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0e8e83c..29472fe 100644 --- a/docs/loesungen.tex +++ b/docs/loesungen.tex @@ -4605,8 +4605,8 @@ Welche Methode auch immer man anwendet hat dies mit \fbox{dem Existenz-Teil} des \uline{\bfseries \punktcref{1}:} Es gilt $\ggT(a,b) - =\max\{d\in\ntrlpos\mid d\divides a,b\} - =\max\{d\in\ntrlpos\mid d\divides b,a\} + =\max\{d\in\ntrlpos : d\divides a,b\} + =\max\{d\in\ntrlpos : d\divides b,a\} =\ggT(b,a)$. \uline{\bfseries \punktcref{2}:} @@ -4614,29 +4614,28 @@ Welche Methode auch immer man anwendet hat dies mit \fbox{dem Existenz-Teil} des dass $d\divides x\Leftrightarrow d\divides|x|$.\\ Darum gilt $\ggT(a,b) - =\max\{d\in\ntrlpos\mid d\divides a,b\} - =\max\{d\in\ntrlpos\mid d\divides |a|,|b|\} + =\max\{d\in\ntrlpos : d\divides a,b\} + =\max\{d\in\ntrlpos : d\divides |a|,|b|\} =\ggT(|a|,|b|)$. \uline{\bfseries \punktcref{3}:} Laut \punktcref{1} reicht es aus \textbf{zu zeigen} $\ggT(a,0)=|a|$.\\ - Es gilt $\ggT(a,0)=\max D$, wobei $D:=\{d\in\ntrl\mid d\divides a,0\}$.\\ - (I) Setze $d_{0}:=|a|$. Offensichtlich gilt $d_{0}\divides a,0$.\\ + Es gilt $\ggT(a,0)=\max D$, wobei $D:=\{d\in\ntrlpos : d\divides a,0\}$.\\ + (I) Setze $d_{0}:=|a|>0$. Offensichtlich gilt $d_{0}\divides a,0$.\\ (II) Für alle $d\in\ntrlpos$ gilt $d\divides a$ $\Rightarrow$ $|\frac{a}{d}|\geq 1$ + (da $a,d\neq 0$) $\Rightarrow$ $d\leq|a|=d_{0}$.\\ - (III) Für alle $d\in\ntrlpos$ gilt $d\divides 0$.\\ - Zusammengefasst, + Daraus ergibt sich, \begin{mathe}[mc]{rcccccl} - d_{0} - &\textoverset{(I)}{\in} + d_{0} &\textoverset{(I)}{\in} &D - &\textoverset{(III)}{=} - &\{d\in\ntrlpos\mid d\divides a\} + &\subseteq + &\{d\in\ntrlpos : d\divides a\} &\textoverset{(II)}{\subseteq} - &\{d\in\ntrlpos\mid d\leq d_{0} + &\{d\in\ntrlpos : d\leq d_{0} \end{mathe} woraus sich ergibt, dass $d_{0}\leq\max D\leq d_{0}$. @@ -4651,12 +4650,12 @@ Welche Methode auch immer man anwendet hat dies mit \fbox{dem Existenz-Teil} des \begin{mathe}[mc]{rcl} d_{1}\divides a,b &\Longleftrightarrow - &\exists{k,j\in\intgr}a=kd_{1}\,\text{und}\,b=jd_{1}\\ + &\exists{k,j\in\intgr:~}a=kd_{1}\,\text{und}\,b=jd_{1}\\ &\Longleftrightarrow - &\exists{k,j\in\intgr}ca=kcd_{1}\,\text{und}\,cb=jcd_{1}\\ + &\exists{k,j\in\intgr:~}ca=kcd_{1}\,\text{und}\,cb=jcd_{1}\\ &\Longleftrightarrow - &\exists{k,j\in\intgr}ca=k|c|d_{1}\,\text{und}\,cb=j|c|d_{1}\\ - &&\text{da manz.