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% --- Basic Settings ---
\def\professor{R. MOSAID}
\def\classname{1BAC.SM}
\def\examtitle{Série: Ensembles}
\def\schoolname{\textbf{Lycée :} Taghzirt}
\def\academicyear{2025/2026}
\def\subject{\wsite}
\def\duration{2h}
\def\secondtitle{\small(By failing to prepare, you are preparing to fail.)}
\def\province{Direction provinciale de\\ Beni Mellal}
\def\logo{\includegraphics[width=\linewidth]{images/logo-men.png}}
\def\wsite{\color{magenta}\texttt{www.mosaid.xyz}}
\def\ddate{\hfill \number\day/\number\month/\number\year~~}
\def\bottommsg{Bonne chance!}
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\begin{document}

% Exercise 1
\printexo{1}{}{
Écrire en extension les ensembles suivants :\\
\begin{tabular}{p{6cm}p{6.5cm}@{}p{6cm}}
     $A = \{ a \in \mathbb{Z} ; |a + 3| \leq 7 \}$
    & $C = \{( p; q) \in \mathbb{N}^2 ; p^2 - q^2 = 24 \}$
    & $E = \{ p \in \mathbb{N} ; \dfrac{2p + 6}{p + 1} \in \mathbb{N} \}$\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     $B = \{( m; n) \in \mathbb{N}^2 ; 0 < 2mn < 7 \}$
    & $D = \left\{ \dfrac{(-1)^k}{k^2 + 1} ; k \in \mathbb{N} \text{ et } 0 \leq k < 3 \right\}$
    & $F = \{( n; m) \in \mathbb{N}^2 ; n + 2m = 11 \}$\\
    \multicolumn{3}{l}{%
    $G = \{( n; m) \in \mathbb{N}^2 ; 2m + 2n = (-1)^{m+n} \}$
    }
\end{tabular}\\
}

% Exercise 3
\printexo{3}{}{
On considère l'ensemble A défini par :
~$
A = \left\{ n \in \mathbb{N} ; \dfrac{n^3 + 5n^2 + 6n + 10}{n + 2} \in \mathbb{N} \right\}
$~
\begin{enumerate}
    \item Montrer que :
    ~$
    \dfrac{n^3 + 5n^2 + 6n + 10}{n + 2} \in \mathbb{N} \iff \dfrac{10}{n + 2} \in \mathbb{N}
    $~
    \item Écrire en extension l'ensemble A
\end{enumerate}
}

% Exercise 4
\printexo{4}{}{
\vspace*{-1cm}
\begin{enumerate}
    \item On considère les ensembles E et F :
    ~$
    E = \left\{ x \in \mathbb{R} ; \left|1 - \dfrac{x}{2}\right| \leq 1 \right\} \quad \text{et} \quad F = [0; 4]
    $~
    Montrer que : $E = F$

    \item A et B deux ensembles tels que :
    ~$
    A = \left\{ \dfrac{\pi}{8} + \dfrac{k\pi}{2}; k \in \mathbb{Z} \right\} \quad \text{et} \quad B = \left\{ -\dfrac{7\pi}{8} + \dfrac{m\pi}{2}; m \in \mathbb{Z} \right\}
    $~\\
    Montrer que $A = B$
\end{enumerate}
}

% Exercise 5
\printexo{5}{}{
On pose dans $\mathbb{N}$ :
~$
A = \left\{ n \in \mathbb{N} \quad;\quad \dfrac{4n^2 - 4n + 10}{2n - 1} \in \mathbb{N} \right\} \quad \text{et} \quad B = \left\{ n \in \mathbb{N} \quad;\quad \dfrac{n + 10}{n - 5} \in \mathbb{N} \right\}
$~
\begin{enumerate}
    \item Montrer que :
    ~$
    (\forall n \in \mathbb{N} - \{ 5 \}) \quad;\quad \dfrac{n + 10}{n - 5} = 1 + \dfrac{15}{n - 5}
    $~
    \item Montrer que :
    ~$
    (\forall n \in \mathbb{N}) \quad;\quad \dfrac{4n^2 - 4n + 10}{2n - 1} = 2n - 1 + \dfrac{9}{2n - 1}
    $~
    \item Déterminer en extension :
    ~$
    A; B; A \cap B; A \cup B; A \setminus B; B \setminus A \text{ et } A \triangle B
    $~
\end{enumerate}
}

