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\begin{document}

\frame{\titlepage}

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\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}

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\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}

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\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 $\widetilde{f}(A)=f(x)$.

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\begin{frame}
\frametitle{Relations d'\'equivalence et ensembles quotients}

\begin{alertblock}{D\'emonstration}
bla-bla
\end{alertblock}

\end{frame}


\end{document}



