From 8c64d4461e17c5b4e2be2183f5a02e4b0e8cc9b3 Mon Sep 17 00:00:00 2001
From: couchot <jf.couchot@gmail.com>
Date: Thu, 17 Oct 2013 21:27:31 +0200
Subject: [PATCH] relations 13 fini

---
 ensembles/relbin13.tex | 105 ++++++++++++++++-----------------
 main13.aux             |  52 +++++++----------
 main13.idx             |   7 +--
 main13.log             | 128 ++++++++++++++++-------------------------
 main13.out             |   7 +--
 main13.pdf             | Bin 324131 -> 320746 bytes
 main13.thm             |  45 ++++++---------
 main13.toc             |   9 ++-
 8 files changed, 148 insertions(+), 205 deletions(-)

diff --git a/ensembles/relbin13.tex b/ensembles/relbin13.tex
index ed68ff3..0ae73ee 100755
--- a/ensembles/relbin13.tex
+++ b/ensembles/relbin13.tex
@@ -28,19 +28,19 @@ que $x$ est \emph{en Relation avec} $y$. On note cela $x{\mathcal{R}}y$.
 Lorsque $E=F$, on parle de relation binaire définie dans l'ensemble $E$.
 Son graphe est une partie de $E^2$.
 Dans ce cas, il est possible que $x \mathcal{R} y$ sans que $y \mathcal{R} x$.
-(Penser à la relation \og est plus agé que \fg{}).
+(Penser à la relation \og est plus âgé que \fg{}).
 \end{Rem}
 
 
 
-\begin{Exo}
-Sur l'ensemble $\Z$ des entiers relatifs, on définit deux relations, notées respectivement $\Sigma$ et $\Delta$, de la façon suivante:
-\begin{itemize}
- \item $x \Sigma y$ quand la somme $x+y$ est paire
-\item  $x \Delta y$ quand la différence $x-y$ est paire
-\end{itemize}
-Sont-elles égales ?
-\end{Exo}
+% \begin{Exo}
+% Sur l'ensemble $\Z$ des entiers relatifs, on définit deux relations $\Sigma$ et $\Delta$:
+% \begin{itemize}
+%  \item $x \Sigma y$ quand la somme $x+y$ est paire
+% \item  $x \Delta y$ quand la différence $x-y$ est paire
+% \end{itemize}
+% Sont-elles égales ?
+% \end{Exo}
 
 
 \section{Relations d'ordre}
@@ -60,7 +60,7 @@ dans un ensemble $E$.
 
 \begin{Def}[Antisymétrie]
  ${\mathcal{R}}$ est dite \emph{antisymétrique} \index{relation!antisymétrique} si, lorsque  $x$ est en relation avec $y$, alors $y$ ne peut pas être en relation avec $x$ (sauf si $x=y$): 
- $\qqs(x,y)\in E^2,\ x {\mathcal{R}} y\ {\rm et}\ y {\mathcal{R}} x\Imp\ x=y$ 
+ $\qqs(x,y)\in E^2,\ x {\mathcal{R}} y\ {\rm et}\ y {\mathcal{R}} x\Imp\ x=y$.
 % ou encore
 % $\qqs x\in E,\ \qqs y\in E,\ (x,y)\in G ~{\rm et}\ (y,x)\in G\ \Imp x=y$
 % %
@@ -82,9 +82,9 @@ $\qqs x\in E,\ \qqs y\in E,\qqs z \in E,\ x {\mathcal{R}} y\ {\rm et}\ y {\mathc
 Les relations suivantes sont-elles réflexives, antisymétriques ou transitives ?
 \begin{enumerate}
 \item $A = \mathbb{R}$ et $x \mathcal{R} y$ si $|x| = |y|$.
-\item $A = \mathbb{R}$ et $x \mathcal{R} y$ si $sin^2 x + \cos^2 y = 1 $.
+\item $A = \mathbb{R}$ et $x \mathcal{R} y$ si $\sin^2 x + \cos^2 y = 1 $.
 \item $A = \mathbb{N}$ et $x \mathcal{R} y$ s'il existe $p$ et $q$ entiers tels que $y = p x^q$.
-\item $A$ est l'ensemble des points du plan, et $x \mathcal{R} y$ si la distance de $x$ à $y$ est inférieure à 52,7 km.
+% \item $A$ est l'ensemble des points du plan, et $x \mathcal{R} y$ si la distance de $x$ à $y$ est inférieure à 52,7 km.
 \end{enumerate}
 \end{Exo}
 
@@ -101,13 +101,13 @@ $a \mathcal{R} b$ si et seulement si $a^b \leq b ^a$.
 
