\section{Abbildungen}

\[ f: A \to B, a \mapsto f(a) = b\]

\subsection{Bild}

Alle Elemente von $B$, die durch $f$ erreicht werden:

\[ im(f) = \{b \in B \mid \exists a \in A: f(a) = b\} \]

\subsection{Urbild}

Alle Elemente von $A$, die auf ein Element von $B$ abgebildet werden:

\[ f^{-1}(b) = \{a \in A \mid f(a) = b\} \]

\subsection{Identität}

Jedes Element wird auf sich selbst abgebildet:

\[ \mathrm{id}_A: A \to A, a \mapsto a \]

\subsection{Eigenschaften}

\subsubsection{Injektiv}

\begin{center}
	\begin{tikzpicture}[scale=0.4,ele/.style={fill=black,circle,minimum width=.8pt,inner sep=1pt},every fit/.style={ellipse,draw,inner sep=-2pt}]
		\node[ele,label=left:$a$] (a1) at (0,4) {};
		\node[ele,label=left:$b$] (a2) at (0,3) {};
		\node[ele,label=left:$c$] (a3) at (0,2) {};
		\node[ele,label=left:$d$] (a4) at (0,1) {};


		\node[ele,,label=right:$1$] (b1) at (4,5) {};
		\node[ele,,label=right:$2$] (b2) at (4,4) {};
		\node[ele,,label=right:$3$] (b3) at (4,3) {};
		\node[ele,,label=right:$4$] (b4) at (4,2) {};
		\node[ele,,label=right:$5$] (b5) at (4,1) {};


		\node[draw,fit= (a1) (a2) (a3) (a4),minimum width=2cm] {} ;
		\node[draw,fit= (b1) (b2) (b3) (b4) (b5),minimum width=2cm] {} ;
		\draw[->,thick,shorten <=2pt,shorten >=2pt] (a1) -- (b4);
		\draw[->,thick,shorten <=2pt,shorten >=2] (a2) -- (b2);
		\draw[->,thick,shorten <=2pt,shorten >=2] (a3) -- (b1);
		\draw[->,thick,shorten <=2pt,shorten >=2] (a4) -- (b3);
	\end{tikzpicture}
\end{center}

\subsubsection{Surjektiv}

\begin{center}
	\begin{tikzpicture}[scale=0.4,ele/.style={fill=black,circle,minimum width=.8pt,inner sep=1pt},every fit/.style={ellipse,draw,inner sep=-2pt}]
		\node[ele,label=left:$a$] (a1) at (0,5) {};
		\node[ele,label=left:$b$] (a2) at (0,4) {};
		\node[ele,label=left:$c$] (a3) at (0,3) {};
		\node[ele,label=left:$d$] (a4) at (0,2) {};
		\node[ele,label=left:$e$] (a5) at (0,1) {};


		\node[ele,,label=right:$1$] (b1) at (4,5) {};
		\node[ele,,label=right:$2$] (b2) at (4,4) {};
		\node[ele,,label=right:$3$] (b3) at (4,3) {};
		\node[ele,,label=right:$4$] (b4) at (4,2) {};



		\node[draw,fit= (a1) (a2) (a3) (a4) (a5),minimum width=2cm] {} ;
		\node[draw,fit= (b1) (b2) (b3) (b4),minimum width=2cm] {} ;
		\draw[->,thick,shorten <=2pt,shorten >=2pt] (a1) -- (b4);
		\draw[->,thick,shorten <=2pt,shorten >=2] (a2) -- (b2);
		\draw[->,thick,shorten <=2pt,shorten >=2] (a3) -- (b1);
		\draw[->,thick,shorten <=2pt,shorten >=2] (a4) -- (b3);
		\draw[->,thick,shorten <=2pt,shorten >=2] (a5) -- (b1);
	\end{tikzpicture}
\end{center}

\subsubsection{Bijektiv}

\begin{center}
	\begin{tikzpicture}[scale=0.4,ele/.style={fill=black,circle,minimum width=.8pt,inner sep=1pt},every fit/.style={ellipse,draw,inner sep=-2pt}]
		\node[ele,label=left:$a$] (a1) at (0,5) {};
		\node[ele,label=left:$b$] (a2) at (0,4) {};
		\node[ele,label=left:$c$] (a3) at (0,3) {};
		\node[ele,label=left:$d$] (a4) at (0,2) {};
		\node[ele,label=left:$e$] (a5) at (0,1) {};


		\node[ele,,label=right:$1$] (b1) at (4,5) {};
		\node[ele,,label=right:$2$] (b2) at (4,4) {};
		\node[ele,,label=right:$3$] (b3) at (4,3) {};
		\node[ele,,label=right:$4$] (b4) at (4,2) {};
		\node[ele,,label=right:$5$] (b5) at (4,1) {};



		\node[draw,fit= (a1) (a2) (a3) (a4) (a5),minimum width=2cm] {} ;
		\node[draw,fit= (b1) (b2) (b3) (b4) (b5),minimum width=2cm] {} ;
		\draw[->,thick,shorten <=2pt,shorten >=2pt] (a1) -- (b3);
		\draw[->,thick,shorten <=2pt,shorten >=2] (a2) -- (b2);
		\draw[->,thick,shorten <=2pt,shorten >=2] (a3) -- (b1);
		\draw[->,thick,shorten <=2pt,shorten >=2] (a4) -- (b5);
		\draw[->,thick,shorten <=2pt,shorten >=2] (a5) -- (b4);
	\end{tikzpicture}
\end{center}
