From f3aff9fdb6cddc46266a46be7325caa3da64b091 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Mon, 29 Apr 2013 19:59:02 +0200 Subject: [PATCH] added much content and fixed minor errors --- .../Diskrete-Mathematik/Handout/Handout.tex | 183 ++++++++++++++++++ .../Diskrete-Mathematik/Handout/Makefile | 8 + .../Diskrete-Mathematik/LaTeX/Ende.tex | 14 +- .../LaTeX/Graphentheorie-I.pdf | Bin 921674 -> 959454 bytes .../LaTeX/Koenigsberger-Brueckenproblem.tex | 3 +- .../LaTeX/Spezielle-Graphen.tex | 25 ++- .../Diskrete-Mathematik/LaTeX/Strukturen.tex | 9 +- .../Diskrete-Mathematik/Plan/Plan.tex | 2 +- 8 files changed, 230 insertions(+), 14 deletions(-) create mode 100644 presentations/Diskrete-Mathematik/Handout/Handout.tex create mode 100644 presentations/Diskrete-Mathematik/Handout/Makefile diff --git a/presentations/Diskrete-Mathematik/Handout/Handout.tex b/presentations/Diskrete-Mathematik/Handout/Handout.tex new file mode 100644 index 0000000..9414cf2 --- /dev/null +++ b/presentations/Diskrete-Mathematik/Handout/Handout.tex @@ -0,0 +1,183 @@ +\documentclass[a4paper,9pt]{scrartcl} +\usepackage{amssymb, amsmath} % needed for math +\usepackage[utf8]{inputenc} % this is needed for umlauts +\usepackage[ngerman]{babel} % this is needed for umlauts +\usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf +\usepackage[margin=2.5cm]{geometry} %layout +\usepackage{hyperref} % links im text +\usepackage{color} +\usepackage{framed} +\usepackage{enumerate} % for advanced numbering of lists +\usepackage{braket} % needed for nice printing of sets +\usepackage{xcolor} +\usepackage{lastpage} +\clubpenalty = 10000 % Schusterjungen verhindern +\widowpenalty = 10000 % Hurenkinder verhindern + +\hypersetup{ + pdfauthor = {Martin Thoma}, + pdfkeywords = {Diskrete Mathematik}, + pdftitle = {Graphentheorie I: Handout} +} + +\usepackage{fancyhdr} +\pagestyle{fancy} +\lhead{Diskrete Mathematik} +\chead{Graphentheorie I (Martin Thoma)} +\rhead{Seite \thepage\ von \pageref{LastPage}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Custom definition style, by % +% http://mathoverflow.net/questions/46583/what-is-a-satisfactory-way-to-format-definitions-in-latex/58164#58164 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\makeatletter +\newdimen\errorsize \errorsize=0.2pt +% Frame with a label at top +\newcommand\LabFrame[2]{% + \fboxrule=\FrameRule + \fboxsep=-\errorsize + \textcolor{FrameColor}{% + \fbox{% + \vbox{\nobreak + \advance\FrameSep\errorsize + \begingroup + \advance\baselineskip\FrameSep + \hrule height \baselineskip + \nobreak + \vskip-\baselineskip + \endgroup + \vskip 0.5\FrameSep + \hbox{\hskip\FrameSep \strut + \textcolor{TitleColor}{\textbf{#1}}}% + \nobreak \nointerlineskip + \vskip 1.3\FrameSep + \hbox{\hskip\FrameSep + {\normalcolor#2}% + \hskip\FrameSep}% + \vskip\FrameSep + }}% +}} +\definecolor{FrameColor}{rgb}{0.25,0.25,1.0} +\definecolor{TitleColor}{rgb}{1.0,1.0,1.0} + +\newenvironment{contlabelframe}[2][\Frame@Lab\ (cont.)]