 
 
-\begin{Exo}
-Soit $\mathcal{R}$ et $\mathcal{S}$ deux relations dans $A$.
-\begin{enumerate}
-\item Montrer que si $\mathcal{R}$ et $\mathcal{S}$ sont transitives alors $\mathcal{R} \cap \mathcal{S}$ est transitive.
-\item Si $\mathcal{R}$ est antisymétrique alors $\mathcal{R} \cap \mathcal{S}$ est antisymétrique.
-\end{enumerate}
-\end{Exo}
+% \begin{Exo}
+% Soit $\mathcal{R}$ et $\mathcal{S}$ deux relations dans $A$.
+% \begin{enumerate}
+% \item Montrer que si $\mathcal{R}$ et $\mathcal{S}$ sont transitives alors $\mathcal{R} \cap \mathcal{S}$ est transitive.
+% \item Si $\mathcal{R}$ est antisymétrique alors $\mathcal{R} \cap \mathcal{S}$ est antisymétrique.
+% \end{enumerate}
+% \end{Exo}
 
 
 \subsection{Relation d'ordre}
@@ -126,7 +126,7 @@ On note $a|b$ si et seulement si $b$ est un multiple de $a$ ($\exists k \in \Net
 C'est une relation d'ordre définie dans $\Net$. En effet, elle est
 
 \begin{description}
-\item[reflexive:] $a=1a$, donc $a|a$ est vrai,
+\item[réflexive:] $a=1a$, donc $a|a$ est vrai,
 \item[antisymétrique:] si $a|b$ et $b|a$, alors $\exists k,k' \in \Net, a=kb$ et $b=k'a$. Donc $a = kk' a$. Comme $a \neq 0$, $kk'=1$. Mais $k,k' \in \Net$, donc $k = k' = 1$, et $a = b$.
 \item[transitive:] si $a|b$ et $b|c$, alors alors $\exists k,k' \in \Net, a=kb$ et $b=k'c$. Donc $a = k k' c$: il existe $k'' \in \Net$ ($k''=k k'$) tel que $a = k''c$: $a|c$.
 \end{description}
@@ -134,10 +134,10 @@ C'est une relation d'ordre définie dans $\Net$. En effet, elle est
 \end{Ex}
 
 
-La structure algébrique constituée par l'ensemble $E$, muni de la
-relation d'ordre ${\mathcal{R}}$, 
-(c'est-à-dire: le couple $(E,{\mathcal{R}})$) est
-celle d'\emph{ensemble ordonné}\index{ensemble!ordonné}.
+% La structure algébrique constituée par l'ensemble $E$, muni de la
+% relation d'ordre ${\mathcal{R}}$, 
+% (c'est-à-dire: le couple $(E,{\mathcal{R}})$) est
+% celle d'\emph{ensemble ordonné}\index{ensemble!ordonné}.
 
 
 
@@ -158,14 +158,15 @@ On considère\ldots
 \begin{enumerate}
 \item $E=\{1\vir 2\vir 3\vir 4\vir 5\vir 6\vir 7\vir 8\vir 9\}$ et on
 définit la relation binaire $\mathcal{R}$ dans $E$ par son graphe
-$G=\{ $ (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9), (2,2), (2,3), (2,4), (2,6), (2,8), (2,9), (3,3), (4,3), (4,4), (4,6), (4,8), (4,9), (5,3), (5,4), (5,5), (5,6), (5,7), (5,8), 
-(5,9), (6,6), (6,8), (6,9), (7,7), (7,8), (7,9), (8,8), (9,9) $\}$
+$G=\{$(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9), (2,2), (2,3), (2,4), (2,6), (2,8), (2,9), (3,3), (4,3), (4,4), (4,6), (4,8), (4,9), (5,3), (5,4), (5,5), (5,6), (5,7), (5,8), 
+(5,9), (6,6), (6,8), (6,9), (7,7), (7,8), (7,9), (8,8), (9,9)$\}$
 (c'est-à-dire: $1{\mathcal{R}}1$, etc\ldots). 
 Justifier que cette relation est une relation d'ordre.
 