\,B. $k$ durch $-k$ ersetzen kann}\\ + &\exists{k,j\in\intgr:~}ca=k|c|d_{1}\,\text{und}\,cb=j|c|d_{1}\\ + &&\text{da man z.\,B. $k$ durch $-k$ ersetzen kann}\\ &\Longleftrightarrow &|c|d_{1}\divides ca,cb\\ \end{mathe} @@ -4728,20 +4727,20 @@ Für jeden Fall berechnen wir $\ggT(a,b)$ mittels des Euklidischen Algorithmus \hline \endhead $1529$ &$170$ &$a = b\cdot q_{1} + r_{1}$ &$1529 = 170\cdot 8 + 169$\\ -&&$b = r_{1}\cdot q_{2} + r_{2}$ &$170 = 169\cdot 1 + \boxed{1}$\\ +&&$b = r_{1}\cdot q_{2} + r_{2}$ &$170 = 169\cdot 1 + \boxed{\mathbf{1}}$\\ &&$r_{1} = r_{2}\cdot q_{3} + r_{3}$ &$169 = 1\cdot 169 + 0$\\ \hline -$13758$ &$21$ &$a = b\cdot q_{1} + r_{1}$ &$13758 = 21\cdot 655 + \boxed{3}$\\ +$13758$ &$21$ &$a = b\cdot q_{1} + r_{1}$ &$13758 = 21\cdot 655 + \boxed{\mathbf{3}}$\\ &&$b = r_{1}\cdot q_{2} + r_{2}$ &$21 = 3\cdot 7 + 0$\\ \hline $210$ &$45$ &$a = b\cdot q_{1} + r_{1}$ &$210 = 45\cdot 4 + 30$\\ -&&$b = r_{1}\cdot q_{2} + r_{2}$ &$45 = 30\cdot 1 + \boxed{15}$\\ +&&$b = r_{1}\cdot q_{2} + r_{2}$ &$45 = 30\cdot 1 + \boxed{\mathbf{15}}$\\ &&$r_{1} = r_{2}\cdot q_{3} + r_{3}$ &$30 = 15\cdot 2 + 0$\\ \hline $1209$ &$102$ &$a = b\cdot q_{1} + r_{1}$ &$1209 = 102\cdot 11 + 87$\\ &&$b = r_{1}\cdot q_{2} + r_{2}$ &$102 = 87\cdot 1 + 15$\\ &&$r_{1} = r_{2}\cdot q_{3} + r_{3}$ &$87 = 15\cdot 5 + 12$\\ -&&$r_{2} = r_{3}\cdot q_{4} + r_{4}$ &$15 = 12\cdot 1 + \boxed{3}$\\ +&&$r_{2} = r_{3}\cdot q_{4} + r_{4}$ &$15 = 12\cdot 1 + \boxed{\mathbf{3}}$\\ &&$r_{3} = r_{4}\cdot q_{5} + r_{5}$ &$12 = 3\cdot 4 + 0$\\ \hline \hline @@ -4755,6 +4754,30 @@ $1209$ &$102$ &$a = b\cdot q_{1} + r_{1}$ &$1209 = 102\cdot 11 + 87$\\ \label{ska:5:ex:7} \let\sectionname\altsectionname +Wir verwenden die Berechnungen aus der Tabelle in \Cref{ska:5:ex:6}. + + \begin{longtable}[mc]{|cc|c|c|} + \hline + \hline + $a$ &$b$ &Rest (symbolisch) &Rest (Werte)\\ + \hline + \endhead +$1529$ &$170$ &$r_{1} = a - 8\cdot b$ &$169 = 1\cdot a + -8\cdot b$\\ +&&$r_{2} = b - 1\cdot r_{1}$ &$\boxed{1 = \mathbf{-1}\cdot a + \mathbf{9}\cdot b}$\\ +\hline +$13758$ &$21$ &$r_{1} = a - 655\cdot b$ &$\boxed{3 = \mathbf{1}\cdot a + \mathbf{-655}\cdot b}$\\ +\hline +$210$ &$45$ &$r_{1} = a - 4\cdot b$ &$30 = 1\cdot a + -4\cdot b$\\ +&&$r_{2} = b - 1\cdot r_{1}$ &$\boxed{15 = \mathbf{-1}\cdot a + \mathbf{5}\cdot