Add Komplexe Zahlen and Images

pull/12/head
Sebastian Preisner 7 years ago
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commit f95e96a94b

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\maketitle % requires ./img/Logo.pdf
% Section
% Mengenlehre
% ----------------------------------------------------------------------
\section{Mengenlehre}
\input{themen/Mengenlehre/definition.tex}
\input{themen/Mengenlehre/operations.tex}
\begin{sectionbox}
\subsection{Definizion}
Ist $E$ eine Eigenschaft, die ein Element haben kann oder auch nicht, so beschreibt man die Menge der $E$ erfüllenden Elemente durch:
A = $\lbrace x \vert x $ hat Eigenschaft $ E \rbrace$
\subsection{Teilmengen}
Sind A und B Mengen, so heißt A Teilmenge oder auch Untermenge von B, wenn jedes Element von A auch Element von B ist.
\begin{cookbox}{Merke zu Teilmengen}
\item Jede Menge A ist Teilmenge von sich selbst, das heißt $A \subset A$
\item Jede Menge A hat die leere Menge als Teilmenge, das heißt: $\emptyset \subset A$
\item Ist $A \subseteq B$ und $B \subseteq C$, so folgt $A \subseteq C$
\item Aus $A \subseteq B$ und $B \subseteq A$ folgt $A = B$
\end{cookbox}
\subsection{Operationen}
\begin{tablebox}{lll}
$A \subseteq B$ & & A ist Teilmenge von B \\
$A \cup B$ & A vereinigt B & $A \cup B = \lbrace x \vert x \in A$ oder $x \in B \rbrace$ \\
$A \cap B$ & A geschnitten B & $A \cap B = \lbrace x \vert x \in A$ und $x \in B \rbrace$ \\
$A \setminus B$ & A ohne B & $A \cup B = \lbrace x \vert x \in A$ und $x \notin B \rbrace$ \\
$\mathcal{P}(A)$ & Potenzmenge A & Potenzmenge der Menge A\\
$A \in B$ & A Element von B & A ist ein Element von B\\
$A \notin B$ & A kein Element von B & A ist nicht in B enthalten \\
\end{tablebox}
\subsection{Potenzmenge}
Die Potenzmenge ist die Menge aller Teilmengen.
\begin{quote}
Es sei A eine Menge. Dann versteht man unter der Potenzmenge $\mathcal{P}(A)$ der Menge A die Menge aller Teilmengen von A. Auch die Menge $\emptyset$ hat eine Teilmenge es gilt: $\mathcal{P}(\emptyset) = \lbrace \emptyset \rbrace$.
\end{quote}
Berechnet wird die Potenzmenge mit Hilfe von $2^{\vert A \vert}$ (Zwei hoch Kardinalität von A)
\subsection{Kardinalität}
Beschreibt die Menge aller Elemente einer Menge.
\begin{quote}
Es sei A eine endliche Menge. Dann versteht man unter der Kardinalität oder auch Mächtigkeit von A die Anzahl der Elemente von A und schreibt dafür $\vert A \vert$, manchmal auch $\#A$. Hat A unendlich viele Elemente, so sagt man, A hat die Kardinalität unendlich, und schreibt $\vert A \vert = \infty$
\end{quote}
\begin{cookbox}{Beispiel}
$M = \lbrace 1, 2\rbrace$ \\
$P\left(M \right) = \lbrace \lbrace \rbrace, \lbrace 1 \rbrace, \lbrace 2 \rbrace, \lbrace 1, 2 \rbrace \rbrace $ \\
Nicht jedoch $\lbrace 2,1 \rbrace$! Es gilt $\lbrace 1,2 \rbrace = \lbrace 2,1 \rbrace$.
\end{cookbox}
\subsection{Komplement}
Das Komplement ist die Differenz zwischen gegebener Menge und Grundmenge.
