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z^`TTYal53OY|;FV$)bp#{Pp7AB|FAZsUwa<4+|705hlh&ZH*NV5XK<^5Zy!uBBAPF z$UO7zyagWoOqOfm*?A0{`6(*r#=UbB*zxmAybz_EHtks#a04`3g`-$N716|jNX!cp zq`Qfc$Um1Y6eVUsE&cB;6o1Fzgw2bqlxOBpojY2Odm|cy{I*CTpMb=NXJlbuVt^$j J6_FE#{XZW93p@Y- diff --git a/docs/loesungen.tex b/docs/loesungen.tex index a9b94c3..0e8e83c 100644 --- a/docs/loesungen.tex +++ b/docs/loesungen.tex @@ -55,6 +55,8 @@ %% | %% — body/ska/ska4.tex; %% | +%% — body/ska/ska5.tex; +%% | %% — body/quizzes/quiz1.tex; %% | %% — body/quizzes/quiz2.tex; @@ -4445,6 +4447,559 @@ für $a,b\in\intgr$. Das heißt das Induktionsargument ist faul, weil der Schritt $1\rightsquigarrow 2$ implizit übersprungen wird. +%% ******************************************************************************** +%% FILE: body/ska/ska5.tex +%% ******************************************************************************** + +\setcounternach{chapter}{5} +\chapter[Woche 5]{Woche 5} + \label{ska:5} + +%% SKA 5-2 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{2} +\section[Aufgabe 2]{} + \label{ska:5:ex:2} +\let\sectionname\altsectionname + +Betrachtet sei die Teilbarkeitsrelation $(\intgr,\divides)$. +Wir prüfen, welche Axiome erfüllt sind und beurteilen aufgrund dessen, +ob es sich um eine Äquivalenzrelation, partielle Ordnung, Abbildung, usw. handelt. + +\begin{kompaktenum}[\rtab][\rtab] + \item[\uwave{{\bfseries Reflexivität:}}] + Sei $a\in\intgr$ beliebig. + \textbf{Zu prüfen:} $a\divides a$?\\ + Es gilt $a=1\cdot a$ und $1\in\intgr$.\\ + Darum gilt $a\divides a$.\\ + Also ist $(\intgr,\divides)$ \fbox{reflexiv}. + \item[\uwave{{\bfseries Symmetrie:}}] + Betrachte bspw. $2,10\in\intgr$.\\ + Es gilt $2\divides 10$ aber $10\ndivides 2$.\\ + Darum ist $(\intgr,\divides)$ \fbox{nicht symmetrisch}. + \item[\uwave{{\bfseries Antisymmetrie:}}] + Betrachte bspw. $2,-2\in\intgr$.\\ + Es gelten $2\divides -2$ und $-2\divides 2$, aber $2\neq -2$.\\ + Darum ist $(\intgr,\divides)$ \fbox{nicht antisymmetrisch}. + \item[\uwave{{\bfseries Transitivität:}}] + Seien, $a,b,c\in\intgr$. + \textbf{Zu prüfen:} ($a\divides b$ und $b\divides c$) $\Rightarrow$ $a\divides c$?\\ + Es gilt: + + \begin{mathe}[mc]{rcl} + a\divides b\,\text{und}\,b\divides c + &\Longleftrightarrow + &\exists{k,j\in\intgr:~}c=kb,\,b=ja\\ + &\Longrightarrow + &\exists{k,j\in\intgr:~}c=(kj)a\\ + &\Longrightarrow + &\exists{m\in\intgr:~}c=ma\\ + &\Longleftrightarrow + &a\divides c.\\ + \end{mathe} + + Also ist $(\intgr,\divides)$ \fbox{transitiv}. + \item[\uwave{{\bfseries Totalität:}}] + Betrachte bspw. $5,7\in\intgr$.