From c9cf9f2e1a026e2e0966d83dd7d299670b36001f Mon Sep 17 00:00:00 2001
From: raj_mathe <raj.dahya@math.uni-leipzig.de>
Date: Mon, 23 Nov 2020 18:04:23 +0100
Subject: [PATCH] master > master: SKA 4-10 direkter Beweis vom Lemma

---
 docs/loesungen.pdf | Bin 393568 -> 396314 bytes
 docs/loesungen.tex | 137 +++++++++++++++++++++++++++++----------------
 2 files changed, 89 insertions(+), 48 deletions(-)

diff --git a/docs/loesungen.pdf b/docs/loesungen.pdf
index 40bf588d5dc48d7265aad850da81cac89e6aa9d3..62404c7efadf914fec6ff0a27f7dbfe7076e0590 100644
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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 $k<n$.
+                    Angenommen, $\Phi(n-1)$ gilt.
 
                 \item[\uwave{{\bfseries Induktionsschritt:}}]
                     Seien $E_{1},E_{2},\ldots,E_{n}$ beliebige endliche Mengen.\\
@@ -3740,62 +3742,101 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion.
 
         \begin{einzug}[\rtab][\rtab]
         \begin{proof}
-            Wir zeigen dies per Induktion über $|Y|$ mit den Fällen $|Y|\leq 1$ als Induktionsanfang.
+            Wir zeigen dies direkt. Seien $m:=|X|$ und $n:=|Y|$.
+            Wegen Endlichkeit liegen $m,n$ in $\ntrlzero$.
+
+            Falls $m=0$ oder $n=0$, so gilt $X=\leer$ oder $Y=\leer$ und damit
+
+                \begin{mathe}[mc]{rcccccccl}
+                    |X\times Y|
+                    &= &|\leer|
+                    &= &0
+                    &= &m\cdot n
+                    &= &|X|\cdot|Y|.\\
+                \end{mathe}
+
+            Beschränken wir uns also auf den Famm $m,n>0$.
+            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}