Add solution for dima u04
This commit is contained in:
Valentin Brandl
parent
b59a93480b
commit
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70
school/di-ma/uebung/04/04_1.tex
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70
school/di-ma/uebung/04/04_1.tex
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@ -0,0 +1,70 @@
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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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\usepackage{tikz}
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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 {2 (Mi 10-12 Andre)}
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\def \qname {Pascal Brackmann}
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\def \qmatrikel {108017113834} %
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\def \uebung {4} %
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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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\noindent \qmatrikel \quad \qname \\
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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 4.1}
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\begin{enumerate}[i)]
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\item Je zwei Koten in $G$ sind durch genau einen Pfad miteinander verbunden.
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$\Rightarrow G$ ist zusammenhängend $\Rightarrow G$ ist ein Baum (Lemma b) aus der Vorlesung)
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$\Rightarrow$ i) $\equiv$ iii)
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\item $G$ ist zusammenhängend und es gilt $|V| = |E| + 1$
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\item $G$ besitzt keinen einfachen Kreis und es gilt $|V| = |E| + 1$
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$|V| = |E| + 1 \Rightarrow |V| - |E| = 1 \Rightarrow G$ hat eine Zusammenhangskomponente $\Rightarrow G$ ist
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zusammenhängend.
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Kreisfrei und zusammenhängend $\Rightarrow G$ ist ein Baum
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$\Rightarrow$ ii) $\equiv$ iii)
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\end{enumerate}
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Da i) $\equiv$ iii) und ii) $\equiv$ iii) $\Rightarrow$ i) $\equiv$ ii) $\equiv$ iii)
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\end{document}
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129
school/di-ma/uebung/04/04_2.tex
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129
school/di-ma/uebung/04/04_2.tex
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@ -0,0 +1,129 @@
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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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\usepackage{tikz}
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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 {2 (Mi 10-12 Andre)}
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\def \qname {Pascal Brackmann}
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\def \qmatrikel {108017113834} %
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\def \uebung {4} %
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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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\noindent \qmatrikel \quad \qname \\
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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 4.2}
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\begin{enumerate}[1.]
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\item Graph $G=(V,E)$ mit\\
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Knotenmenge: $V=\{a,b,c,d,e,f\}$\\
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Kantenmenge: $E=
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\{
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\{a,b\},
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\{a,c\},
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\{a,d\},
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\{a,f\},
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\{b,f\},
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\{b,e\},
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\{c,d\},
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\{d,e\}
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\}
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$
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\\
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Adjazenz-Matrix:\\
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\begin{tabular}{c|ccccccc}
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& a & b & c & d & e & f \\
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\hline
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a & 0 & 1 & 1 & 1 & 0 & 1 \\
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b & 1 & 0 & 0 & 0 & 1 & 1 \\
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c & 1 & 0 & 0 & 1 & 0 & 0 \\
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d & 1 & 0 & 1 & 0 & 1 & 0 \\
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e & 0 & 1 & 0 & 1 & 0 & 0 \\
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f & 1 & 1 & 0 & 0 & 0 & 0 \\
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\end{tabular}
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\\
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\\
