From 2497abf60b295bb735ed64cd2f2f16f0c8b82413 Mon Sep 17 00:00:00 2001 From: couchot Date: Fri, 15 Nov 2013 10:26:46 +0100 Subject: [PATCH] pgcd, euclide,... --- arithmetique/enInfo13.tex | 277 ++++++++ main13.aux | 547 +++++++++------- main13.idx | 78 ++- main13.log | 1251 ++++++++++++------------------------- main13.out | 17 +- main13.pdf | Bin 320746 -> 369126 bytes main13.tex | 36 +- main13.thm | 372 ++++++----- main13.toc | 103 +-- 9 files changed, 1348 insertions(+), 1333 deletions(-) create mode 100644 arithmetique/enInfo13.tex diff --git a/arithmetique/enInfo13.tex b/arithmetique/enInfo13.tex new file mode 100644 index 0000000..9204fcb --- /dev/null +++ b/arithmetique/enInfo13.tex @@ -0,0 +1,277 @@ +\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 diff --git a/main13.aux b/main13.aux index 44666a4..588714e 100644 --- a/main13.aux +++ b/main13.aux @@ -1,8 +1,9 @@ \relax -\catcode`:\active -\catcode`;\active -\catcode`!\active -\catcode`?\active +\providecommand\hyper@newdestlabel[2]{} +\catcode `:\active +\catcode `;\active +\catcode `!\active +\catcode `?\active \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined \global\let\oldcontentsline\contentsline @@ -18,6 +19,7 @@ \global\let\hyper@last\relax \gdef\HyperFirstAtBeginDocument#1{#1} \providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} \select@language{french} \@writefile{toc}{\select@language{french}} \@writefile{lof}{\select@language{french}} @@ -28,262 +30,343 @@ \@writefile{lof}{\addvspace {10\p@ }} 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{\`e}bre de Boole|hyperpage}{4} -\indexentry{fonction!bool\IeC {\'e}enne|hyperpage}{6} +\indexentry{fonction!bool\IeC {\'e}enne|hyperpage}{5} \indexentry{fonction!bool\IeC {\'e}enne!nulle|hyperpage}{6} \indexentry{fonction!r\IeC {\'e}f\IeC {\'e}rentiel|hyperpage}{6} \indexentry{minterme|hyperpage}{6} \indexentry{maxterme|hyperpage}{6} -\indexentry{mon\IeC {\^o}me|hyperpage}{7} +\indexentry{mon\IeC {\^o}me|hyperpage}{6} \indexentry{forme canonique!disjonctive|hyperpage}{7} -\indexentry{forme canonique!conjonctive|hyperpage}{8} -\indexentry{proposition|hyperpage}{12} -\indexentry{principe!de non-contradiction|hyperpage}{12} -\indexentry{principe!du tiers-exclu|hyperpage}{12} -\indexentry{variable propositionnelle|hyperpage}{15} -\indexentry{tautologie|hyperpage}{18} -\indexentry{antilogie|hyperpage}{19} -\indexentry{cons\IeC {\'e}quence logique|hyperpage}{19} -\indexentry{formules \IeC {\'e}quivalentes|hyperpage}{20} -\indexentry{th\IeC {\'e}or\IeC {\`e}me!de substitution|hyperpage}{21} -\indexentry{ensemble|hyperpage}{26} -\indexentry{appartenance|hyperpage}{26} -\indexentry{ensemble!vide|hyperpage}{26} -\indexentry{inclusion|hyperpage}{26} -\indexentry{r\IeC {\'e}union|hyperpage}{28} -\indexentry{r\IeC {\'e}union|hyperpage}{28} -\indexentry{ensemble!compl\IeC {\'e}mentaire|hyperpage}{28} -\indexentry{compl\IeC {\'e}mentation|hyperpage}{28} -\indexentry{involution|hyperpage}{28} -\indexentry{loi de De Morgan|hyperpage}{28} -\indexentry{relation binaire|hyperpage}{30} -\indexentry{relation!r\IeC {\'e}flexive|hyperpage}{30} -\indexentry{relation!antisym\IeC {\'e}trique|hyperpage}{30} -\indexentry{relation!transitive|hyperpage}{30} -\indexentry{relation!d'ordre|hyperpage}{31} -\indexentry{relation!sym\IeC {\'e}trique|hyperpage}{31} -\indexentry{classe d'\IeC {\'e}quivalence|hyperpage}{32} -\indexentry{partition|hyperpage}{32} +\indexentry{forme canonique!conjonctive|hyperpage}{7} +\indexentry{proposition|hyperpage}{11} +\indexentry{principe!de non-contradiction|hyperpage}{11} +\indexentry{principe!du tiers-exclu|hyperpage}{11} +\indexentry{variable propositionnelle|hyperpage}{13} +\indexentry{tautologie|hyperpage}{16} +\indexentry{antilogie|hyperpage}{17} +\indexentry{cons\IeC {\'e}quence logique|hyperpage}{17} +\indexentry{formules \IeC {\'e}quivalentes|hyperpage}{18} +\indexentry{th\IeC {\'e}or\IeC {\`e}me!de substitution|hyperpage}{19} +\indexentry{ensemble|hyperpage}{23} +\indexentry{appartenance|hyperpage}{23} +\indexentry{ensemble!vide|hyperpage}{23} +\indexentry{inclusion|hyperpage}{23} +\indexentry{r\IeC {\'e}union|hyperpage}{24} +\indexentry{r\IeC {\'e}union|hyperpage}{24} +\indexentry{ensemble!compl\IeC {\'e}mentaire|hyperpage}{25} +\indexentry{compl\IeC {\'e}mentation|hyperpage}{25} +\indexentry{involution|hyperpage}{25} +\indexentry{loi de De Morgan|hyperpage}{25} +\indexentry{relation binaire|hyperpage}{27} +\indexentry{relation!r\IeC {\'e}flexive|hyperpage}{27} +\indexentry{relation!antisym\IeC {\'e}trique|hyperpage}{27} +\indexentry{relation!transitive|hyperpage}{27} +\indexentry{relation!d'ordre|hyperpage}{28} +\indexentry{relation!sym\IeC {\'e}trique|hyperpage}{28} +\indexentry{classe d'\IeC {\'e}quivalence|hyperpage}{29} +\indexentry{partition|hyperpage}{29} +\indexentry{r\IeC {\'e}currence!restreinte|hyperpage}{31} +\indexentry{multiple|hyperpage}{31} +\indexentry{diviseur|hyperpage}{31} +\indexentry{nombre!premier|hyperpage}{31} +\indexentry{d\IeC {\'e}composition en facteurs premiers|hyperpage}{31} +\indexentry{plus grand commun diviseur|hyperpage}{32} +\indexentry{PGCD|hyperpage}{32} +\indexentry{PPCM|hyperpage}{32} +\indexentry{plus petit commun multiple|hyperpage}{32} +\indexentry{division euclidienne|hyperpage}{32} +\indexentry{quotient|hyperpage}{32} +\indexentry{reste|hyperpage}{32} +\indexentry{algorithme!d'Euclide|hyperpage}{33} +\indexentry{th\IeC {\'e}or\IeC {\`e}me!de B\IeC {\'e}zout|hyperpage}{34} +\indexentry{algorithme!d'Euclide!g\IeC {\'e}n\IeC 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(./main13.toc LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <12> not available @@ -1138,7 +1146,7 @@ pdfTeX warning (ext4): destination with the same identifier (name{page.1}) has been already used, duplicate ignored \relax -l.48 ...IeC {\'e}quivalence}{32}{subsection.4.3.1} +l.50 ...sembles de nombres entiers}{31}{chapter.5} [1 ]) @@ -1151,7 +1159,7 @@ l.48 ...IeC {\'e}quivalence}{32}{subsection.4.3.1} No file Remerciements.tex. LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <20.74> not available -(Font) Font shape `T1/ptm/b/n' tried instead on input line 229. +(Font) Font shape `T1/ptm/b/n' tried instead on input line 223. [3 @@ -1163,14 +1171,6 @@ LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <14.4> not available (Font) Font shape `T1/ptm/b/n' tried instead on input line 2. Package hyperref Info: bookmark level for unknown Def defaults to 0 on input li ne 6. - -Overfull \hbox (8.67114pt too