% !TEX TS-program = latex %\documentclass[handout]{beamer} %pour compiler sans les pauses \documentclass{beamer} %pour compiler avec les pauses \usetheme{Warsaw} \setbeamertemplate{navigation symbols}{} \title{} \author{} \date{} \newcommand\hmmax{0} \newcommand\bmmax{0} \usepackage[latin1,utf8]{inputenc} \usepackage{amsfonts,amsmath,amssymb,graphicx} \usepackage[french]{babel} \usepackage{tikz} \usepackage{amsthm} \usepackage{amsmath} \usepackage{amssymb} \usepackage{mathrsfs} \usepackage{mathtools} \usepackage{graphicx} \usepackage[T1]{fontenc} \usepackage[latin1]{inputenc} \usepackage{color} \usepackage{amsfonts} \usepackage[all]{xy} \usepackage{stmaryrd} \usepackage{oldgerm} \usepackage{xcolor} \usepackage{mathdots} \usepackage{esvect} % pour les vecteurs longs \usepackage{relsize} \usepackage{tikz,tkz-tab,tikz-3dplot} \usepackage{marginnote} \renewcommand{\tau}{\uptau} \DeclareMathOperator{\Card}{\mathsf{Card}} \DeclareMathOperator{\pgcd}{\mathsf{pgcd}} \DeclareMathOperator{\ppcm}{\mathsf{ppcm}} \DeclareMathOperator{\supp}{\mathsf{supp}} \DeclareMathOperator{\stab}{\mathsf{stab}} \DeclareMathOperator{\Fix}{\mathsf{Fix}} \DeclareMathOperator{\orb}{\mathsf{orb}} \DeclareMathOperator{\N}{\mathsf{N}} \DeclareMathOperator{\tors}{tors} \DeclareMathOperator{\distingue}{\trianglelefteqslant} \DeclareMathOperator{\sign}{\mathsf{sign}} \DeclareMathOperator{\commutant}{\mathsf{C}} \DeclareMathOperator{\NN}{\mathbb{N}} \DeclareMathOperator{\ZZ}{\mathbb{Z}} \DeclareMathOperator{\QQ}{\mathbb{Q}} \DeclareMathOperator{\FF}{\mathbb{F}} \DeclareMathOperator{\RR}{\mathbb{R}} \DeclareMathOperator{\CC}{\mathbb{C}} \DeclareMathOperator{\car}{\mathsf{car}} \DeclareMathOperator{\rad}{\mathsf{rad}} \DeclareMathOperator{\Id}{\mathsf{Id}} \DeclareMathOperator{\I}{I} \DeclareMathOperator{\tr}{\mathsf{Tr}} \DeclareMathOperator{\Mat}{\mathsf{M}} \DeclareMathOperator{\Matrice}{\mathsf{Mat}} \DeclareMathOperator{\GL}{\mathsf{GL}} \DeclareMathOperator{\PGL}{\mathsf{PGL}} \DeclareMathOperator{\SL}{\mathsf{SL}} \DeclareMathOperator{\PSL}{\mathsf{PSL}} \renewcommand{\O}{\mathsf{O}} \DeclareMathOperator{\SO}{\mathsf{SO}} \DeclareMathOperator{\U}{\mathsf{U}} \DeclareMathOperator{\SU}{\mathsf{SU}} \DeclareMathOperator{\GA}{\mathsf{GA}} \DeclareMathOperator{\Aff}{\mathsf{Aff}} \DeclareMathOperator{\ev}{\mathsf{ev}} \DeclareMathOperator{\Imc}{Im} \DeclareMathOperator{\Rec}{Re} \DeclareMathOperator{\Frac}{\mathsf{Frac}} \DeclareMathOperator{\rg}{\mathsf{rg}} \DeclareMathOperator{\Ker}{\mathsf{Ker}} \DeclareMathOperator{\im}{\mathsf{Im}} \DeclareMathOperator{\Coker}{\mathsf{Coker}} \DeclareMathOperator{\isomto}{\overset{\sim}{\to}} \DeclareMathOperator{\from}{\leftarrow} \DeclareMathOperator{\dual}{\mathtt{D}} \DeclareMathOperator{\End}{\mathsf{End}} \DeclareMathOperator{\Hom}{\mathsf{Hom}} \DeclareMathOperator{\Aut}{\mathsf{Aut}} \DeclareMathOperator{\Int