From a0ee5c712b5f4b3e908e2d16da8ff24d414f5f1e Mon Sep 17 00:00:00 2001 From: couchot Date: Thu, 17 Oct 2013 13:47:53 +0200 Subject: [PATCH] =?utf8?q?synth=C3=A8se=20relations?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- ensembles/relbin13.tex | 1095 ++++++---------------------------------- main13.aux | 54 +- main13.idx | 9 + main13.log | 215 +++++++- main13.out | 14 +- main13.pdf | Bin 301042 -> 324131 bytes main13.tex | 4 +- main13.thm | 37 ++ main13.toc | 14 +- 9 files changed, 466 insertions(+), 976 deletions(-) diff --git a/ensembles/relbin13.tex b/ensembles/relbin13.tex index 69a401b..ed68ff3 100755 --- a/ensembles/relbin13.tex +++ b/ensembles/relbin13.tex @@ -1,919 +1,154 @@ -\section{Relations} -On se donne deux ensembles $E$ et $F$. - -\begin{Def}[Relation binaire, graphe] -On dit que: - \begin{itemize} -\item l'on a défini une \emph{relation binaire} \index{relation binaire} ${\mathcal{R}}$ entre ces deux ensembles lorsque l'on s'est donné une partie $G$ de l'ensemble produit $E\times F$ ($G\sse E\times F$). -\item Cette partie est appelée \emph{graphe} \index{graphe} de la relation binaire. -\item Si $x (\in E)$ et $y (\in F)$ sont tels que $(x,y)\in G$, on dit que $x$ est \emph{en relation avec} $y$ par la relation ${\mathcal{R}}$. - \end{itemize} -\end{Def} - - -Pour formaliser cette proposition, on utilise la notation -$x {\mathcal{R}} y$ comme suit: - -\begin{Notation} -$$x {\mathcal{R}} y \Ssi \quad\hbox{[\og x est en relation avec y\fg{} ]}\Ssi -(x,y)\in G\quad\hbox{[$G$: graphe de la relation ${\mathcal{R}}$]}.$$ -\end{Notation} - -\begin{Exo} -On se place dans l'ensemble $E = \{1,2,3,\hdots,20\}$. - -Représenter, dans le plan rapporté à deux axes de coordonnées rectangulaires, les graphes des relations binaires sur $E$ dont les définitions suivent: -\begin{itemize} -\item $x \mathcal{R} y \Longleftrightarrow x \leqslant y$. -\item $x \mathcal{R} y \Longleftrightarrow x | y$: $x$ divise $y$. -\item $x \mathcal{R} y \Longleftrightarrow x \equiv y[3]$: $x$ est congru à $y$ modulo 3. -\item $x \mathcal{R} y \Longleftrightarrow y = x^2$. -\end{itemize} -\end{Exo} - - - - -\begin{Rem} -Lorsque $E=F$, on parle de relation binaire définie dans l'ensemble $E$. -Son graphe est une partie de $E^2$. -\end{Rem} - -\begin{Rem} -Il est possible que $x \mathcal{R} y$ sans que $y \mathcal{R} x$. -\end{Rem} - - -\begin{Exo} -A-t-on $y \mathcal{R} x$ quand $x \mathcal{R} y$, dans les relations binaires définies dans l'exercice précédent ? -\end{Exo} - -\begin{Exo} -Sur l'ensemble des mots de la langue française, on définit la relation: \og le mot M est lié au mot N s'ils coïncident quand on écrit M à l'envers \fg{}. - -Déterminer quelques couples de mots en relation, ainsi que des mots en relation avec eux-mêmes. Comment appelle-t-on de tels mots ? -\end{Exo} - - -\begin{Exoc} -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 ? - -\noindent\textit{\underline{Réponse: }} $x \Sigma y \Leftrightarrow \exists k \in \Z, x+y = 2k \Leftrightarrow \exists k \in \Z, x+y-2y = 2k-2y = 2(k-y).$ - -Donc $x \Sigma y \Leftrightarrow \exists k' \in \Z, x-y = 2k' \Leftrightarrow x \Delta y.