Fragen zum Abbildungsbegriff / Questions regarding the notion of map between two sets.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "In dieser Frage soll Ihr Verständnis des Abbildungsbegriffs an zwei einfachen Beispielen überprüft werden.
", "advice": "Beispiel 1. Die Vorschrift definiert keine Abbildung, denn zum Beispiel für die Zahl $\\frac 13 = \\frac 26$ erhält man unterschiedliche Ergebnisse, je nachdem welche Darstellung der Bruchzahl man verwendet.
\nBeispiel 2. Da für jede Bruchzahl eine eindeutige Darstellung der Form $\\frac ab$ mit $a,b\\in \\mathbb Z$, $b>0$ und ${\\rm ggT}(a,b)=1$ existiert (nämlich die Darstellung als gekürzter Bruch mit positivem Nenner), wird in diesem Fall eine Abbildung von $\\mathbb Q\\to \\mathbb Z$ definiert.
\nZur Berechnung der Funktionswerte: Nach Definition der Abbildungsvorschrift müssen wir zunächst den gekürzten Bruch mit positivem Nenner finden, der die gegebene Zahl darstellt. Danach erhalten wir das Ergebnis, indem wir Zähler und Nenner addieren. Also:
\nEs gilt $\\frac{\\var{aa}}{\\var{bb}} = \\frac{\\var{ax}}{\\var{bx}}$, also $f(\\frac{\\var{aa}}{\\var{bb}}) = f(\\frac{\\var{ax}}{\\var{bx}}) = \\var{res1}$.
\nEs gilt $\\frac{\\var{aa1}}{\\var{bb1}} = \\frac{\\var{ax1}}{\\var{bx1}}$, also $f(\\frac{\\var{aa1}}{\\var{bb1}}) = f(\\frac{\\var{ax1}}{\\var{bx1}}) = \\var{res2}$.
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\n$f(\\frac{\\var{aa}}{\\var{bb}})$ = [[0]]
\n$f(\\frac{\\var{aa1}}{\\var{bb1}})$ = [[1]]
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