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zOSx<)5BghrQR`tDy_8qR^ecAFqBO3#@QjfIC|%a8q<$r3TeqO@sT7s;c6(Zp#-Phi zwPYlV`-;&vRnfK&V9tZpH8&^|LBujyrM-jtyB zWi2pa0#EZJc9Q0o#$o{~jrOEatrg@i4E?$Az6IEXe7hVN*+;O#VqzNZAfZ%Tu!wP( znH>*uMX47rxdh}cTXgZD>JhQhopClr5;t(ufa_5)RuWLyZzHs?Op1?s=^{owPx9l< gY`eB^P;mV!)#PSGC_)5oPA(2sI7&)!c?r1x0e=fl{{R30 diff --git a/docs/loesungen.tex b/docs/loesungen.tex index 9480551..e44ea57 100644 --- a/docs/loesungen.tex +++ b/docs/loesungen.tex @@ -207,7 +207,6 @@ \usepackage{cleveref} % must vor hyperref geladen werden. \pgfplotsset{compat=newest} -\usetikzlibrary{math} \usetikzlibrary{ angles, @@ -217,9 +216,11 @@ decorations, decorations.pathmorphing, decorations.pathreplacing, + math, positioning, patterns, quotes, + snakes, } %% \var ≈ alter Befehl @@ -3805,22 +3806,115 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion. \label{ska:4:ex:11} \let\sectionname\altsectionname - In dem Induktionsschritt + Um ein Argument zurückzuweise, reicht es häufig aus, + das Argument einfach \emph{ausführlich} aufzuschreiben. + Wir nehmen die Ausführung und formalisieren diese: + + \begin{claim*} + Bezeichne mit $G(x)$, dass $x$ ein Goldfisch ist. + Für $n\in\ntrlpos$ bezeichne mit $\Phi(n)$ folgende Aussage + + \begin{kompaktitem}[\rtab][\rtab] + \item + Für alle $n$-elementigen Mengen, $X$, von Fischen, + wenn $\exists{x\in X:~}G(x)$, + dann $\forall{x\in X:~}G(x)$. + \end{kompaktitem} + + Dann $\forall{n\in\ntrlpos:~}\Phi(n)$ + \end{claim*} + + \begin{proof}[ungültiges Argument] + Dies wird per Induktion argumentiert. + + \begin{kompaktenum}[\rtab][\rtab] + \item[\uwave{{\bfseries Induktionsanfang:}}] + Betrachte eine $1$-elementige Menge, $X$, von Fischen.\\ + Angenommen, ein $x_{0}\in X$ mit $G(x_{0})$ existiere.\\ + Da $X$ nur dieses eine Element enthält, + gilt offensichtlich $\forall{x\in X:~}G(x)$. + + \item[\uwave{{\bfseries Induktionsvoraussetzung:}}] + Sei $n\in\ntrlpos$ mit $n>1$. + Angenommen, $\Phi(k)$ gilt für alle $k1$ vorausgesetzt! + + Das heißt das Induktionsargument ist faul, + weil der Schritt $1\rightsquigarrow 2$ implizit übersprungen wird. \setcounternach{part}{3} \part{Quizzes}