New PDF release: Algebra

By Professor Gert Böhme (auth.)

ISBN-10: 3540069860

ISBN-13: 9783540069867

ISBN-10: 3642962351

ISBN-13: 9783642962356

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Extra resources for Algebra

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26). l Satz Ist RcA 2 eine Relation mit Deckung im Vorbereich, die symmetrisch und transitiv ist, so ist Rauch reflexiv. Beweis: Wegen V R metrie) => =A gibt es zu jedem xEA ein yEA mit (x,y) ER 1\ (y,x) ER (Sym- (x,x) ER (Transitivitat). Da (x,x) ER fUr aIle xEA gilt, ist R reflexiv. Beispiel Es sei A = B die Menge der Familienangehorigen. Dann ist die Relation R mit xRy ~ ~ "x ist Schwester von y" transitiv. Sind x, y, z drei Schwestern, so ist dies klar. Sind x,y Schwestern, zein Bruder derselben, so folgt aus (x,y) ER und (y,z) ER ebenfalls (x, z) E R.

Dabei zeigt es sich, daB bereits eine geringe Anzahl von Eigenschaften genugt, urn die wichtigsten Relationentypenherauszuheben. 20 ausschlieBlich Doppelpfeile oder Schlaufen, also keine Einfachpfeile vor (Abb. 18). Stehen x,y fUr Zahlen, so kann man sich auch des ublichen kartesischen Koordinatensystems bedienen und erkennt dann die Symmetrieeigenschaft an der spiegelbildlichen Lage der Punkte bezuglich der Winkelhalbierenden von x- und y-Achse (Abb. 19,20). i In der DDR-Literatur "Relationsgraph" genannt.

Grundlagen der Algebra Diese (diskrete oder kontinuierliche) Punktmenge heiBt Graph von f (Abb. 42). 9. In der DDR-Literatur wird der Abbildungsbegriff anders erkli:irt: nur die dort "eindeutigen Abbildungen von A in (auf) B" sind synonym mit den "Abbildungen von A in (auf) B" in unserem Sinne. 10. 42 zuordnet. Die damit bestimmte Paarmenge deckt sich dann mit unserem FUnktionsbegriff, so daB sich dieser Unterschied in den Erklarungen auf alle folgenden Betrachtungen nicht auswirkt. Bei BOURBAKI wird Funktion als Tripel (A, B, f) definiert und die Paarmenge f als "funktionelle Relation II bezeichnet.

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Algebra by Professor Gert Böhme (auth.)

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