From 0c835cdb00d3bdeaa486099dbc3db19f3602ae1c Mon Sep 17 00:00:00 2001 From: jkarp Date: Thu, 25 Apr 2013 14:43:40 +0000 Subject: [PATCH] =?UTF-8?q?WS1213-MfI1.tex=20hinzugef=C3=BCgt...?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- mathefuerinf1/WS1213-MfI1.tex | 95 +++++++++++++++++++++++++++++++++++ 1 file changed, 95 insertions(+) create mode 100644 mathefuerinf1/WS1213-MfI1.tex diff --git a/mathefuerinf1/WS1213-MfI1.tex b/mathefuerinf1/WS1213-MfI1.tex new file mode 100644 index 0000000..ce65d7b --- /dev/null +++ b/mathefuerinf1/WS1213-MfI1.tex @@ -0,0 +1,95 @@ +\input{../settings/settings} +\usepackage{amsfonts} +\usepackage{fitch_selinger} +\begin{document} + +\klausur{Mathematik für Informatiker 1}{Prof. M. Mendler}{Wintersemester 12/13}{90}{Wörterbuch, Taschenrechner ohne vollständige alphanumerische Tastatur oder Grafikdisplay} +1a) Give an appropriate signature $\Sigma_{archive}$ = (S, OP, REL) for \textit{video archives} with the following properties:\\ +\begin{itemize} +\item You can add single BlueRay discs to an archive or remove them from an archive +\item You can also add single DVDs to an archive or remove them from an archive or replace them by BlueRay discs.\\ +\end{itemize} +Further, introduce +\begin{itemize} +\item a relation that indicates that an archive is \textit{emtpy} +\item a relation that indicates that there are only BlueRay discs in the archive (no DVDs)\\ +\end{itemize} +b) Using the signature $\Sigma_{archive}$ and a $\Sigma_{archive}$-sorted family of variables \textit{X}, formalize the following state ment in first-order predicate logic (FOL). A German translation is given in brackets:\\\\ +"Removing a DVD or a BlueRay disc from an empty archive results in an empty archive."\\\\ +(Das Entfernen einer DVD oder BlueRay aus einem leeren Archiv liefert ein leeres Archiv.)\\\\ +For this, define a $\Sigma_{archive}$-sorted family of variables \textit{x} first! You do not need to prove anything!\\\\ +2) Prove the following argument using the Fitch proof calculus. The Fitch rules are given in the Appendix (Anhang!)\\\\ +% fitch formula here +$ +\begin{nd} +\hypo{1}{\forall x (P(x) \to T(x, k))} +\hypo{2}{\forall y (Q(y) \to \neg T(y, k)} +\have[?]{3}{\forall z (P(z) \to \neg Q(z))} +\end{nd} +$\\\\\\ +3) Use mathematical induction to prove that for all natural numbers n $\in$ $\mathbb{N}$,\\\\ +\begin{center} +$\sum\limits_{i=0}^{n}i 2^i = (2^{n+1} (n-1))+2$\\[5mm] +\end{center} +\newpage +4) Consider the propositional signature $\Sigma$ = (S, OP, REL) with S = OP = $\emptyset$ and REL = \{ A, B, C : Prop\}\\ +Consider the formulas $\varphi_1$ and $\varphi_2$ in $Form_{\Sigma}(\emptyset)$:\\ + \begin{tabbing} + Links \= Mitte \= Rechts \kill +\>\>$\varphi_1$ = $_{df}$ $\neg$(A $\lor$ $\neg$ B) $\to$ (($\neg$A $\wedge$ B) $\wedge$ C)\\ +\>\>$\varphi_2$ = $_{df}$ A $\lor$ ($\neg$B $\lor$ C)\\ + \end{tabbing} +a) Using the laws of Boolean Algebra (see Appendix) shows that $\varphi_1$ = $\varphi_2$ holds.\\ +State for every transformation step the name of the rule which you apply.\\ +Hint: Note that $\phi$ $\to$ $\psi$ can be written as $\neg$ $\phi$ $\lor$ $\psi$ for all formulas $\phi$ and $\psi$!\\\\ +b) Show by structural truth table evaluation that $\varphi_1$ = $\varphi_2$ holds (you find the general truth tables in Appendix)\\\\ +5) Consider the computer system relations of Appendix I\\ +a) Construct a formal expression of relational algebra to answer the following questions:\\\\ +What are the product names (Prod$\_$Name) of the Tablet PCs (Model) with a hard drive size (HD$\_$size) of 1024 and with Doors operating system (OpSys)?\\ +b) Consider the following expression of relational algebra referring to the tables of Appendix I\\\\ +\begin{center} +$\pi$ Price ( $\sigma$Model = Notebook (System * Type * (HardDrive \textbackslash $\sigma_{HD\_Size}$ < 1000 (HardDrive))))\\ +\end{center} +Provide a table (with table header, i.e. attribute names) to the tables of Appendix I.\\ +Explain your workings for instanceby paraphrasing in english or german what this expression means. +\newpage +\textbf{Appendix I} +\begin{center} +Computer Systems\\ +\end{center} +System: +\begin{tabular}{c|l*{2}{|c}|l|r} +CID & Prod$\_$Name & TID & HDID & Op$\_$Sys & Price\\ +\hline +1 & Doll Super 15 & 1 & 1 & Doors & 329\\ +2 & Doll Super 17 & 2 & 1 & Doors & 399\\ +3 & Doll Super 18 & 2 & 2 & Lunix & 399\\ +4 & Doll Spec & 3 & 2 & Lunix & 359\\ +5 & Fishitsu Wonder & 2 & 3 & None & 420\\ +6 & Fishitsu Classic & 3 & 1 & None & 500\\ +7 & Media Monster & 4 & 4 & Doors & 6000\\ +8 & Devil's Eye & 1 & 1 & MockOS & 900\\ +9 & Sugar Play 3000 & 5 & 2 & Doors & 250\\ +\end{tabular} + \\\\\\\\ +Type: +\begin{tabular}{c|l|c|c} +TID & Model & Screen$\_$Size & Touch$\_$Screen\\ +\hline +1 & Notebook & 14 & No\\ +2 & Notebook & 17 & No\\ +3 & Tablet & 15 & Yes\\ +4 & Tablet & 22 & Yes\\ +5 & Desktop & 18 & No\\ +\end{tabular} + \\\\\\\\ +Hard Drive: +\begin{tabular}{c|r} +HDID & HD-Size\\ +\hline +1 & 320\\ +2 & 500\\ +3 & 1024\\ +4 & 2048\\ +\end{tabular} +\end{document} \ No newline at end of file