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a/ws2020/algebra/uebungen/algebra1.tex b/ws2020/algebra/uebungen/algebra1.tex new file mode 100644 index 0000000..c32325c --- /dev/null +++ b/ws2020/algebra/uebungen/algebra1.tex @@ -0,0 +1,166 @@ +\documentclass[uebung]{../../../lecture} + +\usepackage[]{mathrsfs} + +\begin{document} + +\punkte + +\title{Algebra I: Übungsblatt 1} +\author{Christian Merten} + +\begin{aufgabe} + Beh.: $HK = KH \implies HK$ Untergruppe von $G$. + \begin{proof} + Sei $HK = KH$. + \begin{enumerate}[(i)] + \item Es ist $e \in H \cap K$, also $e \in HK$. + \item Seien $a, b \in HK$. Dann ex. $h_1, h_2 \in H$ und $k_1, k_2 \in K$, s.d. + $a = h_1 k_1$ und $b = h_2 k_2$. Dann ist + $ab = h_1 \underbrace{k_1 h_2}_{\in KH} k_2$. Da $KH = HK$ ex. $h_3 \in H$ und $k_3 \in K$, + s.d. $k_1 h_2 = h_3 k_3$. Damit folgt, da $H$ und $K$ Untergruppen: + \[ + ab = h_1 k_1 h_2 k_2 = \underbrace{h_1 h_3}_{\in H} \underbrace{k_3 k_2}_{\in K} + \in HK + .\] + \item Sei $a \in HK$. Dann ex. $h \in H$ und $k \in K$ s.d. $a = hk$. Dann folgt + \[ + a^{-1} = (hk)^{-1} = \underbrace{k^{-1}}_{\in K} \underbrace{h^{-1}}_{\in H} + \in KH = HK + .\] + \end{enumerate} + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $a = \prod_{g \in G, g^2 = e} g$. + \begin{proof} + Es gilt zunächst $\forall g \in G\colon g^2 \neq e$ gilt + $(g^{-1})^2 \neq e$, denn Inverse sind eindeutig in Gruppen. Da + $G$ abelsch folgt durch Umordnung + \[ + \prod_{\substack{g \in G \\ g^2 \neq e}} g = e + .\] + Damit folgt + \begin{salign*} + a = \prod_{g \in G}^{} g + &\stackrel{G \text{ abelsch}}{=} \prod_{\substack{g \in G\\g^2 = e}} g + \cdot \underbrace{\prod_{\substack{g \in G \\ g^2 \neq e}} g}_{= e} + = \prod_{\substack{g \in G\\g^2 = e}} g + .\end{salign*} + \end{proof} + Beh.: $a^2 = e$. + \begin{proof} + Es folgt direkt + \[ + a^2 = \left( \prod_{\substack{g \in G \\ g^2 = e}} g \right)^2 + = \prod_{\substack{g \in G \\ g^2 = e}} g^2 + = e + .\] + \end{proof} + \item Beh.: Für $p$ prim gilt $(p-1)! \equiv -1$ $(\text{mod } p)$. + \begin{proof} + Zunächst ist $\Z / p \Z$ nullteilerfrei, denn $\Z / p \Z$ nach LAI Körper, da $p$ prim. + Damit gilt für $\overline{a} \in \Z/ p\Z$: + \[ + \overline{a}^2 = \overline{1} \iff \overline{a}^2 - \overline{1} = 0 + \iff (\overline{a} + \overline{1})(\overline{a} - \overline{1}) = 0 + \iff \overline{a} = \overline{1} \text{ oder } \overline{a} = \overline{-1} + .\] + Da $p$ prim gilt außerdem + $(\Z / p \Z)^{\times } = (\Z / p \Z) \setminus \{ \overline{0}\} $. Damit + folgt + \[ + \overline{(p-1)!