master > master: SKA5 Diagramm + Quiz4
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@ -62,6 +62,8 @@
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%% — body/quizzes/quiz2.tex;
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%% |
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%% — body/quizzes/quiz3.tex;
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%% |
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%% — body/quizzes/quiz4.tex;
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%% |
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%% — back/index.tex;
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%% |
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@ -201,6 +203,7 @@
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\usepackage{relsize}
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\usepackage{savesym}
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\usepackage{stmaryrd}
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\usepackage{subfigure}
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\usepackage{yfonts} %% <— Altgotische Fonts
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\usepackage{tikz}
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\usepackage{xy}
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@ -1068,35 +1071,10 @@
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\tikzset{
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>=stealth,
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auto,
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node distance=1cm,
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thick,
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main node/.style={
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circle,draw,font=\sffamily\Large\bfseries,minimum size=0pt
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},
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state/.style={minimum size=0pt}
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loop above right/.style={loop,out=30,in=60,distance=0.5cm},
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loop above left/.style={above left,out=150,in=120,loop},
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loop below right/.style={below right,out=330,in=300,loop},
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loop below left/.style={below left,out=240,in=210,loop},
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itria/.style={
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draw,dashed,shape border uses incircle,
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isosceles triangle,shape border rotate=90,yshift=-1.45cm
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},
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rtria/.style={
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draw,dashed,shape border uses incircle,
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isosceles triangle,isosceles triangle apex angle=90,
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shape border rotate=-45,yshift=0.2cm,xshift=0.5cm
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},
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ritria/.style={
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draw,dashed,shape border uses incircle,
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isosceles triangle,isosceles triangle apex angle=110,
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shape border rotate=-55,yshift=0.1cm
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},
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litria/.style={
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draw,dashed,shape border uses incircle,
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isosceles triangle,isosceles triangle apex angle=110,
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shape border rotate=235,yshift=0.1cm
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}
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}
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%% ********************************************************************************
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@ -4828,7 +4806,7 @@ Und für alle anderen rationalen Zahlen, $r\in\rtnl\ohne\{0\}$, wähle
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p_{r} &:= &q_{r}\cdot r\in\intgr.\\
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\end{mathe}
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Da $r$ ration ist, ist $D(r)$ per Definition nicht leer.
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Da $r$ rational ist, ist $D(r)$ per Definition nicht leer.
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Darum ist die Wahl von $q_{r}$ und $p_{r}$ wohldefiniert
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und per Konstruktion gilt $p_{r}/q_{r}=r$.
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(Für $r=0$ gilt ebenfalls offensichtlich $p_{r}/q_{r}=r$.)
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@ -4972,13 +4950,19 @@ Darum entspricht unserer Darstellung der im \cite[Satz 3.5.1]{sinn2020}.
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\label{ska:5:ex:14}
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\let\sectionname\altsectionname
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Die Gruppe von Bijektionen von $\{1,2\}$ auf $\{1,2\}$ entspricht der Permutationsgruppe $S_{2}$.
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Dies hat $2!=2$ Elemente:
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Die Gruppe von Bijektionen von $\{1, 2\}$ auf $\{1, 2\}$ entspricht der Permutationsgruppe $S_{2}$.
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Dies hat $2!=2$ Elemente, die standardgemäß mit folgenden Labels bezeichnet werden:
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\begin{mathe}[mc]{rcl}
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e &:= &\text{Funktion, die alles fixiert}\\
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(1\,2) &:= &\text{Funktion, die $1$ und $2$ tauscht}\\
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\end{mathe}
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\begin{longtable}{|l|l|}
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\hline
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\hline
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Label &Beschreibung des Elements\\
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\hline
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$e$ &Funktion, die alles fixiert\\
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$(1\ 2)$ &Funktion, die 1 und 2 tauscht\\
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\hline
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\hline
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\end{longtable}
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Die Gruppentafel sieht folgendermaßen aus:
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@ -4995,17 +4979,23 @@ Die Gruppentafel sieht folgendermaßen aus:
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\hline
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\end{longtable}
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Die Gruppe von Bijektionen von $\{1,2,3\}$ auf $\{1,2,3\}$ entspricht der Permutationsgruppe $S_{3}$.
