From 381385111635a5d15f8bd60d6666f6c0f6a61bcc Mon Sep 17 00:00:00 2001
From: couchot <jf.couchot@gmail.com>
Date: Mon, 4 Nov 2013 20:26:49 +0100
Subject: [PATCH] controle ensembles
---
partiels/131105S1/controle.aux | 13 +
partiels/131105S1/controle.log | 1128 ++++++++++++++++++++++++++++
partiels/131105S1/controle.pdf | Bin 0 -> 88472 bytes
partiels/131105S1/controle.tex | 135 ++++
partiels/131105S1/controle.tex.bak | 134 ++++
partiels/131105S1/controle.tex~ | 134 ++++
partiels/131105S1/controle.thm | 0
partiels/131105S1/symboles.sty | 1 +
8 files changed, 1545 insertions(+)
create mode 100644 partiels/131105S1/controle.aux
create mode 100644 partiels/131105S1/controle.log
create mode 100644 partiels/131105S1/controle.pdf
create mode 100644 partiels/131105S1/controle.tex
create mode 100644 partiels/131105S1/controle.tex.bak
create mode 100644 partiels/131105S1/controle.tex~
create mode 100644 partiels/131105S1/controle.thm
create mode 120000 partiels/131105S1/symboles.sty
diff --git a/partiels/131105S1/controle.aux b/partiels/131105S1/controle.aux
new file mode 100644
index 0000000..bdcfa3a
--- /dev/null
+++ b/partiels/131105S1/controle.aux
@@ -0,0 +1,13 @@
+\relax
+\catcode `:\active
+\catcode `;\active
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+\catcode `?\active
+\select@language{french}
+\@writefile{toc}{\select@language{french}}
+\@writefile{lof}{\select@language{french}}
+\@writefile{lot}{\select@language{french}}
+\@writefile{toc}{\contentsline {section}{\numberline {1}Cours (3pts) }{1}}
+\@writefile{toc}{\contentsline {section}{\numberline {2}(5 pts) Simplification de formules bool\IeC {\'e}ennes (d'apr\IeC {\`e}s sujet de BTS-IG-2008)}{1}}
+\@writefile{toc}{\contentsline {section}{\numberline {3}(6 pts) D\IeC {\'e}monstrations ensemblistes}{1}}
+\@writefile{toc}{\contentsline {section}{\numberline {4}(6 pts) Relation entre puissances d'\IeC {\'e}l\IeC {\'e}ments}{1}}
diff --git a/partiels/131105S1/controle.log b/partiels/131105S1/controle.log
new file mode 100644
index 0000000..8c27c7d
--- /dev/null
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diff --git a/partiels/131105S1/controle.tex b/partiels/131105S1/controle.tex
new file mode 100644
index 0000000..057bddb
--- /dev/null
+++ b/partiels/131105S1/controle.tex
@@ -0,0 +1,135 @@
+\documentclass[11pt,a4paper,french]{article}
+\usepackage[francais]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{a4}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{framed}
+\usepackage[amsmath,thmmarks,thref,framed]{ntheorem}
+\usepackage[dvips]{graphics}
+\usepackage{epsfig}
+\usepackage{calc}
+\usepackage{tabls}
+\usepackage{times}
+\usepackage{tabularx}
+\usepackage{textcomp}
+\usepackage{pst-all}
+\usepackage[a4paper]{geometry}
+
+\input{symboles.sty}
+\geometry{hmargin=1cm, tmargin=-0.5cm, bmargin=1.5cm}
+
+\title{
+DUT d'informatique. Contrôle de mathématiques discrètes.\\
+Semestre 1 (Novembre 2013). Durée 1h.}
+
+\date{}
+
+\begin{document}
+\maketitle
+
+\vspace{-3em}
+\noindent Seule une fiche manuscrite de format A5 est autorisée.
+Tous les exercices de ce sujet sont indépendants les uns des autres.
+Dans tout ce qui suit, on considère une algèbre de Boole
+munie des opérateurs classiques ``+'', ``.'',
+``$\overline{\begin{array}{l}~\end{array}}$'' et des variables booléennes
+$a$, $b$, $c$, $d$.
