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rotate=-55,yshift=0.1cm - }, - litria/.style={ - draw,dashed,shape border uses incircle, - isosceles triangle,isosceles triangle apex angle=110, - shape border rotate=235,yshift=0.1cm - } } %% ******************************************************************************** @@ -4828,7 +4806,7 @@ Und für alle anderen rationalen Zahlen, $r\in\rtnl\ohne\{0\}$, wähle p_{r} &:= &q_{r}\cdot r\in\intgr.\\ \end{mathe} -Da $r$ ration ist, ist $D(r)$ per Definition nicht leer. +Da $r$ rational ist, ist $D(r)$ per Definition nicht leer. Darum ist die Wahl von $q_{r}$ und $p_{r}$ wohldefiniert und per Konstruktion gilt $p_{r}/q_{r}=r$. (Für $r=0$ gilt ebenfalls offensichtlich $p_{r}/q_{r}=r$.) @@ -4972,13 +4950,19 @@ Darum entspricht unserer Darstellung der im \cite[Satz 3.5.1]{sinn2020}. \label{ska:5:ex:14} \let\sectionname\altsectionname -Die Gruppe von Bijektionen von $\{1,2\}$ auf $\{1,2\}$ entspricht der Permutationsgruppe $S_{2}$. -Dies hat $2!=2$ Elemente: +Die Gruppe von Bijektionen von $\{1, 2\}$ auf $\{1, 2\}$ entspricht der Permutationsgruppe $S_{2}$. +Dies hat $2!=2$ Elemente, die standardgemäß mit folgenden Labels bezeichnet werden: - \begin{mathe}[mc]{rcl} - e &:= &\text{Funktion, die alles fixiert}\\ - (1\,2) &:= &\text{Funktion, die $1$ und $2$ tauscht}\\ - \end{mathe} + \begin{longtable}{|l|l|} + \hline + \hline + Label &Beschreibung des Elements\\ + \hline + $e$ &Funktion, die alles fixiert\\ + $(1\ 2)$ &Funktion, die 1 und 2 tauscht\\ + \hline + \hline + \end{longtable} Die Gruppentafel sieht folgendermaßen aus: @@ -4995,17 +4979,23 @@ Die Gruppentafel sieht folgendermaßen aus: \hline \end{longtable} -Die Gruppe von Bijektionen von $\{1,2,3\}$ auf $\{1,2,3\}$ entspricht der Permutationsgruppe $S_{3}$. -Dies hat $3!=6$ Elemente: +Die Gruppe von Bijektionen von $\{1, 2, 3\}$ auf $\{1, 2, 3\}$ entspricht der Permutationsgruppe $S_{3}$. +Dies hat $3!=6$ Elemente, die standardgemäß mit folgenden Labels bezeichnet werden: - \begin{mathe}[mc]{rcl} - e &:= &\text{Funktion, die alles fixiert}\\ - (1\,2) &:= &\text{Funktion, die $1$ und $2$ tauscht}\\ - (1\,3) &:= &\text{Funktion, die $1$ und $3$ tauscht}\\ - (2\,3) &:= &\text{Funktion, die $2$ und $3$ tauscht}\\ - (1\,2\,3) &:= &\text{Funktion, die $1\mapsto 2\mapsto 3\mapsto 1$ abbildet}\\ - (1\,3\,2) &:= &\text{Funktion, die $1\mapsto 3\mapsto 2\mapsto 1$ abbildet}\\ - \end{mathe} + \begin{longtable}{|l|l|} + \hline + \hline + Label &Beschreibung des Elements\\ + \hline + $e$ &Funktion, die alles fixiert\\ + $(2\ 3)$ &Funktion, die 2 und 3 tauscht\\ + $(1\ 2)$ &Funktion, die 1 und 2 tauscht\\ + $(1\ 2\ 3)$ &Funktion, die $1\mapsto2\mapsto3$ abbildet\\ + $(1\ 3\ 2)$ &Funktion, die $1\mapsto3\mapsto2$ abbildet\\ + $(1\ 3)$ &Funktion, die 1 und 3 tauscht\\ + \hline + \hline + \end{longtable} Die Gruppentafel sieht folgendermaßen aus: @@ -5038,6 +5028,88 @@ Die Gruppentafel sieht folgendermaßen aus: An der Tafel lässt sich leicht erkennen, ob eine Gruppe kommutativ ist: eine Gruppe, $G$, ist genau dann kommutativ, wenn die Gruppentafel symmetrisch ist. +Hierbei sollte man darauf achten, dass die \emph{Labels} der Elemente gar keine Rolle spielen. +Um diese Urteil also leichter treffen zu können ersetzen wir die Elemente durch verschieden gefärbte Quadrate: + + \begin{figure}[h] + \footnotesize + \hraum + \subfigure[$S_{2}$]{ + \begin{tikzpicture}[node distance=8mm, thick] + \pgfmathsetmacro\habst{1} + \pgfmathsetmacro\vabst{1} + \pgfmathsetmacro\rad{8mm} + + \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) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (h_0) at (1*\habst, 0*\vabst) {}; + \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) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (h_1) at (2*\habst, 0*\vabst) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \end{tikzpicture} + } + \hraum + \subfigure[$S_{3}$]{ + \begin{tikzpicture}[node distance=8mm, thick] + \pgfmathsetmacro\habst{1} + \pgfmathsetmacro\vabst{1} + \pgfmathsetmacro\rad{8mm} + + \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) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,64}, draw] (h_0) at (1*\habst, 0*\vabst) {}; + \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) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,102}, draw] (h_1) at (2*\habst, 0*\vabst) {}; + \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) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,140}, draw] (h_2) at (3*\habst, 0*\vabst) {}; + \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) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,178}, draw] (h_3) at (4*\habst, 0*\vabst) {}; + \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) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,216}, draw] (h_4) at (5*\habst, 0*\vabst) {}; + \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) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:white,255}, draw] (h_5) at (6*\habst, 0*\vabst) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \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) {}; + \end{tikzpicture} + } + \hraum + \caption{Gruppentafel mit Elementen durch Farben ersetzt} + \end{figure} Nach den o.\,s. Tafeln ist die erste Gruppe, $S_{2}$, kommutativ und die zweite, $S_{3}$, nicht. @@ -5232,6 +5304,88 @@ wobei Dann $f(f^{-1}(B))=f(f^{-1}(Y))=f(X)=\{1\}\subset Y$ (strikt). \end{enumerate} +%% ******************************************************************************** +%% FILE: body/quizzes/quiz4.tex +%% ******************************************************************************** + +\setcounternach{chapter}{4} +\chapter[Woche 4]{Woche 4} + \label{quiz:4} + +Gegeben seien Mengen $X$, $Y$, $Z$, +und Funktionen $f:X\to Y$ und $g:Y\to Z$. +Wir betrachten die Komposition ${g\circ f:X\to Z}$ + +\hraum +\begin{tikzpicture}[node distance=0.5cm, thick] + \pgfmathsetmacro\habst{3} + \pgfmathsetmacro\vabst{1} + \node[label=below:{$X$}] (SetX) at (0*\habst,0*\vabst) {$\bullet$}; + \node[label=below:{$Y$}] (SetY) at (1*\habst,0*\vabst) {$\bullet$}; + \node[label=below:{$Z$}] (SetZ) at (2*\habst,0*\vabst) {$\bullet$}; + \draw (SetX) edge [->] node [pos=0.5, above] {\footnotesize $f$} (SetY); + \draw (SetY) edge [->] node [pos=0.5, above] {\footnotesize $g$} (SetZ); +\end{tikzpicture} +\hraum + +\begin{enumerate}{\bfseries (a)} + %% QUIZ 4-a + \item + + \begin{claim*} + $g\circ f$ injektiv $\Rightarrow$ $f$ injektiv. + \end{claim*} + + \begin{proof} + Angenommen, $g\circ f$ sei injektiv. + \textbf{Zu zeigen:} $f$ ist injektiv\\ + \textbf{Zu zeigen:} Für alle $x_{1},x_{2}\in X$ gilt $f(x_{1})=f(x_{2})\Rightarrow x_{1}=x_{2}$.\\ + Seien also $x_{1},x_{2}\in X$ beliebig. + Es gilt: + + \begin{mathe}[mc]{rcl} + f(x_{1}) = f(x_{2}) + &\Longrightarrow + &g(f(x_{1})) = g(f(x_{2}))\\ + &\Longrightarrow + &(g\circ f)(x_{1}) = (g\circ f)(x_{2})\\ + &\Longrightarrow + &x_{1} = x_{2}, + \,\text{da $g\circ f$ injektiv}.\\ + \end{mathe} + + Also ist $f$ injektiv. + \end{proof} + + %% QUIZ 4-b + \item + \begin{claim*} + $f,g$ injektiv $\Rightarrow$ $g\circ f$ injektiv. + \end{claim*} + + \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 %% ********************************************************************************