b}$\\ +\hline +$1209$ &$102$ &$r_{1} = a - 11\cdot b$ &$87 = 1\cdot a + -11\cdot b$\\ +&&$r_{2} = b - 1\cdot r_{1}$ &$15 = -1\cdot a + 12\cdot b$\\ +&&$r_{3} = r_{1} - 5\cdot r_{2}$ &$12 = 6\cdot a + -71\cdot b$\\ +&&$r_{4} = r_{2} - 1\cdot r_{3}$ &$\boxed{3 = \mathbf{-7}\cdot a + \mathbf{83}\cdot b}$\\ + \hline + \hline + \end{longtable} + %% SKA 5-8 \let\altsectionname\sectionname \def\sectionname{SKA} @@ -4959,15 +4982,15 @@ Dies hat $2!=2$ Elemente: Die Gruppentafel sieht folgendermaßen aus: - \begin{longtable}{|CC|CC|} + \begin{longtable}{|CL|CC|} \hline \hline - &&\multicolumn{2}{C|}{h}\\ - &gh &e &(1\,2)\\ + &&\multicolumn{2}{c|}{$h$}\\ + &gh &e &(1\ 2)\\ \hline - \multirow{2}{*}{g} - &e &e &(1\,2)\\ - &(1\,2) &(1\,2) &e\\ + \multirow{2}{*}{$g$} + &e &e &(1\ 2)\\ + &(1\ 2) &(1\ 2) &e\\ \hline \hline \end{longtable} @@ -4986,7 +5009,24 @@ Dies hat $3!=6$ Elemente: Die Gruppentafel sieht folgendermaßen aus: -({\itshape Unter Arbeit}) + \begin{longtable}{|CL|CCCCCC|} + \hline + \hline + &&\multicolumn{6}{c|}{$h$}\\ + &gh &e &(2\ 3) &(1\ 2) &(1\ 2\ 3) &(1\ 3\ 2) &(1\ 3)\\ + \hline + \multirow{6}{*}{$g$} + &e &e &(2\ 3) &(1\ 2) &(1\ 2\ 3) &(1\ 3\ 2) &(1\ 3)\\ + &(2\ 3) &(2\ 3) &e &(1\ 3\ 2) &(1\ 3) &(1\ 2) &(1\ 2\ 3)\\ + &(1\ 2) &(1\ 2) &(1\ 2\ 3) &e &(2\ 3) &(1\ 3) &(1\ 3\ 2)\\ + &(1\ 2\ 3) &(1\ 2\ 3) &(1\ 2) &(1\ 3) &(1\ 3\ 2) &e &(2\ 3)\\ + &(1\ 3\ 2) &(1\ 3\ 2) &(1\ 3) &(2\ 3) &e &(1\ 2\ 3) &(1\ 2)\\ + &(1\ 3) &(1\ 3) &(1\ 3\ 2) &(1\ 2\ 3) &(1\ 2) &(2\ 3) &e\\ + \hline + \hline + \end{longtable} + +({\itshape Achtung: Tafel wurde per Code generiert, also ist die Reihenfolge möglicherweise nicht »ästhetisch«.}) %% SKA 5-15 \let\altsectionname\sectionname @@ -4996,9 +5036,10 @@ Die Gruppentafel sieht folgendermaßen aus: \label{ska:5:ex:15} \let\sectionname\altsectionname -Die erste Gruppe, $S_{2}$, ist kommutativ (»abelsch«). Das lässt sich daran erkennen, dass die Tafel symmetrisch ist. +An der Tafel lässt sich leicht erkennen, ob eine Gruppe kommutativ ist: +eine Gruppe, $G$, ist genau dann kommutativ, wenn die Gruppentafel symmetrisch ist. -Die zweite Gruppe, $S_{3}$, ist kommutativ (»abelsch«). Das lässt sich daran erkennen, dass die Tafel nicht symmetrisch ist. +Nach den o.\,s. Tafeln ist die erste Gruppe, $S_{2}$, kommutativ und die zweite, $S_{3}$, nicht. \setcounternach{part}{3} \part{Quizzes}