\\ + Dann $5\ndivides 7$, $7\ndivides 5$, und $5\neq 7$.\\ + Darum ist $(\intgr,\divides)$ \fbox{nicht total}. + \item[\uwave{{\bfseries Linkstotalität:}}] + Sei $a\in\intgr$. + \textbf{Zu prüfen:} $\exists{b\in\intgr:~}a\divides b$?\\ + Wegen Reflexivität gilt nun $a\divides a$.\\ + Also ist $(\intgr,\divides)$ \fbox{linkstotal}. + \item[\uwave{{\bfseries Rechtseindeutigkeit:}}] + Betrachte bspw. $2,10,100\in\intgr$.\\ + Es gilt $2\divides 10$ und $2\divides 100$, aber $10\neq 100$.\\ + Darum ist $(\intgr,\divides)$ \fbox{nicht rechtseindeutig}. +\end{kompaktenum} + +Daraus folgt, dass $(\intgr,\divides)$ +weder + eine Äquivalenzrelation + noch + eine (lineare) Ordnungsrelation + noch + eine partielle Ordnungsrelation +ist. +Und es gibt keine Funktion ${f:\intgr\to\intgr}$, +so dass $\graph(f)=\divides$. + +\textbf{Bemerkung:} Man kann aber zeigen, das die Beschränkungen + $(\ntrlzero,\divides)$ +und $(\ntrlpos,\divides)$ +zusätzlich Antisymmetrie aufweisen, +sodass diese partielle Ordnungsrelationen sind. + +%% SKA 5-3 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{3} +\section[Aufgabe 3]{} + \label{ska:5:ex:3} +\let\sectionname\altsectionname + +Seien $a,b\in\intgr$ mit $b>0$. +Um $a$ durch $b$ (mit Rest) zu teilen, +setzt man + + \begin{mathe}[mc]{rcl} + q &:= &[a/b] \in\intgr\\ + r &:= &a-b\cdot q\in\intgr.\\ + \end{mathe} + +wobei ${[\cdot]:\reell\to\intgr}$ die Gaußklammerfunktion ist, +die reelle Zahlen \emph{abrundet}. +Per Definition gilt + + \begin{mathe}[mc]{rcccl} + q &\leq &a/b &< &q+1\\ + \end{mathe} + +also + + \begin{mathe}[mc]{rcccl} + 0 &\leq &a-b\cdot q &< &b\\ + \end{mathe} + +Darum $r\in\{0,1,\ldots,b-1\}$. + +Um dies aber \emph{per Hand} bzw. im Kopf zu machen, +verwendet man iterative Algorithmen, +die aus Schritten besteht: + $a$ und $b$ durch »einfachere« Zahlen ersetzen; + mit einfacheren Zahlen teilen; + Nachjustieren. + +Welche Methode auch immer man anwendet hat dies mit \fbox{dem Existenz-Teil} des Beweises zu tun. + +%% SKA 5-4 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{4} +\section[Aufgabe 4]{} + \label{ska:5:ex:4} +\let\sectionname\altsectionname + +\begin{claim} + \makelabel{claim:main:ska:5:ex:4} + Es gilt + + \begin{kompaktenum}{\bfseries (i)}[\rtab][\rtab] + \item\punktlabel{1} + $\ggT(a,b)=\ggT(b,a)$; + \item\punktlabel{2} + $\ggT(a,b)=\ggT(|a|,|b|)$; + \item\punktlabel{3} + $\ggT(a,0)=\ggT(0,a)=|a|$, solange $a\neq 0$; + \item\punktlabel{4} + $\ggT(ca,cb)=|c|\ggT(a,b)$, solange $b,c\neq 0$; + \end{kompaktenum} + + für $a,b,c\in\intgr$. +\end{claim} + + \begin{einzug}[\rtab][\rtab] + \begin{proof} + \uline{\bfseries \punktcref{1}:} + Es gilt + $\ggT(a,b) + =\max\{d\in\ntrlpos\mid d\divides a,b\} + =\max\{d\in\ntrlpos\mid d\divides b,a\} + =\ggT(b,a)$. + + \uline{\bfseries \punktcref{2}:} + Sei $d,x\in\intgr$. Dann ist es einfach zu sehen, + dass $d\divides x\Leftrightarrow d\divides|x|$.