wide) in paragraph at lines 6--11 -[]\T1/ptm/m/sl/10.95 On ap-pelle \T1/ptm/m/n/10.95 al-gèbre de Boole[] \T1/ptm/ -m/sl/10.95 la struc-ture al-gébrique $\OT1/lmr/m/n/10.95 (\OMS/lmsy/m/n/10.95 A -\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 +\OML/lmm/m/it/10.95 ; :; []\OT1/lmr/m -/n/10.95 )$ - [] - LaTeX Font Info: External font `lmex10' loaded for size (Font) <10> on input line 36. LaTeX Font Info: External font `lmex10' loaded for size @@ -1181,7 +1181,7 @@ Package hyperref Info: bookmark level for unknown Rem defaults to 0 on input li ne 68. Package hyperref Info: bookmark level for unknown Exo defaults to 0 on input li ne 74. -[4 + [4 ] Package hyperref Info: bookmark level for unknown Th defaults to 0 on input lin @@ -1191,599 +1191,163 @@ line 162. Package hyperref Info: bookmark level for unknown Ex defaults to 0 on input lin e 172. [5] -Overfull \hbox (0.52283pt too wide) in paragraph at lines 327--329 -[]\T1/ptm/m/sl/10.95 Un \T1/ptm/m/n/10.95 monôme[] \T1/ptm/m/sl/10.95 est une f -onc-tion booléenne pro-duit de vari-ables booléennes +Underfull \hbox (badness 2762) in paragraph at lines 309--312 +[]\T1/ptm/m/n/10.95 Les min-termes et max-termes, pour un [] -[6] [7] [8] +[6] [7] Package hyperref Info: bookmark level for unknown Exoc defaults to 0 on input l ine 500. - [9] -Overfull \hbox (0.39145pt too wide) in paragraph at lines 626--627 -[]\T1/ptm/m/it/10.95 Donner la forme canon-ique dis-jonc-tive de la fonc-tion b -ooléeene + [8] +Overfull \hbox (10.98196pt too wide) in paragraph at lines 653--655 +[]\T1/ptm/m/it/10.95 On consi-dère deux fonc-tions boo-léennes $\OML/lmm/m/it/1 +0.95 u$ \T1/ptm/m/it/10.95 et $\OML/lmm/m/it/10.95 v$ \T1/ptm/m/it/10.95 des qu +atres [] -[10]) [11] +[9]) [10] Chapitre 2. -(./logique/Propositions13.texpdfTeX warning (ext4): destination with the same i -dentifier (name{Def.1}) has been already used, duplicate ignored - - \relax -l.34 \begin{Def} - [Proposition]pdfTeX warning (ext4): destination with the same i -dentifier (name{Exo.1}) has been already used, duplicate ignored - - \relax -l.43 \begin{Ex} - [12 +(./logique/Propositions13.tex [11 -]pdfTeX warning (ext4): destination with the same identifier (name{Rem.1}) has -been already used, duplicate ignored - - \relax -l.175 \begin{Rem} - [13]pdfTeX warning (ext4): destination with the same identifi -er (name{Exo.2}) has been already used, duplicate ignored - - \relax -l.197 \begin{Exo} - pdfTeX warning (ext4): destination with the same identifier (n -ame{Rem.2}) has been already used, duplicate ignored - - \relax -l.219 \begin{Rem} - pdfTeX warning (ext4): destination with the same identifier (n -ame{Exo.3}) has been already used, duplicate ignored - - \relax -l.225 \begin{Exoc} - pdfTeX warning (ext4): destination with the same identifier ( -name{Exo.4}) has been already used, duplicate ignored - - \relax -l.256 \begin{Exo} - pdfTeX warning (ext4): destination with the same identifier (n -ame{Exo.5}) has been already used, duplicate ignored - - \relax -l.289 \begin{Exo} - pdfTeX warning (ext4): destination with the same identifier (n -ame{Exo.6}) has been already used, duplicate ignored - - \relax -l.300 \begin{Exo} - [14] -LaTeX Font Info: Try loading font information for TS1+ptm on