+
+
+\vspace{-1em}
+\section{Cours (3pts) }
+
+\begin{enumerate}
+\item Soit $A$ et $B$ deux ensembles d'un
+univers $\Omega$ tels que $A \subseteq B$. Définir $B\setminus A$ en compréhension.
+\item L'ensemble vide appartient-il à tout ensemble? Justifier.
+\end{enumerate}
+
+
+
+
+\vspace{-1em}
+\section{(5 pts) Simplification de formules booléennes (d'après sujet de BTS-IG-2008)}
+La société Jurabois exploite des forêts d'arbres constituées exclusivement de
+feuillus et de résineux. Elle désire simplifier le règlement que les salariés
+doivent appliquer pour la coupe du bois.
+Actuellement, le règlement dit qu'un arbre est à abattre dans les quatre cas
+suivants:
+\begin{itemize}
+\item si c'est un résineux au tronc droit mesurant plus de 20 m de hauteur;
+\item si c'est un feuillus de 50 ans ou plus;
+\item s'il a moins de 50 ans et mesure plus de 20 m de hauteur;
+\item s'il est tordu.
+\end{itemize}
+Pour un arbre quelconque, on définit les variables booléennes suivantes par:
+\begin{itemize}
+\item $a=1$ si l'arbre est un résineux;
+\item $b=1$ si l'arbre a moins de 50 ans;
+\item $c=1$ si l'arbre mesure plus de 20 m de hauteur;
+\item $d=1$ si l'arbre est tordu.
+\end{itemize}
+
+
+\begin{enumerate}
+\item Ãcrire sans justifier
+ la fonction booléenne $f(a,b,c,d)$ qui traduit le règlement
+ actuel d'abattage d'un arbre.
+
+\item Grâce à une bonne gestion des forêts que la société exploite,
+ il n'y a maintenant plus d'arbre tordu.
+\begin{enumerate}
+\item Montrer que le nouveau règlement d'abattage se traduit par la fonction
+$
+g(a,b,c) = ac + \overline{a}\overline{b} + bc
+$.
+\item A l'aide d'une table de Karnaugh, donner sans justifier
+ l'expression $G$ qui
+ est la forme la plus réduite de $g$.
+
+
+\item Ãcrire la nouvelle règle d'abattage d'un arbre sous la forme
+ la plus simple possible.
+\end{enumerate}
+\end{enumerate}
+
+
+
+\vspace{-1em}
+\section{(6 pts) Démonstrations ensemblistes}
+Soit $A$ et $B$ deux sous-ensembles de $\Omega$.
+\begin{enumerate}
+\item Soit $A_0=\{1,2,3\}$ et $B_0=\{3,4\}$.
+Définir en extension $A_0 \cup B_0$, $P(A_0)$, $P(B_0)$, $P(A_0\cup B_0)$ et
+$P(A_0) \cup P(B_0)$.
+\item A-t-on l'égalité $P(A \cup B) = P(A) \cup P(B)$? Justifier.
+\item Montrer que l'on a $P(A \cap B) = P(A) \cap P(B)$.
+\end{enumerate}
+
+
+
+\vspace{-1em}
+\section{(6 pts) Relation entre puissances d'éléments}
+
+
+Sur $\N^*$ on définit la relation
+$a \mathcal{R} b$ si $a^b \leq b ^a$.
+\begin{enumerate}
+\item A-t-on $2 \mathcal{R} 3$? $2 \mathcal{R} 7$? $2 \mathcal{R} 4$? $4 \mathcal{R} 2$? Justifier à chaque fois.
+
+\item La relation est-elle réflexive? Le démontrer.
+
+\item La fonction logarithme népérien $\ln: \R^{+*} \rightarrow \R$ étant
+ croissante, montrer que sur $\N^*$
+ $a \mathcal{R} b$ si $\dfrac{\ln(a)}{a} \leq \dfrac{\ln(b)}{b}$.
+\item La relation est-elle transitive? Pour justifier votre réponse, on
+ pourra utiliser la question précédente.