$ -\end{Exoc} - -\section{Application d'un ensemble dans un autre} - -\subsection{Application et relation fonctionnelle} - - -\begin{Def}[Application] -Une \emph{application}\index{application} de l'ensemble $E$ dans l'ensemble $F$ est une relation binaire particulière ${\mathcal{R}}$ entre $E$ et $F$, dont le graphe $G$ doit posséder les propriétés suivantes: - -\begin{itemize} - \item pour tout élément $x$ de $E$, il doit exister un élément $y$ de $F$ tel que $(x,y)$ soit élément de $G$; - \item cet élément $y$ doit être unique. -\end{itemize} -\end{Def} - - - -\noindent Voici la formalisation (partielle) de ces propositions: - -\begin{itemize} -\item $\qqs x\in E,\ \exi y\in F,\ (x,y)\in G$ -\item $\qqs x\in E,\ \qqs y\in F,\ \qqs y'\in F, [(x,y)\in G\ {\rm et}\ (x,y')\in G \Longrightarrow y=y']$. -\end{itemize} - -\medskip - -Il existe un intermédiaire entre relation et application \ldots - -\begin{Def}[Relation fonctionnelle] -On parle de \emph{relation fonctionnelle}\index{relation!fonctionnelle} quand tout élément de l'ensemble de départ possède au plus une image. -\end{Def} - -\begin{Rem} -Une application est donc une relation fonctionnelle particulière : tout élément de l'ensemble de départ possède exactement une image. -\end{Rem} - - -\begin{Exo} -Parmi les relations suivantes de $\mathbb{R}$ vers $\mathbb{R}$, repérez les relations fonctionnelles, repérez les applications: -\begin{enumerate} -\item $\mathcal{R} = \{ (x,y) \in \mathbb{R}^2, |y| = \sqrt{x} \}$ -\item $\mathcal{R} = \{ (x,y) \in \mathbb{R}^2, xy = 1 \}$ -\item $\mathcal{R} = \{ (x,y) \in \mathbb{R}^2, y-x+2 = 0 \}$ -\end{enumerate} -\end{Exo} - -%\noindent\textit{\underline{Réponse:}} Les deux dernières sont des relations fonctionnelles, et la dernière est la seule application. - -\subsection{Image et antécédent d'un élément} - - -On suppose dorénavant que $\mathcal{R}$ est une application. Pour un $x$ donné de $E$, il lui correspond un et un seul $y$ de $F$ qui est en relation avec lui par ${\mathcal{R}}$. - -\begin{Def} -Cet unique $y$ est alors appelé \emph{image}\index{image} de $x$ par l'application définie par ${\mathcal{R}}$. -\end{Def} - -\begin{Notation} -Si l'on désigne par $f$ cette application, l'expression \og $y$ est l'image de $x$ par $f$\fg{} est formalisée par $y=f(x)$. -De plus, on formalise la proposition \og $f$ est une application de $E$ dans $F$\fg{} par $f: E\vers F$. La proposition \og $y$ est l'image de $x$ par $f$\fg{} peut aussi être traduite par: $f: x \fc y$. -\end{Notation} - - - -\begin{Exo} - Interpréter chacune des situations suivantes au moyen d'une application. Pour cela, on définira deux ensembles $A$ et $B$ ainsi que $f: A\vers B$. Préciser dans chaque cas pourquoi il s'agit bien d'une application. - -\begin{enumerate} -\item Le registre d'un hôtel qui possède 55 chambres. -\item Le numéro d'INSEE. -\item La parité d'un entier naturel. -\end{enumerate} - -\end{Exo} - - -Réciproquement, soit $f$ une application\ldots - -\begin{Def}[Antécédent] -Si $y$ est l'image de $x$ par $f$, alors $x$ est appelé un \emph{antécédent} \index{antécédent} de $y$ par $f$. -\end{Def} - -Un quelconque élément $y$ de $F$ ne possède pas forcément d'antécédant par une application $f : E \longrightarrow F$. Et il n'y a pas forcément unicité quand il en possède : un $y$ de $F$ peut avoir plusieurs antécédants par une application $f$. - -Les cas particuliers où tout $y$ de $F$ possède au plus un antécédant, et au moins un antécédant, sont étudiés dans les deux sections suivantes. - -\subsection{Applications injectives} - - - -\begin{Def}[Injectivité] -L'application $f:E \longrightarrow F$ est dite \emph{injective} \index{application!injective} quand tout $y \in F$ possède au plus un antécédant par $f$. -\end{Def} - - -\begin{Rem} -Le terme injection\index{injection} est synonyme d'\og application injective\fg{}. -\end{Rem} - -L'injectivité d'une application peut se caractériser de la manière suivante. - -\begin{Th}[Caractérisation des fonctions injectives] -$$\hbox{\og $f:E \longrightarrow F$ est une application injective\fg{} } \Longleftrightarrow [ \forall x,x' \in E, f(x)=f(x') \Imp x=x' ]$$ -\end{Th} - -\begin{Exo} -Prouver, en utilisant la caractérisation ci-dessus, que les applications suivantes sont injectives : -\begin{enumerate} -\item $f : \R \longrightarrow \R, x \fc 2x+3$. -\item $f : \mathbb{R}^+ \longrightarrow \R, x \fc 3 x^2 +1$. -\end{enumerate} -\end{Exo} - - -\begin{Exo} -Prouver, en utilisant la caractérisation ci-dessus, que les applications suivantes ne sont pas injectives : -\begin{enumerate} -\item $f : \R \longrightarrow \R, x \fc |x|$. -\item $f : \R \longrightarrow \R, x \fc x^2$. -\end{enumerate} -\end{Exo} - - -\begin{Exo} -Tracez le graphe d'une application qui est injective, et d'une application qui ne l'est pas. Trouver une caractérisation de l'injectivité d'une application $f :E \longrightarrow F$, à partir du nombre d'intersections entre la courbe $\mathcal{C}_f$ et les droites verticales $y=b, b\in F$. -\end{Exo} - - -\begin{Exo} -Soit $f:E \longrightarrow F$ une application injective. Peut-elle perdre ce caractère injectif si on réduit l'ensemble d'arrivée ? Et si l'on réduit l'ensemble de départ ? - -Que se passe-t-il si l'on change «réduit» en «augmente» dans les précédentes questions ? -\end{Exo} - - -\begin{Exo} -On suppose $g o f$ injective. Montrer que $f$ est injective. Est-ce que $g$ est obligatoirement injective ? -\end{Exo} - - -\subsection{Applications surjectives} - -La définition d'une application $f$ exige seulement que chaque élément $x$ de $E$ admette une image (unique) $y$ dans $F$, mais pas que tout élément $y$ de $F$ admette un antécédent dans $E$. -S'il en est néanmoins ainsi, l'application est dite \emph{surjective}: -\begin{Def} - Une application \emph{surjective}\index{application!surjective} $f: E\vers F$ est une application telle que tout $y$ de $F$ admette un antécédent dans $E$. -\end{Def} - -\begin{Rem} -\emph{Surjection} est synonyme d'\og application surjective\fg{}. \index{surjection} -\end{Rem} - - -\begin{Exo} -Tracez le graphe d'une application qui est surjective, et d'une application qui ne l'est pas. -\end{Exo} - -\begin{Exo} -Donnez des exemples (sous forme analytique) de fonctions surjectives, et de fonction qui ne le sont pas. -\end{Exo} - -\subsection{Image d'un ensemble par une application} - -D'une manière générale, on peut considérer l'ensemble des images des éléments de $E$ par une application $f$ de $E$ dans $F$ (ils en ont tous une, et une seule). - -\begin{Def}[Image d'un ensemble par une application] -Cet ensemble, qui est évidemment une partie de $F$, est noté $f$, et est appelé \emph{image de $E$ par $f$}: $$f=\{f(x)\in F \mid x\in E\}$$ -\end{Def} - - -\begin{Rem} -Si tous les éléments de $F$ ont un antécédent dans $E$ ($f$ est surjective), cela signifie que tout élément de $F$ est élément de $f$, donc que $F\sse f$. Comme on a remarqué par ailleurs que $f\sse F$, on a, dans ce cas, $f=F$. -\end{Rem} - - - -Cette dernière remarque permet la formalisation suivante: - -\begin{Th}[Caractérisation de la surjectivité] -$$\hbox{\og $f$ est (une application) surjective\fg{} } \Ssi f=F$$ -\end{Th} - - -\begin{Ex} -Soit l'application \og élévation au carré \fg{} $f: x \longmapsto x^2$ de $\R$ dans $\R$. Elle est: -\begin{itemize} -\item non surjective: $f < \R > = \R^+$, -\item non injective: $f(-2) = f(2) = 4$. -\end{itemize} -\end{Ex} - -\begin{Exo} -On suppose $g o f$ surjective. Montrer que $g$ est surjective. Est-ce que $f$ est obligatoirement surjective ? -\end{Exo} - -\subsection{Applications