\documentclass[10pt,a4paper,oneside,ngerman,numbers=noenddot]{scrartcl}
\usepackage[T1]{fontenc}
\usepackage[utf8x]{inputenc}
\usepackage[ngerman]{babel}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{paralist}
\usepackage{gauss}
\usepackage{pgfplots}
\usepackage[locale=DE,exponent-product=\cdot,detect-all]{siunitx}
\usepackage{tikz}
\usetikzlibrary{automata,matrix,fadings,calc,positioning,decorations.pathreplacing,arrows,decorations.markings,petri}
\usepackage{polynom}
\usepackage{multirow}
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% ensures that paragraphs are separated by empty lines
\parskip 12pt plus 1pt minus 1pt
\parindent 0pt
% define how the sections are rendered
\def\thesection{8.\arabic{section})}
\def\thesubsection{\arabic{subsection}.}
\def\thesubsubsection{(\alph{subsubsection})}
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  \hskip -\arraycolsep
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  \array{#1}}
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    place/.style={
        circle,
        thick,
        draw=black,
        fill=white,
        minimum size=6mm,
        font=\bfseries
    },
    transitionH/.style={
        rectangle,
        thick,
        draw=black,
        fill=white,
        minimum width=8mm,
        inner ysep=4pt,
        font=\bfseries
    },
    transitionV/.style={
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        minimum height=8mm,
        inner xsep=2pt
    }
}

\begin{document}
\author{Benjamin Kuffel, Jim Martens\\Gruppe 6}
\title{Hausaufgaben zum 8. Dezember}
\maketitle

\setcounter{section}{2}
\section{} %8.3
	\subsection{}
	Die Prozesse sind auf den Abbildungen von \fref{fig:831-1} bis \fref{fig:831-18} zu sehen.
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
	\end{tikzpicture}
	\caption{Prozess 1 für 8.3.1}
	\label{fig:831-1}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=0.5 of b2] {b4};
		\node[place] (b5) [below=0.5 of b3] {b5};
		
