From 9c8838c36376cde4dd444cefd90f573cf6175359 Mon Sep 17 00:00:00 2001 From: couchot Date: Wed, 26 Feb 2014 13:16:15 +0100 Subject: [PATCH] typo ensemble13.tex --- ensembles/IntroAuxEnsembles13.tex | 4 +- graphes/S2MD.tex | 64 ++-- logique/Propositions13.tex | 605 ++++++++++++++---------------- main13.aux | 475 +++++++++++------------ main13.idx | 81 ++-- main13.log | 105 +++--- main13.out | 67 ++-- main13.pdf | Bin 370852 -> 358833 bytes main13.thm | 320 ++++++++-------- main13.toc | 93 +++-- 10 files changed, 866 insertions(+), 948 deletions(-) diff --git a/ensembles/IntroAuxEnsembles13.tex b/ensembles/IntroAuxEnsembles13.tex index 6b6bc06..7439ae9 100755 --- a/ensembles/IntroAuxEnsembles13.tex +++ b/ensembles/IntroAuxEnsembles13.tex @@ -219,7 +219,7 @@ La complémentation a plusieurs propriétés remarquables : \begin{Exo} -Pour deux ensembles $A$ et $B$, +Pour deux ensembles $A$ et $B$ inclus dans $E$, on appelle différence symétrique, note $A\Delta B$, l'ensemble défini par $A \Delta B = (A \cup B) \setminus (A \cap B)$ @@ -298,7 +298,7 @@ régulière dans $E$ et que, si $X$ est régulière, il en est de même de $f(X) \begin{Exo}[Fonction caractéristique des parties d'un ensemble] -On appelle fonction caractéristique de la partie $A$ de l'ensemble $E$ $(E\neq\vide$, $A\neq\vide$, $A\sse E$) l'application $f_A:E\imp\{0,1\}$, définie par: +On appelle fonction caractéristique de la partie $A$ de l'ensemble $E$ $(E\neq\vide$, $A\neq\vide$, $A\sse E$) l'application $f_A:E\rightarrow\{0,1\}$, définie par: \begin{itemize} \item $\qqs x\in A,\ f_A(x) = 1$; \item $\qqs x \in E\moins A,\ f_A(x) = 0$. diff --git a/graphes/S2MD.tex b/graphes/S2MD.tex index a86b2b9..0db9d34 100644 --- a/graphes/S2MD.tex +++ b/graphes/S2MD.tex @@ -1050,17 +1050,17 @@ $3$? Les fils de $1$? Les anc \end{exo} -% \begin{exo} -% On considère dans cet exercice des arbres codés par liste d'adjacence comme -% dans l'exercice en Python de la page~\pageref{titi}. On suppose que la -% racine est le premier sommet de la liste $V$. - -% \begin{enumerate} -% \item Écrire une fonction qui compte le nombre de feuille d'un arbre. -% \item Écrire une fonction qui compte le nombre de descendants d'un sommet -% donné. -% \end{enumerate} -% \end{exo} +\begin{exo} +On considère dans cet exercice des arbres codés par liste d'adjacence comme +dans l'exercice en Python de la page~\pageref{titi}. On suppose que la +racine est le premier sommet de la liste $V$. + +\begin{enumerate} +\item Écrire une fonction qui compte le nombre de feuille d'un arbre. +\item Écrire une fonction qui compte le nombre de descendants d'un sommet + donné. +\end{enumerate} +\end{exo} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Parcours d'arbres} @@ -1159,10 +1159,10 @@ Parcours-Suffixe(V,E,x): Que donne le parcours suffixe sur les arbres des figures~\ref{fig:arbre} et~\ref{fig:arbre2}? \end{exo} -% \begin{exo} -% Avec le même codage que précédemment, écrire en Python des fonctions de -% parcours préfixe et suffixe. -% \end{exo} +\begin{exo} +Avec le même codage que précédemment, écrire en Python des fonctions de +parcours préfixe et suffixe. +\end{exo} %%%%%%%%%%%%%%%%%%%%% \subsection{Arbres couvrants} @@ -1297,11 +1297,11 @@ pas $(3)$. En revanche celui de droite est bien un arbre couvrant. \caption{Arbres non couvrant et couvrant}\label{fig:graphcouvrant2} \end{figure} -% \begin{exo} -% Écrire une fonction qui étant donné un graphe et un arbre codé en Python -% comme précédemment, teste si l'arbre est un arbre couvrant du graphe (on ne -% testera pas si c'est bien un arbre). -% \end{exo} +\begin{exo} +Écrire une fonction qui étant donné un graphe et un arbre codé en Python +comme précédemment, teste si l'arbre est un arbre couvrant du graphe (on ne +testera pas si c'est bien un arbre). +\end{exo} L'objectif des algorithmes de parcours de graphes est de construire des @@ -1583,16 +1583,16 @@ la figure~\ref{fig:exo}. \end{exo} -% \begin{exo} -% Écrire en Python l'algorithme de parcours en largeur pour un graphe codé par -% liste d'adjacence. -% \end{exo} +\begin{exo} +Écrire en Python l'algorithme de parcours en largeur pour un graphe codé par +liste d'adjacence. +\end{exo} -% \begin{exo} -% Écrire en Python une fonction qui teste si un graphe donné par liste -% d'adjacence est un arbre dont la racine est le plus petit sommet. -% \end{exo} +\begin{exo} +Écrire en Python une fonction qui teste si un graphe donné par liste +d'adjacence est un arbre dont la racine est le plus petit sommet. +\end{exo} \begin{exo} Les @@ -1761,10 +1761,10 @@ sur le graphe de la figure~\ref{fig:exo}. -% \begin{exo} -% Écrire en Python l'algorithme de parcours en profondeur pour un graphe codé par -% liste d'adjacence. -% \end{exo} +\begin{exo} +Écrire en Python l'algorithme de parcours en profondeur pour un graphe codé par +liste d'adjacence. +\end{exo} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù diff --git a/logique/Propositions13.tex b/logique/Propositions13.tex index c6cf86d..e159edb 100644 --- a/logique/Propositions13.tex +++ b/logique/Propositions13.tex @@ -24,8 +24,7 @@ les $Z$ sont des $Y$, de déduire que tous les $Z$ sont des $X$}~\cite{Dowek07}. L'homme exprime son raisonnement par un discours, et ce discours utilise une langue (une langue naturelle, français, anglais,\ldots). -D'une manière générale, ce discours est articulé en phrases, d'un -niveau de complexité variable, et c'est l'étude de ces \og +Ce discours est articulé en phrases et c'est l'étude de ces \og énoncés\fg{} que se propose de faire la logique. @@ -117,7 +116,7 @@ $$ -D'une manière générale, le calcul propositionnel ne se préoccupe que +Attention, le calcul propositionnel ne se préoccupe que des valeurs de vérité, et pas du tout des liens sémantiques qui peuvent exister entre des propositions. Ces dernières sont reliées entre elles syntaxiquement par des connecteurs comme \og ou\fg{} ou @@ -130,39 +129,38 @@ propositions (\og plus simples\fg{}). \subsection{Tables de vérité des connecteurs logiques} -\centerline{\begin{tabular}{|c|c|c|} +\begin{tabular}{|c|c|c|} \hline $P$ & $Q$ & $P\ou Q$ \\ \hline F & F & F \\ \hline F & V & V \\ \hline V & F & V \\ \hline V & V & V \\ \hline - \end{tabular}} - -\centerline{\begin{tabular}{|c|c|c|} + \end{tabular} +~ +\begin{tabular}{|c|c|c|} \hline $P$ & $Q$ & $P\et Q$ \\ \hline F & F & F \\ \hline F & V & F \\ \hline V & F & F \\ \hline -V & V & V \\ \hline \end{tabular}} - -\centerline{ -\begin{tabular}{|c|c|} \hline +V & V & V \\ \hline \end{tabular} +~ +{ \begin{tabular}{|c|c|} \hline $P$ & $\non P$ \\ \hline F & V \\ \hline V & F \\ \hline \end{tabular}} - -\centerline{\begin{tabular}{|c|c|c|} +~ +{\begin{tabular}{|c|c|c|} \hline $P$ & $Q$ & $P\imp Q$ \\ \hline F & F & V \\ \hline F & V & V \\ \hline V & F & F \\ \hline V & V & V \\ \hline \end{tabular}} - -\centerline{\begin{tabular}{|c|c|c|} +~ +{\begin{tabular}{|c|c|c|} \hline $P$ & $Q$ & $P\eqv Q$ \\ \hline F & F & V \\ \hline @@ -202,27 +200,27 @@ Déterminer la valeur de vérité des propositions suivantes \item \og si le soleil tourne autour de la terre alors la terre est ronde\fg{} \item \og si la terre est ronde alors le soleil tourne autour de la terre\fg{} \item \og si vous étudiez la logique alors $E=m.c^2$\fg{} -\item \og si Napoléon est mort alors il a gagné la bataille de Waterloo\fg{} -\item \og s'il pleut en ce moment alors il pleut en ce moment\fg{} -\item \og si tous les hommes sont passionnés par la logique alors Dieu existe\fg{} -\item \og si le Diable existe alors ceci est un exercice de logique\fg{} +% \item \og si Napoléon est mort alors il a gagné la bataille de Waterloo\fg{} +% \item \og s'il pleut en ce moment alors il pleut en ce moment\fg{} +% \item \og si tous les hommes sont passionnés par la logique alors Dieu existe\fg{} +% \item \og si le Diable existe alors ceci est un exercice de logique\fg{} \end{enumerate} \end{Exo} -La manière de mener un raisonnement qui utilise éventuellement des -propositions qui se présentent sous la forme d'implications logiques -est l'objet de la théorie de la déduction qui sera étudiée plus loin. +% La manière de mener un raisonnement qui utilise éventuellement des +% propositions qui se présentent sous la forme d'implications logiques +% est l'objet de la théorie de la déduction qui sera étudiée plus loin. -\begin{Rem} -Même remarque que pour l'implication logique: l'équivalence logique -de deux propositions fausses est une proposition vraie. -\end{Rem} +% \begin{Rem} +% Même remarque que pour l'implication logique: l'équivalence logique +% de deux propositions fausses est une proposition vraie. +% \end{Rem} -\begin{Exoc} +\begin{Exo} En notant $M$ et $C$ les affirmations suivantes: \begin{itemize} \item $M$ = \og Jean est fort en Maths\fg{}, @@ -237,44 +235,44 @@ usuels. \item \label{it:x1} \og Jean est fort en Maths mais faible en Chimie\fg{} \item \label{it:x2} \og Jean n'est fort ni en Maths ni en Chimie\fg{} \item \label{it:x3} \og Jean est fort en Maths ou il est à la fois fort en Chimie et faible en Maths\fg{} -\item \label{it:x4} \og Jean est fort en Maths s'il est fort en Chimie\fg{} -\item \label{it:x5} \og Jean est fort en Chimie et en Maths ou il est fort en Chimie et faible en Maths\fg{} +% \item \label{it:x4} \og Jean est fort en Maths s'il est fort en Chimie\fg{} +% \item \label{it:x5} \og Jean est fort en Chimie et en Maths ou il est fort en Chimie et faible en Maths\fg{} \end{enumerate} -% AG: je peux ajouter ici un mecanisme d'option qui fait que -% l'etudiant n'a pas la reponse dans son poly, mais le prof l'a. +% % AG: je peux ajouter ici un mecanisme d'option qui fait que +% % l'etudiant n'a pas la reponse dans son poly, mais le prof l'a. -Réponses: +% Réponses: -\ref{it:x1}. $M \et (\non C)$; -\ref{it:x2}. $(\non M) \et (\non C)$; -\ref{it:x3}. $M \ou ( C \et \non M )$; -\ref{it:x4}. $C \imp M$; -\ref{it:x5}. $(M \et C) \ou (\non M \et Q)$. -\end{Exoc} +% \ref{it:x1}. $M \et (\non C)$; +% \ref{it:x2}. $(\non M) \et (\non C)$; +% \ref{it:x3}. $M \ou ( C \et \non M )$; +% \ref{it:x4}. $C \imp M$; +% \ref{it:x5}. $(M \et C) \ou (\non M \et Q)$. +\end{Exo} -\begin{Exo} - En notant $M$, $C$ et $A$ les trois affirmations suivantes: -\begin{itemize} - \item $M$ = \og Pierre fait des Maths\fg{}; -\item $C$ = \og Pierre fait de la Chimie\fg{}; -\item $A$ = \og Pierre fait de l'Anglais\fg{}. -\end{itemize} +% \begin{Exo} +% En notant $M$, $C$ et $A$ les trois affirmations suivantes: +% \begin{itemize} +% \item $M$ = \og Pierre fait des Maths\fg{}; +% \item $C$ = \og Pierre fait de la Chimie\fg{}; +% \item $A$ = \og Pierre fait de l'Anglais\fg{}. +% \end{itemize} -Représenter les affirmations qui suivent sous forme -symbolique, à l'aide des lettres $M$, $C$, $A$ et des connecteurs -usuels. +% Représenter les affirmations qui suivent sous forme +% symbolique, à l'aide des lettres $M$, $C$, $A$ et des connecteurs +% usuels. -\begin{enumerate} - \item \label{ex2:1} \og Pierre fait des Maths et de l'Anglais mais pas de Chimie\fg{} -\item \label{ex2:2} \og Pierre fait des Maths et de la Chimie mais pas à la fois de la Chimie et de l'Anglais\fg{} -\item \label{ex2:3}\og Il est faux que Pierre fasse de l'Anglais sans faire de Maths\fg{} -\item \label{ex2:4} \og Il est faux que Pierre ne fasse pas des Maths et fasse quand même de la chimie\fg{} -\item \label{ex2:5} \og Il est faux que Pierre fasse de l'Anglais ou de la Chimie sans faire des Maths\fg{} -\item \label{ex2:6} \og Pierre ne fait ni Anglais ni Chimie mais il fait des Maths\fg{} -\end{enumerate} +% \begin{enumerate} +% \item \label{ex2:1} \og Pierre fait des Maths et de l'Anglais mais pas de Chimie\fg{} +% \item \label{ex2:2} \og Pierre fait des Maths et de la Chimie mais pas à la fois de la Chimie et de l'Anglais\fg{} +% \item \label{ex2:3}\og Il est faux que Pierre fasse de l'Anglais sans faire de Maths\fg{} +% \item \label{ex2:4} \og Il est faux que Pierre ne fasse pas des Maths et fasse quand même de la chimie\fg{} +% \item \label{ex2:5} \og Il est faux que Pierre fasse de l'Anglais ou de la Chimie sans faire des Maths\fg{} +% \item \label{ex2:6} \og Pierre ne fait ni Anglais ni Chimie mais il fait des Maths\fg{} +% \end{enumerate} -\end{Exo} +% \end{Exo} % Réponses: % \ref{ex2:1}. $M \et A \et (\non C)$; @@ -283,7 +281,7 @@ usuels. % \ref{ex2:4}. $\non ((\non M) \et C)$; % \ref{ex2:5}. $\non ((A \ou C) \et \non M)$; % \ref{ex2:6}. $(\non A) \et (\non C) \et M$. -% \end{Exoc} +% \end{Exo} \begin{Exo} @@ -296,6 +294,21 @@ $P \Rightarrow Q$ et $ \neg( P\Rightarrow Q)$ \item Définir les négations de $ P\land Q$, $ P\lor Q$ et $P \Rightarrow Q$. \end{enumerate} + +\begin{Th} +On a les règles syntaxiques suivantes de simplification de négations: +\begin{itemize} +\item $\neg (A \lor B) = (\neg A) \land (\neg B)$; +\item $\neg (A \land B) = (\neg A) \lor (\neg B)$; +\item $\neg \neg A = A $; +\item $\neg (A \Rightarrow B) = A \land (\neg B)$. +\end{itemize} +On remarque que la troisième règle se déduit des trois autres: +$\neg (A \Rightarrow B) = \neg (\neg A \lor B) = (\neg \neg A) \land (\neg B)$. +\end{Th} + + + \end{Exo} \begin{Exo} \'Enoncer la négation des affirmations suivantes en évitant d'employer l'expression: \og il est faux que\fg{} @@ -305,8 +318,8 @@ $P \Rightarrow Q$ et $ \neg( P\Rightarrow Q)$ \item \og Le nombre 522 n'est pas divisible par 3 mais il est divisible par 7\fg{} \item \og Ce quadrilatère n'est ni un rectangle ni un losange\fg{} \item \og Si Paul ne va pas travailler ce matin il va perdre son emploi\fg{} -\item \og Tout nombre entier impair peut être divisible par 3 ou par 5 mais jamais par 2\fg{} -\item \og Tout triangle équilatéral a ses angles égaux à 60°\fg{} +% \item \og Tout nombre entier impair peut être divisible par 3 ou par 5 mais jamais par 2\fg{} +% \item \og Tout triangle équilatéral a ses angles égaux à 60°\fg{} \end{enumerate} @@ -323,7 +336,7 @@ $P \Rightarrow Q$ et $ \neg( P\Rightarrow Q)$ % \end{enumerate} \end{Exo} -% \begin{Exoc} +% \begin{Exo} % Quelles sont les valeurs de vérité des propositions suivantes ? % \begin{enumerate} % \item \label{it:3:1}$\pi$ vaut 4 et la somme des angles d'un triangle @@ -363,7 +376,7 @@ $P \Rightarrow Q$ et $ \neg( P\Rightarrow Q)$ % \ref{it:3:9} V; % %\ref{it:3:10} F; % \ref{it:3:11} V. -% \end{Exoc} +% \end{Exo} % \begin{Exo} % Partant des deux affirmations $P$ et $Q$, on peut en construire une autre, notée $P \downarrow Q$, bâtie sur le modèle: \og ni $P$, ni $Q$\fg{}. @@ -437,13 +450,13 @@ suivants: \begin{enumerate} \item \og $A$ si $B$\fg{} \item \og $A$ est condition nécessaire pour $B$\fg{} -\item \og $A$ sauf si $B$\fg{} +%\item \og $A$ sauf si $B$\fg{} \item \og $A$ seulement si $B$\fg{} \item \og $A$ est condition suffisante pour $B$\fg{} \item \og $A$ bien que $B$\fg{} \item \og Non seulement $A$, mais aussi $B$\fg{} \item \og $A$ et pourtant $B$\fg{} -\item \og $A$ à moins que $B$\fg{} +%\item \og $A$ à moins que $B$\fg{} \item \og Ni $A$, ni $B$\fg{} \end{enumerate} \end{Exo} @@ -461,11 +474,11 @@ traduction vous paraît impossible, dites-le et expliquez pourquoi): \item C'est seulement si un étudiant travaille qu'il a de bonnes notes. \item Un étudiant n'a de bonnes notes que s'il travaille. \item Pour un étudiant, le travail est une condition nécessaire à l'obtention de bonnes notes. -\item Un étudiant a de mauvaises notes, à moins qu'il ne travaille. +%\item Un étudiant a de mauvaises notes, à moins qu'il ne travaille. \item Malgré son travail, un étudiant a de mauvaises notes. \item Un étudiant travaille seulement s'il a de bonnes notes. -\item \`A quoi bon travailler, si c'est pour avoir de mauvaises notes? -\item Un étudiant a de bonnes notes sauf s'il ne travaille pas. +%\item \`A quoi bon travailler, si c'est pour avoir de mauvaises notes? +%\item Un étudiant a de bonnes notes sauf s'il ne travaille pas. \end{enumerate} \end{Exo} @@ -515,54 +528,54 @@ dans la même proposition, puisqu'il n'y a pas de priorité entre $\ou$ et $\et$: $(A \ou B) \et C \neq A \ou (B \et C)$. \end{Th} -\begin{Rem} -L'implication n'est pas associative: $A \imp (B \imp C) \neq (A \imp -B) \imp C$. Donc les parenthèses sont obligatoires. -Il en est de même pour $\eqv$, et a fortiori quand ces deux opérateurs -sont mélangés dans une même proposition. -\end{Rem} +% \begin{Rem} +% L'implication n'est pas associative: $A \imp (B \imp C) \neq (A \imp +% B) \imp C$. Donc les parenthèses sont obligatoires. +% Il en est de même pour $\eqv$, et a fortiori quand ces deux opérateurs +% sont mélangés dans une même proposition. +% \end{Rem} % AG: On convient en général d'associer à droite quand il n'y a % pas de parenthèses. -\begin{Exoc} - Quelles sont les façons de placer des parenthèses dans $\non P \ou Q - \et \non R$ afin d'obtenir l'expression correcte d'une formule - propositionnelle ? Déterminer la table de vérité de chacune des - formules obtenues. +% \begin{Exo} +% Quelles sont les façons de placer des parenthèses dans $\non P \ou Q +% \et \non R$ afin d'obtenir l'expression correcte d'une formule +% propositionnelle ? Déterminer la table de vérité de chacune des +% formules obtenues. -Réponses: +% Réponses: -1) $\non P \ou (Q \et \non R)$; -2) $(\non P \ou Q ) -\et \non R$; -3) $(\non (P \ou Q)) \et \non R$; -4) $\non (P \ou -(Q \et \non R))$; -5) $\non ((P \ou Q) \et \non R)$. +% 1) $\non P \ou (Q \et \non R)$; +% 2) $(\non P \ou Q ) +% \et \non R$; +% 3) $(\non (P \ou Q)) \et \non R$; +% 4) $\non (P \ou +% (Q \et \non R))$; +% 5) $\non ((P \ou Q) \et \non R)$. -Tables de vérité: +% Tables de vérité: -$$ -\begin{array}{|c|c|c||c|c|c|c|c|} -\hline -P & Q & R & 1 & 2 & 3 & 4 & 5 \\ -\hline -V & V & V & F & F & F & F & V \\ -V & V & F & V & V & F & F & F \\ -V & F & V & F & F & F & F & V \\ -V & F & F & F & F & F & F & F \\ -F & V & V & V & F & F & V & V \\ -F & v & F & V & V & F & F & F \\ -F & F & V & V & F & F & V & V \\ -F & F & F & V & V & V & V & V \\ -\hline -\end{array} -$$ +% $$ +% \begin{array}{|c|c|c||c|c|c|c|c|} +% \hline +% P & Q & R & 1 & 2 & 3 & 4 & 5 \\ +% \hline +% V & V & V & F & F & F & F & V \\ +% V & V & F & V & V & F & F & F \\ +% V & F & V & F & F & F & F & V \\ +% V & F & F & F & F & F & F & F \\ +% F & V & V & V & F & F & V & V \\ +% F & v & F & V & V & F & F & F \\ +% F & F & V & V & F & F & V & V \\ +% F & F & F & V & V & V & V & V \\ +% \hline +% \end{array} +% $$ -\end{Exoc} +% \end{Exo} @@ -570,37 +583,37 @@ $$ -\begin{Exoc} +\begin{Exo} Construire les tables de vérité des formules propositionnelles suivantes: -\begin{enumerate} -\item $ \non P \et Q$ -\item $\non P \imp P \ou Q$ -\item $\non ( \non P \et \non Q)$ -\item $P \et Q \imp \non Q$ -\item $(P \imp Q) \ou (Q \imp P)$ -\item $(P \imp \non Q ) \ou (Q \imp \non P)$ -\item $(P \ou \non Q ) \et (\non P \ou Q)$ -\item $P \imp (\non P \imp P)$ -\end{enumerate} +% \begin{enumerate} +% \item $ \non P \et Q$ +% \item $\non P \imp P \ou Q$ +% \item $\non ( \non P \et \non Q)$ +% \item $P \et Q \imp \non Q$ +% \item $(P \imp Q) \ou (Q \imp P)$ +% \item $(P \imp \non Q ) \ou (Q \imp \non P)$ +% \item $(P \ou \non Q ) \et (\non P \ou Q)$ +% \item $P \imp (\non P \imp P)$ +% \end{enumerate} -Réponse: +% Réponse: -$$\begin{array}{|c|c||c|c|c|c|c|c|c|c|} -\hline -P & Q & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ -\hline -V & V & F & V & V & F & V & F & V & V \\ -V & F & F & V & V & V & V & V & F & V \\ -F & V & V & V & V & V & V & V & F & V \\ -F & F & F & F & F & V & V & V & V & V \\ -\hline -\end{array} -$$ -\end{Exoc} +% $$\begin{array}{|c|c||c|c|c|c|c|c|c|c|} +% \hline +% P & Q & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ +% \hline +% V & V & F & V & V & F & V & F & V & V \\ +% V & F & F & V & V & V & V & V & F & V \\ +% F & V & V & V & V & V & V & V & F & V \\ +% F & F & F & F & F & V & V & V & V & V \\ +% \hline +% \end{array} +% $$ +% \end{Exo} -\begin{Exo} - Faire de même avec +% \begin{Exo} +% Faire de même avec \begin{enumerate} \item $(P \ou Q) \ou (\non R)$ \item $P \ou (\non (Q \et R))$ @@ -754,24 +767,24 @@ suivante: \begin{enumerate} -\item Si $F$ est de la forme $P$, o\`u $P$ est une variable - propositionnelle, alors $\Phi_F(p)=p$. +\item Si c'est une variable $P$, propositionnelle, alors $\Phi_P(p)=p$. -\item Si $F$ est de la forme $\non G$, o\`u $G$ est une formule - propositionnelle, alors $\Phi_F=\overline{\Phi_G}$. +\item Si c'est une négation d'une formule + propositionnelle $G$, alors $\Phi_{\non G}=\overline{\Phi_G}$. -\item Si $F$ est de la forme $G\ou H$, o\`u $G$ et $H$ sont des formules - propositionnelles, alors $\Phi_F=\Phi_G+\Phi_H$. +\item Si c'est une disjonction entre deux formules + propositionnelle $G$ ou $H$, alors $\Phi_{G \lor H}=\Phi_G+\Phi_H$. -\item Si $F$ est de la forme $G\et H$, o\`u $G$ et $H$ sont des formules - propositionnelles, alors $\Phi_F=\Phi_G\cdot \Phi_H$. +\item Si c'est une conjonction entre deux formules + propositionnelle $G$ et $H$, alors $\Phi_{G \land H}=\Phi_G \cdot \Phi_H$. -\item Si $F$ est de la forme $G\imp H$, o\`u $G$ et $H$ sont des - formules propositionnelles, alors $\Phi_F=\overline{\Phi_G}+\Phi_H$. +\item Si elle est de la forme $G\imp H$, o\`u $G$ et $H$ sont des + formules propositionnelles, alors + $\Phi_{G\imp H}=\overline{\Phi_G}+\Phi_H$. -\item \label{item:eqv} Si F est de la forme $G\eqv H$, o\`u $G$ et $H$ sont des formules - propositionnelles, alors - $\Phi_F=\overline{\Phi_G}\cdot\overline{\Phi_H}+\Phi_G\cdot\Phi_H$. +\item \label{item:eqv} Si c'est une équivalence entre +les formules propostionnelles $G$ et $H$, alors + $\Phi_{G \Leftrightarrow H}=\overline{\Phi_G}\cdot\overline{\Phi_H}+\Phi_G\cdot\Phi_H$. \end{enumerate} \end{Def} @@ -801,34 +814,39 @@ suivante: \begin{Ex} -Soit $F=A\ou\non B\eqv(B\imp C)$. On a alors: - -\hfil$\Phi_F(a,b,c)=\overline{a+\overline b}\cdot -\overline{\overline b+c}+(a+\overline b)\cdot(\overline -b+c)=\overline a\cdot b\cdot b\cdot\overline c+\overline b+a\cdot -c=\overline b+\overline a\cdot\overline c+a\cdot -c$.\hfil +Soit $F=(A\ou\non B) \land (B\imp C)$. Cette formule dépend des trois variables +$A$, $B$ et $C$.On a alors: +$$ +\begin{array}{l} +\Phi_F(a,b,c)= \Phi_{(A\ou\non B) \land (B\imp C)}(a,b,c) = \\ +\Phi_{A\ou\non B}(a,b) \cdot \Phi_{B\imp C}(b,c) = +(\Phi_{A}(a) + \Phi_{\non B}(b)) \cdot (\overline{\Phi_{B}(b)}+ \Phi_{C}(c))=\\ +(a+ \overline{\Phi_{B}(b)}) \cdot (\overline{b}+ c)= +(a+ \overline{b}) \cdot (\overline{b}+ c)= \overline{b} + ac +\end{array} +$$ \end{Ex} % AG: Faux pour imp et eqv qui ne sont pas (utiles) dans l'algèbre % et que l'interpretation ramene a ET, OU, NON. A modifier. -\begin{Rem} -Il est clair que les \og tables de vérité\fg{} des connecteurs -logiques sont les mêmes que les tables des opérations booléennes sur -$\{\faux,\vrai\}$ +% \begin{Rem} +% Il est clair que les \og tables de vérité\fg{} des connecteurs +% logiques sont les mêmes que les tables des opérations booléennes sur +% $\{\faux,\vrai\}$ -\begin{itemize} -\item de la négation booléenne (pour la négation logique), -\item de la somme booléenne (pour la disjonction logique), -\item du produit booléen (pour la conjonction logique), -%\item de la fonction booléenne de deux variables appelée \og -% implication\fg{} (pour l'implication logique) -%\item de la fonction booléenne de deux variables appelée \og -% équivalence\fg{} (pour l'équivalence logique). -\end{itemize} +% \begin{itemize} +% \item de la négation booléenne (pour la négation logique), +% \item de la somme booléenne (pour la disjonction logique), +% \item du produit booléen (pour la conjonction logique), +% %\item de la fonction booléenne de deux variables appelée \og +% % implication\fg{} (pour l'implication logique) +% %\item de la fonction booléenne de deux variables appelée \og +% % équivalence\fg{} (pour l'équivalence logique). +% \end{itemize} -Ainsi, la détermination de la valeur de vérité d'une proposition composée se ramène à un simple calcul en algèbre de Boole sur la fonction de vérité de la formule propositionnelle associée. -\end{Rem} +% +La détermination de la valeur de vérité d'une proposition composée se ramène à un simple calcul en algèbre de Boole sur la fonction de vérité de la formule propositionnelle associée. +% \end{Rem} @@ -894,7 +912,7 @@ tautologies; la reconnaissance de cette propriété n'est cependant pas toujours complètement évidente\ldots -% \begin{Exoc} +% \begin{Exo} % Les formules propositionnelles suivantes sont-elles des tautologies ? % \begin{enumerate} @@ -917,7 +935,7 @@ toujours complètement évidente\ldots % \ref{item:taut:7}. et % \ref{item:taut:8}. sont des tautologies. -% \end{Exoc} +% \end{Exo} % AG: Indiquer la méthode de ``preuve'' ou dire ``Montrer'' \begin{Exo} @@ -1070,9 +1088,9 @@ $$\begin{array}{c r l} % 1 & \{P \et Q\} & P\\ % 2 & \{(P \et Q) \ou R\} & P\ \et (Q \ou R) \\ % 3 & \{(P \et Q) \imp R \} & (P \imp R) \et (Q \imp R) \\ -4 & \{P \imp (Q \ou R)\} & (P \imp Q) \ou (P \imp R) \\ -5 & \{A \imp (P \ou Q), \neg S \lor A \} & (\neg P \lor S) \imp Q \\ -5 & \{A \imp (B \et C), \neg C \lor D \lor R, R \imp \neg B \} & +1 & \{P \imp (Q \ou R)\} & (P \imp Q) \ou (P \imp R) \\ +2 & \{A \imp (P \ou Q), \neg S \lor A \} & (\neg P \lor S) \imp Q \\ +3 & \{A \imp (B \et C), \neg C \lor D \lor R, R \imp \neg B \} & (A \land D) \imp \neg R \\ \end{array} $$ @@ -1161,7 +1179,7 @@ $$ %Réponse: oui pour 1, 2, 3, 5, 6, 7, 9, 10, 12. \end{Exo} -\begin{Exoc} +\begin{Exo} Soit $F$ une formule propositionnelle dépendant de trois variables $P, Q, R$ qui possède deux propriétés: \begin{itemize} @@ -1195,60 +1213,60 @@ $ (P \et \non Q \et \non R) \ou (\non P \et \non Q \et R) \ou (\non P \et Q \et \non R)$ -\end{Exoc} +\end{Exo} -\begin{Exo} -Déterminer des formules propositionnelles $F, G, H$ dépendant des -variables $P,Q,R$ qui admettent les tables de vérité: -$$\begin{array}{ccc} -\begin{array}{|ccc|c|} -\hline +% \begin{Exo} +% Déterminer des formules propositionnelles $F, G, H$ dépendant des +% variables $P,Q,R$ qui admettent les tables de vérité: +% $$\begin{array}{ccc} +% \begin{array}{|ccc|c|} +% \hline -P & Q & R & F \\ -\hline -V & V & V & V\\ -V & V & F & F \\ -V & F & V & V \\ -V & F & F & F \\ -F & V & V & F \\ -F & V & F & V \\ -F & F& V & V \\ -F & F & F & V\\ -\hline -\end{array} -& -\begin{array}{|ccc|c|} -\hline -P & Q & R & G \\ -\hline -V & V & V & F\\ -V & V & F & V \\ -V & F & V & V \\ -V & F & F & F \\ -F & V & V & F \\ -F & V & F & V \\ -F & F& V & V \\ -F & F & F & F\\ -\hline -\end{array} -& -\begin{array}{|ccc|c|} -\hline -P & Q & R & H \\ -\hline -V & V & V & V\\ -V & V & F & V \\ -V & F & V & V \\ -V & F & F & F \\ -F & V & V & F \\ -F & V & F & V \\ -F & F& V & V \\ -F & F & F & V\\ -\hline -\end{array} - \end{array} -$$ -\end{Exo} +% P & Q & R & F \\ +% \hline +% V & V & V & V\\ +% V & V & F & F \\ +% V & F & V & V \\ +% V & F & F & F \\ +% F & V & V & F \\ +% F & V & F & V \\ +% F & F& V & V \\ +% F & F & F & V\\ +% \hline +% \end{array} +% & +% \begin{array}{|ccc|c|} +% \hline +% P & Q & R & G \\ +% \hline +% V & V & V & F\\ +% V & V & F & V \\ +% V & F & V & V \\ +% V & F & F & F \\ +% F & V & V & F \\ +% F & V & F & V \\ +% F & F& V & V \\ +% F & F & F & F\\ +% \hline +% \end{array} +% & +% \begin{array}{|ccc|c|} +% \hline +% P & Q & R & H \\ +% \hline +% V & V & V & V\\ +% V & V & F & V \\ +% V & F & V & V \\ +% V & F & F & F \\ +% F & V & V & F \\ +% F & V & F & V \\ +% F & F& V & V \\ +% F & F & F & V\\ +% \hline +% \end{array} +% \end{array} +% $$ +% \end{Exo} % Réponses: @@ -1265,63 +1283,6 @@ $$ \subsection{Simplification du calcul des fonctions de vérité} -\subsubsection{Théorème de substitution} - -\begin{Th}[Théorème de substitution] - \index{théorème!de substitution} -Soit $F$ une formule propositionnelle dans laquelle interviennent les -variables propositionnelles $P_1\,$, $P_2\,$, $P_3\,$,\ldots, $P_n$. -Supposons que l'on remplace ces variables par des formules -propositionnelles $G_1\,$, $G_2\,$, $G_3\,$,\ldots, $G_n$; la nouvelle -formule propositionnelle obtenue est notée $F^*$. - - -Dans ces conditions: si $\tauto F$, alors $\tauto F^*$. -\end{Th} - - - -\begin{Proof} -$F$ étant une tautologie, sa fonction de vérité ne dépend pas des -valeurs de vérité des variables booléennes, qui peuvent donc -être remplacées par n'importe quelle fonction booléenne. -\end{Proof} - -Attention, la réciproque n'est pas vraie\ldots - -\begin{Ex} -Soit $F=A\imp B$ et $F^*=P\et \non P \imp Q$, -obtenue à partir de $F$ en remplaçant $A$ par -$P\et \non P$ et $B$ par $Q$. -Comme $\Phi_{F^*}(p,q)=\overline{p\cdot\overline - p}+q=\overline 0+q=1+q=1$, alors $F^{*}$ est une tautologie. -Cependant de $\Phi_{F}(a,b)$, on ne peut pas dire que $F$ est une tautologie. -\end{Ex} - - -Exemple d'utilisation de ce résultat: - -\begin{Ex} -La formule propositionnelle -$$F^*=((P\imp Q\et\non R)\ou (\non S\eqv T))\imp ((P\imp -Q\et\non R)\ou(\non S\eqv T)),$$ -est compliquée puisqu'elle contient 5 variables propositionnelle. -il y a donc 32 lignes -à calculer pour obtenir les valeurs de la fonction de vérité. -Cependant, il suffit de remarquer que $F^*$ est obtenue à partir de -$F=A \imp A$, qui est une tautologie; donc $F^*$ en est une aussi. -\end{Ex} - - - -Ce résultat peut évidemment être appliqué aussi à des parties de -formules propositionnelles, pour accélérer le calcul de leurs fonctions -de vérité: -si une partie d'une formule propositionnelle constitue à elle seule une -tautologie, la partie correspondante de la fonction de vérité peut -être avantageusement remplacée par 1. - -\subsubsection{Théorème de la validité} \begin{Th}[Théorème de la validité] Soit $\{G_1,G_2,\ldots,G_n\}$ un ensemble de formules propositionnelles @@ -1534,60 +1495,60 @@ Donc Pierre est le meurtrier Marie n'était pas présente? \end{enumerate} \end{Exo} -\subsection{Conclusion} - -Le calcul sur les fonctions de vérité paraît tout-à-fait -satisfaisant et séduisant, lorsqu'il s'agit de calculer des valeurs de -vérité ou d'examiner des conséquences logiques. -Il est vrai qu'il est simple, nécessite un minimum de réflexion (très -important dans le cas des ordinateurs!) et qu'il est très facile à -programmer. - - -Mais, pour une formule propositionnelle qui comporte 10 variables -propositionnelles (ce qui n'est pas beaucoup pour les problèmes que -l'on cherche à programmer!), la table des valeurs de la fonction de -vérité comporte $2^{10}=1024$ lignes. -Celui qui opère à la main a déjà démissionné. -L'ordinateur démissionne un peu plus loin, certes, mais il finit aussi -par avouer son incapacité: -\begin{itemize} -\item Sur les machines modernes, il n'est plus impossible d'envisager - d'écrire et d'exécuter une \og boucle vide\fg{} qui porte sur toutes - les valeurs entières représentables sur 32 bits, donc de 0 à - $2^{32}-1$, le temps d'exécution est récemment devenu raisonnable. -\item Il ne faut cependant pas exiger que ce temps demeure raisonnable - dès qu'il s'agit d'exécuter un algorithme un peu compliqué. Et 32 - variables constituent un nombre -ridiculement petit pour un système expert, dans lequel les expressions -offrent souvent une complexité qui n'a aucune commune mesure avec ce -que l'on peut imaginer de plus compliqué\ldots -\end{itemize} +% \subsection{Conclusion} + +% Le calcul sur les fonctions de vérité paraît tout-à-fait +% satisfaisant et séduisant, lorsqu'il s'agit de calculer des valeurs