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nun die Abbildung + + \begin{mathe}[mc]{rcccl} + F &: &\{g\mid g\,\text{Bij. zw. $X\ohne\{x_{0}\}$ und $Y\ohne\{x_{0}\}$}\} + &\to + &\{f\mid f\,\text{Bij. zw. $X$ und $Y$},\,f(x_{0})=y_{0}\}\\ + &: &g &\mapsto &g\cup\{(x_{0},y_{0})\}.\\ + \end{mathe} + + Das heißt, jede Bijektion ${g:X\ohne\{x_{0}\}\to Y\ohne\{y_{0}\}}$ + wird durch $F$ zu einer Funktion von $X$ nach $Y$ fortgesetzt, + indem das zusätzliche Element, $x_{0}$, auf $y_{0}$ abgebildet wird. + Es ist einfach zu sehen, dass $F$ wohldefiniert ist, + d.\,h. für jede Bijektion ${g:X\ohne\{x_{0}\}\to Y\ohne\{y_{0}\}}$, + es gilt, dass $F(g)$ eine wohldefinierte Funktion zwischen $X$ und $Y$ ist + und weiterhin ist dies eine Bijektion. + Außerdem ist es klar, dass die Abbildung + + \begin{mathe}[mc]{rcccl} + G &: &\{f\mid f\,\text{Bij. zw. $X$ und $Y$},\,f(x_{0})=y_{0}\} + &\to + &\{g\mid g\,\text{Bij. zw. $X\ohne\{x_{0}\}$ und $Y\ohne\{x_{0}\}$}\}\\ + &: &f &\mapsto &f\restr{X\ohne\{x_{0}\}\times Y\ohne\{x_{0}\}}\\ + \end{mathe} + + die Abbildung $F$ nach rechts und links invertiert. + Also ist $G$ eine Bijektion. + Daraus folgt per Definition von Kardinalität + + \begin{mathe}[mc]{rcl} + \eqtag[eq:2:\beweislabel] + |\{f\mid f\,\text{Bij. zw. $X$ und $Y$},\,f(x_{0})=y_{0}\}| + &= &|\{g\mid g\,\text{Bij. zw. $X\ohne\{x_{0}\}$ und $Y\ohne\{x_{0}\}$}\}|\\ + &= &(n-1)!\,\text{laut IV}.\\ + \end{mathe} + + Anderseits ist + ${(\{f\mid f\,\text{Bij. zw. $X$ und $Y$},\,f(x_{0})=y_{0}\})_{y_{0}\in Y}}$ + eine Partition von + ${\{f\mid f\,\text{Bij. zw. $X$ und $Y$}\}}$. + Darum gilt + + \begin{mathe}[mc]{rcl} + |\{f\mid f\,\text{Bij. zw. $X$ und $Y$}\}| + &= &|\bigcup_{y_{0}\in Y}\{f\mid f\,\text{Bij. zw. $X$ und $Y$},\,f(x_{0})=y_{0}\}|\\ + &= &\sum_{y_{0}\in Y}|\{f\mid f\,\text{Bij. zw. $X$ und $Y$},\,f(x_{0})=y_{0}\}|\\ + &&\text{wegen paarweise Disjunktheit}\\ + &\eqcrefoverset{eq:2:\beweislabel}{=} + &\sum_{y_{0}\in Y}(n-1)! + = |Y|\cdot (n-1)! + = n\cdot (n-1)! + = n!.\\ + \end{mathe} + + Also gilt \eqcref{eq:1:\beweislabel}. + \end{kompaktenum} + + Darum gilt $\Phi(n)$ per Induktion für alle $n\in\ntrl$. + \end{proof} + \setcounternach{part}{2} \part{Selbstkontrollenaufgaben}