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z@cR02K>BJ>Lz$pBDt#R#e<;2lG*I%_;_-`@L0Egw$H?ncQ>^qg(;;#M#o_98x?m*F z3F|v3$%5yY^*VMAp*q2ON`;v7*X#df6r~>^4UWzH0Kh(;bzwRPs?@AvWvy33156mrCH`mR(~q8%;(QY zWi0dXPpR1A{#B{0#V6mCDzNy4SEULqe(9!Ek;PX}NEKWB?(U>qn_di~o7@Y?WnegH*L;+a5^OSho9~RIO$Ec1zV+cF*5Z^Dk%jdDk%qqDX9eHD5(ZS zD5(XcCt(^1!ATkcnMs-eaYXlC_q7_N#Qd7jo`@3)hCBm)PwE R9Sk)$G%z&^B_%~qMhf&O>WlyY diff --git a/docs/loesungen.tex b/docs/loesungen.tex index e15f5a9..9480551 100644 --- a/docs/loesungen.tex +++ b/docs/loesungen.tex @@ -3078,16 +3078,16 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion. \label{ska:4:ex:1} \let\sectionname\altsectionname -Seien $X$, $Y$ nicht leere Mengen. -Einer Abbildung, $f:X\to Y$, -können wir eindeutig die Relation - $\graph(f):=\{(x,y)\in X\times Y\mid f(x)=y\}$ -zuordnen. Dies nennt sich der \textbf{Graph von $f$} -(siehe \cite[\S{}2.3]{sinn2020}---dort wird dies mit $\Gamma_{f}$ bezeichnet). -Hier ist $\graph(f)$ also eine Relation auf $X\times Y$. -In der Tat \emph{setzen} manche Werke Funktionen mit ihrem Graphen gleich -(siehe bspw. \cite[S.11]{jech1997}), -aber dies ist streng genommen nicht die ganze Wahrheit. + Seien $X$, $Y$ nicht leere Mengen. + Einer Abbildung, $f:X\to Y$, + können wir eindeutig die Relation + $\graph(f):=\{(x,y)\in X\times Y\mid f(x)=y\}$ + zuordnen. Dies nennt sich der \textbf{Graph von $f$} + (siehe \cite[\S{}2.3]{sinn2020}---dort wird dies mit $\Gamma_{f}$ bezeichnet). + Hier ist $\graph(f)$ also eine Relation auf $X\times Y$. + In der Tat \emph{setzen} manche Werke Funktionen mit ihrem Graphen gleich + (siehe bspw. \cite[S.11]{jech1997}), + aber dies ist streng genommen nicht die ganze Wahrheit. %% SKA 4-2 \let\altsectionname\sectionname @@ -3096,7 +3096,67 @@ aber dies ist streng genommen nicht die ganze Wahrheit. \label{ska:4:ex:2} \let\sectionname\altsectionname -({\itshape Unter Arbeit}) + \textbf{Hinweis:} Hier scheint im Punkt (ii) etwas verwechselt worden zu sein. + + Seien $M$, $N$ Mengen und $R\subseteq M\times N$. + + \begin{claim} + \makelabel{claim:main:ska:4:ex:2} + Angenommen, $R$ erfülle folgende Eigenschaften: + + \begin{kompaktenum}{\bfseries (i)}[\rtab][\rtab] + \item\punktlabel{1} + $\forall{x\in M:~}\exists{y\in N:~}(x,y)\in R$ + \item\punktlabel{2} + $\forall{x\in M:~}\forall{y,y'\in N:~} + (x,y),(x,y')\in R + \Rightarrow y=y'$ + \end{kompaktenum} + + Dann existiert eine (notwendigerweise eindeutige) Funktion, + ${f:M\to N}$, + so dass $\graph(f)=R$. + \end{claim} + + \begin{einzug}[\rtab][\rtab] + \begin{proof} + Wir definieren ${f:M\to N}$ durch + + \begin{mathe}[mc]{rcl} + f(x) &= &y\\ + \end{mathe} + + für $(x,y)\in R$. + Offensichtlich gilt + $\graph(f) + =\{(x,y)\in M\times N\mid f(x)=y\} + =\{(x,y)\in M\times N\mid (x,y)\in R\} + =R$. + + \textbf{Zu zeigen:} + (1) $f$ ist überall definiert; + (2) $f$ ist wohldefiniert. + + \begin{kompaktenum}[\rtab][\rtab] + \item[\uwave{{\bfseries Überall definiert:}}] + Sei $x\in M$. + \textbf{Zu zeigen:} $f(x)=y$ für ein $y\in N$.