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back/index.tex; %% | @@ -1386,7 +1390,7 @@ \noindent \hraum{\footnotesize Raj Dahya}\hraum\\ - \hraum{\small \itshape Fakultät für Mathematik und Informatik/Institut für Philosophie}\hraum\\ + \hraum{\small \itshape Fakultät für Mathematik und Informatik}\hraum\\ \hraum{\small \itshape Universität Leipzig.}\hraum\\ \hraum{\small Wintersemester 2020/2021 }\hraum \end{titlepage} @@ -1398,10 +1402,9 @@ \chapter*{Vorwort} Dieses Dokument enthält Lösungsansätze zu den Übungsserien, Selbstkontrollenaufgaben, und Quizzes. -Diese werden natürlich \emph{nach} Abgabefristen hochgeladen -und dienen \emph{nicht} als Musterlösungen! +Diese werden natürlich \emph{nach} Abgabefristen hochgeladen. Der Zweck dieser Lösungen ist es vielmehr, Ansätze zu präsentieren, -mit denen man seine \emph{eigenen} Versuche vergleichen kann. +mit denen man seine eigenen Versuche vergleichen kann. %% ******************************************************************************** %% FILE: front/contents.tex @@ -3076,11 +3079,6 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion. \chapter[Woche 4]{Woche 4} \label{ueb:4} -\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 4-1 \let\altsectionname\sectionname \def\sectionname{Aufgabe} @@ -3110,13 +3108,13 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve \begin{kompaktenum}[\rtab][\rtab] \item[\uwave{{\itshape Reflexivität:}}] - Sei $(a,b)\in X$ beliebig. + Sei $(a,b)\in X$ beliebig.\\ \textbf{Zu zeigen:} $(a,b)\sim(a,b)$.\\ Offensichtlich gilt $ab=ab$.\\ Per Konstruktion gilt also $(a,b)\sim(a,b)$. \item[\uwave{{\itshape Symmetrie:}}] - Seien $(a,b),(a',b')\in X$ beliebig. + Seien $(a,b),(a',b')\in X$ beliebig.\\ \textbf{Zu zeigen:} ${(a,b)\sim(a',b')\Rightarrow(a',b')\sim(a,b)}$.\\ Es gilt @@ -3185,7 +3183,7 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve für $(a,b),(a',b')\in X$ definiert wird. \begin{claim*} - $(X,\leq)$ ist ist eine Halbordnung aber \fbox{nicht total}. + $(X,\leq)$ ist eine partielle Ordnung aber \fbox{nicht total}. \end{claim*} \begin{proof} @@ -3193,7 +3191,7 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve \begin{kompaktenum}[\rtab][\rtab] \item[\uwave{{\itshape Reflexivität:}}] - Sei $(a,b)\in X$ beliebig. + Sei $(a,b)\in X$ beliebig.\\ \textbf{Zu zeigen:} $(a,b)\leq(a,b)$.\\ Offensichtlich gilt $a\leq a$ und $b\leq b$.\\ Per Konstruktion gilt also $(a,b)\leq(a,b)$. @@ -3254,7 +3252,7 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve \end{mathe} \end{kompaktenum} - Darum erfüllt $(X,\leq)$ die Axiome einer Halbordnung.\\ + Darum erfüllt $(X,\leq)$ die einer partiellen Ordnung.\\ Zum Schluss, beachte, dass $(0,1)$ und $(1,0)$ bzgl. $\leq$ unvergleichbar sind. Darum ist $(X,\leq)$ nicht total. \end{proof} @@ -3405,8 +3403,10 @@ für $a,b\in\intgr$. Seien $q\in\intgr$ und $r\in\{0,1,\ldots,n-1\}$ mit $qn+r=k$. Da $k,r\in\{0,1,\ldots,n-1\}$, gilt $qn=k-r\in\intgr\cap(-n,n)$. - Also muss $q=0$ gelten. - Also $r=k$. + Das heißt, $k-r$ ist eine durch $n$ teilbare ganze Zahl + in $(-n,n)$. + Die einzige solche Zahl ist $0$. + Also $k-r=0$. Also $\modfn(k,n)=r=k$. } Also gilt $k\in\rho(\intgr/\lsim)$. @@ -3429,7 +3429,7 @@ für $a,b\in\intgr$. &= &\{a\in\intgr \mid \modfn(a,n)=\modfn(k,n)\}\\ &= &\{a\in\intgr \mid \modfn(a,n)=k\}\\ &= &\{a\in\intgr \mid \exists{q\in\intgr:~}a=qn+k\}\\ - &= &\{qn+r \mid q\in\intgr\}\\ + &= &\{qn+k \mid q\in\intgr\}\\ &= &\intgr\cdot n + k.