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\def\ntrlzero{\mathbb{N}_{0}} \def\reellNonNeg{\reell_{+}} +\def\imageinh{\imath} +\def\ReTeil{\mathop{\mathfrak{R}\text{\upshape e}}} +\def\ImTeil{\mathop{\mathfrak{I}\text{\upshape m}}} + \def\leer{\emptyset} \def\restr#1{\vert_{#1}} \def\ohne{\setminus} @@ -3535,11 +3549,249 @@ für $a,b\in\intgr$. Darum gilt $\Phi(n)$ per Induktion für alle $n\in\ntrl$. \end{proof} +%% ******************************************************************************** +%% FILE: body/uebung/ueb5.tex +%% ******************************************************************************** + +\setcounternach{chapter}{5} +\chapter[Woche 5]{Woche 5} + \label{ueb:5} + +\textbf{ACHTUNG.} +Diese Lösungen dienen \emph{nicht} als Musterlösungen sondern eher als Referenz. +Hier wird eingehender gearbeitet, als generell verlangt wird. +Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche vergleichen kann. + +%% AUFGABE 5-1 +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 1]{} + \label{ueb:5:ex:1} +\let\sectionname\altsectionname + +Fixiere eine natürliche Zahl $n\in\ntrlzero$. +Sei $a_{i}\in\{0,1,\ldots,10-1\}$ die eindeutige Zahlen, +so dass + + \begin{mathe}[mc]{rcl} + n &= &\sum_{i\in\ntrlzero}a_{i}10^{i}\\ + \end{mathe} + +gilt. + +\begin{enumerate}{\bfseries (a)} + %% AUFGABE 5-1a + \item + \begin{claim*} + $3\divides n$ $\Leftrightarrow$ $3\mid\sum_{i\in\ntrlzero}a_{i}$. + \end{claim*} + + \begin{proof} + Beachte zunächst, dass $10\equiv 1\mod 3$. + Also gilt modulo $3$ + + \begin{mathe}[mc]{rcccl} + n &\equiv &\sum_{i\in\ntrlzero}a_{i}1^{i} &\equiv &\sum_{i\in\ntrlzero}a_{i}.\\ + \end{mathe} + + Folglich gilt + + \begin{mathe}[mc]{rcccccl} + 3\mid n + &\Longleftrightarrow + &n\equiv 0\mod 3 + &\Longleftrightarrow + &\sum_{i\in\ntrlzero}a_{i}\equiv 0\mod 3 + &\Longleftrightarrow + &3\mid\sum_{i\in\ntrlzero}a_{i} + \end{mathe} + + wie behauptet. + \end{proof} + + %% AUFGABE 5-1b + \item + \begin{claim*} + $11\divides n$ $\Leftrightarrow$ $1\mid\sum_{i\in\ntrlzero}(-1)^{i}a_{i}$. + \end{claim*} + + \begin{proof} + Beachte zunächst, dass $10=-1\mod 11$. + Also gilt modulo $11$ + + \begin{mathe}[mc]{rcccl} + n &\equiv &\sum_{i\in\ntrlzero}a_{i}(-1)^{i}.\\ + \end{mathe} + + Folglich gilt + + \begin{mathe}[mc]{rcccccl} + 11\mid n + &\Longleftrightarrow + &n\equiv 0\mod 11 + &\Longleftrightarrow + &\sum_{i\in\ntrlzero}a_{i}\equiv 0\mod 11 + &\Longleftrightarrow + &11\mid\sum_{i\in\ntrlzero}(-1)^{i}a_{i} + \end{mathe} + + wie behauptet. + \end{proof} +\end{enumerate} + +%% AUFGABE 5-2 +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 2]{} + \label{ueb:5:ex:2} +\let\sectionname\altsectionname + +\begin{enumerate}{\bfseries (a)} + %% AUFGABE 5-2a + \item + + Seien $a=142$ und $b=84$. + Wir berechnen $\ggT(a,b)$ mittels des Euklidischen Algorithmus + (siehe \cite[Satz 3.4.7]{sinn2020}). + + \begin{longtable}[mc]{|c|c|} + \hline + \hline + Restberechnung (symbolisch) &Restberechnung (Werte)\\ + \hline + \endhead + $a = b\cdot q_{1} + r_{1}$ &$142 = 84\cdot 1 + 58$\\ + $b = r_{1}\cdot q_{2} + r_{2}$ &$84 = 58\cdot 1 + 26$\\ + $r_{1} = r_{2}\cdot q_{3} + r_{3}$ &$58 = 26\cdot 2 + 6$\\ + $r_{2} = r_{3}\cdot q_{4} + r_{4}$ &$26 = 6\cdot 4 + \boxed{\mathbf{2}}$\\ + $r_{3} = r_{4}\cdot q_{5} + r_{5}$ &$6 = 2\cdot 3 + 0$\\ + \hline + \hline + \end{longtable} + + Darum gilt $\ggT(a,b)=r_{2}=2$. + + %% AUFGABE 5-2b + \item + + \begin{claim} + \makelabel{claim:main:ueb:5:ex:2b} + Seien $a,b,c\in\intgr$ mit $a,b\neq 0$. + Die folgenden Aussagen sind äquivalent: + + \begin{kompaktenum}{\bfseries (i)}[\rtab][\rtab] + \item\punktlabel{1} + $\exists{x,y\in\intgr:~}ax+by=c$ + \item\punktlabel{2} + $\ggT(a,b)\divides c$ + \end{kompaktenum} + + \end{claim} + + \begin{proof} + Fixiere zunächst $d:=\ggT(a,b)$. + Da $a,b\in\intgr\ohne\{0\}$, ist $d\in\ntrl$ eine wohldefinierte positive Zahl. + + \hinRichtung{1}{2} + Angenommen, $ax+by=c$ für ein $x,y\in\intgr$.\\ + Da $x,y\in\intgr$, + erhalten wir $c=ax+by\equiv 0x+0z\equiv 0$ modulo $d$.\\ + Also $\ggT(a,b)=d\divides c$. + + \hinRichtung{2}{1} + Angenommen, $\ggT(a,b)\divides c$.\\ + Dann existiert ein $k\in\intgr$, so dass $c=k\cdot\ggT(a,b)$.\\ + Laut des Lemmas von B\'ezout (siehe \cite[Lemma 3.4.8]{sinn2020}) + existiere nun $u,v\in\intgr$, so dass $\ggT(a,b)=au+bv$.\\ + Daraus folgt ${c=k\cdot\ggT(a,b)=aku+bkv}$.