pgcd, euclide,...

diff --git a/arithmetique/enInfo13.tex b/arithmetique/enInfo13.tex
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+\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

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+\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}

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diff --git a/main13.out b/main13.out
index 9694ee0e914bd43027b0256997f3b09afd3db0ed..583008d67481ee5beb468fbf58c414fa0b68ded3 100644 (file)
--- a/main13.out
+++ b/main13.out
@@ -37,6 +37,17 @@
 \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 [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

diff --git a/main13.pdf b/main13.pdf
index ccd320e47e642d3fc0e257ae23d0ae0070f30c83..cfe3e89265339572c6ceeac85483aac062362718 100644 (file)
Binary files a/main13.pdf and b/main13.pdf differ

diff --git a/main13.tex b/main13.tex
index 2c2069265caf74c3e47844c7dec2f98f9f31ba14..0d1fc498a5e77402f0a40ce8f306410dd601365e 100755 (executable)
--- a/main13.tex
+++ b/main13.tex
@@ -31,7 +31,7 @@
 \input{symboles.sty}
 
 
 \input{symboles.sty}
 
 

-\geometry{hmargin=2.5cm, vmargin=2.5cm }
+\geometry{hmargin=1.5cm, vmargin=1cm }

 
 
 \theoremstyle{plain}
 
 
 \theoremstyle{plain}


@@ -77,30 +77,24 @@
 
 \theoremstyle{plain}
 \theoremheaderfont{\normalfont\bfseries\sc}
 
 \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}
 \theoremseparator{.}
 \newtheorem{Def}{Définition}[chapter]
 
 %\theoremstyle{break}
 \theoremstyle{plain}

-%\theoremheaderfont{\normalfont\bfseries\sc}

 \theoremheaderfont{\normalfont\sc}
 \theorembodyfont{\normalfont}
 \theoremheaderfont{\normalfont\sc}
 \theorembodyfont{\normalfont}

-\theoremsymbol{\ensuremath{}}
-%\theoremsymbol{\rule{1ex}{1ex}}

 \theoremseparator{ :}
 \theoremseparator{ :}

-%\theoremnumbering{roman}
-%\theoremprework{\begin{block}}
-%\theorempostwork{\end{block}}

 \newframedtheorem{Th}{Propriété}[chapter]
 
 \newframedtheorem{Th}{Propriété}[chapter]
 

-\theoremindent0cm
-\theoremsymbol{\ensuremath{\spadesuit}}
+%\theoremindent0cm
+%\theoremsymbol{\ensuremath{\spadesuit}}

 \theoremnumbering{arabic}
 \theoremnumbering{arabic}

-\theoremprework{\begin{block}}
-\theorempostwork{\end{block}}
+%\theoremprework{\begin{block}}
+%\theorempostwork{\end{block}}

 \newtheorem{Corollary}[Th]{Corollary}
 
 
 \newtheorem{Corollary}[Th]{Corollary}
 
 


@@ -248,11 +242,17 @@ showstringspaces=false}     % no special string spaces
 % \chapter{Relations $n$-aires}
 % \input{ensembles/relnaire}
 
 % \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.}
 % \input{arithmetique/representation}
 
 % \chapter{Cryptologie et arithmétique.}

diff --git a/main13.thm b/main13.thm
index 28bfbdc6d494d2a32f0f5b73e36c9fbcc7bfa160..1516ee75901049866a9b7ae812742274e23118d6 100644 (file)
--- a/main13.thm
+++ b/main13.thm
@@ -1,157 +1,215 @@
-\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}
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-\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}
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-\contentsline {Def}{{Définition}{2.{3}}{Tautologie}}{18}{Def.3}
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-\contentsline {Ex}{{Exemple}{2.{16}}{}}{18}{Exo.16}
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-\contentsline {Exo}{{Exercice}{2.{19}}{}}{19}{Exo.19}
-\contentsline {Def}{{Définition}{2.{5}}{Conséquence logique}}{19}{Def.5}
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-\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}
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-\contentsline {Th}{{Propriété}{3.{5}}{Distributivités de $\cup $ et $\cap $}}{28}{Th.5}
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+\contentsline {Def}{{Définition}{1.{1}}{Algèbre de Boole}}{4}{Def.1.1}
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+\contentsline {Th}{{Propriété}{1.{1}}{Suppression de redondance}}{5}{Th.1.1}
+\contentsline {Proof}{{Preuve}{1.{$\alpha $}}{}}{5}{Proof.1.1}
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+\contentsline {Exo}{{Exercice}{1.{6}}{Calcul booléen}}{5}{Exo.1.6}
+\contentsline {Def}{{Définition}{1.{2}}{Fonction booléenne}}{5}{Def.1.2}
+\contentsline {Ex}{{Exemple}{1.{7}}{}}{6}{Exo.1.7}
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+\contentsline {Def}{{Définition}{1.{5}}{Fonction référentiel}}{6}{Def.1.5}
+\contentsline {Def}{{Définition}{1.{6}}{Minterme, maxterme}}{6}{Def.1.6}
+\contentsline {Ex}{{Exemple}{1.{8}}{Minterme à trois variables}}{6}{Exo.1.8}
+\contentsline {Ex}{{Exemple}{1.{9}}{Maxterme à trois variables}}{6}{Exo.1.9}
+\contentsline {Exo}{{Exercice}{1.{10}}{}}{6}{Exo.1.10}
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+\contentsline {Th}{{Propriété}{1.{2}}{Nombre de mintermes et de maxtermes}}{6}{Th.1.2}
+\contentsline {Def}{{Définition}{1.{7}}{Monômes}}{6}{Def.1.7}
+\contentsline {Exo}{{Exercice}{1.{12}}{}}{6}{Exo.1.12}
+\contentsline {Th}{{Propriété}{1.{3}}{}}{6}{Th.1.3}
+\contentsline {Proof}{{Preuve}{1.{$\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}

diff --git a/main13.toc b/main13.toc
index 97f7bb027bf357185b0f523914cd93585e46061a..d78cb9d3fb4ff3584e3ac7d0efe68cbde55d6298 100644 (file)
--- a/main13.toc
+++ b/main13.toc
@@ -3,49 +3,60 @@
 \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 {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}