} = \prod_{\overline{g} \in ( \Z / p\Z)^{\times }} \overline{g} + \stackrel{\text{(a)}}{=} \prod_{\substack{\overline{g} \in (\Z / p \Z)^{\times } \\ \overline{g}^2 = \overline{1}}} \overline{g} = \overline{1} \cdot \overline{-1} = \overline{-1} + .\] Das zeigt die Behauptung. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $Z(G)$ ist Normalteiler in $G$. + \begin{proof} + Zunächst zeige $Z(G)$ ist Untergruppe in $G$. + \begin{enumerate}[(i)] + \item $e \in Z(G)$, denn $e h = h = h e$ $\forall h \in G$. + \item Seien $a, b \in Z(G)$ und $h \in G$ beliebig. Dann ist + \[ + (ab)h \quad \stackrel{b \in Z(G)}{=} \quad a h b + \quad\stackrel{a \in Z(G)}{=} \quad h (a b) + .\] Also $ab \in Z(G)$. + \item Sei $a \in Z(G)$ und $h \in G$. Dann ist + \[ + a^{-1} h = a^{-1} h \cdot a a^{-1} + \quad \stackrel{a \in Z(G)}{=} \quad a^{-1} a h a^{-1} = h a^{-1} + .\] + \end{enumerate} + Bleibt zu zeigen, dass $Z(G)$ Normalteiler. Dazu sei $a \in G$. Dann folgt wegen der + Kommutativität des Zentrums direkt + \[ + a Z(G) a^{-1} = a a^{-1} Z(G) = Z(G) + .\] + \end{proof} + \item Beh.: $G / Z(G)$ zyklisch $\implies$ $G$ abelsch. + \begin{proof} + Sei $G / Z(G)$ zyklisch. Dann ex. $g \in G$, s.d. + $\langle \overline{g} \rangle = G / Z(G)$. Seien nun $a, b \in G$ beliebig. + Dann ist ex. $n, m \in \N$, s.d. $\overline{a} = \overline{g}^{n} = \overline{g^{n}}$ und + $\overline{b} = \overline{g}^{m} = \overline{g^{m}}$. + + Also ex. $h_1, h_2 \in Z(G)$ s.d. $a = h_1 g^{n}$ und $b = h_2 g^{m}$. Damit folgt + unter mehrfacher Ausnutzung der Zentrumseigenschaft für $h_1$ und $h_2$: + \begin{salign*} + a b = h_1 g^{n} h_2 g^{m} = h_1 g^{n} g^{m} h_2 = h_1 g^{m} g^{n} h_2 + = h_2 g^{m} h_1 g^{n} = b a + .\end{salign*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $D_4$ ist eine Untergruppe von $\text{GL}_2(\R)$. + \begin{proof} + \begin{enumerate}[(i)] + \item $E_2 \in D_4$, denn $E_2(\square) = \square$. + \item Seien $A, B \in D_4$. Dann ist + \[ + (A B)(\square) = A ( B (\square)) = A (\square) = \square + .\] Also $AB \in D_4$ + \item Sei $A \in D_4$. Dann ex. $A^{-1} \in \text{GL}_2(\R)$ und es gilt + \[ + \square = (\underbrace{A^{-1}A}_{= E_2})(\square) + = A^{-1} (A (\square)) = A^{-1} (\square) + .\] Also $A^{-1} \in D_4$. + \end{enumerate} + \end{proof} + \item Beh.: $\text{ord}(D_4) = 8$. + \begin{proof} + Zunächst gilt für $A \in D_4$: $A$ ist isometrisch, also ist + nach LAI $A$ immer eine Drehspiegelung um $(0,0)$. + + Damit bleiben für $\square$ genau $4$ erhaltende Rotationen und $4$ Spiegelachsen. Damit folgt + die Behauptung. + \end{proof} + \item Die Punkte von $\square$ sind durch den festen Abstand der Eckpunkte von einander eindeutig + durch ihre zwei benachbarten Ecken festgelegt. + + Damit ex. eine injektive Abbildung $\iota: D_4 \xhookrightarrow{} S_4$. Es reicht also + die Permutation der Eckpunkte anzugeben. Diese sind: + \begin{salign*} + \tau_1 &= \begin{pmatrix} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{pmatrix} = \text{id} \\ + \tau_2 &= \begin{pmatrix} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \end{pmatrix} \\ + \tau_3 &= \begin{pmatrix} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \end{pmatrix} \\ + \tau_4 &= \begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1\end{pmatrix} \\ + \tau_5 &= \begin{pmatrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1\end{pmatrix} \\ + \tau_6 &= \begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{pmatrix} \\ + \tau_7 &= \begin{pmatrix} 1 & 2 & 3 & 4 \\ 3 & 2 & 1 & 4 \end{pmatrix} \\ + \tau_8 &= \begin{pmatrix} 1 & 2 & 3 & 4 \\ 1 & 4 & 3 & 2\end{pmatrix} + .\end{salign*} + \end{enumerate} +\end{aufgabe} + +\end{document} diff --git a/ws2020/ana/uebungen/ana1.pdf b/ws2020/ana/uebungen/ana1.pdf index d9eaef576eb2e71ee5664bb273fe0bc10ad284c6..f0a1f58b2e28d3ec856664b198e87e7f0b2c767e 100644 GIT binary patch delta 125545 zcmZttWl&w+?*IyOaF=p$cXxMpcM23QR^0vI?gxr%ahKxm?oiy_p}4<&e*gE&oqK10 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a/ws2020/ana/uebungen/ana1.tex b/ws2020/ana/uebungen/ana1.tex index 2e0415c..d4ae896 100644 --- a/ws2020/ana/uebungen/ana1.tex +++ b/ws2020/ana/uebungen/ana1.tex @@ -1,5 +1,7 @@ \documentclass[uebung]{../../../lecture} +\usepackage[]{mathrsfs} + \begin{document} \punkte @@ -89,7 +91,203 @@ \end{aufgabe} \begin{aufgabe} - + \begin{enumerate}[a)] + \item Beh.: Es existiert kein Wahrscheinlichkeitsmaß $\nu \colon \mathscr{P}(X) \to \{0, 1\} $ mit + $\nu(A) = 0$ für alle endlichen Mengen $A \subseteq X$. + \begin{proof} + Ang. es existiert ein Wahrscheinlichkeitsmaß $\nu \colon \mathscr{P}(X) \to \{0, 1\} $. Dann + ist $\nu([0,1]) = 1$. Definiere nun induktiv: $I_1 = [0, 1]$. + Für $I_{k+1}$ teile $I_k$ beliebig in zwei disjunkte Teilintervalle $A, B \subseteq I_k$ mit + $A \cap B = \emptyset$, $A \cup B = I_k$ und $A, B \neq \emptyset$. Da $\mu(I_k) = 1$ und + $A$ und $B$ disjunkt folgt mit der Additivität von $\nu$, dass entweder $\nu(A) = 1$ oder + $\nu(B) = 1$. Wähle dann $I_{k+1} = A$ oder $I_{k+1} = B$, s.d. $\nu(I_k) = 1$. Damit + ist $I_{k+1} \subsetneqq I_k$ also $I_k$ monoton fallend und $I_k \searrow \{ x \} $ + für $x \in [0,1]$. Außerdem gilt nach Konstruktion $\nu(I_k) = 1$ $\forall k \in \N$. Damit folgt + nach VL + \[ + \lim_{k \to \infty} \mu(I_k) = 1 \neq 0 = \mu\left( \left\{ x \right\} \right) + \quad \contr + .