bijectives} - - -\begin{Def}[Applications bijectives] -Une application qui est à la fois injective et surjective est dite \emph{bijective} \index{application!bijective}. -\end{Def} - - -\begin{Rem} -Synonyme d'\og application bijective\fg{}: bijection\index{bijection}. -\end{Rem} - -\begin{Exo} -Dans chaque cas, dire si l'application $f:A \vers B$ est injective, surjective ou bijective. - -\begin{enumerate} -\item $A = \R, B = \R, f(x) = x+7$ -\item $A = \R, B = \R, f(x) = x^2+2x-3$ -\item $A = \{ x \in \R | 4 \leqslant x \leqslant 9 \}, B = \{ x \in \R | 21 \leqslant x \leqslant 96 \}, f(x) = x^2+2x-3$ -\item $A = \R, B = \R, f(x) = 3x-2|x|$ -\item $A = \R, B = \R, f(x) = e^x+1$ -\item $A = \N, B = \N, f(x) = x(x+1)$ -\end{enumerate} -\end{Exo} - - -\begin{Th} -Dans le cas d'une bijection, à chaque élément $x$ de $E$ correspond un et un seul élément $y$ de $F$ (définition d'une application) et, réciproquement, à chaque (surjectivité) élément $y$ de $F$ correspond un et un seul (injectivité) élément $x$ de $E$. -\end{Th} - - -Cette dernière proposition est précisément l'affirmation de l'existence d'une application $g$ de $F$ dans $E$, telle que $x = g(y) \Longleftrightarrow f(x) = y $. - - - - -\begin{Def}[Application inverse] -Cette application est appelée \emph{application inverse}\index{application!inverse} de l'application $f$. -\end{Def} - -\begin{Notation} - On la note $f^{-1}$ -\end{Notation} - -\begin{Exo} -Reprendre l'exercice précédent, en trouvant l'application réciproque des applications bijectives. -\end{Exo} - - -% \begin{Rem} -% On peut démontrer que la bijectivité est une condition nécessaire et suffisante pour qu'une application admette une inverse. -% \end{Rem} - -\begin{Ex} -Soit l'application \og multiplication par 2 \fg{} $f: x \longmapsto 2 x$ de $\R$ dans $\R$. Elle est surjective et injective. -Elle admet donc une application inverse: $f^{-1}: x \longmapsto \dfrac{x}{2}$. -\end{Ex} - -\begin{Exo} -Soit $f: \Z \vers \Z$ définie par $f(n) = n + (-1)^n$. -\begin{enumerate} - \item Montrer que $n$ et $f(n)$ sont toujours de parité différente. -\item Montrer que $f$ est bijective. -\item Calculer $f(f(n))$. En déduire une expression de $f^{-1}$ et résoudre l'équation $347 = n+(-1)^n$. -\end{enumerate} - -\end{Exo} - - -\section{Cardinal et puissance d'un ensemble} - -Il s'agit ici de proposer une réflexion sur la notion intuitive de \og nombre d'éléments d'un ensemble\fg{}, nécessaire pour pouvoir aborder ultérieurement les notions de dénombrabilité et de calculabilité, fondamentaux en informatique. - - -\subsection{Cas des ensembles finis} - - -On commence par prendre deux ensembles $E=\{a,b,c,d\}$ et $F=\{1,2,3\}$ et à remarquer que: -\begin{itemize} - \item il est possible de définir une injection de $F$ dans $E$, mais pas une surjection, - \item de définir une surjection de $E$ sur $F$, mais pas d'injection, -\end{itemize} -\ldots tout simplement parce qu'il n'y a \og pas assez\fg{} d'éléments dans $F$ (ou \og trop\fg{} dans $E$). - - -Si l'on veut pouvoir définir une bijection entre deux ensembles, il semble nécessaire et suffisant qu'ils aient le même \og nombre d'éléments\fg{} (on se limite ici au domaine fini). -Cette notion intuitive, résultat immédiat d'une simple opération de comptage, est reliée à la notion mathématique de mise en bijection avec une partie de $\Net$ de la forme $\{1\vir 2\vir\ldots\vir n\}$. - Ainsi, pour $n\in\Net$ donné, les ensembles à $n$ éléments sont tous ceux qui peuvent être mis en bijection avec $\{1\vir 2\vir\ldots\vir n\}$ (et, par convention, $\varnothing$ a 0 