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a/ws2019/ana/lectures/analysis9.tex +++ b/ws2019/ana/lectures/analysis9.tex @@ -1,8 +1,13 @@ \documentclass{../../../lecture} +\usepackage{tikz} +\usepackage{pgfplots} +\usetikzlibrary{quotes, angles} +\usepackage[]{wrapfig} + \begin{document} -Heute: Längstes deutsches Wort! Hi Josua :D +Heute: Längstes deutsches Wort! ,,Intervallschachtelungseigenschaft hat ganze 33 Buchstaben'' meint Kostina. @@ -148,7 +153,138 @@ Heute: Längstes deutsches Wort! Hi Josua :D \begin{bsp} \begin{itemize} \item $A = \{a_1, a_2, a_3\} $ dann $|A| = 3$ + \item $\N, \Q, \R$ sind unendliche Mengen \end{itemize} \end{bsp} +\begin{satz}[Abzählbarkeit] + $\Z$ und $\Q$ sind abzählbar, $\R$ ist überabzählbar. +\end{satz} + +\begin{proof}[$\Z$ Abzählbar] + $\Z$ ist abzählbar, weil $\{z_n \mid n \in \N\} $ mit $z_n = \frac{1}{2}n$ für $n$ gerade + und $z_n = \frac{1}{2}(1-n)$ für $n$ ungerade ist eine Abzählung von $\Z$. +\end{proof} + +\begin{proof}[$\Q$ Abzählbar] + Argumentation nach Cantor + + $p \in \Q$, $q = \frac{n}{m}$ + + \begin{tikzpicture} + %\begin{axis}[grid=both,ymin=-5,ymax=5,xmax=5,xmin=-5,xticklabel=\empty,yticklabel=\empty, + % minor tick num=1,axis lines = middle,xlabel=$x$,ylabel=$y$,label style = + % {at={(ticklabel cs:1.1)}}] + % \draw[-, color=red] (0,0) -- (1000,1000); + % + %\end{axis} + \draw[help lines, color=gray!30, dashed] (-0.5, -0.5) grid (6.9, 6.9); + \draw[->, thick] (-0.5,0) -- (7,0) node[right]{$x$}; + \draw[->, thick] (0, -0.5) -- (0, 7) node[above]{$y$}; + \draw[-, color=red] (1,1) node[below](){$1$} + -- (2,1) node[below](){$2$} + -- (1,2) node[left](){$\frac{1}{2}$} + -- (1,3) node[left](){$\frac{1}{3}$} + -- (3,1) node[below](){$3$} + -- (4,1) node[below](){$4$} + -- (3,2) node[above](){$\frac{3}{2}$} + -- (2,3) node[above](){$\frac{2}{3}$} + -- (1,4) node[left](){$\frac{1}{4}$} + -- (1,5) node[left](){$\frac{1}{5}$} + -- (5,1) node[below](){$5$} + -- (6,1) node[below](){$6$} + -- (5,2) node[above](){$\frac{5}{2}$} + -- (4,3) node[above](){$\frac{4}{3}$} + -- (3,4) node[above](){$\frac{3}{4}$} + -- (2,5) node[above](){$\frac{2}{5}$} + -- (1,6) node[left](){$\frac{1}{6}$} + -- (1,7); + + \end{tikzpicture} + + Hier werden Punkte ausgelassen, für die $n$ und $n$ nicht teilerfremd sind. Die Gitterpunkte + werden durchnummeriert $\implies \{z_n \mid n \in \N\} = \{1, 2, \frac{1}{2}, \frac{1}{3}, 3, \ldots\} $. +\end{proof} + +\begin{proof}[$\R$ ist überabzählbar] + Wir zeigen, dass $[0, 1)$ nicht abzählbar ist. + + Angenommen: $[0,1)$ ist abzählbar, dann sei $\{z_n \mid n \in \N\} $ eine Abzählung, z.B.: + \begin{align*} + z_1 &= 0,d_{11}d_{12}d_{13}\ldots \\ + z_2 &= 0,d_{21}d_{22}d_{23}\ldots \\ + \vdots + \end{align*} + Dann Zahl $y := 0,d_1d_2d_3, \ldots$ mit + \[ + d_n := \begin{cases} + 2 & \text{falls } d_{nn} = 1 \\ + 1 & \text{falls } d_{nn} \neq 1 + \end{cases} + \] + liegt in $[0,1)$, $d_i \neq 9 \forall i$, aber $y \not\in \{z_n \mid n \in \N\} $, denn falls + $y = z_k$ für ein $k \implies$ + \[ + y = 0,d_{k1},d_{k2},d_{k3}, \ldots, d_{kk}, \ldots + .\] aber $d_k \neq d_{kk}$ nach Konstruktion. +\end{proof} + +\subsection{Die Komplexen Zahlen $\C$} +\[ + \C := \R \times \R = \{ z = (x, y) \mid x, y \in \R\} +.\] Addition in $\C$ : $z_1 = (x_1, y_1) \in \C$, $z_2 = (x_2, y_2) \in \C$: +\[ + z_1 + z_2 := (x_1+x_2, y_1+y_2) +.\] Multiplikation in $\C$ : +\[ + z_1\cdot z_2 = (x_1 x_2 - y_1y_2, x_1y_2+x_2y_1) +.\] +\newpage + +\begin{satz}[$\C$ ist ein Körper] + Körperaxiome gelten (nachrechnen!) + + Nullelement $0 := (0, 0)$ \\ + Einselement $1 := (1, 0)$ \\ + Imaginäre Einheit $i := (0,1)$ mit $i^{2} = (0,1)\cdot (0,1) = (-1,0) = -1$ + + Inverse der Addition $-z := (-x, -y)$ \\ + Inverse der Multiplikation $z^{-1} := \left( \frac{x}{\sqrt{x^2 + y^2} } , \frac{-y}{\sqrt{x^2 + y^2} }\right) $ + + Schreibweise / Normaldarstellung\\ + $z = (x, y)$ oder $z = x+iy$ mit $i^{2} = -1$. +\end{satz} + +\begin{bem}[Rechnen in $\C$] + \[ + (x_1+iy_1)(x_2+iy_2) = x_1x_2+i^2y_1y_2 + i y_1x_2 + ix_1y_2 = (x_1x_2-y_1y_2) + i(x_2y_1+x_1y_2) + .\] +\end{bem} + +\begin{definition} + Für $z = (x, y) = x+iy \in \C$ heißt + + $x = \text{Re}(z)$ Realteil von $z$ \\ + $y = \text{Im}(z)$ Imaginärteil von $z$ + + $|z| := \sqrt{x^2 + y^2} $ Betrag von $z$ \\ + $\overline{z} := x - iy = (x, -y)$ zu $z$ konjugierte komplexe Zahl + + \begin{tikzpicture} + \draw[->, thick] (-0.5,0) -- (2,0) coordinate (x) node[right]{$x$}; + \draw[->, thick] (0, -0.5) -- (0, 2) coordinate (y) node[above]{$y$}; + \draw[-] (0, 0) coordinate (origin) -- (1.6, 1.3) coordinate (z) node[right]{$z$}; + \draw[-, dotted] (1.6, 1.3) + -- (1.6, 0) coordinate (re) node[below]{Re($z$)} + pic[solid,draw=black, angle radius=1cm]{angle=re--origin--z}; + \node[] () at (0.7,0.25) {$\varphi$}; + \draw[-, dotted] (1.6, 1.3) -- (0, 1.3) node[left](im){Im($z$)}; + \end{tikzpicture} + + $\text{Re}(z) = |z| \cos(\varphi)$ \\ + $\text{Im}(z) = |z| \sin(\varphi)$ \\ + $\implies z = r (\cos\varphi + i \sin\varphi) = r e^{i\varphi}$ mit $r = |z|$. + +\end{definition} + \end{document}