% Exercise 6
\printexo{6}{}{
A et B deux ensembles tels que :
~$
A = [-1, 1] \times [-1, 1] \quad \text{et} \quad B = \{( x, y) \in \mathbb{R}^2 ; x^2 + y^2 \leq 1 \}
$~
\begin{enumerate}
    \item Montrer que $B \subset A$
    \item Est-ce que $A = B$ ?
\end{enumerate}
}
% Exercise 7
\printexo{7}{}{
On pose :
~$
A = \left\{ x \in \mathbb{R} ~;~ \exists y \in \left[ \dfrac{1}{2}, \dfrac{3}{4} \right] ~;~ xy - x + 2y - 1 = 0 \right\}
$~\\
1.~ Vérifier que $A$ est non vide\\
2.~ Montrer que $A \subset [0; 2]$
}
\newpage
% Exercise 8
\printexo{8}{}{
On pose :
~$
F = \{( x; y) \in \mathbb{R}^2 / x + y = 0 \} \quad \text{et} \quad E = \{( x; y) \in \mathbb{R}^2 / x^2 - 2y^2 - xy = 0 \}
$~
\begin{enumerate}
    \item Montrer que : $E \neq \emptyset$ et $F \subset E$
    \item Déterminer un réel $y$ sachant que $(1; y) \in E$
    \item A-t-on $E = F$ ? Justifier
    \item Déterminer un ensemble $G$ vérifiant $E = F \cup G$
\end{enumerate}
}

% Exercise 9
\printexo{9}{}{
Soient E un ensemble : $A$ et B deux parties de E
\begin{enumerate}
    \item Montrer que : $A = (A \setminus B) \cup (A \cap B)$
    \item Montrer que : $A \cup B = A \Leftrightarrow (B \subset A)$
\end{enumerate}
}

% Exercise 10
\printexo{10}{}{
E un ensemble et A, B et C des parties de E
\begin{enumerate}
    \item Vérifier que : $\nn A \setminus \nn B = B \setminus A$
    \item Montrer que :
    ~$
    (A \cap \nn B) = A \cap \nn C \Rightarrow (A \cap B = A \cap C)
    $~
\end{enumerate}
}

% Exercise 11
\printexo{11}{}{
$A$, $B$ et $C$ trois parties d'un ensemble $E$, Montrer que :
\begin{enumerate}
    \item $(A \cup (A \cap B)) \cap B = A \cap B$
    \item $A \cap (\nn A \cup B) = A \cap B$
    \item $\left[(\nn{A \cap \nn B}) \cap (\nn{A \cap \nn C})\right] \cup A = E$
    \item $(A \Delta B) \cap A = A \setminus (A \cap B)$
\end{enumerate}
}

% Exercise 12
\printexo{12}{}{
Soient E, F, G et H des ensembles. Montrer que :
~$
(E \times F) \cap (G \times H) = (E \cap G) \times (F \cap H)
$~
}

\begin{center}
   \vskip 3pt \hrule height 3pt \vskip 5pt \RL{\arabicfont ﴿بِسْمِ ٱللَّهِ ٱلرَّحْمَـٰنِ ٱلرَّحِيمِ لَآ أُقْسِمُ بِيَوْمِ ٱلْقِيَـٰمَةِ (1)•  وَلَآ أُقْسِمُ بِٱلنَّفْسِ ٱللَّوَّامَةِ (2)•  أَيَحْسَبُ ٱلْإِنسَـٰنُ أَلَّن نَّجْمَعَ عِظَامَهُۥ (3)•  بَلَىٰ قَـٰدِرِينَ عَلَىٰٓ أَن نُّسَوِّىَ بَنَانَهُۥ (4)• ﴾ (القيامة الآيات 1-4) }
\end{center}


\end{document}