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5);
	\end{tikzpicture}
	\caption{Prozess 2 für 8.3.1}
	\label{fig:831-2}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [above right=0.5 and 1 of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (b)
			(b4) edge (b)
			(b) edge (b6);
	\end{tikzpicture}
	\caption{Prozess 3 für 8.3.1}
	\label{fig:831-3}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [right=of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (b)
			(b4) edge (b)
			(b) edge (b6);
	\end{tikzpicture}
	\caption{Prozess 4 für 8.3.1}
	\label{fig:831-4}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [below right=0.5 and 1 of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (c)
			(b5) edge (c)
			(c) edge (b6);
	\end{tikzpicture}
	\caption{Prozess 5 für 8.3.1}
	\label{fig:831-5}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [right=of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (c)
			(b5) edge (c)
			(c) edge (b6);
	\end{tikzpicture}
	\caption{Prozess 6 für 8.3.1}
	\label{fig:831-6}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [right=of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\node[transitionH] (b) [right=of b2] {b};
		\node[place] (b7) [right=of b] {b7};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (c)
			(b5) edge (c)
			(c) edge (b6)
			(b4) edge (b)
			(b3) edge (b)
			(b) edge (b7);
	\end{tikzpicture}
	\caption{Prozess 7 für 8.3.1}
	\label{fig:831-7}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [below right=0.5 and 1 of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\node[transitionH] (b) [above right=0.5 and 1 of b2] {b};
		\node[place] (b7) [right=of b] {b7};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (c)
			(b5) edge (c)
			(c) edge (b6)
			(b4) edge (b)
			(b2) edge (b)
			(b) edge (b7);
	\end{tikzpicture}
	\caption{Prozess 8 für 8.3.1}
	\label{fig:831-8}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [above right=0.5 and 1 of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (b)
			(b4) edge (b)
			(b) edge (b6)
			(b6) edge (d)
			(d) edge (b7);
	\end{tikzpicture}
	\caption{Prozess 9 für 8.3.1}
	\label{fig:831-9}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [right=of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (b)
			(b4) edge (b)
			(b) edge (b6)
			(b6) edge (d)
			(d) edge (b7);
	\end{tikzpicture}
	\caption{Prozess 10 für 8.3.1}
	\label{fig:831-10}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [below right=0.5 and 1 of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (c)
			(b5) edge (c)
			(c) edge (b6)
			(b6) edge (d)
			(d) edge (b7);
	\end{tikzpicture}
	\caption{Prozess 11 für 8.3.1}
	\label{fig:831-11}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [right=of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (c)
			(b5) edge (c)
			(c) edge (b6)
			(b6) edge (d)
			(d) edge (b7);
	\end{tikzpicture}
	\caption{Prozess 12 für 8.3.1}
	\label{fig:831-12}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [above right=0.5 and 1 of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\node[transitionH] (c) [below right=0.5 and 1 of b3] {c};
		\node[place] (b8) [right=of c] {b8};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (b)
			(b4) edge (b)
			(b) edge (b6)
			(b6) edge (d)
			(d) edge (b7)
			(b3) edge (c)
			(b5) edge (c)
			(c) edge (b8);
	\end{tikzpicture}
	\caption{Prozess 13 für 8.3.1}
	\label{fig:831-13}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [right=of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\node[transitionH] (c) [right=of b3] {c};
		\node[place] (b8) [right=of c] {b8};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (b)
			(b4) edge (b)
			(b) edge (b6)
			(b6) edge (d)
			(d) edge (b7)
			(b2) edge (c)
			(b5) edge (c)
			(c) edge (b8);
	\end{tikzpicture}
	\caption{Prozess 14 für 8.3.1}
	\label{fig:831-14}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [below right=0.5 and 1 of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\node[transitionH] (b) [above right=0.5 and 1 of b2] {b};
		\node[place] (b8) [right=of b] {b8};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (c)
			(b5) edge (c)
			(c) edge (b6)
			(b6) edge (d)
			(d) edge (b7)
			(b2) edge (b)
			(b4) edge (b)
			(b) edge (b8);
	\end{tikzpicture}
	\caption{Prozess 15 für 8.3.1}
	\label{fig:831-15}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (c) [right=of b3] {c};
		\node[place] (b6) [right=of c] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\node[transitionH] (b) [right=of b2] {b};
		\node[place] (b8) [right=of b] {b8};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (c)
			(b5) edge (c)
			(c) edge (b6)
			(b6) edge (d)
			(d) edge (b7)
			(b3) edge (b)
			(b4) edge (b)
			(b) edge (b8);
	\end{tikzpicture}
	\caption{Prozess 16 für 8.3.1}
	\label{fig:831-16}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [above right=0.5 and 1 of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\node[transitionH] (c) [below right=0.5 and 1 of b3] {c};
		\node[place] (b8) [right=of c] {b8};
		\node[transitionH] (d2) [right=of b8] {d};
		\node[place] (b9) [right=of d2] {b9};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b2) edge (b)
			(b4) edge (b)
			(b) edge (b6)
			(b6) edge (d)
			(d) edge (b7)
			(b3) edge (c)
			(b5) edge (c)
			(c) edge (b8)
			(b8) edge (d2)
			(d2) edge (b9);
	\end{tikzpicture}
	\caption{Prozess 17 für 8.3.1}
	\label{fig:831-17}
	\end{figure}
	
	\begin{figure}	
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b2) [right=of a] {b2};
		\node[place] (b3) [below=0.25 of b2] {b3};
		\node[place] (b4) [above=of b2] {b4};
		\node[place] (b5) [below=of b3] {b5};
		\node[transitionH] (b) [right=of b2] {b};
		\node[place] (b6) [right=of b] {b6};
		\node[transitionH] (d) [right=of b6] {d};
		\node[place] (b7) [right=of d] {b7};
		\node[transitionH] (c) [right=of b3] {c};
		\node[place] (b8) [right=of c] {b8};
		\node[transitionH] (d2) [right=of b8] {d};
		\node[place] (b9) [right=of d2] {b9};
		\path[->] (b1) edge (a)
			(a) edge (b2)
			(a) edge (b3)
			(a) edge (b4)
			(a) edge (b5)
			(b3) edge (b)
			(b4) edge (b)
			(b) edge (b6)
			(b6) edge (d)
			(d) edge (b7)
			(b2) edge (c)
			(b5) edge (c)
			(c) edge (b8)
			(b8) edge (d2)
			(d2) edge (b9);
	\end{tikzpicture}
	\caption{Prozess 18 für 8.3.1}
	\label{fig:831-18}
	\end{figure}
	