 \item Mêmes questions pour $E'=\{1\vir 2\vir 3\vir 4\vir 5\vir 6\}$ et
-$G'=\{$ (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), 
-(2,2), (2,4), (2,5), (2,6), (3,3), (3,4), (3,6), (4,4), (4,6), (5,5), (5,6), (6,6)$\}$.
+$G'=\{$(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), 
+(2,2), (2,4), (2,5), (2,6), (3,3), (3,4), (3,6), (4,4), (4,6), (5,5), (5,6), 
+(6,6)$\}$.
 \end{enumerate}
 \end{Exo}
 
@@ -182,9 +183,8 @@ On se place encore dans ce paragraphe dans le cas où $E=F$.
 Soit ${\mathcal{R}}$ une relation binaire définie dans un ensemble (non vide) $E$, de graphe $G$.
 
 \begin{Def}[Relation symétrique]
-${\mathcal{R}}$ est dite \emph{symétrique} \index{relation!symétrique} si, dès que $x$ est en relation avec $y$, alors $y$ est en relation avec $x$
-$$\qqs x\in E,\ \qqs y\in E, (x,y)\in G \Imp (y,x)\in G$$
- Ou encore: $\qqs x\in E, \qqs y\in E,\ x {\mathcal{R}} y \Imp y {\mathcal{R}} x$.
+${\mathcal{R}}$ est dite \emph{symétrique} \index{relation!symétrique} si, dès que $x$ est en relation avec $y$, alors $y$ est en relation avec $x$:
+$\qqs x\in E, \qqs y\in E,\ x {\mathcal{R}} y \Imp y {\mathcal{R}} x$.
 \end{Def}
 
 
@@ -214,7 +214,7 @@ alors $y-x=(-k)\cdot n$; or, si $k\in\Z$, $(-k)\in\Z$, donc $y\equiv x\
 [n]$.
 \item[transitive:] si $x\equiv y\ [n]$ et $y\equiv z\ [n]$, $\exi
 k\in\Z$, $x-y=k\cdot n$ et $\exi l\in\Z$, $y-z=l\cdot n$. En
-aditionnant membre à membre ces deux égalités, on obtient
+additionnant membre à membre ces deux égalités, on obtient
 $x-z=(k+l)\cdot n$, or $(k,l)\in\Z^2$, donc $k+l \in\Z$, donc
 $x\equiv z\ [n]$.
 \end{description}
@@ -293,12 +293,9 @@ $x\mathcal{R}y$ ssi   $x.e^{y}= y.e^{x}$.
 
 \begin{Th}
 L'intersection de deux classes d'équivalence distinctes est vide.
+On dit aussi que les classes sont deux à deux disjointes.
 \end{Th}
 
-\begin{Rem}
- On dit aussi que les classes sont deux à deux disjointes.
-\end{Rem}
-
 
 \begin{Pre}
 On considère deux classes, $\dot x$ et $\dot y$, soit $z\in\dot x\inter\dot y$; $\qqs t\in\dot x$, on a $(t,x)\in G$; mais
@@ -355,7 +352,7 @@ Trouver la partition de $A$ induite par $\mathcal{R}$, c'est-à-dire trouver les
 
 
 
-\begin{Exo}[Une relation d'équivalence]
+\begin{Exo}
 On considère l'ensemble des points du plan rapporté à
 deux axes de coordonnées rectangulaires et deux points $P_1$ et
 $P_2$ de coordonnées respectives $(x_1,y_1)$ et $(x_2,y_2)$; on
@@ -378,35 +375,31 @@ $P_1 {\mathcal{R}} P_2 \Ssi x_1y_1=x_2y_2\ {\rm et}\ x_1x_2\geqslant 0$
 % \end{Exo}
 
