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$A\times B$ ist eine Teilmenge des kartesischen Produkts. + + $R \subseteq A\times B$ + +\subsection{Äquivalenzrelation} + Eine Äquivalenzrelation ist eine zweistellige Relation auf einer Ausgangsmenge M mit folgenden drei Eigenschaften: + +\begin{itemize} + +\item \textbf{Reflexivität} + + Jedes Element der Ausgangsmenge M steht sich selbst in Beziehung. + + \begin{quote} + Für alle $a \in M$ gilt $\left( a , a \right) \in R$ + \end{quote} + +\item \textbf{Symmetrie} + + Zu jedem Paar $\left( a , b \right)$ ist auch die Umkehrung in $R$ enthalten. + + \begin{quote} + Wenn $\left( a , b \right) \in R$, dann ist auch $\left( b , a \right) \in R$ + \end{quote} + +\item \textbf{Transitivität} + + Stehen drei Elemente verkettet in Beziehung, dann stehen sie auch direkt in Beziehung. + + \begin{quote} + Wenn $\left( a , b \right) , \left( b , c \right) \in R$ dann ist auch $\left(a , c \right) \in R$ + \end{quote} + +\end{itemize} + \end{sectionbox} % Aussagenlogik -- 2.40.1 From 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zJ^=mynI8`%JdFS_z*B@$qrgWsMu7=625gTav>?!90jUr0!`mMSq$2?blm*C6?+blk{)3|s_-ndD5+abWBUV6Q&_@aq5c~Lp_aRy|97A{(1L`#V zaKi%Z55fNs5JM1XAVm)|1jcyGk$?kA6ye4YpbZf!0zm*aBwsqnM_@3*nQt|9^=-2k`LL$=S%!+1=61oDz@$fk9A`^6<#2$Wi`3aaEr` diff --git a/Formelsammlung.tex b/Formelsammlung.tex index 2aa1d24..c7897e6 100644 --- a/Formelsammlung.tex +++ b/Formelsammlung.tex @@ -82,37 +82,33 @@ Grundlage ist ein Polynom: $a{x}^{2} + bx + c = 0$ \subsection{Potenzrechnung} \begin{minipage}{0.49\textwidth} - \begin{align} + \begin{align*} {a}^{n} \cdot {a}^{m} &= {a}^{n+m} \\ {a}^{n} \cdot {b}^{n} &= {\left(a \cdot b \right)}^{n} \\ \frac{ {a}^{n} }{ {a}^{m} } &= {a}^{n-m} \\ \frac{{a}^{n}}{{b}^{n}} &= {\left(\cfrac{a}{b}\right)}^{n} - \end{align} + \end{align*} \end{minipage} \begin{minipage}{0.49\textwidth} - \begin{align} + \begin{align*} {e}^{lnx} &= x \\ {a}^{-n} &= \frac{1}{ {a}^{n} } \\ {-a}^{-1} &= \cfrac{-1}{a} = \cfrac{{a}^{-1}}{-1} \\ {\left({a}^{m}\right)}^{n} &= {\left({a}^{n}\right)}^{m} = {a}^{m \cdot n} - \end{align} + \end{align*} \end{minipage} \subsection{Wurzelrechnung} -\begin{align} +\begin{align*} \sqrt[n]{{a}^{m}} &= {\left({a}^{m} \right)}^{\frac{1}{n}} = {a}^{\frac{m}{n}} = {\left({a}^{\frac{1}{n}} \right)}^{m} = {\left(\sqrt[n]{a}\right)}^{m} \\ \sqrt[m]{\sqrt[n]{a}} &= \sqrt[m]{{a}^{\frac{1}{n}}} = { \left( {a}^{\frac{1}{n}} \right) }^{ \frac{1}{m} } = {a}^{\frac{1}{m \cdot n}} = \sqrt[m \cdot n]{a} \\ \sqrt[n]{a} \cdot \sqrt[n]{b} &= \left({a}^{\frac{1}{n}} \right) \cdot \left( {b}^{\frac{1}{n}} \right) = { \left( ab \right) }^{ \frac{1}{n} } = \sqrt[n]{ab} \\ \frac{ \sqrt[n]{a} }{ \sqrt[n]{b} } &= \frac{ {a}^{ \frac{1}{n} } }{ {b}^{ \frac{1}{n} } } = { \left( \frac{a}{b} \right) }^{\frac{1}{n}} = \sqrt[n]{ \frac{a}{b} } \text{ wenn } b \neq 0 -\end{align} - - +\end{align*} \end{sectionbox} - -% Weiterführend Allgemeines -% ---------------------------------------------------------------------- +% Manueller Spaltenumbruch \begin{sectionbox} \subsection{Bruchrechnung}\label{bruchrechnung} @@ -171,8 +167,9 @@ $\cfrac{1}{2}\sqrt{2} \cong 0.70710678$ und $\cfrac{1}{2}\sqrt{3} \cong 0.866025 \end{sectionbox} - +% Manueller Spaltenumbruch \begin{sectionbox} + \subsection{Operationen} \begin{tablebox}{lll} $A \subseteq B$ & & A ist Teilmenge von B \\ @@ -246,6 +243,10 @@ $\cfrac{1}{2}\sqrt{2} \cong 0.70710678$ und $\cfrac{1}{2}\sqrt{3} \cong 0.866025 de Morganschen Gesetz & $A \setminus B = A \cap \overline{B}$ \\ \end{tablebox} +\end{sectionbox} +% Manueller Spaltenumbruch +\begin{sectionbox} + \subsection{Kartesisches Produkt} Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten Paare $\left( a , b \right)$ mit $a \in A$ und $b \in B$ @@ -259,24 +260,29 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P \begin{sectionbox} \subsection{Definition} - Eine (zweistellige) Relation R zwischen zwei Mengen $A\times B$ ist eine Teilmenge des kartesischen Produkts. + Eine (zweistellige) Relation $R$ ist eine Teilmenge des kartesischen Produkts zweier Mengen $A$ und $B$. - $R \subseteq A\times B$ + \begin{quote} + $R \subseteq A\times B$ + \end{quote} \subsection{Äquivalenzrelation} - Eine Äquivalenzrelation ist eine zweistellige Relation auf einer Ausgangsmenge M mit folgenden drei Eigenschaften: + Eine Äquivalenzrelation ist eine zweistellige Relation auf einer Ausgangsmenge $M$ mit bestimmten Eigenschaften. -\begin{itemize} + \begin{quote} + $R \subseteq M\times M$ + \end{quote} -\item \textbf{Reflexivität} + \begin{cookbox}{Eigenschaften} + \item \textbf{Reflexivität} - Jedes Element der Ausgangsmenge M steht sich selbst in Beziehung. + Jedes Element der Ausgangsmenge $M$ steht mit sich selbst in Beziehung. - \begin{quote} - Für alle $a \in M$ gilt $\left( a , a \right) \in R$ - \end{quote} + \begin{quote} + Für alle $a \in M$ gilt $\left( a , a \right) \in R$ + \end{quote} -\item \textbf{Symmetrie} + \item \textbf{Symmetrie} Zu jedem Paar $\left( a , b \right)$ ist auch die Umkehrung in $R$ enthalten. @@ -284,7 +290,7 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P Wenn $\left( a , b \right) \in R$, dann ist auch $\left( b , a \right) \in R$ \end{quote} -\item \textbf{Transitivität} + \item \textbf{Transitivität} Stehen drei Elemente verkettet in Beziehung, dann stehen sie auch direkt in Beziehung. @@ -292,7 +298,7 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P Wenn $\left( a , b \right) , \left( b , c \right) \in R$ dann ist auch $\left(a , c \right) \in R$ \end{quote} -\end{itemize} + \end{cookbox} \end{sectionbox} @@ -309,6 +315,9 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P $A \leftrightarrow B$ & A genau dann, wenn B & Äquivalenz oder Bijunktion \\ $A \rightarrow B$ & wenn A dann B & Implikation oder Subjunktion \\ \end{tablebox} + +\end{sectionbox} +\begin{sectionbox} \subsection{Regeln} \begin{tablebox}{ll} @@ -336,7 +345,13 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P \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 + + \end{tablebox} +\end{sectionbox} +% Manueller Spaltenumbruch +\begin{sectionbox} + \begin{tablebox}{ll} + Umwandeln & $A \wedge B = \neg \left( A \rightarrow \neg B \right)$ \\ & $A \vee B = \neg A \rightarrow B $ \\ & $A \rightarrow B = \neg A \vee B$ \\ @@ -349,20 +364,16 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P \end{tablebox} \subsection{Beispiel} - Günter fragt Anna: "Libst du Peter, oder ist es nicht so, dass du Peter oder mich liebst?", darauf Antwortet Anna "Nein". - + 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} - + \begin{quote} + $P \vee \neg \left( P \vee G \right)$ \\ + \end{quote} 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} + \begin{quote} + $\neg \left( P \vee \neg \left( P \vee G \right)\right)$ \\ + \end{quote} \end{sectionbox} @@ -430,11 +441,25 @@ Das kartesische Produkt $A\times B$ (A kreuz B) ist die Menge aller geordneten P \end{sectionbox} +% Manueller Spaltenumbruch +\columnbreak + % Komplexe Zahlen % ---------------------------------------------------------------------- \section{Komplexe Zahlen} \begin{sectionbox} + +\subsection{Notation} +\begin{minipage}{0.39\textwidth} + \textbf{Kartesische Form}\\ + $z = a+b \cdot i$ +\end{minipage} +\begin{minipage}{0.59\textwidth} + \textbf{Trigonometrische Form / Polarform}\\ + $z =\left| z \right| \cdot \left( \cos { \varphi } + i \cdot \sin { \varphi } \right)$ +\end{minipage} + \subsection{Visualisierung} \begin{minipage}{0.49\textwidth} \includegraphics[width=\textwidth]{img/einheitskreis_komplexe_zahlen.png} @@ -457,13 +482,6 @@ $a = 0, b > 0 $ & $\varphi = 90^\circ $ & $\varphi = \cfrac{\pi}{2} $ $a = 0, b < 0 $ & $\varphi = 270^\circ $ & $\varphi = \cfrac{3}{2}\pi $ \\ $a = 0, b = 0 $ & $\varphi = 0^\circ $ & $\varphi = 0 $ \\ \end{tablebox} - -\end{sectionbox} - -\begin{sectionbox} - - - \subsection{Potenzen von i} @@ -498,7 +516,7 @@ $a = 0, b = 0 $ & $\varphi = 0^\circ $ & $\varphi = 0 $ \\ { 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 } + { \varphi }_{ 2 } \right) } +\sin { \left( { \varphi }_{ 1 } + { \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) } \\ @@ -507,6 +525,10 @@ $a = 0, b = 0 $ & $\varphi = 0^\circ $ & $\varphi = 0 $ \\ &=\frac { ac+bd }{ { c }^{ 2 }+{ d }^{ 2 } } +\frac { \left( bc-ad \right) }{ { c }^{ 2 }+{ d }^{ 2 } } i \end{align*} +\end{sectionbox} +% Manueller Spaltenumbruch +\begin{sectionbox} + \textbf{Potenzierung} \begin{align*} { z }^{ n } &={ \left( a+bi \right) }^{ n } \\ @@ -514,25 +536,13 @@ $a = 0, b = 0 $ & $\varphi = 0^\circ $ & $\varphi = 0 $ \\ &={ \left| z \right| }^{ n }\cdot \left( \cos { \left( n\cdot \varphi \right) } +\sin { \left( n\cdot \varphi \right) } i \right) \end{align*} -\textbf{Wurzel} $\lbrace k \in \mathbb{N} \vert k = 0 bis n-1 \rbrace$ +\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. -\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} % Vektoren und Matritzen @@ -685,6 +695,9 @@ $\overrightarrow{p}$ = Stützvektor und $\overrightarrow{u}$,$\overrightarrow{v} \end{sectionbox} +% Manueller Spaltenumbruch +\columnbreak + % Grenzwerte % ---------------------------------------------------------------------- \section{Grenzwerte} -- 2.40.1