\\ + Eigenschaft \punktlabel{1} besagt, dass ein $y\in M$ existiert, + so dass $(x,y)\in R$. + Per Konstruktion erhalten wir, dass $f(x)=y$ gilt. + \item[\uwave{{\bfseries Wohldefiniertheit:}}] + Seien $x\in M$ und $y,y'\in N$. + Angenommen, $f(x)=y$ und $f(x)=y'$.\\ + \textbf{Zu zeigen:} $y=y'$.\\ + Aus $f(x)=y$ und $f(x)=y'$ + folgt $(x,y),(x,y')\in R$ per Konstruktion von $f$.\\ + Eigenschaft \punktlabel{2} besagt, dass $y=y'$. + \end{kompaktenum} + + Darum ist $f$ eine Abbildung zwischen $M$ und $N$ + und $\graph(f)=R$. + \end{proof} + \end{einzug} %% SKA 4-3 \let\altsectionname\sectionname @@ -3105,7 +3165,134 @@ aber dies ist streng genommen nicht die ganze Wahrheit. \label{ska:4:ex:3} \let\sectionname\altsectionname -({\itshape Unter Arbeit}) + Sei $X=\{a,b,c\}$ und betrachte die binäre Relation, + $(\Pot(X),\leq)$, definiert durch + + \begin{mathe}[mc]{rcl} + A\leq B &\Longleftrightarrow &X\ohne A\subseteq X\ohne B\\ + \end{mathe} + + für $A,B\in\Pot(X)$. + + \begin{claim*} + $(\Pot(X),\leq)$ ist eine partielle Ordnung (auch »Halbordnung« genannt). + \end{claim*} + + Es gibt nun 3 Ansätze, um dies zu zeigen. + + \begin{proof}[Ansatz I][Ansatz I] + Beobachte, dass für $A,B\in\Pot(X)$ + + \begin{mathe}[mc]{rcl} + A\leq B + &\textoverset{Defn}{\Longleftrightarrow} + &X\ohne A\subseteq X\ohne B\\ + &\Longrightarrow + &X\ohne (X\ohne A)\supseteq X\ohne (X\ohne B)\\ + &\Longrightarrow + &A\supseteq B, + \,\text{da $A,B\subseteq X$}\\ + &\Longrightarrow + &X\ohne A\subseteq X\ohne B\\ + &\textoverset{Defn}{\Longleftrightarrow} + &A\leq B,\\ + \end{mathe} + + also $A\leq B\Leftrightarrow A\supseteq B$. + Darum kann $(\Pot(X),\leq)$ mit $(\Pot(X),\supseteq)$ + identifiziert werden. + Letzteres ist bekanntermaßen eine Halbordnung. + \end{proof} + + \begin{proof}[Ansatz II][Ansatz II] + Im konkreten Falle von $X=\{a,b,c\}$ können wir die Relation + durch ein \emph{Hasse-Diagramm} skizzieren: + + \hraum + {\footnotesize + \begin{tikzpicture}[node distance=1.5cm, thick] + \pgfmathsetmacro\habst{3} + \pgfmathsetmacro\vabst{2} + + \node[label=below:{$X$}] (Set1) at (0*\habst,0*\vabst) {$\bullet$}; + \node[label=left:{$\{a,b\}$}] (SetAB) at (-1*\habst,1*\vabst) {$\bullet$}; + \node[label=above:{$\{a,c\}$}] (SetAC) at (0*\habst,1*\vabst) {$\bullet$}; + \node[label=right:{$\{b,c\}$}] (SetBC) at (1*\habst,1*\vabst) {$\bullet$}; + \node[label=left:{$\{a\}$}] (SetA) at (-1*\habst,2*\vabst) {$\bullet$}; + \node[label=below:{$\{b\}$}] (SetB) at (0*\habst,2*\vabst) {$\bullet$}; + \node[label=right:{$\{c\}$}] (SetC) at (1*\habst,2*\vabst) {$\bullet$}; + \node[label=above:{$\leer$}] (Set0) at (0*\habst,3*\vabst) {$\bullet$}; + + \draw (Set1) edge [->] (SetAB); + \draw (Set1) edge [->] (SetAC); + \draw (Set1) edge [->] (SetBC); + \draw (SetAB) edge [->] (SetA); + \draw (SetAB) edge [->] (SetB); + \draw (SetAC) edge [->] (SetA); + \draw (SetAC) edge [->] (SetC); + \draw (SetBC) edge [->] (SetB); + \draw (SetBC) edge [->] (SetC); + \draw (SetA) edge [->] (Set0); + \draw (SetB) edge [->] (Set0); + \draw (SetC) edge [->] (Set0); + \end{tikzpicture}} + \hraum + + Man sieht, dass dies einen \emph{Verband} und damit insbesondere eine Halbordnung bildet. + \end{proof} + + \begin{proof}[Ansatz III][Ansatz III] + Wir gehen die Axiome einer Halbordnung durch: + + \begin{kompaktenum}[\rtab][\rtab] + \item[\uwave{{\itshape Reflexivität:}}] + Sei $A\in\Pot(X)$ beliebig. + \textbf{Zu zeigen:} $A\leq A$.\\ + Offensichtlich gilt $X\ohne A\subseteq X\ohne A$.\\ + Per Konstruktion gilt also $A\leq A$. + + \item[\uwave{{\itshape Antisymmetrie:}}] + Seien $ A, A'\in\Pot(X)$ beliebig.\\ + \textbf{Zu zeigen:} $A\leq A'$ und $A'\leq A$ $\Rightarrow$ $A=A'$.\\ + Es gilt + + \begin{mathe}[mc]{rclql} + A\leq A'\,\text{und}\, A'\leq A + &\Longleftrightarrow + &X\ohne A\subseteq X\ohne A' + \,\text{und}\, + X\ohne A'\subseteq X\ohne A + &\text{(per Konstruktion)}\\ + &\Longrightarrow + &X\ohne A=X\ohne A' + &\text{(per Definition von Mengengleichheit)}\\ + &\Longrightarrow + &A=A', + &\text{da $A,A'\subseteq X$}.\\ + \end{mathe} + + \item[\uwave{{\itshape Transitivität:}}] + Seien $A, A',(a'',b'')\in\Pot(X)$ beliebig.\\ + \textbf{Zu zeigen:} $A\leq A'$ und $A'\leq A''$ $\Rightarrow$ $A\leq A''$.\\ + Es gilt + + \begin{mathe}[mc]{rcl} + A\leq A'\,\text{und}\, A'\leq A'' + &\Longleftrightarrow + &X\ohne A\subseteq X\ohne A' + \,\text{und}\, + X\ohne A'\subseteq X\ohne A'' + \,\text{(per Konstruktion)}\\ + &\Longrightarrow + &X\ohne A\subseteq X\ohne A''\\ + &\Longleftrightarrow + &A\leq A'' + \,\text{(per Konstruktion)}.\\ + \end{mathe} + \end{kompaktenum} + + Darum erfüllt $(\Pot(X),\leq)$ die Axiome einer Halbordnung. + \end{proof} %% SKA 4-4 \let\altsectionname\sectionname @@ -3114,7 +3301,71 @@ aber dies ist streng genommen nicht die ganze Wahrheit. \label{ska:4:ex:4} \let\sectionname\altsectionname -({\itshape Unter Arbeit}) + Betrachten wir die Halbordnung aus \cite[Beispiel 2.4.2(2)]{sinn2020}. + Es sei also $C=\{a,b,c\}$ und + die durch folgendes \emph{Hasse-Diagramm} dargestellte Ordnungsrelation auf $Pot(C)$: + + \hraum + {\footnotesize + \begin{tikzpicture}[node distance=1.5cm, thick] + \pgfmathsetmacro\habst{3} + \pgfmathsetmacro\vabst{2} + + \node[label=above:{$C$}] (Set1) at (0*\habst,3*\vabst) {$\bullet$}; + \node[label=left:{$\{a,b\}$}] (SetAB) at (-1*\habst,2*\vabst) {$\bullet$}; + \node[label=below:{$\{a,c\}$}] (SetAC) at (0*\habst,2*\vabst) {$\bullet$}; + \node[label=right:{$\{b,c\}$}] (SetBC) at (1*\habst,2*\vabst) {$\bullet$}; + \node[label=left:{$\{a\}$}] (SetA) at (-1*\habst,1*\vabst) {$\bullet$}; + \node[label=above:{$\{b\}$}] (SetB) at (0*\habst,1*\vabst) {$\bullet$}; + \node[label=right:{$\{c\}$}] (SetC) at (1*\habst,1*\vabst) {$\bullet$}; + \node[label=below:{$\leer$}] (Set0) at (0*\habst,0*\vabst) {$\bullet$}; + + \draw (Set0) edge [->] (SetA); + \draw (Set0) edge [->] (SetB); + \draw (Set0) edge [->] (SetC); + \draw (SetA) edge [->] (SetAB); + \draw (SetA) edge [->] (SetAC); + \draw (SetB) edge [->] (SetAB); + \draw (SetB) edge [->] (SetBC); + \draw (SetC) edge [->] (SetAC); + \draw (SetC) edge [->] (SetBC); + \draw (SetAB) edge [->] (Set1); + \draw (SetAC) edge [->] (Set1); + \draw (SetBC) edge [->] (Set1); + \end{tikzpicture}} + \hraum + + Wenn wir das Element $\leer$ von $\Pot(C)$ entfernen sieht die Struktur folgendermaßen aus + + \hraum + {\footnotesize + \begin{tikzpicture}[node distance=1.5cm, thick] + \pgfmathsetmacro\habst{3} + \pgfmathsetmacro\vabst{2} + + \node[label=above:{$C$}] (Set1) at (0*\habst,3*\vabst) {$\bullet$}; + \node[label=left:{$\{a,b\}$}] (SetAB) at (-1*\habst,2*\vabst) {$\bullet$}; + \node[label=below:{$\{a,c\}$}] (SetAC) at (0*\habst,2*\vabst) {$\bullet$}; + \node[label=right:{$\{b,c\}$}] (SetBC) at (1*\habst,2*\vabst) {$\bullet$}; + \node[label=left:{$\{a\}$}] (SetA) at (-1*\habst,1*\vabst) {$\bullet$}; + \node[label=above:{$\{b\}$}] (SetB) at (0*\habst,1*\vabst) {$\bullet$}; + \node[label=right:{$\{c\}$}] (SetC) at (1*\habst,1*\vabst) {$\bullet$}; + + \draw (SetA) edge [->] (SetAB); + \draw (SetA) edge [->] (SetAC); + \draw (SetB) edge [->] (SetAB); + \draw (SetB) edge [->] (SetBC); + \draw (SetC) edge [->] (SetAC); + \draw (SetC) edge [->] (SetBC); + \draw (SetAB) edge [->] (Set1); + \draw (SetAC) edge [->] (Set1); + \draw (SetBC) edge [->] (Set1); + \end{tikzpicture}} + \hraum + + Offensichtlich hat $(\Pot(C)\ohne\{\leer\},\subseteq)$ kein kleinstes Element. + Die Menge der minimalen Elementen ist $\{\{a\},\{b\},\{c\}\}$, + d.\,h. es gibt $3$ minimale Elemente. %% SKA 4-5 \let\altsectionname\sectionname @@ -3123,7 +3374,35 @@ aber dies ist streng genommen nicht die ganze Wahrheit. \label{ska:4:ex:5} \let\sectionname\altsectionname -({\itshape Unter Arbeit}) + Sei $W$ die Menge aller Wörter und $\Sigma$ die Menge aller Buchstaben. + O.