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Inzidenz-Matrix:\\
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\begin{tabular}{c|ccccccccc}
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a & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
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b & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\
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c & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\
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d & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\
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e & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\
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f & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\
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\end{tabular}
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\item \begin{tikzpicture}
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\def \n {5}
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\def \radius {3cm}
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\def \margin {8} % margin in angles, depends on the radius
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\node[draw, circle] (a) at (3,2) {a};
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\node[draw, circle] (b) at (1,1) {b};
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\node[draw, circle] (c) at (2,1) {c};
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\node[draw, circle] (d) at (3,1) {d};
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\node[draw, circle] (f) at (4,1) {f};
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\node[draw, circle] (e) at (0,0) {e};
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\draw (a)--(b);
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\draw (a)--(c);
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\draw (a)--(d);
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\draw (a)--(f);
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\draw (b)--(e);
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\end{tikzpicture}
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\\
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$G \setminus \{\{b,f\},\{c,d\},\{d,e\}\}$
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\begin{tikzpicture}
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\def \n {5}
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\def \radius {3cm}
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\def \margin {8} % margin in angles, depends on the radius
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\node[draw, circle] (b) at (2,2) {b};
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\node[draw, circle] (a) at (1,1) {a};
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\node[draw, circle] (f) at (2,1) {f};
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\node[draw, circle] (e) at (3,1) {e};
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\node[draw, circle] (c) at (0,0) {c};
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\node[draw, circle] (d) at (2,0) {d};
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\draw (b)--(a);
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\draw (b)--(f);
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\draw (b)--(e);
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\draw (a)--(c);
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\draw (a)--(d);
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\end{tikzpicture}
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\\
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$G \setminus \{\{a,f\},\{c,d\},\{d,e\}\}$
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\end{enumerate}
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\end{document}
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118
school/di-ma/uebung/04/04_3.tex
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118
school/di-ma/uebung/04/04_3.tex
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@ -0,0 +1,118 @@
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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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\usepackage{tikz}
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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 {2 (Mi 10-12 Andre)}
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\def \qname {Pascal Brackmann}
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\def \qmatrikel {108017113834} %
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\def \uebung {4} %
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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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\noindent \qmatrikel \quad \qname \\
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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 4.3}
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\begin{enumerate}[1.]
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\item Graph $G=(V,E)$ mit\\
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Knotenmenge: $V=\{1,2,3,4,5,6\}$\\
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Kantenmenge: $E=
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\{
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(1,2),
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(1,3),
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(2,3),
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(3,4),
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(4,5),
