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z$`z?Ti(Y$wMylVUw;quiu;^x$)SyN8^_DTa8ndgh(y9q4qg0}mOl?4Uv=1nS+JUl1 zg;|Mn04Qg=fzqWHC{y}@l0@f8<;M_EY77Hq#VAlha7>Ps3!If>rNJan2Iz!*9PmRF z^P>~%Cus)ETJ-ySNgDj~!2JhxoV&~lWo~41baG{v`3nXX3o$k}Hwq;sMNdWwsG@iX diff --git a/docs/loesungen.tex b/docs/loesungen.tex index 340faa9..9b6c1ab 100644 --- a/docs/loesungen.tex +++ b/docs/loesungen.tex @@ -1341,6 +1341,8 @@ \def\zerovector{\text{\upshape\bfseries 0}} \def\graph{\mathop{\text{\textup Gph}}} +\def\domain{\mathop{\text{\textup dom}}} +\def\range{\mathop{\text{\textup ran}}} \def\id{\text{\textup id}} \def\modfn{\mathop{\text{\textup mod}}} \def\divides{\mathbin{\mid}} @@ -3706,7 +3708,7 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion. \item[\uwave{{\bfseries Induktionsvoraussetzung:}}] Sei $n>2$. - Angenommen, $\Phi(k)$ gilt für alle $k0$. + Per Definition von Kardinalität (siehe \cite[\S{}3.3, S.54]{sinn2020}) + existieren also Bijektionen + + \begin{mathe}[mc]{rcccl} + f &: &\{0,1,\ldots,m-1\} &\to &X,\\ + g &: &\{0,1,\ldots,n-1\} &\to &Y.\\ + \end{mathe} + + (Wir fangen aus praktischen Gründen mit $0$ statt $1$ an.)\\ + Definiere nun + + \begin{mathe}[mc]{rcccl} + h &: &\{0,1,\ldots,mn-1\} &\mapsto &X\times Y\\ + &: &k &\mapsto &(f(\modfn(k,m)), g([k/m])),\\ + \end{mathe} + + wobei ${[\cdot]:\reell\to\intgr}$ die Gaußklammerfunktion, + die reelle Zahlen \emph{abrundet}.\\ + \textbf{Zu zeigen:} $h$ ist eine wohldefinierte Bijektion. \begin{kompaktenum}[\rtab][\rtab] - \item[\uwave{{\bfseries Induktionsanfang:}}] - Sei $Y$ eine endliche Menge mit $|Y|=0$. - Also $Y=\leer$. - Darum + \item[\uwave{{\bfseries Wohldefiniertheit:}}] + Für alle $k\in\{0,1,\ldots,mn-1\}$ + gilt $i:=\modfn(k,m)\in\{0,1,\ldots,m-1\}=\domain(f)$ + und $j:=[k/m]\in\{0,1,\ldots,n-1\}=\domain(g)$, + sodass $f(i)\in X$ und $g(j)\in Y$ + und damit $h(k)=(f(i),g(j))\in X\times Y$. - \begin{mathe}[mc]{rcccccccccl} - |X\times Y| - &= &|X\times\leer| - &= &|\leer| - &= &0 - &= &|X|\cdot 0 - &= &|X|\cdot|Y|.\\ + \item[\uwave{{\bfseries Injektivität:}}] + Seien $k_{1},k_{2}\in\{0,1,\ldots,mn-1\}$ beliebig. + \textbf{Zu zeigen:} $h(k_{1})=h(k_{2})\Rightarrow k_{1}=k_{2}$.\\ + Nach \cite[Satz 3.4.2]{sinn2020} + existieren (eindeutige) Werte + ${q_{1},q_{2}\in\intgr}$ und + ${r_{1},r_{2}\in\{0,1,\ldots,m-1\}}$, + so dass + + \begin{mathe}[mc]{rcl} + \eqtag[eq:1:\beweislabel] + k_{1} &= &mq_{1}+r_{1},\\ + k_{2} &= &mq_{2}+r_{2}.