\,E. können wir annehmen, dass jedes Wort $w\in W$ der Länge $|w|\geq 2$ ist. + (In Sprachen wie Englisch, Russisch, usw. ist dies nicht der Fall, + aber wir könnten diese trivialen Wörter einfach ausschließen.) + + Betrachten wir die Relation $(W,\sim)$ gegeben durch + + \begin{mathe}[mc]{rcl} + \eqtag[eq:1:ska:4:ex:5] + w\sim w' &:\Longleftrightarrow &f(w)=f(w), + \end{mathe} + + wobei + + \begin{mathe}[mc]{rcccl} + f &: &W &\to &\Sigma\\ + &: &w &\mapsto &\text{1. Buchstabe in $w$}\\ + \end{mathe} + + Dann per Konstruktion \uline{reduziert} $f$ + die Relation $(W,\sim)$ auf $(\Sigma,=)$. + Aufgrund dessen und da $(\Sigma,=)$ eine Äquivalenzrelation ist, + ist $(W,\sim)$ automatisch eine Äquivalenzrelation auch. + + Eigentlich spielt est keine Rolle, wie die Funktion, $f$, aussieht. + Solange die Reduktion \eqcref{eq:1:ska:4:ex:5} gilt, + bleibt $(W,\sim)$ eine Äquivalenzrelation. + Dies gilt also insbesondere ebenfalls, + wenn $f$ den zweitletzten Buchstaben von Wörtern berechnet. %% SKA 4-6 \let\altsectionname\sectionname @@ -3132,7 +3411,34 @@ aber dies ist streng genommen nicht die ganze Wahrheit. \label{ska:4:ex:6} \let\sectionname\altsectionname -({\itshape Unter Arbeit}) +\begin{enumerate}{\bfseries (a)} + %% SKA 4-6a + \item + + \begin{mathe}[mc]{rcl} + \sum_{i=2}^{6}(-1)^{i}i^{2} + &= &(-1)^{2}\cdot 2^{2} + +(-1)^{3}\cdot 3^{2} + +(-1)^{4}\cdot 4^{2} + +(-1)^{5}\cdot 5^{2} + +(-1)^{6}\cdot 6^{2}\\ + &= &4-9+16-25+36 + = 22\\ + \end{mathe} + + %% SKA 4-6b + \item + + \begin{mathe}[mc]{rcl} + \prod_{j=1}^{4}(2j-1) + &= &(2\cdot 1 - 1) + +(2\cdot 2 - 1) + +(2\cdot 3 - 1) + +(2\cdot 4 - 1)\\ + &= &1-3+5-7 + = -4\\ + \end{mathe} +\end{enumerate} %% SKA 4-7 \let\altsectionname\sectionname @@ -3141,7 +3447,94 @@ aber dies ist streng genommen nicht die ganze Wahrheit. \label{ska:4:ex:7} \let\sectionname\altsectionname -({\itshape Unter Arbeit}) + \begin{claim} + \makelabel{claim:main:ska:4:ex:7} + Bezeichne mit $\Phi(n)$ die Aussage + + \begin{mathe}[mc]{rcl} + \eqcref{eq:1:\beweislabel} + \sum_{i=1}^{n}(-1)^{i}i^{2} &= &(-1)^{n}\frac{1}{2}n(n+1).\\ + \end{mathe} + + Dann gilt $\forall{n\in\ntrlpos:~}\Phi(n)$. + \end{claim} + + \begin{einzug}[\rtab][\rtab] + \begin{proof} + Wir zeigen \Cref{\beweislabel} stumpf per Induktion. + + \begin{kompaktenum}[\rtab][\rtab] + \item[\uwave{{\bfseries Induktionsanfang:}}] + Sei $n=1$. Dann + + \begin{mathe}[mc]{rcl} + \sum_{i=1}^{n}(-1)^{i}i^{2} + &= &(-1)^{1}1^{2} = -1\\ + (-1)^{n}\frac{1}{2}n(n+1) + &= &(-1)^{1}\frac{1}{2}\cdot 1\cdot (1+1) = -1\\ + \end{mathe} + + Also gilt \eqcref{eq:1:\beweislabel}. + Also gilt $\Phi(1)$ + + \item[\uwave{{\bfseries Induktionsvoraussetzung:}}] + Sei $n>1$. + Angenommen, $\Phi(n-1)$ gilt. + + \item[\uwave{{\bfseries Induktionsschritt:}}] + \textbf{Zu zeigen:} $\Phi(n)$ gilt, d.\,h. + Gleichung \eqcref{eq:1:\beweislabel} gilt.