élément). - - -\begin{Def}[Cardinal d'un ensemble fini] -L'élément maximum $n$ de la partie finie $P=\{1\vir 2\vir\ldots\vir n\}$ de $\Net$ est un représentant du \emph{cardinal} \index{ensemble!cardinal}\index{cardinal} de tout ensemble en bijection avec $P$. -\end{Def} - - - -Pas de problème donc lorsque l'on s'en tient aux ensembles finis, dont la définition mathématique est: - - -\begin{Def}[Ensemble fini] -Un ensemble est dit fini\index{ensemble!fini} s'il ne peut pas être mis en bijection avec une partie stricte de lui-même. -\end{Def} - - -\subsection{Cas des ensembles infinis} - - - -Lorsque l'on se pose la question du \og nombre d'éléments\fg{} d'un ensemble infini, on peut être tenté, dans un premier temps, de répondre par \og une infinité\fg{}, ce qui n'est pas satisfaisant pour au moins deux raisons: - -\begin{itemize} - \item jusqu'à présent, le \og nombre d'éléments\fg{} était un entier, et l'infini n'est pas un nombre entier, - \item cela laisserait supposer que tous les ensemble infinis ont le même \og nombre d'éléments\fg{} -\end{itemize} - - - -Une réflexion plus appronfondie apparaît comme nécessaire. On décide donc de se calquer sur le domaine fini\ldots - - -\begin{Def}[Puissance d'un ensemble infini] -Deux ensembles ont \emph{même puissance}\index{ensemble!puissance}\index{puissance!d'un ensemble} (\og même nombre d'éléments\fg{}) si l'on peut les mettre en bijection. -\end{Def} - - - -\begin{Ex} -Il existe des bijections entre $\N$ et l'ensemble des entiers pairs ($2\N$) ou impairs ($2\N+1$), qui conduisent à des formulations du style \og il y a autant d'entiers que d'entiers pairs \fg{}. -\end{Ex} - - -\begin{Rem} -$(2\N) \cup (2\N+1) = \N$, bien que ces trois ensembles \og on le même nombre d'éléments \fg{}, les deux premiers étants disjoints \ldots \'A rapprocher de $\infty\plinf=\infty$. -\end{Rem} - - -\begin{Rem} -Pour ce genre de raisons, on a tendance à abandonner ce vocabulaire (sur les nombres d'individus) pour dire simplement que ces ensembles ont même puissance. -\end{Rem} - - - -\subsection{Puissance d'un ensemble infini} - - -Si $E$ est un ensemble infini, le résultat fondamental est: - -\begin{Th} -Il est impossible de définir une surjection de $E$ sur $\mathcal{P}(E)$. -\end{Th} - -\begin{Pre} - Admis -\end{Pre} - -Ce résultat montre que, même si $E$ est un ensemble infini, il ne peut être mis en bijection avec ${\cal P}(E)$, qui a donc \og strictement plus d'éléments\fg{} que $E$. -Conséquemment, $\N$ et ses parties infinies, $\Z$, et même $\Q$ ont tous la même puissance, dite \emph{puissance du dénombrable} \index{puissance!du dénombrable}. Mais\ldots - - -\begin{Def}[Puissance du continu] -${\cal P}(\N)$ n'est pas dénombrable, et est de puissance strictement supérieure, dite \emph{puissance du continu}\index{puissance!du continu}, dite - \emph{puissance de $\R$}. -\end{Def} - - -\noindent De même, ${\cal P}(\R)$ est de puissance strictement supérieure à $\R$, et ainsi de suite\ldots - -\begin{Th} -$\R$ est indénombrable. -\end{Th} - -\begin{Pre}[Démonstration de Cantor] -Supposons que $\mathbb{R}$ soit dénombrable. -On pourrait alors écrire la liste de TOUS les réels, -à partir du premier, en écriture décimale $r_1$, $r_2$, $r_3$, etc. - -On choisit pour les nombres décimaux l'écriture qui ne comporte que des zéros à partir d'un certain rang, s'il y a ambiguïté. -Ainsi, l'on préférera l'écriture 1 à 0,9999999\ldots \ldots (ces nombres sont égaux ; c'est-à-dire que l'on a affaire ici à deux représentations décimales d'un même nombre). - - - Construisons alors le nombre $S$ de la manière suivante: - \begin{enumerate} - \item sa partie entière est 0, - \item et pour sa partie décimale, - \begin{itemize} - \item le premier chiffre est différent du premier chiffre après la virgule de $r_1$, - \item le deuxième chiffre est différent du deuxième chiffre après la virgule de $r_2$, - \item \emph{etc.