\end{sectionbox}
\begin{sectionbox}
\subsection{Lösungsalgorithmus}
\begin{cookbox}{Arbeitsablauf}
\item $\setminus$ entfernen
\item De Morgen Gesetze anwenden
\item Assoziativ- und Distributiv- Gesetze im Wechsel mit dem Vereinfachen
\end{cookbox}
\begin{cookbox}{Arbeitsablauf}
\item $\setminus$ entfernen
\item De Morgen Gesetze anwenden
\item Assoziativ- und Distributiv- Gesetze im Wechsel mit dem Vereinfachen
\end{cookbox}
\subsection{Regeln}
\begin{tablebox}{ll}
Kommutativ & $A \cup B = B \cup A$\\
& $A \cap B = B \cap A$ \\
\ctrule
Assoziativ & $A \cap \left( B \cap C \right) = \left( A \cap B \right) \cap C$ \\
& $A \cup \left( B \cup C \right) = \left( A \cup B \right) \cup C$ \\
\ctrule
Distributiv & $A \cup \left( B \cap C \right) = \left( A \cup B \right) \cap \left(A \cup C \right)$ \\
& $A \cap \left( B \cup C \right) = \left( A \cap B \right) \cup \left(A \cap C \right)$ \\
\ctrule
Adjunktiv & $A \cup \left( A \cap B \right) = A $ \\
& $A \cap \left( A \cup B \right) = A$ \\
\ctrule
de Morganschen Regeln & $A \setminus \left( B \cap C \right) = \left( A \setminus B \right) \cup \left( A \setminus C \right)$ \\
& $A \setminus \left( B \cup C \right) = \left( A \setminus B \right) \cap \left( A \setminus C \right)$ \\
\ctrule
de Morganschen Gesetz & $A \setminus B = A \cap \overline{B}$ \\
\end{tablebox}
\subsection{Vereinfachen}
\begin{tablebox}{lll}
$A \cup A = A$ & $A \cap \emptyset = \emptyset $ & $\overline{\overline{A}} = A$ \\
$A \cap A = A$ & $A \cup \overline{A} = G $ & $\overline{\emptyset} = G $ \\
$A \cup G = G$ & $A \cap \overline{A} = \emptyset $ & $\overline{G} = \emptyset $ \\
$A \cap G = A$ & $\overline{A \cup B} = \overline{A} \cap \overline{B} $ & $\emptyset \neq \lbrace \emptyset \rbrace $!!! \\
$A \cup \emptyset = A $ & $\overline{A \cap B} = \overline{A} \cup \overline{B}$ & $ $ \\
\end{tablebox}
\end{sectionbox}
% Komplexe Zahlen
% ----------------------------------------------------------------------
\section{Komplexe Zahlen}
\begin{sectionbox}
\subsection{Potenzen von i}
\begin{tablebox}{ll}
$i = \sqrt{-1} $ & ${ i }^{ 4 } = 1 $ \\
${ i }^{ 2 } = -1$ & ${ i }^{ 5 } = i$ \\
${ i }^{ 3 } = -i$ & ${ i }^{ 6 } = -1 $... \\
\end{tablebox}
\subsection{Visualisierung}
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{img/einheitskreis_komplexe_zahlen.png}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{img/visualisierung_komplexe_zahlen.png}
\end{minipage}
\begin{tablebox}{lll}
$\vert z \vert = \sqrt{ {a}^{2} + {b}^{2} }$& $\varphi = \arctan \left( \cfrac{b}{a} \right) $ & siehe Tabelle xxx \\
$\tan\left(\varphi\right) = \cfrac{\vert b \vert}{ \vert a \vert}$ & $\cos\left(\varphi\right) = \cfrac{a}{\vert z \vert}$ & $\sin\left(\varphi\right) = \cfrac{b}{\vert z \vert}$ \\