\\ + Es gilt + + \begin{mathe}[mc]{rcl} + \sum_{i=1}^{n}(-1)^{i}i^{2} + &= &\sum_{i=1}^{n-1}(-1)^{i}i^{2} + (-1)^{n}n^{2}\\ + &= &(-1)^{n-1}\frac{1}{2}(n-1)(n-1+1) + (-1)^{n}n^{2}\\ + &&\text{wegen der IV}\\ + &= &(-1)^{n}\cdot(-\frac{1}{2}n(n-1) + n^{2})\\ + &= &(-1)^{n}\cdot(-\frac{1}{2}n^{2} + \frac{1}{2}n + n^{2})\\ + &= &(-1)^{n}\cdot(\frac{1}{2}n^{2} + \frac{1}{2}n)\\ + &= &(-1)^{n}\frac{1}{2}n(n+1).\\ + \end{mathe} + + Also gilt \eqcref{eq:1:\beweislabel}. + Also gilt $\Phi(n)$. + \end{kompaktenum} + + Also gilt $\forall{n\in\ntrlpos:~}\Phi(n)$. + \end{proof} + \end{einzug} + + Für die Summe $\sum_{i=3}^{n}(-1)^{i}i^{2}$ + ist der Ausdruck lediglich + + \begin{mathe}[mc]{rcl} + \sum_{i=3}^{n}(-1)^{i}i^{2} + &= &\sum_{i=1}^{n}(-1)^{i}i^{2}-(-1)^{1}\cdot 1-(-1)^{2}2^{2}\\ + &= &(-1)^{n}\frac{1}{2}n(n+1)-3\\ + \end{mathe} + + für alle $n\geq 3$. + Sollten wir dies per Induktion beweisen wollen, + brauchen wir lediglich im o.\,s. Beweis + den \textbf{Induktionsanfang} auf $n=3$ zu ändern. + Der Rest bleibt erhalten. + +\begin{rem} + Induktion hat mit Deduzieren (»Ableiten«) nichts zu tun. + Induktion ist nur ein Werkzeug, um Aussagen zu \emph{verifizieren}. + Sie hilft uns überhaupt nicht, um \emph{auf die Behauptungen zu kommen}. + In diesem konkreten Falle wurde Vorarbeit geleistet + und \emph{direkt} argumentiert, + um auf den Ausdruck in \eqcref{eq:1:\beweislabel} zu kommen. + In dieser Vorarbeit steckt die eigentliche mathematische Arbeit + und dies bedarf etwas Kreativität, Intuition, usw. + Häufig reicht diese Vorarbeit aber nur, + um auf eine sinnvolle Behauptung zu kommen, + und zum Schluss runden wir dies mit Induktion ab, + um formal die behauptete Aussage zu bestätigen. + Das ist die eigentliche Rolle von Induktion als Beweismittel. +\end{rem} %% SKA 4-8 \let\altsectionname\sectionname @@ -3150,7 +3543,56 @@ aber dies ist streng genommen nicht die ganze Wahrheit. \label{ska:4:ex:8} \let\sectionname\altsectionname -({\itshape Unter Arbeit}) + \uwave{{\bfseries Kurzes Argument:}}\\ + Wenn jede Farbe jeweils auf maximal $1$ Karte vorkommt, + gibt es $\leq 4\cdot 1=4$ Karten. + Aber $5$ Karten wurden gewählt. + + \uwave{{\bfseries Ausführliches Argument:}}\\ + Seien + ${X:=\{\clubsuit, \diamondsuit, \heartsuit, \spadesuit\}}$ + die Menge der Farben und + ${Y:=\{1,2,3,4,5\}}$ + die Indizes der Karten. + Sei ${f:X\to\Pot(Y)}$ die Funktion, + die der Wahl entspricht, d.