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Dies hat $3!=6$ Elemente:
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Die Gruppe von Bijektionen von $\{1, 2, 3\}$ auf $\{1, 2, 3\}$ entspricht der Permutationsgruppe $S_{3}$.
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Dies hat $3!=6$ Elemente, die standardgemäß mit folgenden Labels bezeichnet werden:
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\begin{mathe}[mc]{rcl}
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e &:= &\text{Funktion, die alles fixiert}\\
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(1\,2) &:= &\text{Funktion, die $1$ und $2$ tauscht}\\
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(1\,3) &:= &\text{Funktion, die $1$ und $3$ tauscht}\\
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(2\,3) &:= &\text{Funktion, die $2$ und $3$ tauscht}\\
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(1\,2\,3) &:= &\text{Funktion, die $1\mapsto 2\mapsto 3\mapsto 1$ abbildet}\\
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(1\,3\,2) &:= &\text{Funktion, die $1\mapsto 3\mapsto 2\mapsto 1$ abbildet}\\
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\end{mathe}
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\begin{longtable}{|l|l|}
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\hline
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\hline
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Label &Beschreibung des Elements\\
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\hline
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$e$ &Funktion, die alles fixiert\\
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$(2\ 3)$ &Funktion, die 2 und 3 tauscht\\
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$(1\ 2)$ &Funktion, die 1 und 2 tauscht\\
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$(1\ 2\ 3)$ &Funktion, die $1\mapsto2\mapsto3$ abbildet\\
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$(1\ 3\ 2)$ &Funktion, die $1\mapsto3\mapsto2$ abbildet\\
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$(1\ 3)$ &Funktion, die 1 und 3 tauscht\\
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\hline
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\hline
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\end{longtable}
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Die Gruppentafel sieht folgendermaßen aus:
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@ -5038,6 +5028,88 @@ Die Gruppentafel sieht folgendermaßen aus:
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An der Tafel lässt sich leicht erkennen, ob eine Gruppe kommutativ ist:
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eine Gruppe, $G$, ist genau dann kommutativ, wenn die Gruppentafel symmetrisch ist.
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Hierbei sollte man darauf achten, dass die \emph{Labels} der Elemente gar keine Rolle spielen.
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Um diese Urteil also leichter treffen zu können ersetzen wir die Elemente durch verschieden gefärbte Quadrate:
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\begin{figure}[h]
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\footnotesize
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\hraum
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\subfigure[$S_{2}$]{
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\begin{tikzpicture}[node distance=8mm, thick]
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\pgfmathsetmacro\habst{1}
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\pgfmathsetmacro\vabst{1}
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\pgfmathsetmacro\rad{8mm}
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\node[rectangle, label=left:{$e=:$}, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (g_0) at (0*\habst, -1*\vabst) {};
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\node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (h_0) at (1*\habst, 0*\vabst) {};
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\node[rectangle, label=left:{$(1\ 2)=:$}, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (g_1) at (0*\habst, -2*\vabst) {};
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\node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (h_1) at (2*\habst, 0*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (gh_0_0) at (1*\habst, -1*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (gh_0_1) at (2*\habst, -1*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (gh_1_0) at (1*\habst, -2*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (gh_1_1) at (2*\habst, -2*\vabst) {};
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\end{tikzpicture}
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}
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\hraum
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\subfigure[$S_{3}$]{
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\begin{tikzpicture}[node distance=8mm, thick]
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\pgfmathsetmacro\habst{1}
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\pgfmathsetmacro\vabst{1}
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\pgfmathsetmacro\rad{8mm}
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\node[rectangle, label=left:{$e=:$}, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (g_0) at (0*\habst, -1*\vabst) {};
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\node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (h_0) at (1*\habst, 0*\vabst) {};