{% + % Optional continuation label defaults to the first label plus + \def\Frame@Lab{#2}% + \def\FrameCommand{\LabFrame{#2}}% + \def\FirstFrameCommand{\LabFrame{#2}}% + \def\MidFrameCommand{\LabFrame{#1}}% + \def\LastFrameCommand{\LabFrame{#1}}% + \MakeFramed{\advance\hsize-\width \FrameRestore} +}{\endMakeFramed} +\newcounter{definition} +\newenvironment{definition}[1]{% + \par + \refstepcounter{definition}% + \begin{contlabelframe}{Definition \thedefinition:\quad #1} + \noindent\ignorespaces} +{\end{contlabelframe}} +\makeatother +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Begin document % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} +\begin{definition}{Graph} +Ein Graph ist ein Tupel $(E, K)$, wobei $E \neq \emptyset$ die Eckenmenge und +$K \subseteq E \times E$ die +Kantenmenge bezeichnet. +\end{definition} + +\begin{definition}{Inzidenz} +Sei $e \in E$ und $k = \Set{e_1, e_2} \in K$. + +$e$ heißt \textbf{inzident} zu $k :\Leftrightarrow e = e_1$ oder $e = e_2$ +\end{definition} + +\begin{definition}{Vollständiger Graph} +Sei $G = (E, K)$ ein Graph. + +$G$ heißt \textbf{vollständig} $:\Leftrightarrow K = E \times E \setminus \Set{e \in E: \Set{e, e}}$ +\end{definition} + +Ein vollständiger Graph mit $n$ Ecken wird als $K_n$ bezeichnet. + +\begin{definition}{Bipartiter Graph} +Sei $G = (E, K)$ ein Graph und $A, B \subset E$ zwei disjunkte Eckenmengen mit +$E \setminus A = B$. + +$G$ heißt \textbf{bipartit} $:\Leftrightarrow \forall_{k = \Set{e_1, e_2} \in K}: (e_1 \in A \text{ und } e_2 \in B) \text{ oder } (e_1 \in B \text{ und } e_2 \in A) $ +\end{definition} + +\begin{definition}{Vollständig bipartiter Graph} +Sei $G = (E, K)$ ein bipartiter Graph und $\Set{A, B}$ bezeichne die Bipartition. + +$G$ heißt \textbf{vollständig bipartit} $:\Leftrightarrow A \times B = K$ +\end{definition} + +\begin{definition}{Kantenzug, Länge eines Kantenzuges und Verbindung von Ecken} +Sei $G = (E, K)$ ein Graph. + +Dann heißt eine Folge $k_1, k_2, \dots, k_s$ von Kanten, zu denen es Ecken +$e_0, e_1, e_2, \dots, e_s$ gibt, so dass +\begin{itemize} + \item $k_1 = \Set{e_0, e_1}$ + \item $k_2 = \Set{e_1, e_2}$ + \item \dots + \item $k_s = \Set{e_{s-1}, e_s}$ +\end{itemize} +gilt ein \textbf{Kantenzug}, der $e_0$ und $e_s$ \textbf{verbindet} und $s$ +seine \textbf{Länge}. +\end{definition} + +\begin{definition}{Geschlossener Kantenzug} +Sei $G = (E, K)$ ein Graph und $A = (e_0, e_1, \dots, e_s)$ ein Kantenzug. + +A heißt \textbf{geschlossen} $:\Leftrightarrow e_s = e_0$ . +\end{definition} + +\begin{definition}{Weg} +Sei $G = (E, K)$ ein Graph und $A = (k_1, k_2 \dots, k_s)$ ein Kantenzug. + +A