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(5,6),
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(6,4)
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\}$
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\\
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Adjazenz-Matrix:\\
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\begin{tabular}{c|cccccc}
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& 1 & 2 & 3 & 4 & 5 & 6 \\
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\hline
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1 & 0 & 1 & 1 & 0 & 0& 0 \\
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2 & 0 & 0 & 1 & 0 & 0& 0 \\
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3 & 0 & 0 & 0 & 1 & 0& 0 \\
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4 & 0 & 0 & 0 & 0 & 1& 0 \\
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5 & 0 & 0 & 0 & 0 & 0& 1 \\
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6 & 0 & 0 & 0 & 1 & 0& 0 \\
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\end{tabular}
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\\
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Inzidenz-Matrix:\\
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\begin{tabular}{c|ccccccc}
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1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
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2 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
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3 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
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4 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\
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5 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
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6 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
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\end{tabular}
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\item \begin{tikzpicture}
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\def \n {5}
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\def \radius {3cm}
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\def \margin {8} % margin in angles, depends on the radius
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\node[draw, circle] (1) at (0,1) {1};
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\node[draw, circle] (4) at (2,2) {4};
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\node[draw, circle] (6) at (2,0) {6};
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\node[draw, circle] (3) at (1,0) {3};
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\draw [->]
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(1) edge (3)
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(3) edge (4)
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(6) edge (4);
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\end{tikzpicture}
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\\
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$V'=\{1,3,4,6\}$
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\item \begin{tikzpicture}
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\def \n {5}
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\def \radius {3cm}
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\def \margin {8} % margin in angles, depends on the radius
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\node[draw, circle] (1) at (0,0) {1};
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\node[draw, circle] (3) at (1,0) {3};
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\node[draw, circle] (4) at (2,0) {4};
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\node[draw, circle] (5) at (3,0) {5};
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\draw [->]
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(1) edge (3)
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(3) edge (4)
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(4) edge (5);
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\end{tikzpicture}
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\\
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$V'=\{1,3,4,5\}$
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\end{enumerate}
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\end{document}
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77
school/di-ma/uebung/04/04_4.tex
Normal file
77
school/di-ma/uebung/04/04_4.tex
Normal file
@ -0,0 +1,77 @@
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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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\usepackage{tikz}
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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 {2 (Mi 10-12 Andre)}
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\def \qname {Pascal Brackmann}
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\def \qmatrikel {108017113834} %
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\def \uebung {4} %
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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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\noindent \qmatrikel \quad \qname \\
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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 4.4}
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Zu zeigen: Wenn ein Baum genau $k \geq 1$ Knoten vom Grad 4 enthält (außer Blätter), dann besitzt der Baum mindestens
