Download e-book for iPad: Algebra by Gert Böhme (auth.)

By Gert Böhme (auth.)

ISBN-10: 3540104925

ISBN-13: 9783540104926

ISBN-10: 3642966136

ISBN-13: 9783642966132

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Extra resources for Algebra

Example text

Sie wird der Definition des mathematischen Relationsbegriffes zugrundegelegt werden. Ausdrucklich sei auf den Unterschied zwischen Elementepaar (a, b) und zweielementiger Menge la, bj hingewiesen. " (a=c b=d) 1\ V (a=d 1\ b=c) gilt. Umso interessanter ist der Satz von Kuratowski I Der Paarbegriff kann auf den Mengenbegriff zuruckgefUhrt werden. Beweis: Man setze (a,b) := Ila), la,b)) und zeige die Gultigkeit der Eigenschaften (1) und (2). b,all = llbl, la,b) (b,a) * (a,b). , le,d!! = (c,d). , I a, b !!

CobE[a]. Sei nun c ein beJiebiges Element der Klasse [bJ : c E [bJ. Dann gilt c ~ b, mit b ~ a also c ~ a (Transitivitat) und somit c E [aJ. Also [bJ c [aJ. Nun nehmen wir umgekehrt ein beliebiges dE raJ an: d ~ a raJ C 1\ a [bJ ~ 1\ b co d ~ b '" dE [bJ '" raJ c [b] . ) In Konsequenz dieses Satzes kann man die Aquivalenz zwischen Elementen durch die Gleicheit der entsprechenden Klassen ersetzen. Damit wird ein AbstraktionsprozeB realisiert, der von den Objekten zu den Klassen fUhrt. In der Praxis begegnet uns dieser Sachverhalt uberal!

Wir ki::innen sagen, das Merkmal "im gleichen Bundesland geboren" erzeugt eine Aquivalenzrelation. Zugleich wird an diesem Beispiel deutlich, daB die Relation in elementefremde Teilmengen zerlegt werden kann (die "Bayern" etc. ), die eine Klasseneinteilung von R darstellen. Definition Ist Reine Aquivalenzrelation auf M, so nennen wir zwei in der Beziehung R zueinanderstehende Elemente a qui val en t x~y : ... 2 Relationen 37 und die Menge aller zu einem Element a EM aquivalenten Elemente [a] = I x Ix E M Ax ~ai I eine Aquivalenzklasse mit a als Reprasentanten.

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Algebra by Gert Böhme (auth.)


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