From: couchot Date: Mon, 4 Nov 2013 19:26:49 +0000 (+0100) Subject: controle ensembles X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/cours-maths-dis.git/commitdiff_plain/381385111635a5d15f8bd60d6666f6c0f6a61bcc controle ensembles --- 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 +\catcode `!\active +\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 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+\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 Écrire 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 Écrire 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 Écrire 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 Écrire 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