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\node[rectangle, label=left:{$(2\ 3)=:$}, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,102}, draw] (g_1) at (0*\habst, -2*\vabst) {};
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\node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,102}, draw] (h_1) at (2*\habst, 0*\vabst) {};
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\node[rectangle, label=left:{$(1\ 2)=:$}, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,140}, draw] (g_2) at (0*\habst, -3*\vabst) {};
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\node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,140}, draw] (h_2) at (3*\habst, 0*\vabst) {};
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\node[rectangle, label=left:{$(1\ 2\ 3)=:$}, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,178}, draw] (g_3) at (0*\habst, -4*\vabst) {};
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\node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,178}, draw] (h_3) at (4*\habst, 0*\vabst) {};
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\node[rectangle, label=left:{$(1\ 3\ 2)=:$}, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,216}, draw] (g_4) at (0*\habst, -5*\vabst) {};
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\node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,216}, draw] (h_4) at (5*\habst, 0*\vabst) {};
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\node[rectangle, label=left:{$(1\ 3)=:$}, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (g_5) at (0*\habst, -6*\vabst) {};
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\node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (h_5) at (6*\habst, 0*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (gh_0_0) at (1*\habst, -1*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,102}, draw] (gh_0_1) at (2*\habst, -1*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,140}, draw] (gh_0_2) at (3*\habst, -1*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,178}, draw] (gh_0_3) at (4*\habst, -1*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,216}, draw] (gh_0_4) at (5*\habst, -1*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (gh_0_5) at (6*\habst, -1*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,102}, draw] (gh_1_0) at (1*\habst, -2*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (gh_1_1) at (2*\habst, -2*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,216}, draw] (gh_1_2) at (3*\habst, -2*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (gh_1_3) at (4*\habst, -2*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,140}, draw] (gh_1_4) at (5*\habst, -2*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,178}, draw] (gh_1_5) at (6*\habst, -2*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,140}, draw] (gh_2_0) at (1*\habst, -3*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,178}, draw] (gh_2_1) at (2*\habst, -3*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (gh_2_2) at (3*\habst, -3*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,102}, draw] (gh_2_3) at (4*\habst, -3*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (gh_2_4) at (5*\habst, -3*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,216}, draw] (gh_2_5) at (6*\habst, -3*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,178}, draw] (gh_3_0) at (1*\habst, -4*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,140}, draw] (gh_3_1) at (2*\habst, -4*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (gh_3_2) at (3*\habst, -4*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,216}, draw] (gh_3_3) at (4*\habst, -4*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (gh_3_4) at (5*\habst, -4*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,102}, draw] (gh_3_5) at (6*\habst, -4*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,216}, draw] (gh_4_0) at (1*\habst, -5*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (gh_4_1) at (2*\habst, -5*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,102}, draw] (gh_4_2) at (3*\habst, -5*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (gh_4_3) at (4*\habst, -5*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,178}, draw] (gh_4_4) at (5*\habst, -5*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,140}, draw] (gh_4_5) at (6*\habst, -5*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (gh_5_0) at (1*\habst, -6*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,216}, draw] (gh_5_1) at (2*\habst, -6*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,178}, draw] (gh_5_2) at (3*\habst, -6*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,140}, draw] (gh_5_3) at (4*\habst, -6*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,102}, draw] (gh_5_4) at (5*\habst, -6*\vabst) {};
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\node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (gh_5_5) at (6*\habst, -6*\vabst) {};
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\end{tikzpicture}
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}
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\hraum
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\caption{Gruppentafel mit Elementen durch Farben ersetzt}
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\end{figure}
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Nach den o.\,s. Tafeln ist die erste Gruppe, $S_{2}$, kommutativ und die zweite, $S_{3}$, nicht.
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@ -5232,6 +5304,88 @@ wobei
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Dann $f(f^{-1}(B))=f(f^{-1}(Y))=f(X)=\{1\}\subset Y$ (strikt).
