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  5\usepackage{amsmath}
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 89
 90\newcommand{\hmwkTitle}{Khôlle}
 91\newcommand{\hmwkDueDate}{2024/09/23}
 92\newcommand{\hmwkClass}{Maths}
 93\newcommand{\hmwkClassInstructor}{M. Chiecchio}
 94\newcommand{\hmwkAuthorName}{\textbf{Louis Dalibard}}
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138\begin{document}
139
140\maketitle
141
142\pagebreak
143
144\begin{homeworkProblem}
145    Le but de cet exercice est de démontrer le théorème de Cantor-Bernstein.
146    
147    Considérons $E$ et $F$ deux ensembles infinis.
148    
149    Le théorème de Cantor-Bernstein est le suivant:
150    \[
151        (\exists \Phi : E \longrightarrow F,
152        \exists \Psi : F \longrightarrow E,\quad \Phi \text{ injective et } \Psi \text{ injective} )
153        \implies E \simeq F
154    \]
155    
156    \begin{enumerate}
157        \item Montrer que toute fonction croissante pour $\subseteq$ admet un point fixe. \footnote{\begin{turn}{180} 
158       \textbf{Aide:} Considérer l'ensemble $H = \{x\in P(E) \mid x \subseteq f(x)\}$\end{turn}}
159       \item Supposons:
160        \[
161        \exists \Phi : E \to F,
162        \exists \Psi : F \to E,\quad \Phi \text{ injective et } \Psi \text{ injective} 
163    \]
164       On considere la fonction \begin{align*}
165\gamma\colon \mathcal{P}(E) & \longrightarrow \mathcal{P}(E)\\
166A&\longmapsto E\setminus\Psi(F\setminus\Phi(A)),
167\end{align*}
168\item Montrer que $\gamma$ est croissante pour $\subseteq$
169\item En déduire une bijection de $E$ dans $F$. Conclure.
170
171    \end{enumerate}
172\end{homeworkProblem}
173
174\end{document}