\end{tablebox}
\end{sectionbox}
\begin{sectionbox}
\begin{tablebox}{lll}
\textbf{x,y} & \textbf{(in Grad)} & \textbf{(im Bogenmaß)} \\
$x > 0, y \ge 0$ & $\varphi = \arctan \cfrac{y}{x} $ & $\varphi = arctan \cfrac{y}{x}$ \\
$x < 0$ & $\varphi = \arctan \cfrac{y}{x} + 180^\circ $ & $\varphi = arctan \cfrac{y}{x} + \pi $ \\
$x > 0, y \le 0 $ & $\varphi = \arctan \cfrac{y}{x} + 360^\circ $ & $\varphi = \arctan \cfrac{y}{x} + 2\pi $ \\
$x = 0, y > 0 $ & $\varphi = 90^\circ $ & $\varphi = \cfrac{\pi}{2} $ \\
$x = 0, y < 0 $ & $\varphi = 270^\circ $ & $\varphi = \cfrac{3}{2}\pi $ \\
$x = 0, y = 0 $ & $\varphi = 0^\circ $ & $\varphi = 0 $ \\
\end{tablebox}
\subsection{Formen}
\textbf{Kartesische Form:}
\begin{align*}
{ z }_{ 1 } \cdot { z }_{ 2 } & = \left( a+bi \right) \cdot \left( c+di \right) \\
& = ac+adi+bci+bd{ i }^{ 2 } \\
\end{align*}
\textbf{Trigonometrische Form:}
\begin{align*}
{ z }_{ 1 }\cdot { z }_{ 2 } & =\left| { z }_{ 1 } \right| \left( \cos { \left( { \varphi }_{ 1 } \right) } +\sin { \left( { \varphi }_{ 1 } \right) } i \right) \cdot \left| { z }_{ 2 } \right| \left( \cos { \left( { \varphi }_{ 2 } \right) } \cdot \sin { \left( { \varphi }_{ 2 } \right) } i \right) \\
& =\left| { z }_{ 1 } \right| \cdot \left| { z }_{ 2 } \right| \left( \cos { \left( { \varphi }_{ 1 } \right) \cdot \cos { \left( { \varphi }_{ 2 } \right) } } +\sin { \left( { \varphi }_{ 1 } \right) } \cdot \sin { \left( { \varphi }_{ 2 } \right) } i \right)
\end{align*}
\subsection{Rechenoperationen}
\begin{minipage}{0.49\textwidth}
\textbf{Addition}
\begin{align*}
{ z }_{ 1 } + { z }_{ 2 } &= \left(a + bi\right) + \left(c + di \right)\\
&= a + c + (b + d)i
\end{align*}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\textbf{Subtraktion}
\begin{align*}
{ z }_{ 1 } - { z }_{ 2 } &= \left(a + bi \right) - \left(c + di \right)\\
&= a - c + (b - d)i
\end{align*}
\end{minipage}
\textbf{Multiplikation}
\begin{align*}
{ z }_{ 1 } \cdot { z }_{ 2 } &= \left(a + bi \right) \cdot \left(c + di \right) \\
&= ac + adi + bci + bd { i }^{ 2 }
\end{align*}
\begin{align*}
{ z }_{ 1 }\cdot { z }_{ 2 } & =\left| { z }_{ 1 } \right| \left( \cos { \left( { \varphi }_{ 1 } \right) } +\sin { \left( { \varphi }_{ 1 } \right) } i \right) \cdot \left| { z }_{ 2 } \right| \left( \cos { \left( { \varphi }_{ 2 } \right) } \cdot \sin { \left( { \varphi }_{ 2 } \right) } i \right) \\
& =\left| { z }_{ 1 } \right| \cdot \left| { z }_{ 2 } \right| \left( \cos { \left( { \varphi }_{ 1 } \right) \cdot \cos { \left( { \varphi }_{ 2 } \right) } } +\sin { \left( { \varphi }_{ 1 } \right) } \cdot \sin { \left( { \varphi }_{ 2 } \right) } i \right)
\end{align*}
\textbf{Division}
\begin{align*}
