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