}{\mathsf{Int}} \DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\Sym}{\mathsf{Sym}} \DeclareMathOperator{\Alt}{\mathsf{Alt}} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\trace}{tr} \DeclareMathOperator{\rig}{rig} \DeclareMathOperator{\cont}{\mathsf{c}} \DeclareMathOperator{\an}{an} \DeclareMathOperator{\lin}{lin} \DeclareMathOperator{\com}{\mathsf{com}} \DeclareMathOperator{\Z}{\mathsf{Z}} \DeclareMathOperator{\T}{\mathsf{T}} \DeclareMathOperator{\V}{\mathsf{V}} \DeclareMathOperator{\HOM}{\underline{\mathsf{Hom}}} \DeclareMathOperator{\B}{\mathsf{B}} \DeclareMathOperator{\D}{\mathsf{D}} \DeclareMathOperator{\C}{\mathsf{C}} \DeclareMathOperator{\can}{can} \DeclareMathOperator{\Frob}{\mathsf{F}} \DeclareMathOperator{\Ver}{\mathsf{V}} \DeclareMathOperator{\tg}{\mathsf{tg}} \DeclareMathOperator{\triv}{\mathsf{triv}} \DeclareMathOperator{\Hodge}{\mathsf{h}} \DeclareMathOperator{\mil}{\mathsf{mil}} \DeclareMathOperator{\gp}{\mathsf{gp}} \DeclareMathOperator{\Fil}{\mathsf{Fil}} \DeclareMathOperator{\gr}{\mathsf{gr}} \DeclareMathOperator{\Mod}{\mathbf{Mod}} \DeclareMathOperator{\MF}{\mathbf{MF}} \DeclareMathOperator{\Vect}{\mathsf{Vect}} \DeclareMathOperator{\reg}{reg} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\id}{Id} \DeclareMathOperator{\conv}{\mathsf{conv}} \DeclareMathOperator{\Ass}{\mathsf{Ass}} \DeclareMathOperator{\disc}{\mathsf{disc}} \DeclareMathOperator{\Res}{\mathsf{Res}} \DeclareMathOperator{\grad}{\mathsf{grad}} \DeclareMathOperator{\vol}{\mathsf{vol}} \DeclareMathOperator{\jac}{\mathsf{jac}} \DeclareMathOperator{\Jac}{\mathsf{Jac}} \DeclareMathOperator{\Fr}{Fr} \newcommand{\afortiori}{\textit{a fortiori }} \newcommand{\loccit}{\textit{loc. cit.}} \newcommand{\cf}{\textit{cf }} \newcommand{\ie}{\textit{i.e. }} \newcommand{\idem}{\textit{idem }} \newcommand{\ssi}{si et seulement si } \newcommand{\ops}{on peut supposer } \newcommand{\scal}[2]{\left\langle#1\mid#2\right\rangle} \newcommand{\scalp}[2]{\langle#1\mid#2\rangle} \newcommand{\trans}[1]{\hbox{}^t#1} \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\absp}[1]{\vert#1\vert} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\tnorm}[1]{\big\|#1\big\|} \newcommand{\normp}[1]{\|#1\|} \newcommand{\normtriple}[1]{\left\vvvert#1\right\vvvert} \newcommand{\normtriplep}[1]{\vvvert#1\vvvert} \newcommand{\legendre}[2]{\big(\frac{#1}{#2}\big)} \newcommand{\dlegendre}[2]{\Big(\frac{#1}{#2}\Big)} \newcommand{\interieur}[1]{\mathring{#1}} \def\Iso[#1]{\Hom_{#1\textrm{-}\mathrm{alg}}} %\newcommand{\vv}[1]{\vec{#1}} %{\accentset{\smash{\raisebox{-0.12ex}{$\scriptstyle\circ$}}}{#1}\rule{0pt}{2.3ex}} %\fboxrule0.0001pt \fboxsep0pt % Blank box placeholder for figures (to avoid requiring any % particular graphics capabilities for printing this document). \newcommand{\blankbox}[2]{% \parbox{\columnwidth}{\centering % Set fboxsep to 0 so that the