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$2 \cdot k + 2$ Blätter.\\
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\\
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IA: Ein Baum mit nur einem Knoten von Grad 4 muss 4 Blätter haben
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\begin{tikzpicture}
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\node[draw,circle] (1) at (2,1) {1};
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\node[draw,circle] (2) at (0,0) {2};
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\node[draw,circle] (3) at (1,0) {3};
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\node[draw,circle] (4) at (3,0) {4};
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\node[draw,circle] (5) at (4,0) {5};
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\draw (1)--(2);
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\draw (1)--(3);
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\draw (1)--(4);
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\draw (1)--(5);
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\end{tikzpicture}
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\\
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$A(1) = 2 \cdot 1 + 2 = 4$\\
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\\
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\textbf{IV: $A(k) = 2 \cdot k +2$}\\
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\\
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IS: $k \rightarrow k+1$\\
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Um einen Knoten mit Grad 4 hinzuzufügen, kann man nun eines der Blätter nehmen und drei Blätter anhängen. Man bekommt
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also 3 Blätter hinzu, verliert aber auch eines, da dieses zum neuen Knoten wird.\\
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\\
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$A(k+1) = A(k) + 3 - 1 \overset{(IV)}{=} (2 \cdot k+2)+3-1$
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$= 2 \cdot (k+1)+2$\\
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q.e.d
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\end{document}
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246
school/di-ma/uebung/04/pascal_04.tex
Normal file
246
school/di-ma/uebung/04/pascal_04.tex
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@ -0,0 +1,246 @@
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\documentclass[10pt,a4paper]{article}
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\usepackage[utf8x]{inputenc}
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\usepackage{ucs}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{amssymb}
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\usepackage{graphicx}
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\usepackage{tikz}
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\title{Blatt 04}
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\begin{document}
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\maketitle
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\newpage
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\section{Aufgabe 1}
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TODO
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\section{Aufgabe 2}
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\begin{tikzpicture}
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\def \n {5}
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\def \radius {3cm}
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\def \margin {8} % margin in angles, depends on the radius
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\node[draw, circle] (a) at (0,1) {a};
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\node[draw, circle] (b) at (1,2) {b};
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\node[draw, circle] (c) at (2,2) {c};
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\node[draw, circle] (d) at (3,1) {d};
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\node[draw, circle] (e) at (2,0) {e};
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\node[draw, circle] (f) at (1,0) {f};
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\draw (a)--(b);
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\draw (a)--(c);
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\draw (a)--(d);
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\draw (a)--(f);
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\draw (b)--(f);
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\draw (b)--(e);
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\draw (c)--(d);
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\draw (d)--(e);
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\end{tikzpicture}
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\\
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\subsection{1.)}
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Graph $G=(V,E)$ mit\\
|
||||
Knotenmenge: $V=\{a,b,c,d,e,f\}$\\
|
||||
Kantenmenge: $E=
|
||||
\{
|
||||
\{a,b\},
|
||||
\{a,c\},
|
||||
\{a,d\},
|
||||
\{a,f\},
|
||||
\{b,f\},
|
||||
\{b,e\},
|
||||
\{c,d\},
|
||||
\{d,e\}
|
||||
\}
|
||||
$
|
||||
\\
|
||||
Adjazens-Matrix:\\
|
||||
\begin{tabular}{c|ccccccc}
|
||||
& a & b & c & d & e & f \\
|
||||
\hline
|
||||
a & 0 & 1 & 1 & 1 & 0 & 1 \\
|
||||
b & 1 & 0 & 0 & 0 & 1 & 1 \\
|
||||
c & 1 & 0 & 0 & 1 & 0 & 0 \\
|
||||
d & 1 & 0 & 1 & 0 & 1 & 0 \\
|
||||
e & 0 & 1 & 0 & 1 & 0 & 0 \\
|
||||
f & 1 & 1 & 0 & 0 & 0 & 0 \\
|
||||
\end{tabular}
|
||||
\\
|
||||
\\