\] + \end{proof} + \item Beh.: $\mathcal{A}$ ist eine $\sigma$-Algebra. + \begin{proof} + \begin{enumerate}[(i)] + \item $X \in \mathcal{A}$, denn $X^{c} = \emptyset$ endlich. $\emptyset$ selbst endlich. + \item Sei $A \in \mathcal{A}$. Falls $A$ höchstens abzählbar, dann ist + $\left(A^{c}\right)^{c} = A$ höchstens abzählbar, analog für $A^{c}$ höchstens + abzählbar. Also $A^{c} \in \mathcal{A}$. + \item Sei $A_i \in \mathcal{A}$ $\forall i \in \N$. Falls $A_i$ höchstens + abzählbar $\forall i \in \N$ ist $\bigcup_{i \in \N} A_i$ ebenfalls + abzählbar, also $\bigcup_{i \in \N} A_i \in \mathcal{A}$. + + Falls $\exists j \in \N$, s.d. $A_j \in \mathcal{A}$ überabzählbar, dann ist + $\mathcal{A}^{c}$ höchstens abzählbar nach Definition. Damit folgt + \[ + \left(\bigcup_{i \in \N} A_i\right)^{c} = \bigcap_{i \in \N} A_i ^{c} + \subseteq A_j^{c} + .\] Also $\left( \bigcup_{i \in \N} A_i \right)^{c}$ höchstens abzählbar. + \end{enumerate} + \end{proof} + Beh.: $\mu$ definiert ein Maß auf $\mathcal{A}$. + \begin{proof} + \begin{enumerate}[(i)] + \item Es ist $\mu(\emptyset) = 0$, denn $\emptyset$ endlich. + \item Sei $A_i \in \mathcal{A}$ mit $A_i \cap A_j = \emptyset$ für $i \neq j$. + Falls $\forall i \in \N$ $A_i$ höchstens abzählbar, dann ist $\bigcup_{i \in \N} A_i$ + auch höchstens abzählbar also folgt + \[ + \mu\left( \bigcup_{i \in \N} A_i \right) = 0 = \sum_{i \in \N} \mu(A_i) + .\] + Falls ein $i \in \N$ existiert, s.d. $A_i$ überabzählbar, dann gilt + $\mu(A_i) = 1$, $A_i^{c}$ nach Definition von $\mathcal{A}$ abzählbar + und $\bigcup_{k \in \N} A_k $ überabzählbar. + Da $A_i$ paarweise disjunkt, folgt $\forall j \in \N\colon A_j \subseteq A_{i}^{c}$, + also $A_j$ höchstens abzählbar. Damit folgt + \[ + \sum_{k \in \N} \mu(A_k) = \mu(A_i) + \sum_{k \in \N, k \neq i} \mu(A_k) + = 1 + 0 = 1 = \mu\left( \bigcup_{k \in \N} A_k \right) + .\] + \end{enumerate} + \end{proof} + \item In (a) ist ein Wahrscheinlichkeitsmaß auf dem gesamten Potenzraum gefordert, in (b) + nur auf der Untermenge $\mathcal{A} \subsetneqq \mathscr{P}(X)$, denn beispielsweise + weder $\left[0,\frac{1}{2}\right] \subseteq \R$ noch $\left[ 0, \frac{1}{2} \right]^{c} \subseteq \R$ sind höchstens + abzählbar. Deswegen liegt kein Widerspruch vor. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Beh.: $\mathscr{D}$ $\pi$-System $\implies$ $\mathscr{D}$ $\sigma$-Algebra. + \begin{proof} + Sei $\mathscr{D}$ ein $\pi$-System. Dann gilt + \begin{enumerate}[(i)] + \item $X \in \mathscr{D}$ klar, da $\mathscr{D}$ Dynkinsystem. + \item $A \in \mathscr{D} \implies A^{c} \in \mathscr{D}$ klar, da $\mathscr{D}$ Dynkinsystem. + \item Sei nun $A_i \in \mathscr{D}$ $\forall i \in \N$. Da für zwei + Mengen $A, B \subseteq \mathscr{D}$ gilt $A \setminus B = A \cap B^{c}$. + Da $\mathscr{D}$ Dynkinsystem und + $\pi$-System ist, folgt + $A \cap B^{c} \in \mathscr{D}$. + Also folgt insgesamt $A \setminus B \in \mathscr{D}$. + + Konstruiere nun induktiv $B_1 \coloneqq A_1$ und + $B_{k+1} \coloneqq A_{k+1} \setminus \bigcup_{i=1}^{k} B_i$. Es ist wegen + oben $B_k \in \mathscr{D}$ und nach Konstruktion $B_i \cap B_j = \emptyset$ + $\forall i, j \in \N$ mit $i \neq j$. Damit folgt wegen + $\mathcal{D}$ Dynkinsystem + \[ + \bigcup_{i \in \N} A_i = \mathop{\dot{\bigcup_{i \in \N}}} B_i \in \mathscr{D} + .\] + \end{enumerate} + \end{proof} + \item Beh.: $\mathscr{H}(D)$ ist Dynkinsystem. + \begin{proof} + \begin{enumerate}[(i)] + \item Es ist $\emptyset \in \mathscr{H}(D)$, denn + $\emptyset \cap D = \emptyset \in \mathscr{D}_0$, + da $\mathscr{D}_0$ Dynkinsystem. Ebenfalls + ist $X \in \mathscr{H}(D)$, denn $X \cap D = D \in \mathscr{D}_0$. + \item Sei $A \in \mathscr{H}(D)$. Dann ist $A \cap D \in \mathscr{D}_0$. Da + $\mathscr{D}_0$ Dynkinsystem folgt: + \begin{align*} + A^{c} \cap D = (X \setminus A) \cap D = (X \cap D) \setminus (A \cap D) + = \left( (X \cap D)^{c} \mathop{\dot{\cup}} (A \cap D) \right)^{c} + \in \mathscr{D}_0 + .\end{align*} + \item Sei $A_i \in \mathscr{H}(D)$ $\forall i \in \N$ mit $A_i \cap A_j = \emptyset$ + $\forall i, j \in \N, i \neq j$. Dann folgt direkt, da die $A_i$ paarweise + disjunkt sind und $\mathscr{D}_0$ Dynkinsystem: + \[ + \left( \bigcup_{i \in \N} A_i \right) \cap D + = \mathop{\dot{\bigcup_{i \in \N}}} (\underbrace{A_i \cap D}_{ \in \mathscr{D}_0}) + \in \mathscr{D}_0 + .\] + \end{enumerate} + \end{proof} + \item Beh.: $\mathscr{H}(D) = \mathscr{D}_0$ für alle $D \in \mathscr{D}_0$. + \begin{proof} + \begin{enumerate}[(1)] + \item Z.z.: $H(K) = \mathscr{D}_0$ $\forall K \in \mathscr{K}$. + + Sei $K \in \mathscr{K}$. + Dann ist $\mathscr{K} \subseteq \mathscr{H}(K)$, denn für $A \in \mathscr{K}$ gilt + $A \cap K \in \mathscr{K} \subseteq \mathscr{D}_0$, da $\mathscr{K}$ $\pi$-System. + + Da wegen (b) $\mathscr{H}(K)$ Dynkinsystem und $\mathscr{D}_0$ kleinstes + Dynkinsystem, das $\mathscr{K}$ enthält, folgt $\mathscr{D}_0 \subseteq \mathscr{H}(K)$. + Außerdem ist nach Definition $\mathscr{H}(K) \subseteq \mathscr{D}_0$, also + folgt $\mathscr{H}(K) = \mathscr{D}_0$. + \item Z.z.: $\mathscr{K} \subseteq \mathscr{H}(D)$ $\forall D \in \mathscr{D}_0$. + + Sei $D \in \mathscr{D}_0$ und $K \in \mathscr{K}$ beliebig. Da + $D \in \mathscr{D}_0 = \mathscr{H}(K)$ (wg. 1) folgt $K \cap D \in \mathscr{D}_0$. Also + auch $K \in \mathscr{H}(D)$. + \item Sei nun $D \in \mathscr{D}_0.