From: couchot Date: Mon, 4 Nov 2013 19:29:32 +0000 (+0100) Subject: ajout en vrac de tous les partiels X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/cours-maths-dis.git/commitdiff_plain/333558f52e52a5792ac99d50e89d3a9d837b9d67 ajout en vrac de tous les partiels --- diff --git a/partiels/130311S2/main.aux b/partiels/130311S2/main.aux new file mode 100644 index 0000000..d30543e --- /dev/null +++ b/partiels/130311S2/main.aux @@ -0,0 +1,11 @@ +\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}} +\newlabel{item1}{{1}{2}}[None] +\newlabel{item2}{{2}{2}}[None] diff --git a/partiels/130311S2/main.log b/partiels/130311S2/main.log new file mode 100644 index 0000000..6fdc401 --- /dev/null +++ b/partiels/130311S2/main.log @@ -0,0 +1,1173 @@ +This is pdfTeX, Version 3.1415926-2.4-1.40.13 (TeX Live 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mode 100644 index 0000000..159b58b Binary files /dev/null and b/partiels/131003S1/controle.pdf differ diff --git a/partiels/131003S1/controle.tex b/partiels/131003S1/controle.tex new file mode 100644 index 0000000..0435c0c --- /dev/null +++ b/partiels/131003S1/controle.tex @@ -0,0 +1,101 @@ +\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, vmargin=1.5cm} + +\title{ +DUT d'informatique. Contrôle de mathématiques discrètes.\\ +Semestre 1 (Octobre 2013). Durée 1h.} + +\date{} + +\begin{document} +\maketitle + +\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 +$a$, $b$, $c$, $d$. + + +\section{Cours} + +\begin{enumerate} +\item La première règle de suppression de la redondance est: ``dans une somme booléenne, tout terme absorbe ses multiples''. Démontrer cette règle (sans se servir des règles de suppression de redondance naturellement). +\item Donner la définition de \og la forme cannonique disjonctive d'une expression\fg{}. +\item Démontrer qu'il y a 1024 mintermes à 10 variables. +Si vous utilisez un théorème pour faire la preuve, il faudra le prouver. +\item Que dire des colonnes adjacentes d'une table de Karnaugh? Et les lignes ? + +\end{enumerate} + + + + +\section{Forme canonique disjonctive} +Donner la forme canonique disjonctive de l'expression suivante en détaillant les calculs: +$$ +(a + b + \overline{c}).\overline{(\overline{a} + bc)}. +$$ + +\section{Table de Karnaugh} +On considère les deux expressions booléennes $E_1$ et $E_2$ suivantes: + +$$ +\begin{array}{rcl} +E_1 &= &\overline{a}.\overline{b} + a.c+ a.\overline{b}.\overline{c},\\ +E_2 &=& a.\overline{b} + a.c.d + a.b.c +\overline{a}.\overline{b}.\overline{c}. +\end{array} +$$ + +\begin{enumerate} +\item A l'aide d'une table de Karnaugh donner l'expression $K_1$ qui + serait la forme la plus réduite de $E_1$. + +\item A l'aide d'une autre table de Karnaugh donner l'expression + $K_2$ qui serait la forme la plus réduite de $E_2$. + +\item Justifier algébriquement que l'on a l'égalité $E_1 = K_1$. + +\item Justifier algébriquement que l'on a l'égalité $E_2 = K_2$. +\end{enumerate} + +\section{Opérateur de Peirce} +On considère l'opérateur binaire de Peirce +défini pour toute paire de variables booléennes $(a,b)$ par +$$ +a \downarrow b = \overline{a}.\overline{b} +$$. + +\begin{enumerate} +\item Que valent +$a \downarrow 0$ puis $0 \downarrow 0$ et enfin $1 \downarrow 1$? +\item Que valent $(a \downarrow 0) \downarrow (b \downarrow 0)$ et + $(a \downarrow b ) \downarrow 0$ ? +\item Réécrire l'expression $\overline{a}.(b+c)$ sans utiliser +les opérateurs ``+'', ``.'' et +``$\overline{\begin{array}{l}~\end{array}}$'' mais en +n'utilisant que l'opérateur de Peirce et des parenthèses. +\item L'opérateur de Peirce est-il associatif? Le prouver. +\end{enumerate} + +\end{document} diff --git a/partiels/131003S1/controle.tex.bak b/partiels/131003S1/controle.tex.bak new file mode 100644 index 0000000..97e056d --- /dev/null +++ b/partiels/131003S1/controle.tex.bak @@ -0,0 +1,102 @@ +\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, vmargin=1.5cm} + +\title{ +DUT d'informatique. Contrôle de mathématiques discrètes.\\ +Semestre 1 (Octobre 2013). Durée 1h\\ +} + +\date{} + +\begin{document} +\maketitle + +\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 +$a$, $b$, $c$, $d$\ldots + + +\section{Cours} + +\begin{enumerate} +\item La première règle de suppression de la redondance est: ``Dans une somme booléenne, tout terme absorbe ses multiples: $a + a.b = a$.''. Démontrer cette règles (sans se servir des règles de suppression de redondance naturellement). +\item Donner la définition de \og la forme cannonique disjonctive d'une expression\fg{}. +\item Démontrer qu'il y a 1024 mintermes à 10 variables. +Si vous utilisez un théorème pour faire la preuve, il faudra le prouver. +\item Que dire des colonnes adjacentes d'une table de Karnaugh? Et les lignes ? + +\end{enumerate} + + + + +\section{Forme canonique disjonctive} +Donner la forme canonique disjonctive de l'expression suivante en détaillant les calculs: +$$ +(a + b + \overline{c}).