\\ + Da $ku,kv\in\intgr$, haben wir \punktcref{1} bewiesen. + \end{proof} +\end{enumerate} + +%% AUFGABE 5-3 +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 3]{} + \label{ueb:5:ex:3} +\let\sectionname\altsectionname + +\begin{enumerate}{\bfseries (a)} + %% AUFGABE 5-3a + \item + + Sei $H:=\intgr/2\intgr$ die (abelsche) Gruppe von Restklassen modulo $2$ unter Addition.\\ + Sei $G:=H\times H$ mit Neutralelement $e=([0],[0])$ und versehen mit der Produktstruktur.\\ + Als Produkt von (abelschen) Gruppen ist $G$ automatisch eine (abelsche) Gruppe. + Und offensichtlich hat $G$ genau $|G|=|H\times H|=|H|\cdot|H|=2\cdot 2=4$ Elemente.\\ + Es bleibt \textbf{zu zeigen}, dass $\forall{a\in G:~}a\ast a=e$.\\ + Sei also $a=([k],[j])\in H\times H=G$ ein beliebiges Element. + Es gilt + + \begin{mathe}[mc]{rcl} + a\ast a + &= &([k],[j])\ast([k],[j])\\ + &= &([k]+[k],[j]+[j])\\ + &= &([k+k],[j+j])\\ + &= &([2k],[2j]) + =([0],[0]) + =e, + \end{mathe} + + da $2\equiv 0\mod 2$.\\ + Also ist unsere Konstruktion von $G$ ein passendes Beispiel. + + %% AUFGABE 5-3b + \item + \begin{claim*} + Sei $(G,\ast,e)$ eine Gruppe. + Angenommen, $\forall{a\in G:~}a\ast a=e$. + Dann ist $G$ kommutativ. + \end{claim*} + + \begin{proof} + Beachte zunächst, dass wegen Eindeutigkeit des Inverses + die Annahme zu + + \begin{mathe}[mc]{c} + \eqtag[eq:1:ueb:5:ex:3] + \forall{a\in G:~}a^{-1}=a + \end{mathe} + + äquivalent ist.\\ + \textbf{Zu zeigen:} Für alle $a,b\in G$ gilt $a\ast b=b\ast a$.\\ + Seien also $a,b\in G$ beliebige Elemente. + Es gilt + + \begin{mathe}[mc]{rcccccl} + a\ast b + &\eqcrefoverset{eq:1:ueb:5:ex:3}{=} + &a^{-1}\ast b^{-1} + &= &(b\ast a)^{-1} + &\eqcrefoverset{eq:1:ueb:5:ex:3}{=} + &b\ast a.\\ + \end{mathe} + + Also ist $G$ eine kommutative Gruppe. + \end{proof} +\end{enumerate} + \setcounternach{part}{2} \part{Selbstkontrollenaufgaben} \def\chaptername{SKA Blatt} +%% ******************************************************************************** +%% FILE: body/ska/ska1.tex +%% ******************************************************************************** + +%% ******************************************************************************** +%% FILE: body/ska/ska2.tex +%% ******************************************************************************** + +%% ******************************************************************************** +%% FILE: body/ska/ska3.tex +%% ******************************************************************************** + %% ******************************************************************************** %% FILE: body/ska/ska4.tex %% ******************************************************************************** @@ -5040,14 +5292,14 @@ Um diese Urteil also leichter treffen zu können ersetzen wir die Elemente durch \pgfmathsetmacro\vabst{1} 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size=0.9*\rad, fill={rgb,255:white,64}, draw] (gh_1_1) at (2*\habst, -2*\vabst) {}; + \node[rectangle, label=left:{$e=:$}, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,64;blue,200}, draw] (g_0) at (0*\habst, -1*\vabst) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,64;blue,200}, 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:red,64;green,64;blue,200}, draw] (g_1) at (0*\habst, -2*\vabst) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,200}, draw] (h_1) at (2*\habst, 0*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,64;blue,200}, draw] (gh_0_0) at (1*\habst, -1*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,200}, draw] (gh_0_1) at (2*\habst, -1*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,200}, draw] (gh_1_0) at (1*\habst, -2*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,64;blue,200}, draw] (gh_1_1) at (2*\habst, -2*\vabst) {}; \end{tikzpicture} } \hraum @@ -5057,54 +5309,54 @@ Um diese Urteil also leichter treffen zu können ersetzen wir die Elemente durch \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) {}; - 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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) {}; + \node[rectangle, label=left:{$e=:$}, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,64;blue,200}, draw] (g_0) at (0*\habst, -1*\vabst) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,64;blue,200}, 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:red,200;green,200;blue,200}, draw] (g_1) at (0*\habst, -2*\vabst) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,200;blue,200}, 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:red,64;green,64;blue,64}, draw] (g_2) at (0*\habst, -3*\vabst) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,64}, 