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zW)t&?_CRgmh6hg58LElUMr!+QC;Bg{IwQgqY4UfPC{45mdW#5ppc;fKatni6e^w~L z(F5qY*;7{$T+@TEXtvotJ(_JMFTHBB?bEV%l-tyoPp!pD(knaKrRLpJSJV@&=WTKk zh;)x%>B(FY0Sp?RW+|1bi%n)^FIuj~(Sf&US=#FhhNyYk>s4xz^W?{R#mPWpXx)Ps z;IlNel%2i1((?Ko%oeK}2(2`&x3&9k*V8NdDmb?R#>Fq&Cy)Z)TCm#K8Z=+%nLf@r zhCMtwxU`zr0C8zgRcS0rN_t~Tdwoi8qUk1^$BX6f<(BYoATu6D`2X{1@9bjeOPDlZ28{{YOQ^B({J diff --git a/MafIA1/mafia.tex b/MafIA1/mafia.tex index cf9cc9a..49103ea 100644 --- a/MafIA1/mafia.tex +++ b/MafIA1/mafia.tex @@ -113,14 +113,14 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! \section{Abbildungen} \subsection{Definition} Eine Abbildung $ f $ einer Menge $ U \subset M $ auf eine Menge $ N $ ist eine Vorschrift, die jedem $ x \in M $ (Urbild) genau ein Element $ y \in N $ (Bild von $x$) zuordnet. - Dabei ist $ U $ die \idx{Urbildmenge} und $ N $ die \idx{Bildmenge}. Weiter ist $ U $ die Domain(\idx{Definitionsbereich}) von $ f $ in $ M $ ($ U = dom(f) $)\\\\ + Dabei ist $ U $ die \idx{Urbildmenge} und $ N $ die \idx{Bildmenge}. Weiter ist $ U $ die Domain (\idx{Definitionsbereich}) von $ f $ in $ M $ ($ U = dom(f) $)\\\\ Eine Abbildung $ f $ von einer Menge $ M $ in eine Menge $ Y \subset N $ ist gegeben durch eine Relation:\\ $ R_f $ zwischen $ M $ und $ N $, bei der jedes $ x \in M $ mit genau einem $ y \in N $ in Relation steht:\\ $ (x,y) \in R_f $ und $ (x,\tilde{y}) \in R_f \Rarr y = \tilde{y} $. Wir schreiben in diesem Fall $ f(x) = y $ \subsubsection{Verknüpfte Abbildungen} \begin{align} - (f \circ g)(x) = f(g(x)) \text{ wobei } - g: M \rarr N \text{ und } f: N \rarr P \Rarr f \circ g: M \rarr P + (f \circ g)(x) = f(g(x)) \text{\ wobei } + g: M \rarr N \text{\ und } f: N \rarr P \Rarr f \circ g: M \rarr P \end{align} \subsection{Eigenschaften}\label{abb:eigenschaften} @@ -174,7 +174,7 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! \end{align*} Daraus resultiert, dass $f$ surjektiv ist. \end{proof} - Da $f$ surjektiv und injektiv ist, folgt auch die \emph{Surjektivität} + Da $f$ surjektiv und injektiv ist, folgt auch die \emph{Bijektivität} für $f$. \subsubsection{Regeln für Abbildungen} diff --git a/env/commands.tex b/env/commands.tex index c695865..f91970a 100644 --- a/env/commands.tex +++ b/env/commands.tex @@ -17,7 +17,7 @@ \newcommand{\Larr}{\Leftarrow} \newcommand{\Rarr}{\Rightarrow} -\newcommand{\ex}{\emph{Beispiel: }} +\newcommand*{\ex}{\emph{Beispiel: }} \newcommand{\anm}{\emph{Anmerkung: }} \newcommand{\prac}{\textbf{\textcolor{red}{Übung }}}