+\item La relation est-elle antisymétrique? Le justifier.
+\end{enumerate}
+
+
+
+
+
+\end{document}
diff --git a/partiels/131105S1/controle.tex.bak b/partiels/131105S1/controle.tex.bak
new file mode 100644
index 0000000..40a3f55
--- /dev/null
+++ b/partiels/131105S1/controle.tex.bak
@@ -0,0 +1,134 @@
+\documentclass[11pt,a4paper,french]{article}
+\usepackage[francais]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{a4}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{framed}
+\usepackage[amsmath,thmmarks,thref,framed]{ntheorem}
+\usepackage[dvips]{graphics}
+\usepackage{epsfig}
+\usepackage{calc}
+\usepackage{tabls}
+\usepackage{times}
+\usepackage{tabularx}
+\usepackage{textcomp}
+\usepackage{pst-all}
+\usepackage[a4paper]{geometry}
+
+\input{symboles.sty}
+\geometry{hmargin=1cm, tmargin=-0.5cm, bmargin=1.5cm}
+
+\title{
+DUT d'informatique. Contrôle de mathématiques discrètes.\\
+Semestre 1 (Novembre 2013). Durée 1h.}
+
+\date{}
+
+\begin{document}
+\maketitle
+
+\vspace{-3em}
+\noindent Seule une fiche manuscrite de format A5 est autorisée.
+Tous les exercices de ce sujet sont indépendants les uns des autres.
+Dans tout ce qui suit, on considère une algèbre de Boole
+munie des opérateurs classiques ``+'', ``.'',
+``$\overline{\begin{array}{l}~\end{array}}$'' et des variables booléennes
+$a$, $b$, $c$, $d$.
+
+
+\vspace{-1em}
+\section{Cours (3pts) }
+
+\begin{enumerate}
+\item Soit $A$ et $B$ deux ensembles d'un
+univers $\Omega$ tels que $A \subseteq B$. Définir $B\setminus A$ en compréhension.
+\item L'ensemble vide appartient-il à tout ensemble? Justifier.
+\end{enumerate}
+
+
+
+
+\vspace{-1em}
+\section{(6pts) Simplification de formules booléennes (d'après sujet de BTS-IG-2008)}
+La société Jurabois exploite des coupes d'arbres constitués exclusivement de
+feuillus et de résineux. Elle désire simplifier le réglement que les salariés
+doivent appliquerr pour la coupe du bois.
+Actuellement, le réglement dit qu'un arbre est à abattre dans les quatre cas
+suivants:
+\begin{itemize}
+\item si c'est un résineux au tronc droit mesurant plus de 20 m de hauteur;
+\item si c'est un feuillus de 50 ans ou plus;
+\item s'il a moins de 50 ans et mesure plus de 20 m de hauteur;
+\item s'il est tordu.
+\end{itemize}
+Pour un arbre quelconque, on définit les variables booléennes suivantes par:
+\begin{itemize}
+\item $a=1$ si l'arbre est un résineux;
+\item $b=1$ si l'arbre a moins de 50 ans;
+\item $c=1$ si l'arbre mesure plus de 20 m de hauteur;
+\item $d=1$ si l'arbre est tordu.
+\end{itemize}
+
+
+\begin{enumerate}
+\item Ecrire la fonction booléenne $f(a,b,c,d)$ qui traduit le réglement
+ actuel d'abattage d'un arbre.
+
+\item Grâce à une bonne gestion des forêts que la société exploite,
+ il n'y a maintenant plus d'arbre tordu.
+\begin{enumerate}
+\item Montrer que le nouveau réglement d'abattage se traduit par la fonction
+$
+g(a,b,c) = ac + \overline{a}\overline{b} + bc
+$.
+\item A l'aide d'une table de Karnaugh donner l'expression $G$ qui
+ serait la forme la plus réduite de $g$.
+
+\item Justifier algébriquement que l'on a l'égalité $G = g$.
+
+\item Ecrire la nouvelle règle d'abattage d'un arbre sous la forme
+ la plus simple possible.