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\end{enumerate}
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%% ********************************************************************************
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%% FILE: body/quizzes/quiz4.tex
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%% ********************************************************************************
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\setcounternach{chapter}{4}
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\chapter[Woche 4]{Woche 4}
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\label{quiz:4}
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Gegeben seien Mengen $X$, $Y$, $Z$,
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und Funktionen $f:X\to Y$ und $g:Y\to Z$.
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Wir betrachten die Komposition ${g\circ f:X\to Z}$
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\hraum
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\begin{tikzpicture}[node distance=0.5cm, thick]
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\pgfmathsetmacro\habst{3}
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\pgfmathsetmacro\vabst{1}
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\node[label=below:{$X$}] (SetX) at (0*\habst,0*\vabst) {$\bullet$};
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\node[label=below:{$Y$}] (SetY) at (1*\habst,0*\vabst) {$\bullet$};
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\node[label=below:{$Z$}] (SetZ) at (2*\habst,0*\vabst) {$\bullet$};
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\draw (SetX) edge [->] node [pos=0.5, above] {\footnotesize $f$} (SetY);
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\draw (SetY) edge [->] node [pos=0.5, above] {\footnotesize $g$} (SetZ);
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\end{tikzpicture}
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\hraum
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\begin{enumerate}{\bfseries (a)}
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%% QUIZ 4-a
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\item
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\begin{claim*}
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$g\circ f$ injektiv $\Rightarrow$ $f$ injektiv.
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\end{claim*}
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\begin{proof}
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Angenommen, $g\circ f$ sei injektiv.
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\textbf{Zu zeigen:} $f$ ist injektiv\\
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\textbf{Zu zeigen:} Für alle $x_{1},x_{2}\in X$ gilt $f(x_{1})=f(x_{2})\Rightarrow x_{1}=x_{2}$.\\
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Seien also $x_{1},x_{2}\in X$ beliebig.
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Es gilt:
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\begin{mathe}[mc]{rcl}
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f(x_{1}) = f(x_{2})
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&\Longrightarrow
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&g(f(x_{1})) = g(f(x_{2}))\\
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&\Longrightarrow
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&(g\circ f)(x_{1}) = (g\circ f)(x_{2})\\
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&\Longrightarrow
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&x_{1} = x_{2},
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\,\text{da $g\circ f$ injektiv}.\\
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\end{mathe}
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Also ist $f$ injektiv.
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\end{proof}
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%% QUIZ 4-b
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\item
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\begin{claim*}
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$f,g$ injektiv $\Rightarrow$ $g\circ f$ injektiv.
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\end{claim*}
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||||
|
||||
\begin{proof}
|
||||
Angenommen, $f,g$ seien injektiv.
|
||||
\textbf{Zu zeigen:} $g\circ f$ ist injektiv\\
|
||||
\textbf{Zu zeigen:} Für alle $x_{1},x_{2}\in X$ gilt $(g\circ f)(x_{1})=(g\circ f)(x_{2})\Rightarrow x_{1}=x_{2}$.\\
|
||||
Seien also $x_{1},x_{2}\in X$ beliebig.
|
||||
Es gilt:
|
||||
|
||||
\begin{mathe}[mc]{rcl}
|
||||
(g\circ f)(x_{1}) = (g\circ f)(x_{2})
|
||||
&\Longrightarrow
|
||||
&g(f(x_{1})) = g(f(x_{2}))\\
|
||||
&\Longrightarrow
|
||||
&f(x_{1}) = f(x_{2}),
|
||||
\,\text{da $g$ injektiv}\\
|
||||
&\Longrightarrow
|
||||
&x_{1} = x_{2},
|
||||
\,\text{da $f$ injektiv}.\\
|
||||
\end{mathe}
|
||||
|
||||
Also ist $g\circ f$ injektiv.
|
||||
\end{proof}
|
||||
\end{enumerate}
|
||||
|
||||
%% ********************************************************************************
|
||||
%% FILE: back/index.tex
|
||||
%% ********************************************************************************
|
||||
|
Loading…
Reference in New Issue
Block a user