From c4de942f915d94fb4023dc188292a26b9c98fb20 Mon Sep 17 00:00:00 2001 From: Valentin Brandl Date: Tue, 20 Nov 2018 15:28:53 +0100 Subject: [PATCH] Add dima u05 --- school/di-ma/uebung/05/05_1.tex | 67 ++++++++++++++++++++++ school/di-ma/uebung/05/05_2.tex | 72 +++++++++++++++++++++++ school/di-ma/uebung/05/05_3.tex | 92 ++++++++++++++++++++++++++++++ school/di-ma/uebung/05/05_4.tex | 67 ++++++++++++++++++++++ school/di-ma/uebung/05/a1.png | Bin 0 -> 119614 bytes school/di-ma/uebung/05/pascal.pdf | Bin 0 -> 183781 bytes 6 files changed, 298 insertions(+) create mode 100644 school/di-ma/uebung/05/05_1.tex create mode 100644 school/di-ma/uebung/05/05_2.tex create mode 100644 school/di-ma/uebung/05/05_3.tex create mode 100644 school/di-ma/uebung/05/05_4.tex create mode 100644 school/di-ma/uebung/05/a1.png create mode 100644 school/di-ma/uebung/05/pascal.pdf diff --git a/school/di-ma/uebung/05/05_1.tex b/school/di-ma/uebung/05/05_1.tex new file mode 100644 index 0000000..5d7c3a4 --- /dev/null +++ b/school/di-ma/uebung/05/05_1.tex @@ -0,0 +1,67 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\usepackage{tikz} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \qname {Pascal Brackmann} +\def \qmatrikel {108017113834} % +\def \uebung {5} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\noindent \qmatrikel \quad \qname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 5.1} + + +Party mit Daisy und Donald. Daisy bekommt 9 verschiedene Antworten, Partner begrüßen sich nicht gegenseitig. Insgesamt +gibt es 10 Personen und somit 9 verschiedene Möglichkeiten, wie viele Leute begrüßt werden können.\\ +\\ +\noindent +Nach dem Schubfachprinzip müssen nun 2 Personen die gleiche Anzahl an Personen begrüßt haben. Da Daisy auf Nachfrage 9 +verschiedene Antworten bekommt, muss Daisy die gleiche Anzahl an Begrüßungen haben wie jemand anderes.\\ +\\ +\noindent +Um die korrekte Anzahl an Begrüßungen zu erhalten, hat der Partner von der Person mit 8 Begrüßungen dann 0 Begrüßungen. +Die weiteren Paare haben 7 und 1, 6 und 2, 5 und 3 sowie 4 und 4.\\ +\begin{figure}[h!] +\centering +\includegraphics[scale=0.5]{school/di-ma/uebung/05/a1.png} +\caption{Graph mit Begrüßungen (Label ist Anzahl Begrüßungen)} +\end{figure} +\noindent +Da Daisy die gleiche Anzahl an Begrüßungen hat und es in dem Graphen (der komb. Argumentation) nur ein Paar gibt mit +gleichen Begrüßungsanzahlen gibt, muss Daisy zu diesem Paar gehören. Demzufolge hat Donald ebenfalls die gleiche Anzahl +und beide haben 4 Begrüßungen. + +\end{document} diff --git a/school/di-ma/uebung/05/05_2.tex b/school/di-ma/uebung/05/05_2.tex new file mode 100644 index 0000000..0183c26 --- /dev/null +++ b/school/di-ma/uebung/05/05_2.tex @@ -0,0 +1,72 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\usepackage{tikz} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \qname {Pascal Brackmann} +\def \qmatrikel {108017113834} % +\def \uebung {5} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\noindent \qmatrikel \quad \qname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 5.2} + + +Zu zeigen: Ein zusammenhängender Graph $G=(V,E)$ mit $|E|=|V|-1$ ist ein Baum.\\ +\noindent +Nach Definition ist ein Baum ein kreisfreier zusammenhängender Graph. +Es gilt für G zu zeigen:, dass +\begin{itemize} + \item G ist kreisfrei + \item G ist zusammenhängend (nach Aufgabenstellung bereits erfüllt) +\end{itemize} +\noindent +\textbf{Beweis durch Widerspruch}\\ +Annahme: Es gibt einen Kreis (1,...,k) in G, wobei $k \in \mathbb{N} $ und $1\leq k \leq n$ $(V=\{1,...,n\})$.\\ +\noindent +Die Knoten 1 bis k sind durch einen Pfad mit (k-1) Knoten verbunden. Eine weitere Kante besteht zwischen k und 1, da es +sich um einen Kreis handelt. Da der Graph zusammenhängend ist, gibt es weitere (n-k) Knoten, die nicht im Kreis liegen +und durch eine Kante mit dem Graphen verbunden sind.\\ +\\ +\noindent +$\Rightarrow |E|=(k-1)+1+(n-k)=n=|V|$\\ +Dies ist aber ein Widerspruch dazu, dass gilt $|E|=|V|-1$. +$\Rightarrow$ G ist somit Kreisfrei.\\ +\\ +\noindent +Insgesamt folgt damit, dass $G=(V,E)$ mit $|E|=|V|-1$ kreisfrei und zusammenhängend ist. Somit ist G ein Baum.