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LrK;-c@5TiHXY4dy diff --git a/Formelsammlung.tex b/Formelsammlung.tex index 3c14e15..0a2d951 100644 --- a/Formelsammlung.tex +++ b/Formelsammlung.tex @@ -89,10 +89,6 @@ \subsection{Komplement} Das Komplement ist die Differenz zwischen gegebener Menge und Grundmenge. - -\end{sectionbox} - -\begin{sectionbox} \subsection{Lösungsalgorithmus} \begin{cookbox}{Arbeitsablauf} \item $\setminus$ entfernen @@ -100,11 +96,19 @@ \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} +\end{sectionbox} + + +\begin{sectionbox} + +\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} \subsection{Regeln} \begin{tablebox}{ll} @@ -125,15 +129,136 @@ \ctrule de Morganschen Gesetz & $A \setminus B = A \cap \overline{B}$ \\ \end{tablebox} - -\subsection{Vereinfachen} + +\end{sectionbox} + +% Aussagenlogik +% ---------------------------------------------------------------------- +\section{Aussagenlogik} + +\begin{sectionbox} + +\subsection{Operationen} \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} + $A \wedge B$ & A und B & Konjunktion\\ + $A \vee B$ & A oder B & Disjunktion \\ + $A \leftrightarrow B$ & A genau dann, wenn B & Äquivalenz oder Bijunktion \\ + $A \rightarrow B$ & wenn A dann B & Implikation oder Subjunktion \\ + \end{tablebox} + +\subsection{Regeln} + \begin{tablebox}{ll} + Kommutativ & $A \wedge B = B \wedge A$\\ + & $A \vee B = B \vee A$ \\ + & $A \leftrightarrow B = B \leftrightarrow A$ \\ + \ctrule + Assoziativ & $A \wedge \left( B \wedge C \right) = \left( A \wedge B \right) \wedge C$ \\ + & $A \vee \left( B \vee C \right) = \left( A \vee B \right) \vee C$ \\ + & $A \leftrightarrow \left( B \leftrightarrow C \right) = \left( A \leftrightarrow B \right) \leftrightarrow C$ \\ + \ctrule + Distributiv & $A \wedge \left( B \vee C \right) = \left( A \wedge B \right) \vee \left(A \wedge C \right)$ \\ + & $A \vee \left( B \wedge C \right) = \left( A \vee B \right) \wedge \left(A \vee C \right)$ \\ + & $A \rightarrow \left( B \vee C \right) = \left( A \rightarrow B \right) \vee \left(A \rightarrow C \right)$ \\ + & $A \rightarrow \left( B \wedge C \right) = \left( A \rightarrow B \right) \wedge \left(A \rightarrow C \right)$ \\ + & $\left( A \vee B \right) \rightarrow C = \left( A \rightarrow C \right) \wedge \left(B \rightarrow C \right)$ \\ + & $\left( A \wedge B \right) \rightarrow C = \left( A \rightarrow C \right) \vee \left(B \rightarrow C \right)$ \\ + \ctrule + Adjunktiv (Absorbtion) & $A \wedge \left( A \vee B \right) = A $ \\ + & $A \vee \left( A \wedge B \right) = A$ \\ + \ctrule + Klammerntausch & $A \rightarrow \left( B \rightarrow C \right) = \left( A \wedge B \right) \rightarrow C $ \\ + \ctrule + Kontraposition & $A \rightarrow B = \neg B \rightarrow \neg A $ \\ + \ctrule + de Morganschen Regeln & $\neg \left( A \vee B \right) = \neg A \wedge \neg B$ \\ + & $\neg \left( A \wedge B \right) = \neg A \vee \neg B$ \\ + \ctrule + Umwandeln & $A \wedge B = \neg \left( A \rightarrow \neg B \right)$ \\ + & $A \vee B = \neg A \rightarrow B $ \\ + & $A \leftrightarrow B = \left( A \wedge B \right) \vee \left(\neg A \wedge \neg B \right)$ \\ + & $A \leftrightarrow B = \left( \neg A \vee B \right) \wedge \left(A \vee \neg B \right)$ \\ + \ctrule + \end{tablebox} + +\end{sectionbox} + +\begin{sectionbox} +\subsection{Beispiel} + Günter fragt Anna: "Libst du Peter, oder ist es nicht so, dass du Peter oder mich