de +% vérité ou d'examiner des conséquences logiques. +% Il est vrai qu'il est simple, nécessite un minimum de réflexion (très +% important dans le cas des ordinateurs!) et qu'il est très facile à +% programmer. + + +% Mais, pour une formule propositionnelle qui comporte 10 variables +% propositionnelles (ce qui n'est pas beaucoup pour les problèmes que +% l'on cherche à programmer!), la table des valeurs de la fonction de +% vérité comporte $2^{10}=1024$ lignes. +% Celui qui opère à la main a déjà démissionné. +% L'ordinateur démissionne un peu plus loin, certes, mais il finit aussi +% par avouer son incapacité: +% \begin{itemize} +% \item Sur les machines modernes, il n'est plus impossible d'envisager +% d'écrire et d'exécuter une \og boucle vide\fg{} qui porte sur toutes +% les valeurs entières représentables sur 32 bits, donc de 0 à +% $2^{32}-1$, le temps d'exécution est récemment devenu raisonnable. +% \item Il ne faut cependant pas exiger que ce temps demeure raisonnable +% dès qu'il s'agit d'exécuter un algorithme un peu compliqué. Et 32 +% variables constituent un nombre +% ridiculement petit pour un système expert, dans lequel les expressions +% offrent souvent une complexité qui n'a aucune commune mesure avec ce +% que l'on peut imaginer de plus compliqué\ldots +% \end{itemize} -Les \og raccourcis\fg{} qui viennent d'être étudiés et qui permettent -d'accélérer, voire de supprimer totalement, le calcul d'une fonction -de vérité, sont plus utiles lorsque l'on opère \og à la main\fg{} que -pour la programmation d'algorithmes de logique. +% Les \og raccourcis\fg{} qui viennent d'être étudiés et qui permettent +% d'accélérer, voire de supprimer totalement, le calcul d'une fonction +% de vérité, sont plus utiles lorsque l'on opère \og à la main\fg{} que +% pour la programmation d'algorithmes de logique. -Il faut donc garder en réserve la méthode des fonctions de vérité: -celle-ci peut être très utile dans certains cas, essentiellement -lorsque le problème peut être résolu \og à la main\fg{}, mais il faut -aussi trouver une autre méthode pour songer à aborder des problèmes -plus complexes. +% Il faut donc garder en réserve la méthode des fonctions de vérité: +% celle-ci peut être très utile dans certains cas, essentiellement +% lorsque le problème peut être résolu \og à la main\fg{}, mais il faut +% aussi trouver une autre méthode pour songer à aborder des problèmes +% plus complexes. -Cette méthode, qui supprime toute référence aux valeurs de vérité, -fait l'objet du -chapitre suivant. +% Cette méthode, qui supprime toute référence aux valeurs de vérité, +% fait l'objet du +% chapitre suivant. -\gsaut +% \gsaut \centerline{\x{Fin du Chapitre}} diff --git a/main13.aux b/main13.aux index 588714e..9eb2c17 100644 --- a/main13.aux +++ b/main13.aux @@ -94,279 +94,258 @@ \@writefile{thm}{\contentsline {Rem}{{Remarque}{2.{1}}{}}{12}{Rem.2.1}} \global\def\markivRemii{\ensuremath {}} \@writefile{thm}{\contentsline {Exo}{{Exercice}{2.{2}}{}}{12}{Exo.2.2}} -\@writefile{thm}{\contentsline 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-\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} +\indexentry{tautologie|hyperpage}{15} +\indexentry{antilogie|hyperpage}{15} +\indexentry{cons\IeC {\'e}quence logique|hyperpage}{15} +\indexentry{formules \IeC {\'e}quivalentes|hyperpage}{16} +\indexentry{ensemble|hyperpage}{20} +\indexentry{appartenance|hyperpage}{20} +\indexentry{ensemble!vide|hyperpage}{20} +\indexentry{inclusion|hyperpage}{20} +\indexentry{r\IeC {\'e}union|hyperpage}{21} +\indexentry{r\IeC {\'e}union|hyperpage}{21} +\indexentry{ensemble!compl\IeC {\'e}mentaire|hyperpage}{22} +\indexentry{compl\IeC {\'e}mentation|hyperpage}{22} +\indexentry{involution|hyperpage}{22} +\indexentry{loi de De Morgan|hyperpage}{22} +\indexentry{relation binaire|hyperpage}{24} +\indexentry{relation!r\IeC {\'e}flexive|hyperpage}{24} +\indexentry{relation!antisym\IeC {\'e}trique|hyperpage}{24} +\indexentry{relation!transitive|hyperpage}{24} +\indexentry{relation!d'ordre|hyperpage}{25} +\indexentry{relation!sym\IeC {\'e}trique|hyperpage}{25} +\indexentry{classe d'\IeC {\'e}quivalence|hyperpage}{26} +\indexentry{partition|hyperpage}{26} +\indexentry{r\IeC {\'e}currence!restreinte|hyperpage}{28} +\indexentry{multiple|hyperpage}{28} +\indexentry{diviseur|hyperpage}{28} +\indexentry{nombre!premier|hyperpage}{28} +\indexentry{d\IeC {\'e}composition en facteurs premiers|hyperpage}{28} +\indexentry{plus grand commun diviseur|hyperpage}{29} +\indexentry{PGCD|hyperpage}{29} +\indexentry{PPCM|hyperpage}{29} +\indexentry{plus petit commun multiple|hyperpage}{29} +\indexentry{division euclidienne|hyperpage}{29} +\indexentry{quotient|hyperpage}{29} +\indexentry{reste|hyperpage}{29} +\indexentry{algorithme!d'Euclide|hyperpage}{30} +\indexentry{th\IeC {\'e}or\IeC {\`e}me!de B\IeC {\'e}zout|hyperpage}{31} +\indexentry{algorithme!d'Euclide!g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}|hyperpage}{32} +\indexentry{congru|hyperpage}{33} +\indexentry{modulo|hyperpage}{33} +\indexentry{pseudo-premier|hyperpage}{35} diff --git a/main13.log b/main13.log index 7d9a72e..da137fa 100644 --- a/main13.log +++ b/main13.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2013.11.13) 24 JAN 2014 11:19 +This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2013.11.13) 12 FEB 2014 09:14 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -1146,7 +1146,7 @@ pdfTeX warning (ext4): destination with the same identifier (name{page.1}) has been already used, duplicate ignored \relax -l.50 ...sembles de nombres entiers}{31}{chapter.5} +l.49 ...ine {II}Nombres premiers}{28}{section.5.2} [1 ]) @@ -1209,27 +1209,16 @@ atres Chapitre 2. (./logique/Propositions13.tex [11 -] -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 -File: ts1ptm.fd 2001/06/04 font definitions for TS1/ptm. -) [13] -Underfull \hbox (badness 2717) in paragraph at lines 489--491 +] [12] +Underfull \hbox (badness 2717) in paragraph at lines 502--504 []\T1/ptm/m/n/10.95 Les conven-tions de prio-rité des [] -[14] [15] +[13] [14] Package hyperref Info: bookmark level for unknown Notation defaults to 0 on inp -ut line 857. - [16] [17] [18] -Overfull \hbox (20.57855pt too wide) in paragraph at lines 1352--1361 +ut line 