106 lines
3.6 KiB
TeX
106 lines
3.6 KiB
TeX
\section{Abbildungen}
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\[ f: A \to B, a \mapsto f(a) = b\]
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\subsection{Bild}
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Alle Elemente von $B$, die durch $f$ erreicht werden:
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\[ im(f) = \{b \in B \mid \exists a \in A: f(a) = b\} \]
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\subsection{Urbild}
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Alle Elemente von $A$, die auf ein Element von $B$ abgebildet werden:
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\[ f^{-1}(b) = \{a \in A \mid f(a) = b\} \]
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\subsection{Identität}
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Jedes Element wird auf sich selbst abgebildet:
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\[ \mathrm{id}_A: A \to A, a \mapsto a \]
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\subsection{Eigenschaften}
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\subsubsection{Injektiv}
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\begin{center}
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\begin{tikzpicture}[scale=0.4,ele/.style={fill=black,circle,minimum width=.8pt,inner sep=1pt},every fit/.style={ellipse,draw,inner sep=-2pt}]
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\node[ele,label=left:$a$] (a1) at (0,4) {};
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\node[ele,label=left:$b$] (a2) at (0,3) {};
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\node[ele,label=left:$c$] (a3) at (0,2) {};
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\node[ele,label=left:$d$] (a4) at (0,1) {};
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\node[ele,,label=right:$1$] (b1) at (4,5) {};
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\node[ele,,label=right:$2$] (b2) at (4,4) {};
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\node[ele,,label=right:$3$] (b3) at (4,3) {};
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\node[ele,,label=right:$4$] (b4) at (4,2) {};
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\node[ele,,label=right:$5$] (b5) at (4,1) {};
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\node[draw,fit= (a1) (a2) (a3) (a4),minimum width=2cm] {} ;
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\node[draw,fit= (b1) (b2) (b3) (b4) (b5),minimum width=2cm] {} ;
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\draw[->,thick,shorten <=2pt,shorten >=2pt] (a1) -- (b4);
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\draw[->,thick,shorten <=2pt,shorten >=2] (a2) -- (b2);
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\draw[->,thick,shorten <=2pt,shorten >=2] (a3) -- (b1);
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\draw[->,thick,shorten <=2pt,shorten >=2] (a4) -- (b3);
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\end{tikzpicture}
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\end{center}
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\subsubsection{Surjektiv}
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\begin{center}
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\begin{tikzpicture}[scale=0.4,ele/.style={fill=black,circle,minimum width=.8pt,inner sep=1pt},every fit/.style={ellipse,draw,inner sep=-2pt}]
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\node[ele,label=left:$a$] (a1) at (0,5) {};
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\node[ele,label=left:$b$] (a2) at (0,4) {};
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\node[ele,label=left:$c$] (a3) at (0,3) {};
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\node[ele,label=left:$d$] (a4) at (0,2) {};
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\node[ele,label=left:$e$] (a5) at (0,1) {};
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\node[ele,,label=right:$1$] (b1) at (4,5) {};
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\node[ele,,label=right:$2$] (b2) at (4,4) {};
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\node[ele,,label=right:$3$] (b3) at (4,3) {};
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\node[ele,,label=right:$4$] (b4) at (4,2) {};
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\node[draw,fit= (a1) (a2) (a3) (a4) (a5),minimum width=2cm] {} ;
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\node[draw,fit= (b1) (b2) (b3) (b4),minimum width=2cm] {} ;
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\draw[->,thick,shorten <=2pt,shorten >=2pt] (a1) -- (b4);
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\draw[->,thick,shorten <=2pt,shorten >=2] (a2) -- (b2);
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\draw[->,thick,shorten <=2pt,shorten >=2] (a3) -- (b1);
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\draw[->,thick,shorten <=2pt,shorten >=2] (a4) -- (b3);
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\draw[->,thick,shorten <=2pt,shorten >=2] (a5) -- (b1);
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\end{tikzpicture}
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\end{center}
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\subsubsection{Bijektiv}
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\begin{center}
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\begin{tikzpicture}[scale=0.4,ele/.style={fill=black,circle,minimum width=.8pt,inner sep=1pt},every fit/.style={ellipse,draw,inner sep=-2pt}]
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\node[ele,label=left:$a$] (a1) at (0,5) {};
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\node[ele,label=left:$b$] (a2) at (0,4) {};
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\node[ele,label=left:$c$] (a3) at (0,3) {};
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\node[ele,label=left:$d$] (a4) at (0,2) {};
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\node[ele,label=left:$e$] (a5) at (0,1) {};
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\node[ele,,label=right:$1$] (b1) at (4,5) {};
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\node[ele,,label=right:$2$] (b2) at (4,4) {};
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\node[ele,,label=right:$3$] (b3) at (4,3) {};
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\node[ele,,label=right:$4$] (b4) at (4,2) {};
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\node[ele,,label=right:$5$] (b5) at (4,1) {};
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\node[draw,fit= (a1) (a2) (a3) (a4) (a5),minimum width=2cm] {} ;
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\node[draw,fit= (b1) (b2) (b3) (b4) (b5),minimum width=2cm] {} ;
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\draw[->,thick,shorten <=2pt,shorten >=2pt] (a1) -- (b3);
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\draw[->,thick,shorten <=2pt,shorten >=2] (a2) -- (b2);
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\draw[->,thick,shorten <=2pt,shorten >=2] (a3) -- (b1);
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\draw[->,thick,shorten <=2pt,shorten >=2] (a4) -- (b5);
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\draw[->,thick,shorten <=2pt,shorten >=2] (a5) -- (b4);
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\end{tikzpicture}
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\end{center}
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