\frac { z_{ 1 } }{ z_{ 2 } } &=\frac { a+bi }{ c+di } \quad =\frac { \left( a+bi \right) }{ \left( c+di \right) } \cdot \frac { \left( c-di \right) }{ \left( c-di \right) } \\
&=\frac { ac\quad -\quad adi\quad +\quad bci\quad -\quad bd{ i }^{ 2 } }{ { c }^{ 2 }-{ \left( di \right) }^{ 2 } } \\
&=\frac { ac+bd+\left( bc-ad \right) i }{ { c }^{ 2 }+{ d }^{ 2 } } \\
&=\frac { ac+bd }{ { c }^{ 2 }+{ d }^{ 2 } } +\frac { \left( bc-ad \right) }{ { c }^{ 2 }+{ d }^{ 2 } }
\end{align*}
\textbf{Potenzierung}
\begin{align*}
{ z }^{ n } &={ \left( a+bi \right) }^{ n } \\
&={ \left( \left| z \right| \cdot \left( \cos { \varphi } +\sin { \varphi } i \right) \right) }^{ n } \\
&={ \left| z \right| }^{ n }\cdot \left( \cos { \left( n\cdot \varphi \right) } +\sin { \left( n\cdot \varphi \right) } i \right)
\end{align*}
\textbf{Wurzel} $\lbrace k \in \mathbb{N} \vert k = 0 bis n-1 \rbrace$
\begin{align*}
\sqrt[n]{z} &= \sqrt[n]{ a+bi } \\
{ z }_{ k } &= \sqrt[n]{\vert z \vert} \cdot \left( \cos{ \left( \cfrac{ \varphi + k \cdot 360}{n} \right) } +\sin{\left( \cfrac{\varphi + k \cdot 360}{n} \right)} i \right)
\end{align*}
Es gibt immer $n$ Ergebnisse die in ${ z }_{ k } $ für $k= 0$ bis $k= n-1$ berechnet werden.
\end{sectionbox}
% Tipps und Tricks
% ----------------------------------------------------------------------
\section{Tipps}
\begin{sectionbox}
\subsection{Sinus \& Cosinus}
\begin{tablebox}{ll}
$\cos 0^\circ = 1$ & $\sin 0^\circ = 0$ \\
$\cos 30^\circ = \cfrac{1}{2}\sqrt{3} \cong 0.8660254$ & $\sin 30^\circ = \cfrac{1}{2}$ \\
$\cos 45^\circ = \cfrac{1}{2}\sqrt{2} \cong 0.70710678$ & $\sin 45^\circ = \cfrac{1}{\sqrt{2}} \cong 0.70710678$ \\
$\cos 90^\circ = 0$ & $\sin 90^\circ = 1$ \\
\end{tablebox}
\end{sectionbox}
% ======================================================================
% End
% ======================================================================

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# LaTeX4EI Template
Write beautiful latex cheat sheets with minimal effort.
## How to use
put `latex4ei` folder in `./template-files/` into the same folder as
your latex file. Specify \documentclass{latex4ei/latex4ei_sheet} and compile your code.
See the documentation for further details.
## Permanent installation
Copy the `latex4ei` folder into your tex-distribution directory
### Windows
Copy files to `C:\texlive\XXXX\texmf-dist\tex\latex\latex4ei`
### Linux
```bash
sudo ln -s ./pkg /usr/share/texlive/texmf-dist/tex/latex/latex4ei
sudo mktexlsr
```
### Mac OS X
```bash
ln -s ./pkg /usr/local/texlive/texmf-local/tex/latex/latex4ei
```
If you want to share your documents please refer to the license.txt
Read the changelog.txt and known_bugs.txt and check [www.latex4ei.de](http://latex4ei.de) for updates.
© 2011-2016, LaTeX4EI
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