+\end{enumerate}
+\end{enumerate}
+
+
+
+\vspace{-1em}
+\section{(5pts) Démonstrations ensemblistes}
+Soit $A$ et $B$ deux sous-ensembles de $\Omega$.
+\begin{enumerate}
+\item Soit $A_0=\{1,2,3\}$ et $B_0=\{3,4\}$.
+Définir $A_0 \cup B_0$, $P(A_0)$, $P(B_0)$, $P(A_0\cup B_0)$ et
+$P(A_0) \cup P(B_0)$.
+\item A-t-on l'égalité $P(A \cup B) = P(A) \cup P(B)$? Justifier.
+\item Montrer que l'on a $P(A \cap B) = P(A) \cap P(B)$.
+\end{enumerate}
+
+
+
+\vspace{-1em}
+\section{(6 pts) Relation entre puissances d'éléments}
+
+
+Sur $\N^*$ on définit la relation
+$a \mathcal{R} b$ si $a^b \leq b ^a$.
+\begin{enumerate}
+\item A-t-on $2 \mathcal{R} 3$? $2 \mathcal{R} 7$? $2 \mathcal{R} 4$? $3 \mathcal{R} 3$? $4 \mathcal{R} 2$? Justifier à chaque-fois.
+
+\item La relation est-elle réflexive? Le démontrer.
+
+\item La fonction logarithme népérien $\ln: \R^{+*} \rightarrow \R$ étant
+ croissante, montrer que sur $\N^*$
+ $a \mathcal{R} b$ si $\dfrac{\ln(a)}{a} \leq \dfrac{\ln(b)}{b}$.
+\item La relation est-elle transitive? Pour justifier votre réponse, on
+ pourra utiliser la question précédente.
+\item La relation est-elle antisymétrique? Le justifier.
+\end{enumerate}
+
+
+
+
+
+\end{document}
diff --git a/partiels/131105S1/controle.tex~ b/partiels/131105S1/controle.tex~
new file mode 100644
index 0000000..efd326f
--- /dev/null
+++ b/partiels/131105S1/controle.tex~
@@ -0,0 +1,134 @@
+\documentclass[11pt,a4paper,french]{article}
+\usepackage[francais]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{a4}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{framed}
+\usepackage[amsmath,thmmarks,thref,framed]{ntheorem}
+\usepackage[dvips]{graphics}
+\usepackage{epsfig}
+\usepackage{calc}
+\usepackage{tabls}
+\usepackage{times}
+\usepackage{tabularx}
+\usepackage{textcomp}
+\usepackage{pst-all}
+\usepackage[a4paper]{geometry}
+
+\input{symboles.sty}
+\geometry{hmargin=1cm, tmargin=-0.5cm, bmargin=1.5cm}
+
+\title{
+DUT d'informatique. Contrôle de mathématiques discrètes.\\
+Semestre 1 (Novembre 2013). Durée 1h.}
+
+\date{}
+
+\begin{document}
+\maketitle
+
+\vspace{-3em}
+\noindent Seule une fiche manuscrite de format A5 est autorisée.
+Tous les exercices de ce sujet sont indépendants les uns des autres.
+Dans tout ce qui suit, on considère une algèbre de Boole
+munie des opérateurs classiques ``+'', ``.'',
+``$\overline{\begin{array}{l}~\end{array}}$'' et des variables booléennes
+$a$, $b$, $c$, $d$.
+
+
+\vspace{-1em}
+\section{Cours (3pts) }
+
+\begin{enumerate}
+\item Soit $A$ et $B$ deux ensembles d'un
+univers $\Omega$ tels que $A \subseteq B$. Définir $B\setminus A$ en compréhension.
+\item L'ensemble vide appartient-il à tout ensemble? Justifier.
+\end{enumerate}
+
+
+
+
+\vspace{-1em}
+\section{(6pts) Simplification de formules booléennes (d'après sujet de BTS-IG-2008)}
+La société Jurabois exploite des coupes d'arbres constitués exclusivement de
+feuillus et de résineux. Elle désire simplifier le règlement que les salariés
+doivent appliquer pour la coupe du bois.