heißt \textbf{Weg} $:\Leftrightarrow \forall_{i, j \in 1, \dots, s}: i \neq j \Rightarrow k_i \neq k_j$ . +\end{definition} + +\begin{definition}{Einfacher Kreis} +Sei $G = (E, K)$ ein Graph und $A = (k_1, k_2 \dots, k_s)$ ein Kantenzug. + +A heißt \textbf{Kreis} $:\Leftrightarrow A$ ist geschlossen und ein Weg. +\end{definition} + +\begin{definition}{Zusammenhängender Graph} +Sei $G = (E, K)$ ein Graph. + +$G$ heißt \textbf{zusammenhängend} $:\Leftrightarrow \forall e_1, e_2 \in E: $ +Es ex. ein Kantenzug, der $e_1$ und $e_2$ verbindet +\end{definition} + +\begin{definition}{Grad einer Ecke} +Der \textbf{Grad} einer Ecke ist die Anzahl der Kanten, die von dieser Ecke +ausgehen. +\end{definition} + +\begin{definition}{Isolierte Ecke} +Hat eine Ecke den Grad 0, so nennt man ihn \textbf{isoliert}. +\end{definition} + +\begin{definition}{Eulerscher Kreis} +Sei $G$ ein Graph und $A$ ein Kreis in $G$. + +$A$ heißt \textbf{eulerscher Kreis} $:\Leftrightarrow \forall_{e \in E}: e \in A$. +\end{definition} + +\begin{definition}{Eulerscher Graph} +Ein Graph heißt \textbf{eulersch}, wenn er einen eulerschen Kreis enthält. +\end{definition} + +\begin{definition}{Offene eulersche Linie} +Sei $G$ ein Graph und $A$ ein Weg, der kein Kreis ist. + +$A$ heißt \textbf{offene eulersche Linie} von $G :\Leftrightarrow$ Jede Kante +in $G$ kommt genau ein mal in $A$ vor. +\end{definition} + +\end{document} diff --git a/presentations/Diskrete-Mathematik/Handout/Makefile b/presentations/Diskrete-Mathematik/Handout/Makefile new file mode 100644 index 0000000..5a4f1ad --- /dev/null +++ b/presentations/Diskrete-Mathematik/Handout/Makefile @@ -0,0 +1,8 @@ +SOURCE = Handout +make: + pdflatex $(SOURCE).tex -output-format=pdf + pdflatex $(SOURCE).tex -output-format=pdf + make clean + +clean: + rm -rf $(TARGET) *.class *.html *.log *.aux *.out diff --git a/presentations/Diskrete-Mathematik/LaTeX/Ende.tex b/presentations/Diskrete-Mathematik/LaTeX/Ende.tex index 16536c5..5d58a6b 100644 --- a/presentations/Diskrete-Mathematik/LaTeX/Ende.tex +++ b/presentations/Diskrete-Mathematik/LaTeX/Ende.tex @@ -1,3 +1,10 @@ +\subsection{Weitere Aufgaben} +\begin{frame}{Aufgabe 3} +Zeigen Sie: Ein Kreis ist genau dann bipartit, wenn er gerade Länge hat. +\end{frame} + +% TODO + \subsection{Bildquelle} \begin{frame}{Bildquelle} \begin{itemize} @@ -6,4 +13,9 @@ \end{itemize} \end{frame} - +\subsection{Literatur} +\begin{frame}{Literatur} +\begin{itemize} + \item A. Beutelspacher: \textit{Diskrete Mathematik für Einsteiger}, 4. Auflage, ISBN 978-3-8348-1248-3 +\end{itemize} +\end{frame} diff --git a/presentations/Diskrete-Mathematik/LaTeX/Graphentheorie-I.pdf b/presentations/Diskrete-Mathematik/LaTeX/Graphentheorie-I.pdf index 415f4ec55e73277985469ebb7435eaf8bf4e7af8..f280f0aa74b59508555ca4bfd68cae8beb56ad60 100644 GIT