 
-\begin{Exo}
-Définir, par leurs graphes, les relations d'équivalence dans $E$ qui comportent respectivement le moins et le plus possible de points.
-Que peut-on dire de ces relations? 
-\end{Exo}
 
 
-\subsection{Ensemble-quotient}
+% \subsection{Ensemble-quotient}
 
-\begin{Def}[Ensemble-quotient]
-Il s'agit de l'ensemble des classes d'équivalence de tous les éléments de $E$.
-\end{Def}
+% \begin{Def}[Ensemble-quotient]
+% Il s'agit de l'ensemble des classes d'équivalence de tous les éléments de $E$.
+% \end{Def}
 
 
-\begin{Notation}
-$E/{\mathcal{R}}$.
-\end{Notation}
+% \begin{Notation}
+% $E/{\mathcal{R}}$.
+% \end{Notation}
 
 
-Pour parler aisément d'une classe, on choisit un de ses éléments, 
-et cet élément, surmonté d'un point, sert à représenter la classe en question.
-Une fois que ce choix est fait, il est définitif, et il n'est plus question d'évoquer les autres éléments de cette classe, il faut
-se tenir, sous peine d'incohérence, au choix qui a été fait.
+% Pour parler aisément d'une classe, on choisit un de ses éléments, 
+% et cet élément, surmonté d'un point, sert à représenter la classe en question.
+% Une fois que ce choix est fait, il est définitif, et il n'est plus question d'évoquer les autres éléments de cette classe, il faut
+% se tenir, sous peine d'incohérence, au choix qui a été fait.
 
 
 
-\begin{Ex}[Congruence modulo 4]
-On choisit pour représentants les entiers $<4$, donc 0, 1, 2 et 3.
-L'ensemble-quotient est $\Z/4\Z=\{\dot 0,\dot 1,\dot 2,\dot 3\}$.
-\end{Ex}
+% \begin{Ex}[Congruence modulo 4]
+% On choisit pour représentants les entiers $<4$, donc 0, 1, 2 et 3.
+% L'ensemble-quotient est $\Z/4\Z=\{\dot 0,\dot 1,\dot 2,\dot 3\}$.
+% \end{Ex}
 
 
 
diff --git a/main13.aux b/main13.aux
index 2be045c..44666a4 100644
--- a/main13.aux
+++ b/main13.aux
@@ -245,53 +245,45 @@
 \@writefile{thm}{\contentsline {Def}{{Définition}{4.{1}}{Relation binaire}}{30}{Def.1}}
 \@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{1}}{}}{30}{Exo.1}}
 \@writefile{thm}{\contentsline {Rem}{{Remarque}{4.{1}}{}}{30}{Rem.1}}
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diff --git a/main13.idx b/main13.idx
index 4f9ac36..4aa9057 100644
--- a/main13.idx
+++ b/main13.idx
@@ -29,9 +29,8 @@
 \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}{31}
+\indexentry{relation!transitive|hyperpage}{30}
 \indexentry{relation!d'ordre|hyperpage}{31}
-\indexentry{ensemble!ordonn\IeC {\'e}|hyperpage}{31}
-\indexentry{relation!sym\IeC {\'e}trique|hyperpage}{32}
+\indexentry{relation!sym\IeC {\'e}trique|hyperpage}{31}
 \indexentry{classe d'\IeC {\'e}quivalence|hyperpage}{32}
-\indexentry{partition|hyperpage}{33}
+\indexentry{partition|hyperpage}{32}
diff --git a/main13.log b/main13.log
index ee44dc5..113a226 100644
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 l.124 \begin{Ex}
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 l.145 \begin{Exo}
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 l.156 \begin{Exo}
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 l.193 \begin{Def}
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 l.199 \begin{Ex}
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 l.205 \begin{Ex}
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 l.225 \begin{Exo}
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 l.250 \begin{Notation}
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                    \relax 
-l.406 \begin{Ex}
-                [Congruence modulo 4] [33]) [34] [35
+l.355 \begin{Exo}
+                 ) [33] [34
 
 ]
 \openout2 = `PPN.aux'.
 