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:red,64;green,200;blue,200}, draw] (g_3) at (0*\habst, -4*\vabst) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,200}, 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:red,64;green,64;blue,200}, draw] (g_4) at (0*\habst, -5*\vabst) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,200}, 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:red,64;green,200;blue,64}, draw] (g_5) at (0*\habst, -6*\vabst) {}; + \node[rectangle, line width=2pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,64}, draw] (h_5) at (6*\habst, 0*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,64;blue,200}, draw] (gh_0_0) at (1*\habst, -1*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,200;blue,200}, draw] (gh_0_1) at (2*\habst, -1*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,64}, draw] (gh_0_2) at (3*\habst, -1*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,200}, draw] (gh_0_3) at (4*\habst, -1*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,200}, draw] (gh_0_4) at (5*\habst, -1*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,64}, draw] (gh_0_5) at (6*\habst, -1*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,200;blue,200}, draw] (gh_1_0) at (1*\habst, -2*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,64;blue,200}, draw] (gh_1_1) at (2*\habst, -2*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,200}, draw] (gh_1_2) at (3*\habst, -2*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,64}, draw] (gh_1_3) at (4*\habst, -2*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,64}, draw] (gh_1_4) at (5*\habst, -2*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,200}, draw] (gh_1_5) at (6*\habst, -2*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,64}, draw] (gh_2_0) at (1*\habst, -3*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,200}, draw] (gh_2_1) at (2*\habst, -3*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,64;blue,200}, draw] (gh_2_2) at (3*\habst, -3*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,200;blue,200}, draw] (gh_2_3) at (4*\habst, -3*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,64}, draw] (gh_2_4) at (5*\habst, -3*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,200}, draw] (gh_2_5) at (6*\habst, -3*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,200}, draw] (gh_3_0) at (1*\habst, -4*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,64}, draw] (gh_3_1) at (2*\habst, -4*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,64}, draw] (gh_3_2) at (3*\habst, -4*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,200}, draw] (gh_3_3) at (4*\habst, -4*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,64;blue,200}, draw] (gh_3_4) at (5*\habst, -4*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,200;blue,200}, draw] (gh_3_5) at (6*\habst, -4*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,200}, draw] (gh_4_0) at (1*\habst, -5*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,64}, draw] (gh_4_1) at (2*\habst, -5*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,200;blue,200}, draw] (gh_4_2) at (3*\habst, -5*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,64;blue,200}, draw] (gh_4_3) at (4*\habst, -5*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,200}, draw] (gh_4_4) at (5*\habst, -5*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,64}, draw] (gh_4_5) at (6*\habst, -5*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,64}, draw] (gh_5_0) at (1*\habst, -6*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,200}, draw] (gh_5_1) at (2*\habst, -6*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,200;blue,200}, draw] (gh_5_2) at (3*\habst, -6*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,64;green,64;blue,64}, draw] (gh_5_3) at (4*\habst, -6*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,200;blue,200}, draw] (gh_5_4) at (5*\habst, -6*\vabst) {}; + \node[rectangle, line width=0.5pt, minimum size=0.9*\rad, fill={rgb,255:red,200;green,64;blue,200}, draw] (gh_5_5) at (6*\habst, -6*\vabst) {}; \end{tikzpicture} } \hraum @@ -5386,6 +5638,47 @@ Wir betrachten die Komposition ${g\circ f:X\to Z}$ \end{proof} \end{enumerate} +%% ******************************************************************************** +%% FILE: body/quizzes/quiz5.tex +%% ******************************************************************************** + +\setcounternach{chapter}{5} +\chapter[Woche 5]{Woche 5} + \label{quiz:5} + +\begin{claim*} + Seien $n\in\ntrlpos$ und $p\in\mathbb{P}$ mit $n