\\ \end{mathe} @@ -6046,11 +6046,11 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$. Wir haben etwas ausführlicher gezeigt, dass die Menge $X'$ mindestens $n-1$ Goldfische enthält. Wenn wir $\tilde{x}:=x_{0}$ wählen entspricht dies der Größe des Schnitts $X_{0}\cap X_{1}$. Das \uline{Diagramm} mag andeuten, dass dieser Schnitt nicht leer ist, aber das Diagramm täuscht. - Im Induktionsschritt setzen wir nur voraus, dass $n\geq 1$. - Darum ist $n-1>0$ nur garantiert, wenn stattdessen $n'\geq 2$ vorausgesetzt wird. + Wir brauchen $n\geq 2$, damit der Schnitt nicht leer ist. + Aber im Induktionsschritt wird nur $n\geq 1$ vorausgesetzt. - Das heißt das Induktionsargument ist faul, - weil der Schritt $1\rightsquigarrow 2$ implizit übersprungen wird. + Kurzgesagt, das heißt das Induktionsargument ist faul, + \fbox{weil der Schritt $1\rightsquigarrow 2$ implizit übersprungen wird}. %% ******************************************************************************** %% FILE: body/ska/ska5.tex @@ -7612,6 +7612,222 @@ Wir betrachten die Komposition ${g\circ f:X\to Z}$ \end{proof} \end{enumerate} +%% ******************************************************************************** +%% FILE: body/quizzes/quiz7.tex +%% ******************************************************************************** + +\setcounternach{chapter}{7} +\chapter[Woche 7]{Woche 7} + \label{quiz:7} + +\begin{enumerate}{\bfseries 1.} + \item + \begin{defn*} + Sei $V$ ein Vektorraum über einem Körper $K$. + Seien $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}\in V$. + Die Vektoren, $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$, + heißen dann + \textbf{linear unabhängig}, + wenn für alle Skalare, $c_{1},c_{2},\ldots,c_{n}\in K$, + + \begin{mathe}[mc]{rcl} + \sum_{i=1}^{n}c_{i}\cdot\mathbf{v}_{i}=\zerovector + &\Longrightarrow + &\forall{i\in\{1,2,\ldots,n\}:~}c_{i}=0_{K} + \end{mathe} + + gilt. + \end{defn*} + \item + \begin{claim*} + Seien + + \begin{mathe}[mc]{rclqrclqrcl} + \mathbf{v}_{1} &= &\begin{svector}1\\2\\2\\\end{svector} + &\mathbf{v}_{2} &= &\begin{svector}0\\2\\1\\\end{svector} + &\mathbf{v}_{3} &= &\begin{svector}2\\1\\1\\\end{svector}\\ + \end{mathe} + + Vektoren im Vektorraum $\mathbb{F}_{5}^{3}$ über dem Körper $\mathbb{F}_{5}$. + Diese Vektoren sind linear unabhängig. + \end{claim*} + + \begin{proof} + Laut Satz \cite[Satz~5.2.4]{sinn2020} sind die Vektoren, + $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}$, + dann linear unabhängig, wenn $A\mathbf{x}=\zerovector$ + nur die Triviallösung hat, wobei + + \begin{mathe}[mc]{rcl} + A &= &\begin{matrix}{ccc} +1 &0 &2\\ +2 &2 &1\\ +2 &1 &1\\ +\end{matrix}.\\ + \end{mathe} + + Wir lösen also das homogene System: + + \begin{algorithm}[\rtab][\rtab] + Zeilenoperationen + ${Z_{2}\leftsquigarrow Z_{2} - 2\cdot Z_{1}}$ + und + ${Z_{3}\leftsquigarrow Z_{3} - 2\cdot Z_{1}}$ + anwenden:\footnotemark[ft:1:aufg:2:quiz:8] + + \begin{mathe}[mc]{rcl} + \begin{matrix}{ccc} +1 &0 &2\\ +0 &2 &2\\ +0 &1 &2\\ +\end{matrix}.\\ + \end{mathe} + + Zeilenoperation + ${Z_{3}\leftsquigarrow 2\cdot Z_{3} - Z_{2}}$ + anwenden: + + \begin{mathe}[mc]{rcl} + \begin{matrix}{ccc} +1 &0 &2\\ +0 &2 &2\\ +0 &0 &2\\ +\end{matrix}.\\ + \end{mathe} + \end{algorithm} + + \footnotetext[ft:1:aufg:2:quiz:8]{ + Beachte, dass wir im Körper $\mathbb{F}_{5}$ d.