  (./PPN.tex
 Chapitre 5.
-) [36
+) [35
 
 
 ]
 No file main13.ind.
-(./main13.bbl) [37
+(./main13.bbl) [36
 
 
 ]
@@ -1834,7 +1802,7 @@ Overfull \hbox (11.59552pt too wide) in paragraph at lines 13--14
 []\T1/ptm/m/n/10.95 ] : Pour un pub-lic aver-tis, souhai-
  []
 
-) [38
+) [37
 
 
 ]
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 Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 336.
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-Output written on main13.pdf (39 pages, 324131 bytes).
+Output written on main13.pdf (38 pages, 320746 bytes).
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diff --git a/main13.out b/main13.out
index a5d50b7..9694ee0 100644
--- a/main13.out
+++ b/main13.out
@@ -37,7 +37,6 @@
 \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.4.3.2}{Ensemble-quotient}{section.4.3}% 40
-\BOOKMARK [-1][]{part.3}{III Annexes}{}% 41
-\BOOKMARK [0][]{chapter.5}{Programme P\351dagogique National 2005 \(PPN\)}{part.3}% 42
-\BOOKMARK [0][]{chapter.5}{Index}{part.3}% 43
+\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
diff --git a/main13.pdf b/main13.pdf
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diff --git a/main13.thm b/main13.thm
index 3a5d32c..28bfbdc 100644
--- a/main13.thm
+++ b/main13.thm
@@ -128,37 +128,30 @@
 \contentsline {Def}{{Définition}{4.{1}}{Relation binaire}}{30}{Def.1}
 \contentsline {Exo}{{Exercice}{4.{1}}{}}{30}{Exo.1}
 \contentsline {Rem}{{Remarque}{4.{1}}{}}{30}{Rem.1}
-\contentsline {Exo}{{Exercice}{4.{2}}{}}{30}{Exo.2}
 \contentsline {Def}{{Définition}{4.{2}}{Réflexivité}}{30}{Def.2}
 \contentsline {Def}{{Définition}{4.{3}}{Antisymétrie}}{30}{Def.3}
-\contentsline {Def}{{Définition}{4.{4}}{Transitivité}}{31}{Def.4}
+\contentsline {Def}{{Définition}{4.{4}}{Transitivité}}{30}{Def.4}
+\contentsline {Exo}{{Exercice}{4.{2}}{}}{30}{Exo.2}
 \contentsline {Exo}{{Exercice}{4.{3}}{}}{31}{Exo.3}
-\contentsline {Exo}{{Exercice}{4.{4}}{}}{31}{Exo.4}
-\contentsline {Exo}{{Exercice}{4.{5}}{}}{31}{Exo.5}
 \contentsline {Def}{{Définition}{4.{5}}{Relation d'ordre}}{31}{Def.5}
-\contentsline {Ex}{{Exemple}{4.{6}}{}}{31}{Exo.6}
-\contentsline {Ex}{{Exemple}{4.{7}}{Relation de divisibilité}}{31}{Exo.7}
-\contentsline {Exo}{{Exercice}{4.{8}}{}}{31}{Exo.8}
-\contentsline {Exo}{{Exercice}{4.{9}}{Diagrammes de transitivité}}{31}{Exo.9}
-\contentsline {Def}{{Définition}{4.{6}}{Relation symétrique}}{32}{Def.6}
-\contentsline {Def}{{Définition}{4.{7}}{Relation d'équivalence}}{32}{Def.7}
-\contentsline {Ex}{{Exemple}{4.{10}}{}}{32}{Exo.10}
-\contentsline {Ex}{{Exemple}{4.{11}}{Relation de congruence modulo $n$ dans $\Z $}}{32}{Exo.11}
-\contentsline {Exo}{{Exercice}{4.{12}}{}}{32}{Exo.12}