actual size of the box will match the % given measurements more closely. \setlength{\fboxsep}{0pt}% \fbox{\raisebox{0pt}[#2]{\hspace{#1}}}% }% } \newcommand{\udots}{\mbox{\larger[-2]{\reflectbox{$\ddots$}}}} %petits points en diagonale montante \def \ooverline #1#2#3% {\mkern #1mu \overline{\mkern -#1mu #3 \mkern -#2mu }\mkern #2mu } \def \Abar {\ooverline40A{\mkern 1mu}{}} \def \Bbar {\ooverline40B{\mkern 1mu}{}} \def \Cbar {\ooverline40C{\mkern 1mu}{}} \def \Dbar {\ooverline40D{\mkern 1mu}{}} \def \Ebar {\ooverline40E{\mkern 1mu}{}} \def \Fbar {\ooverline40F{\mkern 1mu}{}} \def \Gbar {\ooverline40G{\mkern 1mu}{}} \def \Hbar {\ooverline41H{\mkern 1mu}{}} \def \Ibar {\ooverline40I{\mkern 1mu}{}} \def \Jbar {\ooverline60J{\mkern 1mu}{}} \def \Kbar {\ooverline40K{\mkern 1mu}{}} \def \Lbar {\ooverline40L{\mkern 1mu}{}} \def \Mbar {\ooverline41M{\mkern 1mu}{}} \def \Nbar {\ooverline41N{\mkern 1mu}{}} \def \Obar {\ooverline40O{\mkern 1mu}{}} \def \Pbar {\ooverline40P{\mkern 1mu}{}} \def \Qbar {\ooverline40Q{\mkern 1mu}{}} \def \Rbar {\ooverline40R{\mkern 1mu}{}} \def \Sbar {\ooverline40S{\mkern 1mu}{}} \def \Tbar {\ooverline10T{\mkern 1mu}{}} \def \Ubar {\ooverline31U{\mkern 1mu}{}} \def \Vbar {\ooverline21V{\mkern 1mu}{}} \def \Wbar {\ooverline21W{\mkern 1mu}{}} \def \Xbar {\ooverline30X{\mkern 1mu}{}} \def \Ybar {\ooverline11Y{\mkern 1mu}{}} \def \Zbar {\ooverline40Z{\mkern 1mu}{}} \def \uunderline #1#2#3% {\mkern #1mu \underline{\mkern -#1mu #3 \mkern -#2mu }\mkern #2mu } \def \Xline {\uunderline04X{\mkern 1mu}{}\!} \def \Yline {\uunderline04Y{\mkern 1mu}{}} \def \Sline {\uunderline04S{\mkern 1mu}{}} \def \Tline {\uunderline04T{\mkern 1mu}{}} \def \Dline {\uunderline04D{\mkern 1mu}{}} \def \Uline {\uunderline04U{\mkern 1mu}{}} %\hypersetup{ % %backref=true, % %pagebackref=true, % %hyperindex=true, % colorlinks=true, % breaklinks=true, % urlcolor= blue, % linkcolor= blue, % %bookmarks=true, % bookmarksopen=true, % pdftitle={Structures alg\'ebriques 2}, % pdfauthor={Brinon}, % %pdfsubject={} % } % grand isomorphismes sous xypic \newcommand{\eq}[1][r] {\ar@<-3pt>@{-}[#1] \ar@<-1pt>@{}[#1]|<{}="gauche" \ar@<+0pt>@{}[#1]|-{}="milieu" \ar@<+1pt>@{}[#1]|>{}="droite" \ar@/^1.25pt/@{-}"gauche";"milieu" \ar@/_1.25pt/@{-}"milieu";"droite"} %inclusions dans xypic \newcommand{\incl}[1][r] {\ar@<-0.2pc>@{^(-}[#1] \ar@<+0.2pc>@{-}[#1]} \newcommand\attention{\textcolor{red}{\fontencoding{U}\fontfamily{futs}\selectfont\char 66\relax}% } %%%%%%%%%%%%%%%%%%%%%%%%%%%% entourer un caractere de facon bien centree %%%%%%%%%%%%%%%%%%%%%% \newcommand*\circled[1]{\tikz[baseline=(char.base)]{ \node[shape=circle,draw,inner sep=.2pt] (char) {#1};}} \setbeamercolor{alerted text}{fg=yellow} \begin{document} \frame{\titlepage} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \frametitle{Relations d'\'equivalence et ensembles quotients} \begin{block}{D\'efinition} Soit $X$ un ensemble. Une \emph{relation d'\'equivalence} sur $X$ est une relation binaire $\mathcal{R}$ sur $X$ v\'erifiant les propri\'et\'es suivantes : \pause \begin{itemize} \item<2->[$\bullet$] $(\forall x\in X)\,x\mathcal{R}x$ (r\'eflexivit\'e) ; \item<3->[$\bullet$] $(\forall x,y\in X)\,x\mathcal{R}y\Rightarrow y\mathcal{R}x$ (sym\'etrie) ; \item<4->[$\bullet$] $(\forall x,y,z\in X)\,(x\mathcal{R}y\textrm{ et }y\mathcal{R}z)\Rightarrow x\mathcal{R}z$ (transitivit\'e). \end{itemize} \onslide<5->La \emph{classe d'\'equivalence} de $x\in X$ est alors $[x]:=\{y\in X\,;\,x\mathcal{R}y\}$. \onslide<6->Les classes d'\'equivalence forment une partition de $X$. \onslide<7->L'\emph{ensemble quotient} $X/\mathcal{R}$ est la partie de $\mathscr{P}(X)$ constitu\'ee par les classes d'\'equivalences. Si $A\in X/\mathcal{R}$, on a $A=[x]$ pour tout $x\in A$ : un tel \'el\'ement $x$ s'appelle un \emph{repr\'esentant} de $A$. \end{block} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \frametitle{Relations d'\'equivalence et ensembles quotients} \begin{exampleblock}{Exemple} Soit $f\colon X\to Y$ une application. On d\'efinit une relation d'\'equivalence $\mathcal{R}\!_f$ sur $X$ en posant $x_1\mathcal{R}\!_fx_2\Leftrightarrow f(x_1)=f(x_2)$. Les classes d'\'equivalence sont les pr\'eimages non vides des singletons. \end{exampleblock} \pause \begin{block}{D\'efinition} Si $\mathcal{R}$ est une relation d'\'equivalence sur un ensemble $X$, on dispose de la \emph{surjection canonique} \begin{align*} \pi_{\mathcal{R}}\colon X &\to X/\mathcal{R}\\ x &\mapsto [x]. \end{align*} \pause Un \emph{syst\`eme (complet) de repr\'esentants} est une partie $T\subset X$ telle que la restriction de $\pi_{\mathcal{R}}$ \`a $T$ induise une bijection $T\isomto X/\mathcal{R}$. \pause Cela signifie que pour tout $A\in X/\mathcal{R}$, il existe un unique $t\in T$ tel que $A=[t]$, \ie tel que $t$ soit un repr\'esentant de $A$. \end{block} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \frametitle{Relations d'\'equivalence et ensembles quotients} \begin{block}{Proposition - Propri\'et\'e universelle} Soient $\mathcal{R}$ une relation d'\'equivalence sur un ensemble $X$ et $f\colon X\to Y$ une application. \pause Supposons que $x_1\mathcal{R}x_2\Rightarrow f(x_1)=f(x_2)$. \pause Alors il existe une unique application $\widetilde{f}\colon X/\mathcal{R}\to Y$ telle que $f=\widetilde{f}\circ\pi_{\mathcal{R}}$. \end{block} \pause $$\xymatrix{X\ar[r]^f\ar[d]_{\pi_{\mathcal{R}}} & Y\\ X/\mathcal{R} \only<5->{\ar@{-->}[ru]_{\widetilde{f}}} & }$$ \onslide<6->Si $A\in X/\mathcal{R}$ et $x\in A$, on a n\'ecessairement $\widetilde{f}(A)=f(x)$. \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \frametitle{Relations d'\'equivalence et ensembles quotients} \begin{alertblock}{D\'emonstration} bla-bla \end{alertblock} \end{frame} \end{document}