+Actuellement, le règlement dit qu'un arbre est à abattre dans les quatre cas
+suivants:
+\begin{itemize}
+\item si c'est un résineux au tronc droit mesurant plus de 20 m de hauteur;
+\item si c'est un feuillus de 50 ans ou plus;
+\item s'il a moins de 50 ans et mesure plus de 20 m de hauteur;
+\item s'il est tordu.
+\end{itemize}
+Pour un arbre quelconque, on définit les variables booléennes suivantes par:
+\begin{itemize}
+\item $a=1$ si l'arbre est un résineux;
+\item $b=1$ si l'arbre a moins de 50 ans;
+\item $c=1$ si l'arbre mesure plus de 20 m de hauteur;
+\item $d=1$ si l'arbre est tordu.
+\end{itemize}
+
+
+\begin{enumerate}
+\item Ãcrire la fonction booléenne $f(a,b,c,d)$ qui traduit le règlement
+ actuel d'abattage d'un arbre.
+
+\item Grâce à une bonne gestion des forêts que la société exploite,
+ il n'y a maintenant plus d'arbre tordu.
+\begin{enumerate}
+\item Montrer que le nouveau règlement d'abattage se traduit par la fonction
+$
+g(a,b,c) = ac + \overline{a}\overline{b} + bc
+$.
+\item A l'aide d'une table de Karnaugh donner l'expression $G$ qui
+ serait la forme la plus réduite de $g$.
+
+\item Justifier algébriquement que l'on a l'égalité $G = g$.
+
+\item Ãcrire la nouvelle règle d'abattage d'un arbre sous la forme
+ la plus simple possible.
+\end{enumerate}
+\end{enumerate}
+
+
+
+\vspace{-1em}
+\section{(5pts) Démonstrations ensemblistes}
+Soit $A$ et $B$ deux sous-ensembles de $\Omega$.
+\begin{enumerate}
+\item Soit $A_0=\{1,2,3\}$ et $B_0=\{3,4\}$.
+Définir $A_0 \cup B_0$, $P(A_0)$, $P(B_0)$, $P(A_0\cup B_0)$ et
+$P(A_0) \cup P(B_0)$.
+\item A-t-on l'égalité $P(A \cup B) = P(A) \cup P(B)$? Justifier.
+\item Montrer que l'on a $P(A \cap B) = P(A) \cap P(B)$.
+\end{enumerate}
+
+
+
+\vspace{-1em}
+\section{(6 pts) Relation entre puissances d'éléments}
+
+
+Sur $\N^*$ on définit la relation
+$a \mathcal{R} b$ si $a^b \leq b ^a$.
+\begin{enumerate}
+\item A-t-on $2 \mathcal{R} 3$? $2 \mathcal{R} 7$? $2 \mathcal{R} 4$? $3 \mathcal{R} 3$? $4 \mathcal{R} 2$? Justifier à chaque fois.
+
+\item La relation est-elle réflexive? Le démontrer.
+
+\item La fonction logarithme népérien $\ln: \R^{+*} \rightarrow \R$ étant
+ croissante, montrer que sur $\N^*$
+ $a \mathcal{R} b$ si $\dfrac{\ln(a)}{a} \leq \dfrac{\ln(b)}{b}$.
+\item La relation est-elle transitive? Pour justifier votre réponse, on
+ pourra utiliser la question précédente.
+\item La relation est-elle antisymétrique? Le justifier.
+\end{enumerate}
+
+
+
+
+
+\end{document}
diff --git a/partiels/131105S1/controle.thm b/partiels/131105S1/controle.thm
new file mode 100644
index 0000000..e69de29
diff --git a/partiels/131105S1/symboles.sty b/partiels/131105S1/symboles.sty
new file mode 120000
index 0000000..a4e1afa
--- /dev/null
+++ b/partiels/131105S1/symboles.sty
@@ -0,0 +1 @@
+/home/couchot/enseignement/stats/partiels/111103/symboles.sty
\ No newline at end of file
--
2.39.5