From 8c64d4461e17c5b4e2be2183f5a02e4b0e8cc9b3 Mon Sep 17 00:00:00 2001 From: couchot 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}} 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{\'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 --- a/main13.log +++ b/main13.log @@ -1,8 +1,8 @@ -This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.4.28) 17 OCT 2013 13:42 +This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 2012/Debian) (format=pdflatex 2013.4.28) 17 OCT 2013 21:26 entering extended mode restricted \write18 enabled. %&-line parsing enabled. -**main13 +**main13.tex (./main13.tex LaTeX2e 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identifier (name{Exo.8}) has b +een already used, duplicate ignored \relax l.199 \begin{Ex} pdfTeX warning (ext4): destination with the same identifier (na -me{Exo.11}) has been already used, duplicate ignored +me{Exo.9}) has been already used, duplicate ignored \relax l.205 \begin{Ex} - [Relation de congruence modulo $n$ dans $\Z$]pdfTeX warning (ex -t4): destination with the same identifier (name{Exo.12}) has been already used, - duplicate ignored + [Relation de congruence modulo $n$ dans $\Z$] [31]pdfTeX warnin +g (ext4): destination with the same identifier (name{Exo.10}) has been already +used, duplicate ignored \relax l.225 \begin{Exo} @@ -1745,12 +1734,12 @@ ame{Notation.1}) has been already used, duplicate ignored \relax l.250 \begin{Notation} pdfTeX warning (ext4): destination with the same identifi -er (name{Exo.13}) has been already used, duplicate ignored +er (name{Exo.11}) has been already used, duplicate ignored \relax l.255 \begin{Exo} pdfTeX warning (ext4): destination with the same identifier (n -ame{Exo.14}) has been already used, duplicate ignored +ame{Exo.12}) has been already used, duplicate ignored \relax l.264 \begin{Exo} @@ -1759,63 +1748,42 @@ ame{Th.1}) has been already used, duplicate ignored \relax l.294 \begin{Th} - [32]pdfTeX warning (ext4): destination with the same identifie -r (name{Rem.2}) has been already used, duplicate ignored - - \relax -l.298 \begin{Rem} - + Package hyperref Info: bookmark level for unknown Pre defaults to 0 on input li -ne 304. +ne 301. pdfTeX warning (ext4): destination with the same identifier (name{Th.2}) has be en already used, duplicate ignored \relax -l.321 \begin{Th} - pdfTeX warning (ext4): destination with the same identifier (na -me{Exo.15}) has been already used, duplicate ignored +l.318 \begin{Th} + [32]pdfTeX warning (ext4): destination with the same identifie +r (name{Exo.13}) has been already used, duplicate ignored \relax -l.331 \begin{Ex} +l.328 \begin{Ex} pdfTeX warning (ext4): destination with the same identifier (na -me{Exo.16}) has been already used, duplicate ignored - - \relax -l.346 \begin{Exo} - pdfTeX warning (ext4): destination with the same identifier (n -ame{Exo.17}) has been already used, duplicate ignored - - \relax -l.358 \begin{Exo} - [Une relation d'équivalence]pdfTeX warning (ext4): destinatio -n with the same identifier (name{Exo.18}) has been already used, duplicate igno -red +me{Exo.14}) has been already used, duplicate ignored \relax -l.381 \begin{Exo} +l.343 \begin{Exo} pdfTeX warning (ext4): destination with the same identifier (n -ame{Notation.2}) has been already used, duplicate ignored - - \relax -l.394 \begin{Notation} - pdfTeX warning (ext4): destination with the same identifi -er (name{Exo.19}) has been already used, duplicate ignored +ame{Exo.15}) has been already used, duplicate ignored \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 ] @@ -1847,17 +1815,17 @@ Package atveryend Info: Empty hook `AfterLastShipout' on input line 336. Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 336. Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 336. Package rerunfilecheck Info: File `main13.out' has not changed. -(rerunfilecheck) Checksum: BDAE9B6D6DE61206219E655EA40146AA;3248. +(rerunfilecheck) Checksum: B510A10B4C2809C1121D155E1D0BF426;3178. Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 336. ) Here is how much of TeX's memory you used: - 11796 strings out of 495059 - 159536 string characters out of 3182030 + 11789 strings out of 495059 + 159454 string characters out of 3182030 286421 words of memory out of 3000000 - 14358 multiletter control sequences out of 15000+200000 + 14355 multiletter control sequences out of 15000+200000 97094 words of font info for 97 fonts, out of 3000000 for 9000 14 hyphenation exceptions out of 8191 - 30i,13n,32p,465b,618s stack positions out of 5000i,500n,10000p,200000b,50000s + 30i,13n,32p,469b,618s stack positions out of 5000i,500n,10000p,200000b,50000s {/usr/share/texlive/texmf-dist/fonts/enc/dvips/base/8r.enc}{/usr/share/texmf/ fonts/enc/dvips/lm/lm-mathex.enc}{/usr/share/texmf/fonts/enc/dvips/lm/lm-mathsy .enc}{/usr/share/texmf/fonts/enc/dvips/lm/lm-mathit.enc}{/usr/share/texmf/fonts @@ -1876,10 +1844,10 @@ urw/times/utmb8a.pfb> -Output written on main13.pdf (39 pages, 324131 bytes). +Output written on main13.pdf (38 pages, 320746 bytes). 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