82 lines
2.6 KiB
TeX
82 lines
2.6 KiB
TeX
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\documentclass[12pt,a4paper,german]{article}
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\usepackage{url}
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%\usepackage{graphics}
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\usepackage{times}
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\usepackage[T1]{fontenc}
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\usepackage{ngerman}
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\usepackage{float}
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\usepackage{diagbox}
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\usepackage[utf8]{inputenc}
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\usepackage{geometry}
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\usepackage{csquotes}
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\usepackage{graphicx}
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\usepackage{epsfig}
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\usepackage{paralist}
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\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
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%%%%%%%%%% Fill out the the definitions %%%%%%%%%
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\def \name {Valentin Brandl} %
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\def \matrikel {108018274494} %
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\def \pname {Marvin Herrmann} %
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\def \pmatrikel {108018265436} %
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\def \gruppe {Mi 10-12 (Andre)}
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\def \uebung {1} %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DO NOT MODIFY THIS HEADER
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\newcommand{\hwsol}{
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\vspace*{-2cm}
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\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
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\noindent \pmatrikel \quad \pname \\
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\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
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}
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\begin{document}
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%Import header
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\hwsol
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\section*{Aufgabe 1.3}
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\begin{enumerate}[(1)]
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\item
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$2^n$ beschreibt die Potenzmenge, also die Anzahl aller m\"oglichen Teilmengen einer Menge mit $n$ Elementen.
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$\binom{n}{k}$ beschreibt die Anzahl der $k$-elementigen Teilmengen einer $n$-elementigen Menge.
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$\sum\limits_{k=0}^{n} \binom{n}{k}$ beschreibt also alle m\"oglichen Teilmengen einer $n$-elementigen Menge.
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$\Rightarrow \sum\limits_{k=0}^n \binom{n}{k} = 2^n$
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\item
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$A_n$: Menge aller durch $n$ teilbaren Zahlen $\leq 100$
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$M = \{1,\ldots,100\}, |M| = 100$
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\begin{eqnarray*}
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|A_2 \cup A_3 \cup A_5 \cup A_7 | &=& |A_2| + |A_3| + |A_5| + |A_7| \\
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&& - |A_2 \cap A_3| - |A_2 \cap A_5| - |A_2 \cap A_7| - |A_3 \cap A_5| -
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|A_3 \cap A_7| - |A_5 \cap A_7| \\
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&&+ |A_2 \cap A_3 \cap A_5| + |A_2 \cap A_3 \cap A_7|
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+ |A_2 \cap A_5 \cap A_7| + |A_3 \cap A_5 \cap A_7| \\
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&&- |A_2 \cap A_3 \cap A_5 \cap A_7| \\
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&=& 50 + 33 +20 +14 -16 -10 -7 -6 -4 -2 + 3 + 2 + 1 + 1 - 1 \\
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&=& 78
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\end{eqnarray*}
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78 Zahlen, die durch 2, 3, 5 oder 7 teilbar sind, dabei sind 2, 3, 5 und 7 mitgez\"ahlt $\Rightarrow 78 - 4 =
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74$ nicht-Primzahlen $\leq 100$.
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\begin{eqnarray*}
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\Rightarrow \overline{|A_2 \cup A_3 \cup A_5 \cup A_7|} &=& |\bar{A_2} \cap \bar{A_3} \cap \bar{A_5} \cap
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\bar{A_7}| \\
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&=& |M \setminus (A_2 \cup A_3 \cup A_5 \cup A_7)| \\
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&=& |M| - |A_2 \cup A_3 \cup A_5 \cup A_7| \\
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&=& 100 - 77 \\
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&=& 26
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\end{eqnarray*}
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Da Primzahlen $\geq 2$ sind und bis jetzt die 1 mitgez\"ahlt wurde: $\Rightarrow 26 - 1 = 25$.
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\end{enumerate}
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\end{document}
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