\\ + Darum gilt + $\ggT(a,b) + =\max\{d\in\ntrlpos\mid d\divides a,b\} + =\max\{d\in\ntrlpos\mid d\divides |a|,|b|\} + =\ggT(|a|,|b|)$. + + \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 $d_{0}\divides a,0$.\\ + (II) Für alle $d\in\ntrlpos$ gilt + $d\divides a$ + $\Rightarrow$ $|\frac{a}{d}|\geq 1$ + $\Rightarrow$ $d\leq|a|=d_{0}$.\\ + (III) Für alle $d\in\ntrlpos$ gilt $d\divides 0$.\\ + Zusammengefasst, + + \begin{mathe}[mc]{rcccccl} + d_{0} + &\textoverset{(I)}{\in} + &D + &\textoverset{(III)}{=} + &\{d\in\ntrlpos\mid d\divides a\} + &\textoverset{(II)}{\subseteq} + &\{d\in\ntrlpos\mid d\leq d_{0} + \end{mathe} + + woraus sich ergibt, dass $d_{0}\leq\max D\leq d_{0}$. + Also $\ggT(a,0)=\max D=d_{0}=|a|$. + + \uline{\bfseries \punktcref{4}:} + Setze $d_{1}:=\ggT(a,b)$ und $d_{2}:=\ggT(ca,cb)$. + \textbf{Zu zeigen:} $d_{2}=|c|d_{1}$.\\ + Wir zeigen dies durch zwei Ungleichungen.\\ + Es gilt + + \begin{mathe}[mc]{rcl} + d_{1}\divides a,b + &\Longleftrightarrow + &\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}\\ + &\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}\\ + &\Longleftrightarrow + &|c|d_{1}\divides ca,cb\\ + \end{mathe} + + Per Maximalität von $d_{2}$ unter den positiven Teilern, + folgt \fbox{$|c|d_{1}\leq d_{2}$}.\\ + Andererseits existieren nach dem \emph{Lemma von B\'ezout} + $u,v\in\intgr$, so dass + + \begin{mathe}[mc]{rcl} + d_{1}=\ggT(a,b) &= &ua+vb\\ + \end{mathe} + + woraus sich ergibt, dass + + \begin{mathe}[mc]{rcl} + \eqtag[eq:1:\beweislabel] + \frac{|c|d_{1}}{d_{2}} &= &\underbrace{% + \textstyle\pm u\frac{ca}{d_{2}}+\pm v\frac{cb}{d_{2}} + }_{=:w}\\ + \end{mathe} + + Da $d_{2}\divides ca,cb$ ist die rechte Seite von \eqcref{eq:1:\beweislabel} in $\intgr$. + Und da $|c|,d_{1},d_{2}>0$, ist die linke Seite von \eqcref{eq:1:\beweislabel} strikt positiv, + sodass $w\geq 1$ gilt. + Darum \fbox{$|c|d_{1}=w\cdot d_{2}\geq 1\cdot d_{2}$}. + \end{proof} + \end{einzug} + +\textbf{Bemerkung.