$ Da $\mathscr{H}(D)$ Dynkinsystem, das + $\mathscr{K}$ enthält folgt wie in (1), dass $\mathscr{H}(D) = \mathscr{D}_0$. + \end{enumerate} + \end{proof} + \item Beh.: $\sigma(\mathscr{K}) \subseteq \mathscr{D}$. + \begin{proof} + $\mathscr{D}_0$ ist $\pi$-System, denn $\mathscr{D}_0 \neq \emptyset$ + und für $A, B \in \mathscr{D}_0$ betrachte: + $A \in \mathscr{H}(B) = D_0$. Damit folgt $A \cap B \in \mathscr{D_0}$. + + Mit (a) ist $\mathscr{D}_0$ also $\sigma$-Algebra, die mit (c) $\mathscr{K}$ enthält. + Da $\mathscr{D}_0$ kleinstes Dynkinsystem ist, das $\mathscr{K}$ enthält, und jede + $\sigma$-Algebra auch Dynkinsystem ist und $\mathscr{D}_0$ selber $\sigma$-Algebra, folgt + \[ + \delta(\mathscr{K}) = \mathscr{D}_0 = \sigma(\mathscr{K}) + .\] Da bereits $\mathscr{D} \supseteq \mathscr{K}$, folgt + $\sigma(\mathscr{K}) \subseteq \mathscr{D}$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe}[Zusatzaufgabe] + \begin{enumerate}[a)] + \item Beh.: $A_{*} = \{ x \in X \colon x \in A_k \text{ f.f.a. } k \in \N\} $. + \begin{proof} + Sei $x \in X$. Dann gilt + \begin{align*} + x \in A_{*} &\iff x \in \bigcup_{n \ge 1} \bigcap_{m \ge n} A_m \\ + &\iff \exists n \in \N\colon \forall m \ge n\colon x \in A_m \\ + &\iff x \in A_k \text{ für fast alle } k \in \N + .\end{align*} + \end{proof} + Beh.: $A^{*} = \{ x \in X \colon x \in A_k \text{ für unendlich viele } k \in \N\} $. + \begin{proof} + Sei $x \in X$. Dann folgt + \begin{align*} + x \in A^{*} &\iff x \in \bigcap_{n \ge 1} \bigcup_{m \ge n} A_m \\ + &\iff \forall n \in \N\colon \exists m \in \N, n \ge m \colon x \in A_m \\ + &\iff x \in A_k \text{ für unendlich viele } k \in \N + .\end{align*} + \end{proof} + \item Verwendet werden im Folgenden die Charakterisierungen aus (a). + + Beh.: $\chi_{A_{*}} = \liminf_{k \to \infty} \chi_{A_k} $. + \begin{proof} + Sei $x \in X$. Falls $x \in A_{*}$: Dann ex. ein $k \in \N$ s.d. $\forall n \ge k$: + $x \in A_n$, also $\chi_{A_k}(x) = 1$ also + \[ + \liminf_{n \to \infty} \chi_{A_n} (x) + = \sup_{n \in \N} \inf_{m \ge n} \chi_{A_k}(x) = 1 = \chi_{A_*}(x) + .\] Falls $x \not\in A_{*}$: $\forall n \in \N\colon \exists k \in \N, k \ge n\colon x\not\in A_k \implies \chi_{A_k}(x) = 0$, also + \[ + \liminf_{n \to \infty} \chi_{A_n} (x) = 0 = \chi_{A_*}(x) + .\] + \end{proof} + Die Behauptung für den limes superior funktioniert exakt analog. + \item Beh.: $A_* = \emptyset$ und $A^{*} = [0,1)$. + \begin{proof} + Bemerke: Für $x \in [0,1)$ existieren $\infty$-viele $A_k$ mit $x \in A_k$ und + $\infty$-viele $A_k$ mit $x \not\in A_k$, denn die $A_k$ bilden immer feinere + Unterteilungen des Intervalls $[0, 1)$. + + Damit folgt die Behauptung aus den Charakterisierungen aus (a). + \end{proof} + \end{enumerate} \end{aufgabe} \end{document}