liebst?", darauf Antwortet Anna "Nein". + + Für die Aussage Anna liebt Peter setzen wir P und für Anna liebt Günther G. Die Frage lautet somit "Gilt P, oder gilt nicht P $\wedge$ G?". Formal bedeutet das: + + \begin{equation} + P \vee \neg \left( P \vee G \right) + \end{equation} + + + Da Anna mit "Nein" Antwortet muss der ganze Block negativiert werden. + + \begin{equation} + \neg \left( P \vee \neg \left( P \vee G \right)\right) \\ + \end{equation} + + +\subsection{Wahrheitstafeln} +\begin{minipage}{0.49\textwidth} + \textbf{Konjunkiton} (UND) + \begin{tablebox}{|l|l|l|} + \hline + $A $ & $B$ & $A \wedge B$ \\ \hline + $0$ & $0$ & $0$ \\ \hline + $0$ & $1$ & $0$ \\ \hline + $1$ & $0$ & $0$ \\ \hline + $1$ & $1$ & $1$ \\ \hline + \end{tablebox} +\end{minipage} +\begin{minipage}{0.49\textwidth} + \textbf{Disjunktion} (ODER) + \begin{tablebox}{|l|l|l|} + \hline + $A $ & $B$ & $A \vee B$ \\ \hline + $0$ & $0$ & $0$ \\ \hline + $0$ & $1$ & $1$ \\ \hline + $1$ & $0$ & $1$ \\ \hline + $1$ & $1$ & $1$ \\ \hline + \end{tablebox} +\end{minipage} +\begin{minipage}{0.49\textwidth} + \textbf{Bijunktion} (ist richtig wenn beide gleich sind) + \begin{tablebox}{|l|l|l|} + \hline + $A $ & $B$ & $A \leftrightarrow B$ \\ \hline + $0$ & $0$ & $1$ \\ \hline + $0$ & $1$ & $0$ \\ \hline + $1$ & $0$ & $0$ \\ \hline + $1$ & $1$ & $1$ \\ \hline + \end{tablebox} +\end{minipage} +\begin{minipage}{0.49\textwidth} + \textbf{Implikation} (aus A folgt B) + \begin{tablebox}{|l|l|l|} + \hline + $A $ & $B$ & $A \rightarrow B$ \\ \hline + $0$ & $0$ & $1$ \\ \hline + $0$ & $1$ & $1$ \\ \hline + $1$ & $0$ & $0$ \\ \hline + $1$ & $1$ & $1$ \\ \hline + \end{tablebox} +\end{minipage} + +\begin{tablebox}{|l|l|l|l|l|l|l|} + \hline + $A $ & $B$ & $C$ & $A \wedge B$ & $A \vee B$ & $A \wedge B \rightarrow A \vee B$ & $G$\\ \hline + $0$ & $0$ & $0$ & $0 $ & $0 $ & $1 $ & $0 $ \\ \hline + $0$ & $0$ & $1$ & $0 $ & $0 $ & $1 $ & $1 $ \\ \hline + $0$ & $1$ & $0$ & $0 $ & $1 $ & $1 $ & $0 $ \\ \hline + $0$ & $1$ & $1$ & $0 $ & $1 $ & $1 $ & $1 $ \\ \hline + $1$ & $0$ & $0$ & $0 $ & $1 $ & $1 $ & $0 $ \\ \hline + $1$ & $0$ & $1$ & $0 $ & $1 $ & $1 $ & $1 $ \\ \hline + $1$ & $1$ & $0$ & $1 $ & $1 $ & $1 $ & $0 $ \\ \hline + $1$ & $1$ & $1$ & $1 $ & $1 $ & $1 $ & $1 $ \\ \hline + \end{tablebox} + \end{sectionbox} % Komplexe Zahlen @@ -141,14 +266,6 @@ \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} @@ -166,6 +283,7 @@ \begin{sectionbox} + \begin{tablebox}{l|l|l} \textbf{x,y} & \textbf{(in Grad)} & \textbf{(im Bogenmaß)} \\ \hline $a > 0, b \ge 0$ & $\varphi = \arctan \cfrac{b}{a} $ & $\varphi = arctan \cfrac{b}{a}$ \\ @@ -176,20 +294,13 @@ $a = 0, b < 0 $ & $\varphi = 270^\circ $ & $\varphi = \cfrac{3}{2}\pi $a = 0, b = 0 $ & $\varphi = 0^\circ $ & $\varphi = 0 $ \\ \end{tablebox} +\subsection{Potenzen von i} - -\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*} +\begin{tablebox}{ll} + $i = \sqrt{-1} $ & ${ i }^{ 4 } = 1 $ \\ + ${ i }^{ 2 } = -1$ & ${ i }^{ 5 } = i$ \\ + ${ i }^{ 3 } = -i$ & ${ i }^{ 6 } = -1 $... \\ + \end{tablebox} \subsection{Rechenoperationen} \begin{minipage}{0.49\textwidth} @@ -241,6 +352,21 @@ Es gibt immer $n$ Ergebnisse die in ${ z }_{ k } $ für $k= 0$ bis $k= n-1$ bere \end{sectionbox} +\begin{sectionbox} +\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*} +\end{sectionbox} + % Tipps und Tricks % ---------------------------------------------------------------------- \section{Tipps}