} - \end{itemize} - \end{enumerate} - -Le nombre $S$ ne fait alors pas partie de la liste des réels, ce qui est contradictoire: l'hypothèse $\mathbb{R}$ dénombrable (on peut numéroter tous les réels avec $\mathbb{N}$) est erronée. -\end{Pre} -\section{Relations d'ordre} -Dans ce paragraphe, on se place dans le cas où $E=F$. -Soit ${\mathcal{R}}$ une relation binaire définie dans un ensemble $E$, de graphe $G$. - -\subsection{Réflexivité, antisymétrie, transitivité} - - -\begin{Def}[Réflexivité] - ${\mathcal{R}}$ est dite \emph{réflexive} \index{relation!réflexive} quand tout élément de $E$ est en relation avec lui-même: -$$\qqs x\in E,\ (x,x)\in G$$ -\end{Def} - - -\begin{Rem} - C'est-à-dire $\qqs x\in E,\ x {\mathcal{R}} x$, ou encore: la diagonale de $E^2$ est incluse dans $G$. -\end{Rem} - - - -\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 $x=y$): - -$$\qqs x\in E,\ \qqs y\in E,\ (x,y)\in G ~{\rm et}\ (y,x)\in G\ \Imp x=y$$ -\end{Def} - - -\begin{Rem} - C'est-à-dire $\qqs(x,y)\in E^2,\ x {\mathcal{R}} y\ {\rm et}\ y {\mathcal{R}} x\Imp\ x=y$. -\end{Rem} - - - -\begin{Def}[Transitivité] - ${\mathcal{R}}$ est dite \emph{transitive} \index{relation!transitive} lorsque, si $x$ est en relation avec $y$, et si $y$ l'est avec $z$, alors $x$ est en relation avec $z$: -$$\qqs x\in E,\qqs y\in E,\ \qqs z\in E,\ (x,y)\in G\ {\rm et}\ (y,z)\in G -\Imp (x,z)\in G$$ -\end{Def} - - -\begin{Rem} -C'est-à-dire: $\qqs x\in E,\ \qqs y\in E,\qqs z \in E,\ x {\mathcal{R}} y\ {\rm et}\ y {\mathcal{R}} z \Imp x {\mathcal{R}} z$. -\end{Rem} - - -\begin{Exo} -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 $A = \mathbb{N}$ 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. -\end{enumerate} -\end{Exo} - - -\begin{Exo} -Sur un ensemble à $n$ éléments, combien y a-t-il de relations\ldots -\begin{enumerate} -\item réflexives~? -\item symétriques~? -\end{enumerate} -\end{Exo} - - - -\begin{Exo} -Déterminer quand une relation $\mathcal{R}$ dans un ensemble $A$ est -\begin{enumerate} -\item non réflexive, -\item non symétrique, -\item non transitive, -\item non antisymétrique. -\end{enumerate} -\end{Exo} - - -\begin{Exo} -Donner des exemples de relation $\mathcal{R}$ dans $\{1,2,3\}$ ayant les propriétés suivantes: -\begin{enumerate} -\item $\mathcal{R}$ est symétrique, -\item $\mathcal{R}$ n'est ni symétrique ni antisymétrique, -\item $\mathcal{R}$ est transitive mais $\mathcal{R} \cup \mathcal{R}^{-1}$ n'est pas transitive. -\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} - -\begin{Def}[Relation d'ordre] -${\mathcal{R}}$ est une \emph{relation d'ordre}\index{relation!d'ordre} lorsqu'elle est réflexive, antisymétrique et transitive. -\end{Def} - - -\begin{Ex}[Exemples de relations d'ordre] -Quelques relations d'ordre: -\begin{itemize} -\item $(\R,\leqslant)$ -\item $({\cal P}(E),\sse)$ -\end{itemize} -\end{Ex} - -\begin{Ex}[Relation de divisibilité] -On note $a|b$ si et seulement si $b$ est un multiple de $a$ ($\exists k \in \Net, b=ka$). - -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[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} - -\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}. - - - -\subsection{Ordre partiel, ordre total} - -Une relation d'ordre définie dans un ensemble $E$ peut posséder une propriété supplémentaire, celle selon laquelle tous les éléments de $E$ sont comparables entre eux. -On formalise comme suit: - - -\begin{Def}[Relation d'ordre totale] -Une relation d'ordre qui possède cette dernière propriété est dite relation d'\emph{ordre total}\index{relation!d'ordre!totale}, et la structure algébrique correspondante est celle d'\emph{ensemble totalement ordonné}\index{ensemble!totalement ordonné}. -\end{Def} - +Dans tout ce chapitre, on se donne deux ensembles $E$ et $F$. -\begin{Rem} -Cette propriété est aussi équivalente à: $$\qqs x\in E,\ \qqs y\in E,\ x\ {\mathcal{R}}\ y\ {\rm ou}\ y {\mathcal{R}} x$$ -\noindent ou encore: \og si $x$ n'est pas en relation avec $y$, alors $y$ est en relation avec $x$ \fg{}. -\end{Rem} +\section{Relations} -\begin{Def}[Relation d'ordre partiel] -Dans le cas contraire, il existe des éléments qui ne sont pas comparables: on parle alors d'\emph{ordre partiel} \index{relation!d'ordre!partiel}. +\begin{Def}[Relation binaire] +On définit une \emph{relation binaire} \index{relation binaire} entre +les deux ensembles $E$ et $F$ lorsqu'on construit une partie $G$ +de l'ensemble produit $E\times F$ +($G\sse E\times F$). +Si $x$ dans $E$ et $y$ dans $F$ sont tels que $(x,y)\in G$, on dit +que $x$ est \emph{en Relation avec} $y$. On note cela $x{\mathcal{R}}y$. \end{Def} - -\begin{Ex} -$\leqslant$ est une relation d'ordre totale dans $\R$. -\end{Ex} - -\begin{Ex} -$\subset$ dans $\mathcal{P}(E)$, et $|$ dans $\N$ sont des relations d'ordre partiels. -\end{Ex} - - - \begin{Exo} -Parmi les relations suivantes sur l'ensemble $E$, repérez les relations d'ordre, les relations d'ordre total: \begin{enumerate} -\item $E = \mathbb{Z}$, $x \mathcal{R} y \Leftrightarrow \exists k \in \mathbb{N}: x = y^k$. -\item $E = \mathbb{N}$, $x \mathcal{R} y \Leftrightarrow x Babel and hyphenation patterns for english, dumylang, nohyphenation, lo @@ -1138,7 +1138,7 @@ pdfTeX warning (ext4): destination with the same identifier (name{page.1}) has been already used, duplicate ignored \relax -l.44 \contentsline {chapter}{Index}{32}{chapter.4} +l.48 ...IeC {\'e}quivalence}{32}{subsection.4.3.1} [1 ]) @@ -1626,19 +1626,196 @@ ame{Exo.17}) has been already used, duplicate ignored \relax l.300 \begin{Exo} [Fonction caractéristique des 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b/main13.thm index 1fae668..3a5d32c 100644 --- a/main13.thm +++ b/main13.thm @@ -125,3 +125,40 @@ \contentsline {Exo}{{Exercice}{3.{15}}{}}{29}{Exo.15} \contentsline {Exo}{{Exercice}{3.{16}}{}}{29}{Exo.16} \contentsline {Exo}{{Exercice}{3.{17}}{Fonction caractéristique des parties d'un ensemble}}{29}{Exo.17} +\contentsline {Def}{{Définition}{4.{1}}{Relation binaire}}{30}{Def.1} +\contentsline {Exo}{{Exercice}{4.{1}}{}}{30}{Exo.1} +\contentsline {Rem}{{Remarque}{4.{1}}{}}{30}{Rem.1} +\contentsline {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 {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 {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 {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} diff --git a/main13.toc b/main13.toc index c1b588f..6578952 100644 --- a/main13.toc +++ b/main13.toc @@ -39,6 +39,14 @@ \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 {part}{III\hspace {1em}Annexes}{30}{part.3} -\contentsline {chapter}{\numberline {4}Programme P\IeC {\'e}dagogique National 2005 (PPN)}{31}{chapter.4} -\contentsline {chapter}{Index}{32}{chapter.4} +\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}{32}{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} -- 2.39.5