\,h. + + \begin{mathe}[mc]{rcl} + f(x) &= &\{y\in Y\mid\text{Karte $y$ hat Farbe $x$}\}\\ + \end{mathe} + + für alle Farben $x\in X$. + + Nun, jede Karte, $y\in Y$, hat eine Farbe, sodass $y\in f(x)$ für ein $x\in X$. + Also $Y\subseteq\bigcup_{x\in X}f(x)$. + Und per Definition $f(x)\subseteq Y$ für alle $x\in X$. + Darum $\bigcup_{x\in X}f(x)\subseteq Y$. + Also + + \begin{mathe}[mc]{rcl} + Y &= &\bigcup_{x\in X}f(x)\\ + \end{mathe} + + Andererseits sind die Mengen $(f(x))_{x\in X}$ paarweise disjunkt, + da jede Karte höchstens eine Farbe hat. + Also ist $(f(x))_{x\in X}$ eine \emph{Partition} von $Y$. + Darum + + \begin{mathe}[mc]{ll} + &|Y| = |\bigcup_{x\in X}f(x)| + = \sum_{x\in X}|f(x)| + \leq |X|\cdot\max_{x\in X}|f(x)|\\ + \Longrightarrow + &\max_{x\in X}|f(x)| \geq |Y|/|X| = 5/4 > 1\\ + \Longrightarrow + &\exists{x\in X:~}|f(x)|>1\\ + \Longrightarrow + &\exists{x\in X:~}|f(x)|\geq 2\\ + \end{mathe} + + Nach der Definition von $f$ heißt dies, + es gibt eine Farbe, $x\in\{\clubsuit, \diamondsuit, \heartsuit, \spadesuit\}$, + so dass mindestens $2$ der gezogenen Karten die Farbe $x$ haben. %% SKA 4-9 \let\altsectionname\sectionname @@ -3159,7 +3601,58 @@ aber dies ist streng genommen nicht die ganze Wahrheit. \label{ska:4:ex:9} \let\sectionname\altsectionname -({\itshape Unter Arbeit}) + \uwave{{\bfseries Kurzes Argument:}}\\ + Wenn jeder Kalendartag jeweils von maximal $17$ Studierenden gefeiert wird, + gibt es $\leq 366\cdot 17=6222$ Studierende. + Aber es gibt $\geq 7000$ Studierende. + + \uwave{{\bfseries Ausführliches Argument:}}\\ + Seien + ${X=\{\text{1.~Jan},\,\text{2.~Jan},\,\ldots,\,\text{31.~Dez}\}}$ + die Menge der Kalendartage + und + ${Y=\{x\mid x\,\text{ein/e Studierende/r an der Uni Leipzig}\}}$. + Sei ${f:X\to\Pot(Y)}$ die Funktion, + die der Wahl entspricht, d.\,h. + + \begin{mathe}[mc]{rcl} + f(x) &= &\{y\in Y\mid\text{Studierende/r $y$ hat am Tag $x$ Geburtstag}\}\\ + \end{mathe} + + für alle Kalendartage $x\in X$. + + Nun, jede/r Studierende/r, $y\in Y$, hat einen Geburtstag, + sodass $y\in f(x)$ für ein $x\in X$. + Also $Y\subseteq\bigcup_{x\in X}f(x)$. + Und per Definition $f(x)\subseteq Y$ für alle $x\in X$. + Darum $\bigcup_{x\in X}f(x)\subseteq Y$. + Also + + \begin{mathe}[mc]{rcl} + Y &= &\bigcup_{x\in X}f(x)\\ + \end{mathe} + + Andererseits sind die Mengen $(f(x))_{x\in X}$ paarweise disjunkt, + da jede/r Studierende/r höchstens einen Geburtstag hat. + Also ist $(f(x))_{x\in X}$ eine \emph{Partition} von $Y$. + Darum + + \begin{mathe}[mc]{ll} + &|Y| = |\bigcup_{x\in X}f(x)| + = \sum_{x\in X}|f(x)| + \leq |X|\cdot\max_{x\in X}|f(x)|\\ + \Longrightarrow + &\max_{x\in X}|f(x)| \geq |Y|/|X| \geq 7000/366 > 19\\ + \Longrightarrow + &\exists{x\in X:~}|f(x)|>19\\ + \Longrightarrow + &\exists{x\in X:~}|f(x)|\geq 20\\ + \end{mathe} + + Nach der Definition von $f$ heißt dies, + es gibt einen Kalendartag, ${x\in\{\text{1.~Jan},\,\text{2.~Jan},\,\ldots,\,\text{31.