\\ +q.e.d. + +\end{document} diff --git a/school/di-ma/uebung/05/05_3.tex b/school/di-ma/uebung/05/05_3.tex new file mode 100644 index 0000000..4539480 --- /dev/null +++ b/school/di-ma/uebung/05/05_3.tex @@ -0,0 +1,92 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\usepackage{tikz} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \qname {Pascal Brackmann} +\def \qmatrikel {108017113834} % +\def \uebung {5} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\noindent \qmatrikel \quad \qname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 5.3} + +\begin{figure}[h!] +\begin{tabular}{c|cccc} + & A & B & C & D \\\hline + 1 & \underline{2} & 3 & 3 & 2 \\ + 2 & 3 & 4 & \underline{4} & 3 \\ + 3 & 3 & \underline{4} & 4 & 3 \\ + 4 & 2 & 3 & 3 & \underline{2} +\end{tabular} +\caption{Anzahl der von einem Feld aus erreichbaren Felder} +\end{figure} + +\begin{itemize} + + \item Gegenüberliegende Ecken teilen sich die beiden Felder, die von der Ecke aus erreichbar sind. Für die + gegenüberliegenden Ecken $A1$ und $D4$ teilen sich die Felder $B3$ und $C2$ als mögliche Ein- und + Ausgangafelder. + + \item Wenn man auf einem der inneren Felder beginnt und die jeweils gegenüberliegenden Ecken erreichen will, ist es + unvermeidbar, ein Feld 2 mal zu besuchen. Im Beispiel oben: $(B3, A1, C2, D4)$. Anschließend können nur bereits + besuchte Felder erreicht werden. + +\end{itemize} + +Fallunterscheidung: + +\begin{enumerate}[1.] + + \item Fall: Starte nicht in einer Ecke + + Um eine Ecke zu erreichen, muss man eines der beiden inneren Felder besuchen, von denen aus man in die Ecke + kommt. Danach ist nur noch eines der beiden Innenfelder erreichbar. Möchte man die gegenüberliegende Ecke auch + besuchen, muss man von der Ecke über das zweite Innenfeld in die gegenüberliegende Ecke springen. Jetzt sind + beide Felder, die von der Ecke erreichbar sind bereits besucht und man kann nicht weiter ziehen. + + \item Fall: Starte in einer Ecke + + Jetzt ist es möglich die Start Ecke und die gegenüberliegende Ecke zu besuchen. Wie in Fall 1 gezeigt, ist es + jetzt aber nicht mehr möglich, die dritte und vierte Ecke zu besuchen. + +\end{enumerate} + +$\Rightarrow$ Man muss also in einer Ecke starten, um die gegenüberliegende Ecke besuchen zu können. Da man aber 2 Paare an +gegenüberliegenden Ecken hat, kann die Bedingung nur für eines der beiden Paare zutreffen und es ist nicht möglich, +beide Ecken des anderen Paares zu besuchen. + +\end{document} + diff --git a/school/di-ma/uebung/05/05_4.tex b/school/di-ma/uebung/05/05_4.tex new file mode 100644 index 0000000..966d276 --- /dev/null +++ b/school/di-ma/uebung/05/05_4.tex @@ -0,0 +1,67 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\usepackage{tikz} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% Fill out the the definitions %%%%%%%%% +\def \name {Valentin Brandl} % +\def \matrikel {108018274494} % +\def \pname {Marvin Herrmann} % +\def \pmatrikel {108018265436} % +\def \gruppe {2 (Mi 10-12 Andre)} +\def \qname {Pascal Brackmann} +\def \qmatrikel {108017113834} % +\def \uebung {5} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % DO NOT MODIFY THIS HEADER +\newcommand{\hwsol}{ +\vspace*{-2cm} +\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\ +\noindent \pmatrikel \quad \pname \\ +\noindent \qmatrikel \quad \qname \\ +\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center} +} + +\begin{document} +%Import header +\hwsol + +\section*{Aufgabe 5.4} + +\begin{itemize} + + \item Betrachtung des Problems analog zur korrekten Klammerung + + \item Für jede \enquote{Stufe der Pyramide}, die man hochsteigt, muss auch wieder eine Stufe runter gestiegen + werden. Z.B. für die Basis mit 3 Münzen: $HRHRHR = ()()()$ + + \item Stapelt man jetzt Münzen, z.B. 4 Basismünzen und ganz links 2 Münzen nebeneinander + + $HHRHRRHR = (()())()$ + + \item Da jeder Münzturm mit $n$ Münzen zur Basis genau auf eine korrekte Klammerung mit $2n$ Klammern abbildet (und + natürlich anders herum), sind die Mengen (aller Münztürme mit $n$ Basismünzen, alle korrekten Klammerungen mit + $2n$ Klammern) gleich + +\end{itemize} + +$\Rightarrow C_n = \binom{2n}{n} - \binom{2n}{n+1}$ + + +\end{document} + diff --git a/school/di-ma/uebung/05/a1.png b/school/di-ma/uebung/05/a1.png new file mode 100644 index 0000000000000000000000000000000000000000..95cfdc95f07a04950e23f34658063b3ed82cd3ea GIT binary patch literal 119614 zcmYhibyQuw6E;e5cjsUQin|@$-Cc{j7A*%WQrz9$t+;!UgTuk4KyfSXetCc2ckjA? zWbKu`c6O4PXC|2^lL%F188l>KWGE;oG&xyGbttG$u^)vS5&q-N1roy8$K#Wmx{Nr~ zuPMOMM+MeeOi2t1sy+_o#T4$Nj`U4d&kYI+ec-?H3G7^M0R{D5A}1-P>1Fiq)jFA4 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