\\ \end{mathe} - \item[] - Sei $Y$ eine $1$-elementige Menge. - Dann $Y=\{y\}$ für ein Objekt, $y$. - Es ist einfach zu sehen, dass - ${x\in X\mapsto (x,y)\in X\times Y}$ - eine Bijektion ist. - Folglich sind $X$ und $X\times Y$ gleichmächtig. - D.\,h. $|X\times Y|=|X|=|X|\cdot 1=|X|\cdot|Y|$. - \item[\uwave{{\bfseries Induktionsvoraussetzung:}}] - Sei $n>1$. - Angenommen, $|X\times Y'|=|X|\cdot |Y'|$ - für alle $n-1$-elementigen Mengen, $Y'$. + Daraus lässt sich ableiten, + dass + $\modfn(k_{1},m)=r_{1}$, + $\modfn(k_{2},m)=r_{2}$, + $[k_{1}/m]=q_{1}$, + und + $[k_{2}/m]=q_{2}$. + Darum gilt - \item[\uwave{{\bfseries Induktionsschritt:}}] - Sei $Y$ eine beliebige $n$-elementige Menge.\\ - \textbf{Zu zeigen:} $|X\times Y|=|X|\cdot|Y|$ gilt.\\ - Da $n>0$, können wir ein beliebiges $y_{0}\in Y$ fixieren.\\ - Setze $Y':=Y\ohne\{y_{0}\}$. - Da $Y'$ $n-1$-elementig ist, gilt per Induktionsvoraussetzung - $|X\times Y'|=|X|\cdot|Y'|=|X|\cdot(n-1)$.\\ - Wegen Disjunktheit von $Y'$ und $\{y_{0}\}$, - sind $X\times Y'$ und $X\times\{y_{0}\}$ ebenfalls disjunkt. - Es folgt + \begin{mathe}[mc]{rcl} + h(k_{1})=h(k_{2}) + &\textoverset{Defn}{\Longleftrightarrow} + &(f(r_{1}),g(q_{1}))=(f(r_{2}),g(q_{2}))\\ + &\Longrightarrow + &f(r_{1})=f(r_{2})\,\text{und}\,g(q_{1})=g(q_{2})\\ + &\Longrightarrow + &r_{1}=r_{2}\,\text{und}\,q_{1}=q_{2}\\ + &&\text{da $f,g$ injektiv sind}\\ + &\eqcrefoverset{eq:1:\beweislabel}{\Longrightarrow} + &k_{1}=mq_{1}+r_{1}=mq_{2}+r_{2}=k_{2}.\\ + \end{mathe} - \begin{longmathe}[mc]{RCLqL} - |X\times Y| - &= &|X\times (Y'\cup\{y_{0}\}|\\ - &= &|(X\times Y')\cup (X\times\{y_{0}\})|\\ - &= &|X\times Y'| + |X\times\{y_{0}\}| - &\text{wegen Disjunktheit}\\ - &= &|X|\cdot(n-1) + |X|\cdot 1 - &\text{wegen Fall für $1$-elem. Mengen}\\ - &= &|X|\cdot n - &\text{wegen rekursiver Defn von Multiplikation}\\ - &= &|X|\cdot |Y|.\\ - \end{longmathe} + \item[\uwave{{\bfseries Surjektivität:}}] + Sei $(x,y)\in X\times Y$. + \textbf{Zu zeigen:} $(x,y)\in\range(h)$.\\ + Wegen der Surjektivität von $f,g$ existieren nun + $i\in\{0,1,\ldots,m-1\}$ und + $j\in\{0,1,\ldots,n-1\}$, + so dass $f(i)=x$ und $g(j)=y$.\\ + Setze nun $k:=mj+i$.\\ + Dann $\modfn(k,m)=i$ und $[k/m]=j$, + sodass $h(k)=(f(i),g(j))=(x,y)$.\\ + Also gilt $(x,y)\in\range(h)$. \end{kompaktenum} - Darum gilt $|X\times Y|=|X|\cdot|Y|$ für alle Mengen $X,Y$. + Darum ist $h$ ist eine wohldefinierte Bijektion, + woraus sich per Definition von Kardinalität direkt ergibt, + dass $|X\times Y|=mn=|X|\cdot|Y|$. \end{proof} \end{einzug}