} +Ohne das \emph{Lemma von B\'ezout} ist ein Beweis vom Letzten Punkt praktisch unmachbar. + +%% SKA 5-5 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{5} +\section[Aufgabe 5]{} + \label{ska:5:ex:5} +\let\sectionname\altsectionname + +Seien $a=57$ und $b=21$. +Dann gilt $a=qb+r$, wobei $q=2$ und \fbox{$r=15$}. +Es gilt + $\ggT(a,b)=3$\footnote{% + weil $r=3\cdot 5$ und $3,5\in\mathbb{P}$ und nur $3\divides 57$ gilt. + } +und + $\ggT(b,r)=3$\footnote{% + weil $r=3\cdot 5$ und $3,5\in\mathbb{P}$ und nur $3\divides 21$ gilt. + } +Also gilt $\ggT(a,b)=\ggT(b,\modfn(a,b))$, +genau wie \cite[Lemma 3.4.5]{sinn2020} allgemein besagt. + +%% SKA 5-6 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{6} +\section[Aufgabe 6]{} + \label{ska:5:ex:6} +\let\sectionname\altsectionname + +Für jeden Fall berechnen wir $\ggT(a,b)$ mittels des Euklidischen Algorithmus +(siehe \cite[Satz 3.4.7]{sinn2020}). + + \begin{longtable}[mc]{|cc|c|c|} + \hline + \hline + $a$ &$b$ &Restberechnung (symbolisch) &Restberechnung (Werte)\\ + \hline + \endhead +$1529$ &$170$ &$a = b\cdot q_{1} + r_{1}$ &$1529 = 170\cdot 8 + 169$\\ +&&$b = r_{1}\cdot q_{2} + r_{2}$ &$170 = 169\cdot 1 + \boxed{1}$\\ +&&$r_{1} = r_{2}\cdot q_{3} + r_{3}$ &$169 = 1\cdot 169 + 0$\\ +\hline +$13758$ &$21$ &$a = b\cdot q_{1} + r_{1}$ &$13758 = 21\cdot 655 + \boxed{3}$\\ +&&$b = r_{1}\cdot q_{2} + r_{2}$ &$21 = 3\cdot 7 + 0$\\ +\hline +$210$ &$45$ &$a = b\cdot q_{1} + r_{1}$ &$210 = 45\cdot 4 + 30$\\ +&&$b = r_{1}\cdot q_{2} + r_{2}$ &$45 = 30\cdot 1 + \boxed{15}$\\ +&&$r_{1} = r_{2}\cdot q_{3} + r_{3}$ &$30 = 15\cdot 2 + 0$\\ +\hline +$1209$ &$102$ &$a = b\cdot q_{1} + r_{1}$ &$1209 = 102\cdot 11 + 87$\\ +&&$b = r_{1}\cdot q_{2} + r_{2}$ &$102 = 87\cdot 1 + 15$\\ +&&$r_{1} = r_{2}\cdot q_{3} + r_{3}$ &$87 = 15\cdot 5 + 12$\\ +&&$r_{2} = r_{3}\cdot q_{4} + r_{4}$ &$15 = 12\cdot 1 + \boxed{3}$\\ +&&$r_{3} = r_{4}\cdot q_{5} + r_{5}$ &$12 = 3\cdot 4 + 0$\\ + \hline + \hline + \end{longtable} + +%% SKA 5-7 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{7} +\section[Aufgabe 7]{} + \label{ska:5:ex:7} +\let\sectionname\altsectionname + +%% SKA 5-8 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{8} +\section[Aufgabe 8]{} + \label{ska:5:ex:8} +\let\sectionname\altsectionname + +Das Lemma von B\'ezout wird mittels des \fbox{Euklidischen Algorithmus} bewiesen.\\ +Korollar 3.4.10 baut darauf und charakterisiert, wann zwei Zahlen teilerfremd sind.\\ +Lemma 3.4.12 baut darauf und zeigt $\forall{i:~}b,a_{i}\,\text{teilerfremd}\Rightarrow b,\prod_{i=1}^{n}a\,\text{teilerfremd}$.