binary patch delta 385354 zcmZsBb9AOL6K~(z-7U7ZZQFLYwr%UJx7~KDx3+EDHnz5H`|kIB_x^X!lV_6i%gH1; z$;@OX8O!0?9On5>t|TTw&&6(vXJbz$hJZ&0FefWLfYSnHU2r)O zyRNFYm3hJ`00Pyi6a!8RMa1n62#>~W^DiN&lR4^XWrmt)?VjeOG9(Zjn8a6!7; z5U|d*vJ@@z=T&r5D?xCtqWLZNI7kqYx-=#Ml@#YmdOt{J>h3HK;%5)IBcDRIk0Um= zDq>w}OR@iKi%>-(vMr&K)o=r8@s>p3Qcuk*X7K2r)u7giJgR5|qGW={BVr_+CY3vvalzCIp!PT~9(F88kyRz+EG-1}|LL3AEiW0Zo= zWYDyX8x4L&G7+UJ`moi7yrHc1U&igFL=@i zgrO+T)nS8C^_|o;3?LZ*h$uBD#V4&K6ozf4REC;zuo-`ol!hqAF8{aS{J5sPox|yc zNqT!nIPEAWY1LK4hFi6XS#%mhBqeuuQ#6ejiM$03q2a7UP9`ttqb**tuLPm8X9XI3{z~i8F~7SD2fz-=0aC)0p{eY4Ngp*Ky<3dVXVd 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z^=+uajAtrS#|)5U|G^B{4U-i1!4@I6oBi^ckZWYENi)Ce36xh!ax2i=$PAcfebVcM zCLz)-TM>8NR|-?v{Ls)EY2KT|H441shI)tjM@MQ@d5@0QJQ&##QnQT*2`E@^DINMe}h5-#9= z6axJ^SifhF#ry36j#|ZtNmjP(`F6#oWThqprLMguE1xGl98943nW|+QRC}M=KR!7Y h3cfRjN=yn)P)H&qgoUFljZ6(KOi-Gd4jzuE{{fWkV!r?Y diff --git a/presentations/Diskrete-Mathematik/LaTeX/Koenigsberger-Brueckenproblem.tex b/presentations/Diskrete-Mathematik/LaTeX/Koenigsberger-Brueckenproblem.tex index ae4d111..6708147 100644 --- a/presentations/Diskrete-Mathematik/LaTeX/Koenigsberger-Brueckenproblem.tex +++ b/presentations/Diskrete-Mathematik/LaTeX/Koenigsberger-Brueckenproblem.tex @@ -102,7 +102,8 @@ TODO \begin{block}{Offene eulersche Linie} Sei $G$ ein Graph und $A$ ein Weg, der kein Kreis ist. -$A$ heißt \textbf{offene eulersche Linie} von $G :\Leftrightarrow$ Jede Kante in $G$ kommt genau ein mal in $A$ vor. +$A$ heißt \textbf{offene eulersche Linie} von $G :\Leftrightarrow$ Jede Kante +in $G$ kommt genau ein mal in $A$ vor. \end{block} Ein Graph kann genau dann "`in einem Zug"' gezeichnet werden, wenn er eine diff --git a/presentations/Diskrete-Mathematik/LaTeX/Spezielle-Graphen.tex b/presentations/Diskrete-Mathematik/LaTeX/Spezielle-Graphen.tex index f46f828..fb984df 100644 --- a/presentations/Diskrete-Mathematik/LaTeX/Spezielle-Graphen.tex +++ b/presentations/Diskrete-Mathematik/LaTeX/Spezielle-Graphen.tex @@ -3,7 +3,7 @@ \begin{block}{Vollständiger Graph} Sei $G = (E, K)$ ein Graph. -$G$ heißt \textbf{vollständig} $:\Leftrightarrow = E \times E \setminus \Set{e \in E: \Set{e, e}}$ +$G$ heißt \textbf{vollständig} $:\Leftrightarrow K = E \times E \setminus \Set{e \in E: \Set{e, e}}$ \end{block} Ein vollständiger Graph mit $n$ Ecken wird als $K_n$ bezeichnet. @@ -21,9 +21,9 @@ Ein vollständiger Graph mit $n$ Ecken wird als $K_n$ bezeichnet. \end{gallery} \end{frame} -\begin{frame}{Bipartite Graphen} -\begin{block}{Bipartite Graphen} -Sei $G = (E, K)$ ein Graph und $A, B \subset V$ zwei disjunkte Eckenmengen mit +\begin{frame}{Bipartiter Graph} +\begin{block}{Bipartiter Graph} +Sei $G = (E, K)$ ein Graph und $A, B \subset E$ zwei disjunkte Eckenmengen mit $E \setminus A = B$. $G$ heißt \textbf{bipartit} $:\Leftrightarrow \forall_{k = \Set{e_1, e_2} \in K}: (e_1 \in A \text{ und } e_2 \in B) \text{ oder } (e_1 \in B \text{ und } e_2 \in A) $ @@ -41,11 +41,11 @@ $G$ heißt \textbf{bipartit} $:\Leftrightarrow \forall_{k = \Set{e_1, e_2} \in K \end{gallery} \end{frame} -\begin{frame}{Vollständig bipartite Graphen} -\begin{block}{Vollständig bipartite Graphen} +\begin{frame}{Vollständig bipartiter Graph} +\begin{block}{Vollständig bipartiter Graph} Sei $G = (E, K)$ ein bipartiter Graph und $\Set{A, B}$ bezeichne die Bipartition. -$G$ heißt \textbf{vollständig bipartit} $:\Leftrightarrow \Set{\Set{a, b} | a \in A \land b \in B} = K$ +$G$ heißt \textbf{vollständig bipartit} $:\Leftrightarrow A \times B = K$ \end{block} \begin{gallery} @@ -75,3 +75,14 @@ bezeichnet man mit $K_{|A|, |B|}$. \galleryimage[Green]{vollstaendig-bipartit/k-5-5} \end{gallery} \end{frame} + +\begin{frame}{Aufgabe 2} +Wie viele Ecken und wie viele Kanten hat der $K_{m, n}$? + +\visible<2>{ + \begin{align} + \text{Ecken: } &m+n\\ + \text{Kanten: } &m\cdot n + \end{align} +} +\end{frame} diff --git a/presentations/Diskrete-Mathematik/LaTeX/Strukturen.tex b/presentations/Diskrete-Mathematik/LaTeX/Strukturen.tex index c72c7b9..6809efe 100644 --- a/presentations/Diskrete-Mathematik/LaTeX/Strukturen.tex +++ b/presentations/Diskrete-Mathematik/LaTeX/Strukturen.tex @@ -1,6 +1,6 @@ \subsection{Strukturen in Graphen} -\begin{frame}{Kantenzug} -\begin{block}{Kantenzug} +\begin{frame}{Kantenzug, Länge eines Kantenzuges und Verbindung von Ecken} +\begin{block}{Kantenzug, Länge eines Kantenzuges und Verbindung von Ecken} Sei $G = (E, K)$ ein Graph. Dann heißt eine Folge $k_1, k_2, \dots, k_s$ von Kanten, zu denen es Ecken @@ -111,7 +111,8 @@ Manchmal wird das auch "`einfacher Kreis"' genannt. \begin{block}{Zusammenhängender Graph} Sei $G = (E, K)$ ein Graph. -$G$ heißt \textbf{zusammenhängend} $:\Leftrightarrow \forall e_1, e_2 \in E: $ Es ex. ein Kantenzug, der $e_1$ und $e_2$ verbindet +$G$ heißt \textbf{zusammenhängend} $:\Leftrightarrow \forall e_1, e_2 \in E: $ +Es ex. ein Kantenzug, der $e_1$ und $e_2$ verbindet \end{block} \begin{gallery} @@ -132,7 +133,7 @@ Der \textbf{Grad} einer Ecke ist die Anzahl der Kanten, die von dieser Ecke ausgehen. \end{block} -\begin{block}{Isolierte Ecken} +\begin{block}{Isolierte Ecke} Hat eine Ecke den Grad 0, so nennt man ihn \textbf{isoliert}. \end{block} diff --git a/presentations/Diskrete-Mathematik/Plan/Plan.tex b/presentations/Diskrete-Mathematik/Plan/Plan.tex index 024ab8c..aa97a71 100644 --- a/presentations/Diskrete-Mathematik/Plan/Plan.tex +++ b/presentations/Diskrete-Mathematik/Plan/Plan.tex @@ -13,7 +13,7 @@ \hypersetup{ pdfauthor = {Martin Thoma}, - pdfkeywords = {Lineare Algebra}, + pdfkeywords = {Diskrete Mathematik}, pdftitle = {Vortrag Graphentheorie I: Tafelbild + Text} }