875. + [15] [16] +Overfull \hbox (20.57855pt too wide) in paragraph at lines 1313--1322 \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 $\OML/lmm/ @@ -1237,13 +1226,13 @@ m/it/10.95 H$ \T1/ptm/m/n/10.95 est une cons /10.95 f\OML/lmm/m/it/10.95 G[]; G[]; [] ; G[]\OMS/lmsy/m/n/10.95 g$ [] -[19] [20]) [21] [22 +[17]) [18] [19 ] Chapitre 3. -(./ensembles/IntroAuxEnsembles13.tex [23 +(./ensembles/IntroAuxEnsembles13.tex [20 -] [24] +] [21] 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 $\OML/lmm @@ -1255,16 +1244,16 @@ mr/m/n/10.95 = \OMS/lmsy/m/n/10.95 f\OML/lmm/m/it/10.95 x \OMS/lmsy/m/n/10.95 2 .95 g$\T1/ptm/m/it/10.95 . [] -[25] +[22] 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- [] -) [26] +) [23] Chapitre 4. -(./ensembles/relbin13.tex [27 +(./ensembles/relbin13.tex [24 -] [28] +] [25] Package hyperref Info: bookmark level for unknown Pre defaults to 0 on input li ne 301. @@ -1294,7 +1283,7 @@ 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 :$ [] -) [29] [30 +) [26] [27 ] Chapitre 5. @@ -1305,7 +1294,7 @@ Overfull \hbox (11.42195pt too wide) in paragraph at lines 134--134 10.95 a[]b[]c[] []$\T1/ptm/m/n/10.95 , [] -[31 +[28 ] LaTeX Font Info: External font `lmex10' loaded for size @@ -1321,7 +1310,12 @@ Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): 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] +[29] [30] [31] [32] +LaTeX Font Info: Try loading font information for TS1+ptm on input line 702. + + (/usr/share/texlive/texmf-dist/tex/latex/psnfss/ts1ptm.fd +File: ts1ptm.fd 2001/06/04 font definitions for TS1/ptm. +) Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 710. @@ -1330,24 +1324,24 @@ Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 710. -[36] +[33] Underfull \hbox (badness 10000) in paragraph at lines 842--843 [] -[37]) [38] [39 +[34]) [35] [36 ] \openout2 = `PPN.aux'. 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[0][]{chapter.3}{Introduction \340 la th\351orie des ensembles}{part.2}% 22 -\BOOKMARK [1][]{section.3.1}{Rappels de th\351orie des ensembles}{chapter.3}% 23 -\BOOKMARK [2][]{subsection.3.1.1}{Notion premi\350re d'ensemble}{section.3.1}% 24 -\BOOKMARK [2][]{subsection.3.1.2}{R\350gles de fonctionnement}{section.3.1}% 25 -\BOOKMARK [2][]{subsection.3.1.3}{Sous-ensembles, ensemble des parties}{section.3.1}% 26 -\BOOKMARK [1][]{section.3.2}{Op\351rations sur les ensembles}{chapter.3}% 27 -\BOOKMARK [2][]{subsection.3.2.1}{\311galite de deux ensembles}{section.3.2}% 28 -\BOOKMARK [2][]{subsection.3.2.2}{R\351union, intersection}{section.3.2}% 29 -\BOOKMARK [2][]{subsection.3.2.3}{Compl\351mentation}{section.3.2}% 30 -\BOOKMARK [2][]{subsection.3.2.4}{Produit cart\351sien}{section.3.2}% 31 -\BOOKMARK [1][]{section.3.3}{Exercices suppl\351mentaires}{chapter.3}% 32 -\BOOKMARK [0][]{chapter.4}{Relations binaires entre ensembles}{part.2}% 33 -\BOOKMARK [1][]{section.4.1}{Relations}{chapter.4}% 34 -\BOOKMARK [1][]{section.4.2}{Relations d'ordre}{chapter.4}% 35 -\BOOKMARK [2][]{subsection.4.2.1}{R\351flexivit\351, antisym\351trie, transitivit\351}{section.4.2}% 36 -\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 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 +\BOOKMARK [-1][]{part.2}{II Th\351orie des ensembles}{}% 20 +\BOOKMARK [0][]{chapter.3}{Introduction \340 la th\351orie des ensembles}{part.2}% 21 +\BOOKMARK [1][]{section.3.1}{Rappels de th\351orie des ensembles}{chapter.3}% 22 +\BOOKMARK [2][]{subsection.3.1.1}{Notion premi\350re d'ensemble}{section.3.1}% 23 +\BOOKMARK [2][]{subsection.3.1.2}{R\350gles de fonctionnement}{section.3.1}% 24 +\BOOKMARK [2][]{subsection.3.1.3}{Sous-ensembles, ensemble des parties}{section.3.1}% 25 +\BOOKMARK [1][]{section.3.2}{Op\351rations sur les ensembles}{chapter.3}% 26 +\BOOKMARK [2][]{subsection.3.2.1}{\311galite de deux ensembles}{section.3.2}% 27 +\BOOKMARK [2][]{subsection.3.2.2}{R\351union, intersection}{section.3.2}% 28 +\BOOKMARK [2][]{subsection.3.2.3}{Compl\351mentation}{section.3.2}% 29 +\BOOKMARK [2][]{subsection.3.2.4}{Produit cart\351sien}{section.3.2}% 30 +\BOOKMARK [1][]{section.3.3}{Exercices suppl\351mentaires}{chapter.3}% 31 +\BOOKMARK [0][]{chapter.4}{Relations binaires entre ensembles}{part.2}% 32 +\BOOKMARK [1][]{section.4.1}{Relations}{chapter.4}% 33 +\BOOKMARK [1][]{section.4.2}{Relations d'ordre}{chapter.4}% 34 +\BOOKMARK [2][]{subsection.4.2.1}{R\351flexivit\351, antisym\351trie, transitivit\351}{section.4.2}% 35 +\BOOKMARK [2][]{subsection.4.2.2}{Relation d'ordre}{section.4.2}% 36 +\BOOKMARK [1][]{section.4.3}{Relations d'\351quivalence}{chapter.4}% 37 +\BOOKMARK [2][]{subsection.4.3.1}{Classes d'\351quivalence}{section.4.3}% 38 +\BOOKMARK [-1][]{part.3}{III Arithm\351tique}{}% 39 +\BOOKMARK [0][]{chapter.5}{Ensembles de nombres entiers}{part.3}% 40 +\BOOKMARK [1][]{section.5.1}{Principe de r\351currence }{chapter.5}% 41 +\BOOKMARK [1][]{section.5.2}{Nombres premiers}{chapter.5}% 42 +\BOOKMARK [1][]{section.5.3}{Division euclidienne dans Z et applications}{chapter.5}% 43 +\BOOKMARK [1][]{section.5.4}{Algorithmes d'Euclide}{chapter.5}% 44 +\BOOKMARK [2][]{subsection.5.4.1}{L'algorithme initial}{section.5.4}% 45 +\BOOKMARK [2][]{subsection.5.4.2}{Algorithme d'Euclide g\351n\351ralis\351}{section.5.4}% 46 +\BOOKMARK [2][]{subsection.5.4.3}{L'algorithme.}{section.5.4}% 47 +\BOOKMARK [2][]{subsection.5.4.4}{Exemple.}{section.5.4}% 48 +\BOOKMARK [1][]{section.5.5}{Arithm\351tique modulo n}{chapter.5}% 49 +\BOOKMARK [-1][]{part.4}{IV Annexes}{}% 50 +\BOOKMARK [0][]{chapter.6}{Programme P\351dagogique National 2005 \(PPN\)}{part.4}% 51 +\BOOKMARK [0][]{chapter.6}{Index}{part.4}% 52 diff --git a/main13.pdf b/main13.pdf index 401ddcb0232d03603eb555a49b9de395e594961b..40466c7690d2b733595376e3710de88bc48a686d 100644 GIT binary patch delta 106105 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{www.bibmath.net})}}{30}{Exo.5.11} +\contentsline {Exo}{{Exercice}{5.{12}}{}}{30}{Exo.5.12} +\contentsline {Rem}{{Remarque}{5.{2}}{}}{31}{Rem.5.2} +\contentsline {Exo}{{Exercice}{5.{13}}{}}{31}{Exo.5.13} +\contentsline {Exo}{{Exercice}{5.{14}}{}}{31}{Exo.5.14} +\contentsline {Exo}{{Exercice}{5.{15}}{}}{31}{Exo.5.15} +\contentsline {Th}{{Propriété}{5.{4}}{Théorème de Bézout}}{31}{Th.5.4} +\contentsline {Proof}{{Preuve}{5.