{33}{Exo.5.11}
+\contentsline {Exo}{{Exercice}{5.{12}}{}}{33}{Exo.5.12}
+\contentsline {Rem}{{Remarque}{5.{2}}{}}{34}{Rem.5.2}
+\contentsline {Exo}{{Exercice}{5.{13}}{}}{34}{Exo.5.13}
+\contentsline {Exo}{{Exercice}{5.{14}}{}}{34}{Exo.5.14}
+\contentsline {Exo}{{Exercice}{5.{15}}{}}{34}{Exo.5.15}
+\contentsline {Th}{{Propriété}{5.{4}}{Théorème de Bézout}}{34}{Th.5.4}
+\contentsline {Proof}{{Preuve}{5.{$\alpha $}}{}}{34}{Proof.5.1}
+\contentsline {Rem}{{Remarque}{5.{3}}{}}{34}{Rem.5.3}
+\contentsline {Proof}{{Preuve}{5.{$\beta $}}{}}{34}{Proof.5.2}
+\contentsline {Exo}{{Exercice}{5.{16}}{Application de l'algorithme d'Euclide et de Bézout}}{34}{Exo.5.16}
+\contentsline {Pre}{{Preuve}{3}{}}{35}{Pre.3}
+\contentsline {Ex}{{Exemple}{5.{17}}{}}{35}{Exo.5.17}
+\contentsline {Rem}{{Remarque}{5.{4}}{}}{35}{Rem.5.4}
+\contentsline {Exo}{{Exercice}{5.{18}}{}}{35}{Exo.5.18}
+\contentsline {Th}{{Propriété}{5.{5}}{Théorème de Gauss}}{35}{Th.5.5}
+\contentsline {Exo}{{Exercice}{5.{19}}{}}{36}{Exo.5.19}
+\contentsline {Exo}{{Exercice}{5.{20}}{}}{36}{Exo.5.20}
+\contentsline {Exo}{{Exercice}{5.{21}}{}}{36}{Exo.5.21}
+\contentsline {Exo}{{Exercice}{5.{22}}{}}{36}{Exo.5.22}
+\contentsline {Def}{{Définition}{5.{7}}{Congruence modulo $n$}}{36}{Def.5.7}
+\contentsline {Exo}{{Exercice}{5.{23}}{}}{36}{Exo.5.23}
+\contentsline {Th}{{Propriété}{5.{6}}{}}{36}{Th.5.6}
+\contentsline {Proof}{{Preuve}{5.{$\gamma $}}{}}{36}{Proof.5.3}
+\contentsline {Ex}{{Exemple}{5.{24}}{}}{37}{Exo.5.24}
+\contentsline {Th}{{Propriété}{5.{7}}{}}{37}{Th.5.7}
+\contentsline {Notation}{{Notation}{5.{1}}{}}{37}{Notation.5.1}
+\contentsline {Ex}{{Exemple}{5.{25}}{}}{37}{Exo.5.25}
+\contentsline {Def}{{Définition}{5.{8}}{}}{37}{Def.5.8}
+\contentsline {Th}{{Propriété}{5.{8}}{}}{37}{Th.5.8}
+\contentsline {Proof}{{Preuve}{5.{$\delta $}}{}}{37}{Proof.5.4}
+\contentsline {Def}{{Définition}{5.{9}}{}}{37}{Def.5.9}
+\contentsline {Ex}{{Exemple}{5.{26}}{}}{37}{Exo.5.26}
+\contentsline {Rem}{{Remarque}{5.{5}}{}}{38}{Rem.5.5}
+\contentsline {Exo}{{Exercice}{5.{27}}{}}{38}{Exo.5.27}
+\contentsline {Exo}{{Exercice}{5.{28}}{Systèmes de congruences}}{38}{Exo.5.28}
+\contentsline {Exo}{{Exercice}{5.{29}}{}}{38}{Exo.5.29}
+\contentsline {Exo}{{Exercice}{5.{30}}{}}{38}{Exo.5.30}
\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}{31}{section.4.3}
-\contentsline {subsection}{\numberline {III.1}Classes d'\IeC {\'e}quivalence}{32}{subsection.4.3.1}
-\contentsline {part}{III\hspace {1em}Annexes}{34}{part.3}
-\contentsline {chapter}{\numberline {5}Programme P\IeC {\'e}dagogique National 2005 (PPN)}{35}{chapter.5}
-\contentsline {chapter}{Index}{36}{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}{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}