|
||||
Inz.-Matrix:\\
|
||||
\begin{tabular}{c|ccccccccc}
|
||||
a & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
|
||||
b & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\
|
||||
c & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\
|
||||
d & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\
|
||||
e & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\
|
||||
f & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\
|
||||
\end{tabular}
|
||||
|
||||
\subsection{2.)}
|
||||
\begin{tikzpicture}
|
||||
\def \n {5}
|
||||
\def \radius {3cm}
|
||||
\def \margin {8} % margin in angles, depends on the radius
|
||||
\node[draw, circle] (a) at (3,2) {a};
|
||||
\node[draw, circle] (b) at (1,1) {b};
|
||||
\node[draw, circle] (c) at (2,1) {c};
|
||||
\node[draw, circle] (d) at (3,1) {d};
|
||||
\node[draw, circle] (f) at (4,1) {f};
|
||||
\node[draw, circle] (e) at (0,0) {e};
|
||||
|
||||
\draw (a)--(b);
|
||||
\draw (a)--(c);
|
||||
\draw (a)--(d);
|
||||
\draw (a)--(f);
|
||||
\draw (b)--(e);
|
||||
\end{tikzpicture}
|
||||
\\
|
||||
$G \setminus \{\{b,f\},\{c,d\},\{d,e\}\}$
|
||||
|
||||
\begin{tikzpicture}
|
||||
\def \n {5}
|
||||
\def \radius {3cm}
|
||||
\def \margin {8} % margin in angles, depends on the radius
|
||||
\node[draw, circle] (b) at (2,2) {b};
|
||||
\node[draw, circle] (a) at (1,1) {a};
|
||||
\node[draw, circle] (f) at (2,1) {f};
|
||||
\node[draw, circle] (e) at (3,1) {e};
|
||||
\node[draw, circle] (c) at (0,0) {c};
|
||||
\node[draw, circle] (d) at (2,0) {d};
|
||||
|
||||
\draw (b)--(a);
|
||||
\draw (b)--(f);
|
||||
\draw (b)--(e);
|
||||
\draw (a)--(c);
|
||||
\draw (a)--(d);
|
||||
|
||||
\end{tikzpicture}
|
||||
\\
|
||||
$G \setminus \{\{a,f\},\{c,d\},\{d,e\}\}$
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
\section{Aufgabe 3}
|
||||
|
||||
|
||||
\begin{tikzpicture}
|
||||
\def \n {5}
|
||||
\def \radius {3cm}
|
||||
\def \margin {8} % margin in angles, depends on the radius
|
||||
\node[draw, circle] (1) at (0,1) {1};
|
||||
\node[draw, circle] (2) at (1,2) {2};
|
||||
\node[draw, circle] (4) at (2,2) {4};
|
||||
\node[draw, circle] (5) at (3,1) {5};
|
||||
\node[draw, circle] (6) at (2,0) {6};
|
||||
\node[draw, circle] (3) at (1,0) {3};
|
||||
|
||||
\draw [->]
|
||||
(1) edge (2)
|
||||
(1) edge (3)
|
||||
(2) edge (3)
|
||||
(3) edge (4)
|
||||
(4) edge (5)
|
||||
(5) edge (6)
|
||||
(6) edge (4);
|
||||
\end{tikzpicture}
|
||||
|
||||
\subsection{1.)}
|
||||
Graph $G=(V,E)$ mit\\
|
||||
Knotenmenge: $V=\{1,2,3,4,5,6\}$\\
|
||||
Kantenmenge: $E=
|
||||
\{
|
||||
(1,2),
|
||||
(1,3),
|
||||
(2,3),
|
||||
(3,4),
|
||||
(4,5),
|
||||
(5,6),
|
||||
(6,4)
|
||||
\}$
|
||||
\\
|
||||
|
||||
Adjazens-Matrix:\\
|
||||
\begin{tabular}{c|cccccc}
|
||||
& 1 & 2 & 3 & 4 & 5 & 6 \\
|
||||
\hline
|
||||
1 & 0 & 1 & 1 & 0 & 0& 0 \\
|
||||
2 & 0 & 0 & 1 & 0 & 0& 0 \\
|
||||
3 & 0 & 0 & 0 & 1 & 0& 0 \\
|
||||
4 & 0 & 0 & 0 & 0 & 1& 0 \\
|
||||
5 & 0 & 0 & 0 & 0 & 0& 1 \\
|
||||
6 & 0 & 0 & 0 & 1 & 0& 0 \\
|
||||
\end{tabular}
|
||||
\\
|
||||
Inzid.-Matrix:\\
|
||||
\begin{tabular}{c|ccccccc}
|
||||
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
|
||||
2 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
|
||||
3 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
|
||||
4 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\
|
||||
5 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
|
||||
6 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
|
||||
\end{tabular}
|
||||
\subsection{2.)}
|
||||
\begin{tikzpicture}
|
||||
\def \n {5}
|
||||
\def \radius {3cm}
|
||||
\def \margin {8} % margin in angles, depends on the radius
|
||||
\node[draw, circle] (1) at (0,1) {1};
|
||||
\node[draw, circle] (4) at (2,2) {4};
|
||||
\node[draw, circle] (6) at (2,0) {6};
|
||||
\node[draw, circle] (3) at (1,0) {3};
|
||||
\draw [->]
|
||||
(1) edge (3)
|
||||
(3) edge (4)
|
||||
(6) edge (4);
|
||||
\end{tikzpicture}
|
||||
\\
|
||||
$V'=\{1,3,4,6\}$\\
|
||||
\\
|
||||
\subsection{3.)}
|
||||
\begin{tikzpicture}
|
||||
\def \n {5}
|
||||
\def \radius {3cm}
|
||||
\def \margin {8} % margin in angles, depends on the radius
|
||||
\node[draw, circle] (1) at (0,0) {1};
|
||||
\node[draw, circle] (3) at (1,0) {3};
|
||||
\node[draw, circle] (4) at (2,0) {4};
|
||||
\node[draw, circle] (5) at (3,0) {5};
|
||||
\draw [->]
|
||||
(1) edge (3)
|
||||
(3) edge (4)
|
||||
(4) edge (5);
|
||||
\end{tikzpicture}
|
||||
\\
|
||||
$V'=\{1,3,4,5\}$\\
|
||||
\section{Aufgabe 4}
|
||||
|
||||
Zu zeigen: Wenn ein Baum genau $k \geq 1$ Knoten vom Grad 4 enthält (außer Blätter), dann besitzt der Baum mindestenz $2 \cdot k + 2$ Blätter.\\
|
||||
\\
|
||||
IA: Ein Baum mit nur einem Knoten von Grad 4 muss 4 Blätter haben
|
||||
|
||||
\begin{tikzpicture}
|
||||
\node[draw,circle] (1) at (2,1) {1};
|
||||
\node[draw,circle] (2) at (0,0) {2};
|
||||
\node[draw,circle] (3) at (1,0) {3};
|
||||
\node[draw,circle] (4) at (3,0) {4};
|
||||
\node[draw,circle] (5) at (4,0) {5};
|
||||
|
||||
\draw (1)--(2);
|
||||
\draw (1)--(3);
|
||||
\draw (1)--(4);
|
||||
\draw (1)--(5);
|
||||
\end{tikzpicture}
|
||||
\\
|
||||
$A(1) = 2 \cdot 1 + 2 = 4$\\
|
||||
\\
|
||||
\textbf{IV: $A(k) = 2 \cdot k +2$}\\
|
||||
\\
|
||||
IS: $k \rightarrow k+1$\\
|
||||
Um einen Knoten mit Grad 4 hinzuzufügen, kann man nun eines der Blätter nehmen und drei Blätter anhängen. Man bekommt also 3 Blätter hinzu, verliert aber auch eines, da dieses zum neuen Knoten wird.\\
|
||||
\\
|
||||
$A(k+1) = A(k) + 3 - 1 \overset{(IV)}{=} (2 \cdot k+2)+3-1$
|
||||
$= 2 \cdot (k+1)+2$\\
|
||||
q.e.d
|
||||
\end{document}
|
||||
Loading…
Reference in New Issue
Block a user