{$\alpha $}}{}}{31}{Proof.5.1} +\contentsline {Rem}{{Remarque}{5.{3}}{}}{31}{Rem.5.3} +\contentsline {Proof}{{Preuve}{5.{$\beta $}}{}}{31}{Proof.5.2} +\contentsline {Exo}{{Exercice}{5.{16}}{Application de l'algorithme d'Euclide et de Bézout}}{31}{Exo.5.16} +\contentsline {Pre}{{Preuve}{3}{}}{32}{Pre.3} +\contentsline {Ex}{{Exemple}{5.{17}}{}}{32}{Exo.5.17} +\contentsline {Rem}{{Remarque}{5.{4}}{}}{32}{Rem.5.4} +\contentsline {Exo}{{Exercice}{5.{18}}{}}{32}{Exo.5.18} +\contentsline {Th}{{Propriété}{5.{5}}{Théorème de Gauss}}{32}{Th.5.5} +\contentsline {Exo}{{Exercice}{5.{19}}{}}{33}{Exo.5.19} +\contentsline {Exo}{{Exercice}{5.{20}}{}}{33}{Exo.5.20} +\contentsline {Exo}{{Exercice}{5.{21}}{}}{33}{Exo.5.21} +\contentsline {Exo}{{Exercice}{5.{22}}{}}{33}{Exo.5.22} +\contentsline {Def}{{Définition}{5.{7}}{Congruence modulo $n$}}{33}{Def.5.7} +\contentsline {Exo}{{Exercice}{5.{23}}{}}{33}{Exo.5.23} +\contentsline {Th}{{Propriété}{5.{6}}{}}{33}{Th.5.6} +\contentsline {Proof}{{Preuve}{5.{$\gamma $}}{}}{33}{Proof.5.3} +\contentsline {Ex}{{Exemple}{5.{24}}{}}{34}{Exo.5.24} +\contentsline {Th}{{Propriété}{5.{7}}{}}{34}{Th.5.7} +\contentsline {Notation}{{Notation}{5.{1}}{}}{34}{Notation.5.1} +\contentsline {Ex}{{Exemple}{5.{25}}{}}{34}{Exo.5.25} +\contentsline {Def}{{Définition}{5.{8}}{}}{34}{Def.5.8} +\contentsline {Th}{{Propriété}{5.{8}}{}}{34}{Th.5.8} +\contentsline {Proof}{{Preuve}{5.{$\delta $}}{}}{34}{Proof.5.4} +\contentsline {Def}{{Définition}{5.{9}}{}}{34}{Def.5.9} +\contentsline {Ex}{{Exemple}{5.{26}}{}}{34}{Exo.5.26} +\contentsline {Rem}{{Remarque}{5.{5}}{}}{35}{Rem.5.5} +\contentsline {Exo}{{Exercice}{5.{27}}{}}{35}{Exo.5.27} +\contentsline {Exo}{{Exercice}{5.{28}}{Systèmes de congruences}}{35}{Exo.5.28} +\contentsline {Exo}{{Exercice}{5.{29}}{}}{35}{Exo.5.29} +\contentsline {Exo}{{Exercice}{5.{30}}{}}{35}{Exo.5.30} diff --git a/main13.toc b/main13.toc index d78cb9d..acab1ef 100644 --- a/main13.toc +++ b/main13.toc @@ -12,51 +12,48 @@ \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} +\contentsline {section}{\numberline {III}S\IeC {\'e}mantique du calcul propositionnel}{14}{section.2.3} +\contentsline {subsection}{\numberline {III.1}Fonctions de v\IeC {\'e}rit\IeC {\'e}}{14}{subsection.2.3.1} +\contentsline {subsection}{\numberline {III.2}Formules propositionnelles particuli\IeC {\`e}res}{14}{subsection.2.3.2} +\contentsline {subsubsection}{\numberline {III.2.1}Tautologies}{15}{subsubsection.2.3.2.1} +\contentsline {subsubsection}{\numberline {III.2.2}Antilogies}{15}{subsubsection.2.3.2.2} +\contentsline {subsection}{\numberline {III.3}Cons\IeC {\'e}quences logiques}{15}{subsection.2.3.3} +\contentsline {subsection}{\numberline {III.4}Formules \IeC {\'e}quivalentes}{16}{subsection.2.3.4} +\contentsline {subsection}{\numberline {III.5}Simplification du calcul des fonctions de v\IeC {\'e}rit\IeC {\'e}}{17}{subsection.2.3.5} +\contentsline {part}{II\hspace {1em}Th\IeC {\'e}orie des ensembles}{19}{part.2} +\contentsline {chapter}{\numberline {3}Introduction \IeC {\`a} la th\IeC {\'e}orie des ensembles}{20}{chapter.3} +\contentsline {section}{\numberline {I}Rappels de th\IeC {\'e}orie des ensembles}{20}{section.3.1} +\contentsline {subsection}{\numberline {I.1}Notion premi\IeC {\`e}re d'ensemble}{20}{subsection.3.1.1} +\contentsline {subsection}{\numberline {I.2}R\IeC {\`e}gles de fonctionnement}{20}{subsection.3.1.2} +\contentsline {paragraph}{Relation d'appartenance.}{20}{section*.2} +\contentsline {paragraph}{Objets distincts.}{20}{section*.3} +\contentsline {paragraph}{Ensemble vide.}{20}{section*.4} +\contentsline {paragraph}{Derni\IeC {\`e}re r\IeC {\`e}gle de fonctionnement des ensembles.}{20}{section*.5} +\contentsline {subsection}{\numberline {I.3}Sous-ensembles, ensemble des parties}{20}{subsection.3.1.3} +\contentsline {section}{\numberline {II}Op\IeC {\'e}rations sur les ensembles}{21}{section.3.2} +\contentsline {subsection}{\numberline {II.1}\'Egalite de deux ensembles}{21}{subsection.3.2.1} +\contentsline {subsection}{\numberline {II.2}R\IeC {\'e}union, intersection}{21}{subsection.3.2.2} +\contentsline {subsection}{\numberline {II.3}Compl\IeC {\'e}mentation}{22}{subsection.3.2.3} +\contentsline {subsection}{\numberline {II.4}Produit cart\IeC {\'e}sien}{22}{subsection.3.2.4} +\contentsline {section}{\numberline {III}Exercices suppl\IeC {\'e}mentaires}{23}{section.3.3} +\contentsline {chapter}{\numberline {4}Relations binaires entre ensembles}{24}{chapter.4} +\contentsline {section}{\numberline {I}Relations}{24}{section.4.1} +\contentsline {section}{\numberline {II}Relations d'ordre}{24}{section.4.2} +\contentsline {subsection}{\numberline {II.1}R\IeC {\'e}flexivit\IeC {\'e}, antisym\IeC {\'e}trie, transitivit\IeC {\'e}}{24}{subsection.4.2.1} +\contentsline {subsection}{\numberline {II.2}Relation d'ordre}{25}{subsection.4.2.2} +\contentsline {section}{\numberline {III}Relations d'\IeC {\'e}quivalence}{25}{section.4.3} +\contentsline {subsection}{\numberline {III.1}Classes d'\IeC {\'e}quivalence}{26}{subsection.4.3.1} +\contentsline {part}{III\hspace {1em}Arithm\IeC {\'e}tique}{27}{part.3} +\contentsline {chapter}{\numberline {5}Ensembles de nombres entiers}{28}{chapter.5} +\contentsline {section}{\numberline {I}Principe de r\IeC {\'e}currence }{28}{section.5.1} +\contentsline {section}{\numberline {II}Nombres premiers}{28}{section.5.2} +\contentsline {section}{\numberline {III}Division euclidienne dans ${\mathbb Z}$ et applications}{29}{section.5.3} +\contentsline {section}{\numberline {IV}Algorithmes d'Euclide}{30}{section.5.4} +\contentsline {subsection}{\numberline {IV.1}L'algorithme initial}{30}{subsection.5.4.1} +\contentsline {subsection}{\numberline {IV.2}Algorithme d'Euclide g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}}{32}{subsection.5.4.2} +\contentsline {subsection}{\numberline {IV.3}L'algorithme.}{32}{subsection.5.4.3} +\contentsline {subsection}{\numberline {IV.4}Exemple.}{32}{subsection.5.4.4} +\contentsline {section}{\numberline {V}Arithm\IeC {\'e}tique modulo $n$}{33}{section.5.5} +\contentsline {part}{IV\hspace {1em}Annexes}{36}{part.4} +\contentsline {chapter}{\numberline {6}Programme P\IeC {\'e}dagogique National 2005 (PPN)}{37}{chapter.6} +\contentsline {chapter}{Index}{38}{chapter.6} -- 2.39.5