Add Komplexe Zahlen and Images

Formelsammlung.tex

@@ -33,14 +33,227 @@
 \maketitle   % requires ./img/Logo.pdf
 
 
-% Section
+% Mengenlehre
 % ----------------------------------------------------------------------
 \section{Mengenlehre}
 
-\input{themen/Mengenlehre/definition.tex}
+\begin{sectionbox}
-\input{themen/Mengenlehre/operations.tex}
 
+\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/README.md

@@ -0,0 +1,29 @@
+# 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