~Dez}\}}$, + so dass mindestens $20$ Studierende $x$ als Geburtstag feiern. + Insbesondere gibt es $18$ Menschen, die den gleichen Geburtstag feiern. %% SKA 4-10 \let\altsectionname\sectionname @@ -3168,7 +3661,142 @@ aber dies ist streng genommen nicht die ganze Wahrheit. \label{ska:4:ex:10} \let\sectionname\altsectionname -({\itshape Unter Arbeit}) + \begin{claim} + \makelabel{claim:main:ska:4:ex:10} + Bezeichne mit $\Phi(n)$ die Aussage + + \begin{kompaktitem}[\rtab][\rtab] + \item + Für alle endlichen Mengen, $E_{1},E_{2},\ldots,E_{n}$, + gilt $|\prod_{i=1}^{n}E_{i}|=\prod_{i=1}^{n}|E_{i}|$. + \end{kompaktitem} + Dann gilt $\forall{n\in\ntrlpos:~}\Phi(n)$. + \end{claim} + + \begin{einzug}[\rtab][\rtab] + \begin{proof} + Wir zeigen \Cref{\beweislabel} per Induktion. + Als Induktionsanfang widmen wir uns den Fällen $n\leq 2$. + + \begin{kompaktenum}[\rtab][\rtab] + \item[\uwave{{\bfseries Induktionsanfang:}}] + Sei $n=1$. Dann für alle Mengen, $E_{1}$ + + \begin{mathe}[mc]{rcccl} + |\prod_{i=1}^{1}E_{i}| + &= &|E_{1}| + &= &\prod_{i=1}^{1}|E_{i}|\\ + \end{mathe} + + Also gilt $\Phi(1)$. + \item[] + Sei $n=2$. Dann gilt für alle endlichen Mengen $E_{1},E_{2}$ + + \begin{mathe}[mc]{rcccccl} + |\prod_{i=1}^{2}E_{i}| + &= &|E_{1}\times E_{2}| + &= &|E_{1}|\cdot|E_{2}| + &= &\prod_{i=1}^{2}|E_{i}|.\\ + \end{mathe} + + (Dieses Resultat haben wir in \Cref{lemm:1:ska:4:ex:10} ausgelagert.)\\ + Also gilt $\Phi(2)$. + + \item[\uwave{{\bfseries Induktionsvoraussetzung:}}] + Sei $n>2$. + Angenommen, $\Phi(k)$ gilt für alle $k1$. + Angenommen, $|X\times Y'|=|X|\cdot |Y'|$ + für alle $k$-elementigen Mengen, $Y'$ + und für alle $k0$, 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} + |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 des Falls für $1$-elementigen Mengen}\\ + &= &|X|\cdot n\\ + &&\text{wegen der rekursiven Definition von Multiplikation}\\ + &= &|X|\cdot |Y|,\\ + \end{mathe} + \end{kompaktenum} + + Darum gilt $|X\times Y|=|X|\cdot|Y|$ für alle Mengen $X,Y$. + \end{proof} + \end{einzug} %% SKA 4-11 \let\altsectionname\sectionname @@ -3177,7 +3805,22 @@ aber dies ist streng genommen nicht die ganze Wahrheit. \label{ska:4:ex:11} \let\sectionname\altsectionname -({\itshape Unter Arbeit}) + In dem Induktionsschritt + + \begin{quote} + Jetzt können wir aber auch einen der Goldfische rausnehmen + und haben wieder ein Aquarium mit $n$ Fischen \uline{und mindestens einem} Golfisch. + \end{quote} + + Dieser Teil des Arguments voraus, dass unter der zweiten Auswahl von $n$ Fischen + ein Goldfisch vorhanden ist. + In \emph{dieser} Auswahl kommt aber der zuerst rausgezogene Fisch vor + und dieser war kein Goldfisch. + Darum muss ein Goldfisch unter den $n-1$ anderen Fischen. + Aber das ist nur möglich, wenn $n-1\geq 1$, + also wenn $n\geq 2$. + + Das heißt, das Induktionsargument überspringt den Fall $n=2$! \setcounternach{part}{3} \part{Quizzes}