	\subsection{}
	Die \(<\)-Relation kann auf \fref{fig:832-l} gesehen werden. Die \(\lessdot \)-Relation kann auf \fref{fig:832-ld} gesehen werden. Die \textbf{li}-Relation kann auf \fref{fig:832-li} gesehen werden. Die \textbf{co}-Relation kann auf \fref{fig:832-co} gesehen werden.
	\begin{figure}
	\begin{tikzpicture}[node distance=2cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b3) [right=of a] {b3};
		\node[place] (b4) [below=of b3] {b4};
		\node[transitionH] (c) [below right=of b4] {c};
		\node[place] (b2) [left=3 of c] {b2};
		\node[place] (b5) [right=of c] {b5};
		
		\path[->] (b1) edge (a)
			(a) edge (b3)
			(a) edge (b4)
			(b4) edge (c)
			(b2) edge (c)
			(c) edge (b5)
			(b1) edge[bend left] (b3)
			(b1) edge[bend right] (b4)
			(b1) edge[bend right] (c)
			(b1) edge[bend left=90] (b5)
			(a) edge[bend left] (c)
			(a) edge (b5)
			(b2) edge[bend right] (b5)
			(b4) edge (b5);
	\end{tikzpicture}
	\caption{\(<\)-Relation}
	\label{fig:832-l}
	\end{figure}
	\begin{figure}
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b3) [right=of a] {b3};
		\node[place] (b4) [below=of b3] {b4};
		\node[transitionH] (c) [below right=of b4] {c};
		\node[place] (b2) [left=3 of c] {b2};
		\node[place] (b5) [right=of c] {b5};
		
		\path[->] (b1) edge (a)
			(a) edge (b3)
			(a) edge (b4)
			(b4) edge (c)
			(b2) edge (c)
			(c) edge (b5);
	\end{tikzpicture}
	\caption{\(\lessdot \)-Relation}
	\label{fig:832-ld}
	\end{figure}
	
	\begin{figure}
	\begin{tikzpicture}[node distance=2cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b3) [right=of a] {b3};
		\node[place] (b4) [below=of b3] {b4};
		\node[transitionH] (c) [below right=of b4] {c};
		\node[place] (b2) [left=3 of c] {b2};
		\node[place] (b5) [right=of c] {b5};
		
		\path (b1) edge (a)
			(a) edge (b3)
			(a) edge (b4)
			(b4) edge (c)
			(b2) edge (c)
			(c) edge (b5)
			(b1) edge[bend left] (b3)
			(b1) edge[bend right] (b4)
			(b1) edge[bend right] (c)
			(b1) edge[bend left=90] (b5)
			(a) edge[bend left] (c)
			(a) edge (b5)
			(b2) edge[bend right] (b5)
			(b4) edge (b5);
	\end{tikzpicture}
	\caption{\textbf{li}-Relation}
	\label{fig:832-li}
	\end{figure}
	
	\begin{figure}
	\begin{tikzpicture}[node distance=1cm]
		\node[place] (b1) {b1};
		\node[transitionH] (a) [right=of b1] {a};
		\node[place] (b3) [right=of a] {b3};
		\node[place] (b4) [below=of b3] {b4};
		\node[transitionH] (c) [below right=of b4] {c};
		\node[place] (b2) [left=3 of c] {b2};
		\node[place] (b5) [right=of c] {b5};
		
		\path (b1) edge (b2)
			(b3) edge (b5)
			(b3) edge (b2)
			(b2) edge (b4)
			(a) edge (b2)
			(b3) edge (b4)
			(b3) edge (c);
	\end{tikzpicture}
	\caption{\textbf{co}-Relation}
	\label{fig:832-co}
	\end{figure}
	
	\subsection{}
	P-Schnitt: b2, b4, b3 und T-Schnitt: a
\section{} %8.4
\section{} %8.5
\end{document}
