Blatt7_Loesung - Ubungen zur Mathematik I f r Studierende...

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¨ Ubungen zur Mathematik I f¨ur Studierende Informatik und Wirtschaftsinformatik (Diskrete Mathematik) im Wintersemester 2014/2015 Fachbereich Mathematik, Stefan Geschke B: Hausaufgaben zum 4. und 5. Dezember 2014 1. Sei f : A B eine Funktion, A 1 , A 2 A und B 1 , B 2 B . Zeigen Sie: (a) f - 1 [ B 1 B 2 ] = f - 1 [ B 1 ] f - 1 [ B 2 ] (b) f [ f - 1 [ B 1 ]] B 1 osung. (a) Sei a f - 1 [ B 1 B 2 ]. Dann gilt f ( a ) B 1 B 2 . Also ist f ( a ) B 1 oder f ( a ) B 2 . Damit gilt a f - 1 [ B 1 ] oder a f - 1 [ B 2 ]. Das zeigt a f - 1 [ B 1 ] f - 1 [ B 2 ]. Also gilt f - 1 [ B 1 B 2 ] f - 1 [ B 1 ] f - 1 [ B 2 ]. Sei nun a f - 1 [ B 1 ] f - 1 [ B 2 ]. Dann ist f ( a ) B 1 oder f ( a ) B 2 . Also gilt f ( a ) B 1 B 2 und damit a f - 1 [ B 1 B 2 ]. Es folgt f - 1 [ B 1 ] f - 1 [ B 2 ] f - 1 [ B 1 B 2 ]. (b) Sei b f [ f - 1 [ B 1 ]]. Dann existiert a f - 1 [ B 1 ] mit f ( a ) = b . Wegen a f - 1 [ B 1 ] gilt b = f ( a ) B 1 . Das zeigt f [ f - 1 [ B 1 ]] B 1 . 2. Wir suchen Beispiele daf¨ur, dass in Satz 4.25 aus dem Skript in (1), (5) und (6) tats¨ achlich nicht die Gleichheit gelten muss.
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