input line 309. +] +Overfull \hbox (0.7556pt too wide) in paragraph at lines 220--222 +[]\T1/ptm/m/n/10.95 Même re-marque que pour l'im-pli-ca-tion lo-gique : l'équi- +va-lence lo-gique de deux pro-po-si-tions fausses + [] +[12] +LaTeX Font Info: Try loading font information for TS1+ptm on input line 309. -(/usr/share/texlive/texmf-dist/tex/latex/psnfss/ts1ptm.fd + (/usr/share/texlive/texmf-dist/tex/latex/psnfss/ts1ptm.fd File: ts1ptm.fd 2001/06/04 font definitions for TS1/ptm. -)pdfTeX warning (ext4): destination with the same identifier (name{Th.1}) has b -een already used, duplicate ignored - - \relax -l.400 \begin{Th} - pdfTeX warning (ext4): destination with the same identifier (na -me{Rem.3}) has been already used, duplicate ignored - - \relax -l.420 \begin{Rem} - pdfTeX warning (ext4): destination with the same identifier (n -ame{Exo.7}) has been already used, duplicate ignored - - \relax -l.431 \begin{Exo} - pdfTeX warning (ext4): destination with the same identifier (n -ame{Exo.8}) has been already used, duplicate ignored - - \relax -l.452 \begin{Exo} - [15]pdfTeX warning (ext4): destination with the same identifi -er (name{Exo.9}) has been already used, duplicate ignored - - \relax -l.474 \begin{Exo} - -Overfull \hbox (2.21603pt too wide) in paragraph at lines 482--487 -[]\T1/ptm/m/n/10.95 Lorsqu'on rem-place, dans une for-mule propo-si-tion-nelle, - les vari-ables propo-si-tion-nelles par des propo- +) [13] +Underfull \hbox (badness 2717) in paragraph at lines 489--491 +[]\T1/ptm/m/n/10.95 Les conven-tions de prio-rité des [] -pdfTeX warning (ext4): destination with the same identifier (name{Th.2}) has be -en already used, duplicate ignored - - \relax -l.488 \begin{Th} - [Règles de priorité des connecteurs logiques] -Underfull \hbox (badness 10000) in paragraph at lines 489--491 -[]\T1/ptm/m/n/10.95 Les con-ven-tions de - 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- \relax -l.958 \begin{Ex} - pdfTeX warning (ext4): destination with the same identifier (na -me{Exo.19}) has been already used, duplicate ignored - - \relax -l.969 \begin{Exo} - pdfTeX warning (ext4): destination with the same identifier (n -ame{Def.5}) has been already used, duplicate ignored - - \relax -l.1005 \begin{Def} - [Conséquence logique] -Overfull \hbox (9.10168pt too wide) in paragraph at lines 1015--1017 -[]\T1/ptm/m/n/10.95 On note ce ré-sul-tat : $\OMS/lmsy/m/n/10.95 f\OML/lmm/m/it -/10.95 F[]; [] ; F[]\OMS/lmsy/m/n/10.95 g j[]\OT1/lmr/m/n/10.95 = \OML/lmm/m/it -/10.95 A$ \T1/ptm/m/n/10.95 (se lit : $\OML/lmm/m/it/10.95 A$ \T1/ptm/m/n/10.95 - est con-séquence logique de $\OMS/lmsy/m/n/10.95 f\OML/lmm/m/it/10.95 F[]; [] -; F[]\OMS/lmsy/m/n/10.95 g$\T1/ptm/m/n/10.95 ). $$ + [16] [17] [18] +Overfull \hbox (20.57855pt too wide) in paragraph at lines 1352--1361 +\OMS/lmsy/m/n/10.95 f\OML/lmm/m/it/10.95 G[]; G[]; [] ; G[]\OMS/lmsy/m/n/10.95 +g$ \T1/ptm/m/n/10.95 sont vraies, $\OML/lmm/m/it/10.95 G[] \OMS/lmsy/m/n/10.95 +) \OML/lmm/m/it/10.95 H$ \T1/ptm/m/n/10.95 est vraie. Re-gar-dons si $\OML/lmm/ +m/it/10.95 H$ \T1/ptm/m/n/10.95 est une consé-quence lo-gique de $\OMS/lmsy/m/n +/10.95 f\OML/lmm/m/it/10.95 G[]; G[]; [] ; G[]\OMS/lmsy/m/n/10.95 g$ [] -pdfTeX warning (ext4): destination with the same