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Bezeichne mit $\Phi(n)$ die Aussage \begin{mathe}[mc]{rcl} - \eqcref{eq:1:\beweislabel} + \eqtag[eq:1:\beweislabel] \sum_{i=1}^{n}(-1)^{i}i^{2} &= &(-1)^{n}\frac{1}{2}n(n+1).\\ \end{mathe} @@ -3519,13 +3519,16 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion. 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}. + Man merkt, dass Induktion mit Deduzieren (»Ableiten«) nichts zu tun hat. + Induktion ist schließlich nur ein Werkzeug, + um Behauptungen zu \emph{verifizieren}. + Sie verschafft uns aber keine Mittel, + 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 + Ohne diese Arbeit wären wir auf diesen Ausdruck gar nicht gekommen. + In dieser Vorarbeit steckt also die eigentliche mathematische Arbeit und dies bedarf etwas Kreativität, Intuition, usw. Häufig reicht diese Vorarbeit aber nur, um auf eine sinnvolle Behauptung zu kommen, @@ -3543,8 +3546,8 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion. \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. + gibt es $\leq 4\cdot 1$ Karten. + Aber $5$ Karten werden gewählt. \uwave{{\bfseries Ausführliches Argument:}}\\ Seien @@ -3590,7 +3593,7 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion. 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. + so dass $\geq 2$ der gezogenen Karten der Farbe $x$ sind. %% SKA 4-9 \let\altsectionname\sectionname