\,h. modulo 5 berechnen. + } + + Aus der Zeilenstufenform geht hervor, + dass es keine freien unbekannten gibt und insbesondere, + dass das homogene System nur die Triviallösung besitzt. + Darum sind die Vektoren linear unabhängig. + \end{proof} +\end{enumerate} + +%% ******************************************************************************** +%% FILE: body/quizzes/quiz8.tex +%% ******************************************************************************** + +\setcounternach{chapter}{8} +\chapter[Woche 8]{Woche 8} + \label{quiz:8} + +\begin{claim*} + Sei $V$ ein Vektorraum über Körper $K$ + und seien $U_{1},U_{2}\subseteq V$ lineare Unterräume. + Dann gilt + + \begin{mathe}[bc]{rcl} + U_{1}+U_{2} = U_{2} + &\Longleftrightarrow + &U_{1}\subseteq U_{2}.\\ + \end{mathe} + + \nvraum{1} +\end{claim*} + + \begin{einzug}[\rtab][\rtab] + \begin{proof} + \hinRichtung Angenommen, + + \begin{mathe}[bc]{rcl} + \eqtag[eq:1:quiz:8] + U_{1}+U_{2} &= &U_{2}.\\ + \end{mathe} + + \textbf{Zu zeigen:} $U_{1}\subseteq U_{2}$. + Da jeder lineare Unterraum den Nullvektor enthält + (siehe \cite[Lemma~1.4.2]{sinn2020}), gilt + + \begin{mathe}[bc]{rcccccl} + U_{1} + &= + &U_{1} + +\underbrace{ + \{\zerovector\} + }_{\subseteq U_{2}} + &\subseteq + &U_{1}+U_{2} + &\eqcrefoverset{eq:1:quiz:8}{=} + &U_{2}.\\ + \end{mathe} + + \herRichtung Angenommen, $U_{1}\subseteq U_{2}$. + \textbf{Zu zeigen:} $U_{1}+U_{2}=U_{2}$.\\ + Für Mengengleichheit teilen wir dies in zwei Teile auf: + + \BeweisRichtung[$\supseteq$] \textbf{Zu zeigen:} $U_{2}\subseteq U_{1}+U_{2}$. + Da jeder lineare Unterraum den Nullvektor enthält, + + \begin{mathe}[bc]{rcccl} + U_{2} + &= + &\underbrace{ + \{\zerovector\} + }_{\subseteq U_{1}} + +U_{2} + &\subseteq + &U_{1}+U_{2}\\ + \end{mathe} + + \BeweisRichtung[$\subseteq$] \textbf{Zu zeigen:} $U_{1}+U_{2}\subseteq U_{2}$. + Es gilt: + + \begin{mathe}[bc]{rcl} + U_{1}+U_{2} + &\subseteq &U_{2}+U_{2}, + \quad\text{da per Annahme $U_{1}\subseteq U_{2}$}\\ + &\subseteq + &U_{2}, + \quad\text{da als Vektorraum $U_{2}$ unter Addition stabil ist}.\\ + \end{mathe} + \end{proof} + \end{einzug} + +\begin{rem*} +In diesem Beweis haben wir mit mengenweise Operationen gearbeitet. +Z.\,B. aus $\{\zerovector\}\subseteq U_{1}$ +folgt $\{\zerovector\}+U_{2}\subseteq U_{1}+U_{2}$; +und aus $U_{1}\subseteq U_{2}$ +folgt $U_{1}+U_{2}\subseteq U_{2}+U_{2}$. +Diese Implikationen haben nichts mit linearer Algebra zu tun. +Sie sind rein mengentheoretische Ergebnisse und lassen sich +allgemein folgendermaßen zeigen: + + \begin{mathe}[mc]{rcccccl} + A\star B + &= &\{a\star b\mid a\in A,\,b\in B\} + &\subseteq &\{a\star b\mid a\in A',\,b\in B\} + &= &A'\star B\\ + \end{mathe} + +für alle Mengen, $A,A',B$ mit $A\subseteq A'$ +und alle Operationen $\star$ definiert auf $A'\times B$. + +Gebrauch machten wir auch von + + \begin{mathe}[mc]{rcccccccl} + U + \{\zerovector\} + &= &\{u+v\mid u\in U,\,v\in\{\zerovector\}\} + &= &\{u+\zerovector\mid u\in U\} + &= &\{u\mid u\in U\} + &= &U\\ + \end{mathe} + +für alle Teilmengen, $U\subseteq V$, und von + + \begin{mathe}[mc]{rcccccccl} + U+U + &= &\{\underbrace{u+v}_{\in U}\mid u\in U,\,v\in U\} + &\subseteq &U\\ + \end{mathe} + +für alle linearen Unterräume, $U\subseteq V$. +\end{rem*} + %% ******************************************************************************** %% FILE: back/index.tex %% ******************************************************************************** @@ -7627,12 +7843,7 @@ Wir betrachten die Komposition ${g\circ f:X\to Z}$ %% FILE: ./back/quelle.bib %% ******************************************************************************** -\begin{thebibliography}{EFT18} - -\bibitem[EFT18]{ebbinghaus2018} -Heinz-Dieter Ebbinghaus, J\"org Flum, and Wolfgang Thomas. -\newblock {\em {Einf\"uhrung in die mathematische Logik}}. -\newblock 2018. +\begin{thebibliography}{Wal16} \bibitem[Jec97]{jech1997} Thomas Jech.