\\ +Satz 3.4.14 baut darauf und zeigt $p\divides\prod_{i=1}^{n}a_{i}\Rightarrow \exists{i:~}p\divides a_{i}$ für $p$ prim. + +Dieses letzte Ergebnis wird im Induktionsargument instrumentalisiert, +um Primzerlegungen der Länge $k,l$ auf Primfaktorzerlegungen der Länge $k-1,l-1$ zu reduzieren, +um das Induktionsargument voranzubringen. + +\begin{rem} + In der Algebra gibt es zwei Begriffe, die bei gewöhnlichen Primzahlen, sich anwenden lassen: + \emph{Irriduzibilität} und \emph{prim}. + Die Definition in abstrakten Kontexten von \emph{prim} entspricht der Eigenschaft in \cite[Satz 3.4.14]{sinn2020}, + während \emph{Irriduzibilität} eher sich auf die Teilbarkeit bezieht. + Etwas »Zufälligerweise« handelt es sich bei $\intgr$ um eine Art von Struktur, + in der diese zwei Konzepte zusammenfallen. + Wie in fast allen technischen Bereichen sollte man auf solche »Zufälligkeiten« achten: + Irgendwann befindet man sich in einer Situation, + wo man feiner unterscheiden muss und es nicht mehr selbstverständlich ist, + zwei Konzepte als identisch zu behandeln. +\end{rem} + +%% SKA 5-10 +\let\altsectionname\sectionname +\def\sectionname{SKA} +\setcounternach{section}{10} +\section[Aufgabe 10]{} + \label{ska:5:ex:10} +\let\sectionname\altsectionname + +Siehe \cite[Satz 3.5.1]{sinn2020}. +Hier eine Alternative:\\ +Für $r=0$ setze man $q_{r}:=1$ und $p_{r}:=0$. +Und für alle anderen rationalen Zahlen, $r\in\rtnl\ohne\{0\}$, wähle + + \begin{mathe}[mc]{rcl} + q_{r} &:= &\min\overbrace{% + \{n\in\ntrlpos\mid q_{r}\cdot r\in\intgr\} + }^{D(r)}\in\ntrlpos\\ + p_{r} &:= &q_{r}\cdot r\in\intgr.\\ + \end{mathe} + +Da $r$ ration 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 $p_{r}/q_{r}=r$.) +Damit haben wir die Existenz einer kanonischen Darstellung begründet. + +Stimmt dies mit der Konstruktion im \cite[Satz 3.5.1]{sinn2020} überein? + +Für $r=0$ gilt offensichtlich $\ggT(p_{r},q_{r})=1$. +Für $r\in\rtnl\ohne\{0\}$ gilt $d:=\ggT(p_{r},q_{r})=1$, +denn sonst wäre $\frac{q_{r}}{d}$ eine positive natürliche Zahl in $D(r)$, +da $\frac{q_{r}}{d}\cdot r=\frac{q_{r}\cdot r}{d}=\frac{p_{r}}{d}\in\intgr$, +während $\frac{q_{r}}{d}0$. + Angenommen, wir haben bereits + $((a_{k},b_{k}))_{k=0}^{n-1}$ + konstruiert. + Aus $(\frac{a_{n}}{b_{n}})^{2}=p$ + folgt nun $a_{n}^{2}=pb_{n}^{2}$. + Daraus folgt + ${p\divides a_{n}\cdot a_{n}}$ + und damit gilt (vgl. \cite[Satz 3.4.14]{sinn2020}) + ${p\divides a_{n}}$, + weil $p$ prim ist. + Da $b_{n}^{2}=p\cdot(\frac{a_{n}}{p})^{2}$ + und da $\frac{a_{n}}{p}\in\intgr$, + erhalten wir ebenfalls + ${p\divides b_{n}\cdot b_{n}}$ + und wiederum ${p\divides b_{n}}$. + + Setze also $a_{n+1}:=\frac{a_{n}}{p}$ und $b_{n+1}:=\frac{b_{n}}{p}$. + Dann wie oben gezeigt wurde, gilt $a_{n+1}\in\intgr$ + und $b_{n+1}\in\ntrlpos$. + Offensichtlich gilt $(\frac{a_{n+1}}{b_{n+1}})^{2}=(\frac{a_{n}}{b_{n}})^{2}=p$. + Und, da $p>1$, gilt $b_{n+1}