identifier (name{Exo.20}) has -been already used, duplicate ignored - - \relax -l.1023 \begin{Ex} - pdfTeX warning (ext4): destination with the same identifier (n -ame{Exo.21}) has been already used, duplicate ignored - - \relax -l.1065 \begin{Exo} - [19]pdfTeX warning (ext4): destination with the same identif -ier (name{Exo.22}) has been already used, duplicate ignored - - \relax -l.1086 \begin{Exo} - pdfTeX warning (ext4): destination with the same identifier ( -name{Def.6}) has been already used, duplicate ignored - - \relax -l.1119 \begin{Def} - [Formules équivalentes] -Overfull \hbox (12.18623pt too wide) in paragraph at lines 1120--1124 -[]\T1/ptm/m/sl/10.95 Si la for-mule propo-si-tion-nelle $\OML/lmm/m/it/10.95 G$ - \T1/ptm/m/sl/10.95 est con-séquence logique - [] +[19] [20]) [21] [22 -[20]pdfTeX warning (ext4): destination with the same identifier (name{Th.4}) ha -s been already used, duplicate ignored - - \relax -l.1270 \begin{Th} - [Théorème de substitution] -Underfull \hbox (badness 1552) in paragraph at lines 1272--1277 -[]\T1/ptm/m/n/10.95 Soit $\OML/lmm/m/it/10.95 F$ \T1/ptm/m/n/10.95 une for-mule - propo-si-tion-nelle dans +] +Chapitre 3. +(./ensembles/IntroAuxEnsembles13.tex [23 + +] [24] +Overfull \hbox (37.774pt too wide) in paragraph at lines 186--187 +[]\T1/ptm/m/it/10.95 Faire la réunion des en-sembles $\OML/lmm/m/it/10.95 A$ \T +1/ptm/m/it/10.95 et $\OML/lmm/m/it/10.95 B$\T1/ptm/m/it/10.95 , quand $\OML/lmm +/m/it/10.95 A \OT1/lmr/m/n/10.95 = \OMS/lmsy/m/n/10.95 f\OML/lmm/m/it/10.95 x \ +OMS/lmsy/m/n/10.95 2 \U/msb/m/n/10.95 N\OMS/lmsy/m/n/10.95 j\OML/lmm/m/it/10.95 + x[]\OMS/lmsy/m/n/10.95 g$\T1/ptm/m/it/10.95 , et $\OML/lmm/m/it/10.95 B \OT1/l +mr/m/n/10.95 = \OMS/lmsy/m/n/10.95 f\OML/lmm/m/it/10.95 x \OMS/lmsy/m/n/10.95 2 + \U/msb/m/n/10.95 N\OMS/lmsy/m/n/10.95 j\OML/lmm/m/it/10.95 x[]\OMS/lmsy/m/n/10 +.95 g$\T1/ptm/m/it/10.95 . [] -pdfTeX warning (ext4): destination with the same identifier (name{Proof.1}) has - been already used, duplicate ignored - - \relax -l.1284 \begin{Proof} - [21]pdfTeX warning (ext4): destination with the same ident -ifier (name{Th.5}) has been already used, duplicate ignored - - \relax -l.1326 \begin{Th} - [Théorème de la validité]pdfTeX warning (ext4): destination - with the same identifier (name{Proof.2}) has been already used, duplicate igno -red - - \relax -l.1338 \begin{Proof} - [22] -Overfull \hbox (1.48752pt too wide) in paragraph at lines 1490--1492 -[]\T1/ptm/m/it/10.95 Après une in-tense réflex-ion, l'in-specteur fait sor-tir -l'un des deux de l'hôpi-tal. 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- \relax -l.160 \begin{Def} - [Intersection]\index{réunion} -Overfull \hbox (1.73936pt too wide) in paragraph at lines 161--163 -[]\T1/ptm/m/sl/10.95 L'\T1/ptm/m/n/10.95 intersection \T1/ptm/m/sl/10.95 des de -ux en-sem-bles $\OML/lmm/m/it/10.95 A$ \T1/ptm/m/sl/10.95 et $\OML/lmm/m/it/10. -95 B$\T1/ptm/m/sl/10.95 , notée $\OML/lmm/m/it/10.95 A \OMS/lmsy/m/n/10.95 \ \O -ML/lmm/m/it/10.95 B$\T1/ptm/m/sl/10.95 , est l'ensem- +Overfull \hbox (350.24652pt too wide) in paragraph at lines 344--350 +[]\T1/ptm/m/it/10.95 Soit $\OMS/lmsy/m/n/10.95 R$ \T1/ptm/m/it/10.95 la re-la-t +ion d'équi-va-lence sui-vante dans l'en-semble $\OML/lmm/m/it/10.95 A \OT1/lmr/ +m/n/10.95 = \OMS/lmsy/m/n/10.95 f\OT1/lmr/m/n/10.95 1\OML/lmm/m/it/10.95 ; \OT1 +/lmr/m/n/10.95 2\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 3\OML/lmm/m/it/10.95 ; + \OT1/lmr/m/n/10.95 4\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 5\OML/lmm/m/it/10 +.95 ; \OT1/lmr/m/n/10.95 6\OMS/lmsy/m/n/10.95 g$ \T1/ptm/m/it/10.95 : $\OMS/lms +y/m/n/10.95 R \OT1/lmr/m/n/10.95 = \OMS/lmsy/m/n/10.95 f\OT1/lmr/m/n/10.95 (1\O +ML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 1)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10. +95 (1\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 5)\OML/lmm/m/it/10.95 ; \OT1/lmr/ +m/n/10.95 (2\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 2)\OML/lmm/m/it/10.95 ; \O +T1/lmr/m/n/10.95 (2\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 3)\OML/lmm/m/it/10. +95 ; \OT1/lmr/m/n/10.95 (2\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 6)\OML/lmm/m +/it/10.95 ; \OT1/lmr/m/n/10.95 (3\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 2)\OM +L/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 (3\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.9 +5 3)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 (3\OML/lmm/m/it/10.95 ; \OT1/lmr/m +/n/10.95 6)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 (4\OML/lmm/m/it/10.95 ; \OT +1/lmr/m/n/10.95 4)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 (5\OML/lmm/m/it/10.9 +5 ; \OT1/lmr/m/n/10.95 1)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 (5\OML/lmm/m/ +it/10.95 ; \OT1/lmr/m/n/10.95 5)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 (6\OML +/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 2)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 + (6\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 3)\OML/lmm/m/it/10.95 ; \OT1/lmr/m/ +n/10.95 (6\OML/lmm/m/it/10.95 ; \OT1/lmr/m/n/10.95 6)\OMS/lmsy/m/n/10.95 g\OML/ +lmm/m/it/10.95 :$ [] -pdfTeX warning (ext4): destination with the same identifier (name{Th.4}) has be -en already used, duplicate ignored - - \relax -l.171 \begin{Th} - [Propriétés de la réunion et de l'intersection] -Underfull \hbox (badness 1496) in paragraph at lines 172--173 -[]\T1/ptm/m/n/10.95 La réu-nion de deux +) [29] [30 + +] +Chapitre 5. +(./arithmetique/entiersNaturels13.tex +Overfull \hbox (11.42195pt too wide) in paragraph at lines 134--134 +[]\T1/ptm/m/n/10.95 L'écriture d'un en-tier $\OML/lmm/m/it/10.95 n$ \T1/ptm/m/n +/10.95 sous la forme $\OML/lmm/m/it/10.95 n \OT1/lmr/m/n/10.95 = \OML/lmm/m/it/ +10.95 a[]b[]c[] []$\T1/ptm/m/n/10.95 , [] -pdfTeX warning (ext4): destination with the same identifier (name{Exo.10}) has -been already used, duplicate ignored - - \relax -l.181 \begin{Exo} - pdfTeX warning (ext4): destination with the same identifier (n -ame{Exo.11}) has been already used, duplicate ignored - - \relax -l.185 \begin{Exo} - pdfTeX warning (ext4): destination with the same identifier (n -ame{Th.5}) has been already used, duplicate ignored - - \relax -l.190 \begin{Th} - [Distributivités de $\cup$ et $\cap$]pdfTeX warning (ext4): de -stination with the same identifier (name{Exo.12}) has been already used, duplic -ate ignored - 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