+\contentsline {Ex}{{Exemple}{4.{4}}{}}{31}{Exo.4}
+\contentsline {Ex}{{Exemple}{4.{5}}{Relation de divisibilité}}{31}{Exo.5}
+\contentsline {Exo}{{Exercice}{4.{6}}{}}{31}{Exo.6}
+\contentsline {Exo}{{Exercice}{4.{7}}{Diagrammes de transitivité}}{31}{Exo.7}
+\contentsline {Def}{{Définition}{4.{6}}{Relation symétrique}}{31}{Def.6}
+\contentsline {Def}{{Définition}{4.{7}}{Relation d'équivalence}}{31}{Def.7}
+\contentsline {Ex}{{Exemple}{4.{8}}{}}{31}{Exo.8}
+\contentsline {Ex}{{Exemple}{4.{9}}{Relation de congruence modulo $n$ dans $\Z $}}{32}{Exo.9}
+\contentsline {Exo}{{Exercice}{4.{10}}{}}{32}{Exo.10}
 \contentsline {Def}{{Définition}{4.{8}}{Classe d'équivalence}}{32}{Def.8}
 \contentsline {Notation}{{Notation}{4.{1}}{}}{32}{Notation.1}
-\contentsline {Exo}{{Exercice}{4.{13}}{}}{32}{Exo.13}
-\contentsline {Exo}{{Exercice}{4.{14}}{}}{32}{Exo.14}
-\contentsline {Th}{{Propriété}{4.{1}}{}}{33}{Th.1}
-\contentsline {Rem}{{Remarque}{4.{2}}{}}{33}{Rem.2}
-\contentsline {Pre}{{Preuve}{1}{}}{33}{Pre.1}
-\contentsline {Def}{{Définition}{4.{9}}{Partition d'un ensemble}}{33}{Def.9}
+\contentsline {Exo}{{Exercice}{4.{11}}{}}{32}{Exo.11}
+\contentsline {Exo}{{Exercice}{4.{12}}{}}{32}{Exo.12}
+\contentsline {Th}{{Propriété}{4.{1}}{}}{32}{Th.1}
+\contentsline {Pre}{{Preuve}{1}{}}{32}{Pre.1}
+\contentsline {Def}{{Définition}{4.{9}}{Partition d'un ensemble}}{32}{Def.9}
 \contentsline {Th}{{Propriété}{4.{2}}{}}{33}{Th.2}
 \contentsline {Pre}{{Preuve}{2}{}}{33}{Pre.2}
-\contentsline {Ex}{{Exemple}{4.{15}}{}}{33}{Exo.15}
-\contentsline {Exo}{{Exercice}{4.{16}}{}}{33}{Exo.16}
-\contentsline {Exo}{{Exercice}{4.{17}}{Une relation d'équivalence}}{33}{Exo.17}
-\contentsline {Exo}{{Exercice}{4.{18}}{}}{33}{Exo.18}
-\contentsline {Def}{{Définition}{4.{10}}{Ensemble-quotient}}{34}{Def.10}
-\contentsline {Notation}{{Notation}{4.{2}}{}}{34}{Notation.2}
-\contentsline {Ex}{{Exemple}{4.{19}}{Congruence modulo 4}}{34}{Exo.19}
+\contentsline {Ex}{{Exemple}{4.{13}}{}}{33}{Exo.13}
+\contentsline {Exo}{{Exercice}{4.{14}}{}}{33}{Exo.14}
+\contentsline {Exo}{{Exercice}{4.{15}}{}}{33}{Exo.15}
diff --git a/main13.toc b/main13.toc
index 6578952..97f7bb0 100644
--- a/main13.toc
+++ b/main13.toc
@@ -44,9 +44,8 @@
 \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}{32}{section.4.3}
+\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 {subsection}{\numberline {III.2}Ensemble-quotient}{34}{subsection.4.3.2}
-\contentsline {part}{III\hspace {1em}Annexes}{35}{part.3}
-\contentsline {chapter}{\numberline {5}Programme P\IeC {\'e}dagogique National 2005 (PPN)}{36}{chapter.5}
-\contentsline {chapter}{Index}{37}{chapter.5}
+\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}
-- 
2.39.5