\overline{(\overline{a} + bc)}. +$$ + +\section{Table de Karnaugh} +On considère les deux expressions booléennes $E_1$ et $E_2$ suivantes: + +$$ +\begin{array}{rcl} +E_1 &= &\overline{a}.\overline{b} + a.c+ a.\overline{b}.\overline{c},\\ +E_2 &=& a.\overline{b} + a.c.d + a.b.c +\overline{a}.\overline{b}.\overline{c}. +\end{array} +$$ + +\begin{enumerate} +\item A l'aide d'une table de Karnaugh donner l'expression $K_1$ qui + serait la forme la plus réduite de $E_1$. + +\item A l'aide d'une autre table de Karnaugh donner l'expression + $K_2$ qui serait la forme la plus réduite de $E_2$. + +\item Justifier algébriquement que l'on a l'égalité $E_1 = K_1$. + +\item Justifier algébriquement que l'on a l'égalité $E_2 = K_2$. +\end{enumerate} + +\section{Opérateur de Peirce} +On condidère l'opérateur binaire de Peirce +défini pour toute paire de variables booléenne $(a,b)$ par +$$ +a \downarrow b = \overline{a}.\overline{b} +$$. + +\begin{enumerate} +\item Que valent +$a \downarrow 0$ puis $0 \downarrow 0$ et enfin $1 \downarrow 1$? +\item Que valent $(a \downarrow 0) \downarrow (b \downarrow 0)$ et + $(a \downarrow b ) \downarrow 0$ ? +\item Réécrire l'expression $\overline{a}.(b+c)$ sans utiliser +les opérateurs ``+'', ``.'' et +``$\overline{\begin{array}{l}~\end{array}}$'' mais en +n'utilisant que l'opérateur de Peirce et des parenthèses. +\item L'opéreteur de Peirce est-il associatif? Le prouver. +\end{enumerate} + +\end{document} diff --git a/partiels/131003S1/controle.tex~ b/partiels/131003S1/controle.tex~ new file mode 100644 index 0000000..7c54d59 --- /dev/null +++ b/partiels/131003S1/controle.tex~ @@ -0,0 +1,102 @@ +\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, vmargin=1.5cm} + +\title{ +DUT d'informatique. Contrôle de mathématiques discrètes.\\ +Semestre 1 (Octobre 2013). Durée 1h\\ +} + +\date{} + +\begin{document} +\maketitle + +\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 +$a$, $b$, $c$, $d$\ldots + + +\section{Cours} + +\begin{enumerate} +\item La première règle de suppression de la redondance est: ``Dans une somme booléenne, tout terme absorbe ses multiples: $a + a.b = a$.''. Démontrer cette règles (sans se servir des règles de suppression de redondance naturellement). +\item Donner la définition de \og la forme cannonique disjonctive d'une expression\fg{}. +\item Démontrer qu'il y a 1024 mintermes à 10 variables. +Si vous utilisez un théorème pour faire la preuve, il faudra le prouver. +\item Que dire des colonnes adjacentes d'une table de Karnaugh? Et les lignes ? + +\end{enumerate} + + + + +\section{Forme canonique disjonctive} +Donner la forme canonique disjonctive de l'expression suivante en détaillant les calculs: +$$ +(a + b + \overline{c}).\overline{(\overline{a} + bc)}. +$$ + +\section{Table de Karnaugh} +On considère les deux expressions booléennes $E_1$ et $E_2$ suivantes: + +$$ +\begin{array}{rcl} +E_1 &= &\overline{a}.\overline{b} + a.c+ a.\overline{b}.\overline{c},\\ +E_2 &=& a.\overline{b} + a.c.d + a.b.c +\overline{a}.\overline{b}.\overline{c}. +\end{array} +$$ + +\begin{enumerate} +\item A l'aide d'une table de Karnaugh donner l'expression $K_1$ qui + serait la forme la plus réduite de $E_1$. + +\item A l'aide d'une autre table de Karnaugh donner l'expression + $K_2$ qui serait la forme la plus réduite de $E_2$. + +\item Justifier algébriquement que l'on a l'égalité $E_1 = K_1$. + +\item Justifier algébriquement que l'on a l'égalité $E_2 = K_2$. +\end{enumerate} + +\section{Opérateur de Peirce} +On considère l'opérateur binaire de Peirce +défini pour toute paire de variables booléenne $(a,b)$ par +$$ +a \downarrow b = \overline{a}.\overline{b} +$$. + +\begin{enumerate} +\item Que valent +$a \downarrow 0$ puis $0 \downarrow 0$ et enfin $1 \downarrow 1$? +\item Que valent $(a \downarrow 0) \downarrow (b \downarrow 0)$ et + $(a \downarrow b ) \downarrow 0$ ? +\item Réécrire l'expression $\overline{a}.(b+c)$ sans utiliser +les opérateurs ``+'', ``.'' et +``$\overline{\begin{array}{l}~\end{array}}$'' mais en +n'utilisant que l'opérateur de Peirce et des parenthèses. +\item L'opérateur de Peirce est-il associatif? Le prouver. +\end{enumerate} + +\end{document} diff --git a/partiels/131003S1/controle.thm b/partiels/131003S1/controle.thm new file mode 100644 index 0000000..e69de29 diff --git a/partiels/131003S1/symboles.sty b/partiels/131003S1/symboles.sty new file mode 120000 index 0000000..a4e1afa --- /dev/null +++ b/partiels/131003S1/symboles.sty @@ -0,0 +1 @@ +/home/couchot/enseignement/stats/partiels/111103/symboles.sty \ No newline at end of file