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Extra resources for [Article] The Effects of an Homologous Series of Amines on the Mobilities of Ions in Hydrogen Gas

Example text

2. L. 5. Sind a und b zwei relative ganze Zahlen (pos. ), so gelte aRb, wenn a - b durch 2 teilbar ist. Zeigen Sie, dl& es sich hier urn eine Aquivalenzrelation handelt, und bestimmen Sie die Klassen. 6. Man hat in diesem Kapitel gesehen, dl& jede Aquivalenzrelation auf einer Menge eine Zerlegung dieser Menge bestimmt. Gilt auch die Umkehrung? 7. Zeigen Sie, dl& die Relation ~ (nicht strenge Inklusion) zwischen Untermengen einer Menge eine partielle Ordnungsrelation ist. 8. Zeigen Sie, dl& eine partielle Ordnungsrelation < auf einer endlichen Menge durch ein Hasse-Diagramm dargestellt werden kann.

Auf gleiche Weise erhiilt man • mit (3) und (1): (A II B) U (X II B) = B = (X U B) II (A U B) ; • mit (1) und (0): (X II B) U (X II B) = X = (X U B) II (X U B) ; (A II B) U (X II B) = (A U B) II (X U B) ; • mit (3) und (0): • mit (2) und (1): (A II B) U (X II B) = (X U B) II (A U B). 54 3. Definitionen und Eigenschaften der Booleschen Algebra Weisen wir zum Beispiel im letzten Fall die oben angegebene Regel nach: a) Minterme, die nicht in der Vereinigung (3) und (0) vorkommen; b) Komplement von (3): (0) U (1) U (2) Komplement von (0): (1) U (2) U (3); der Durchschnitt beider Komplemente ergibt (1) U (2); c) aber: Xu B = (0) U (1) U (0) U (2) = (0) U (1) U (2) und AU B = (1) U (2) U (3), woraus die Aquivalenz zwischen (A n B) U (X n B) und (X U B) n (A U B) folgt.

Wichtige Eigenschaften und Anwendungen 2. Betrachten wir nun das Komplement X von A. Es bedarf keiner ErHiuterung, d~ A U A = 1 und A Ii X = 0 . 3. Zwei Klassen A und B sind gleich, wenn gleichzeitig gilt AC B und BCA. 4. Betrachten wir A U A und A Ii A. Wir wissen, d~ A C Bauch wie folgt geschrieben werden kann: A Ii B = A oder A U B = B. Da gilt: A C A, haben wir A Ii A = A und A U A = A . Man hat dieser Eigenschaft den Namen Idempotenz vorbehalten. Sie ist wichtig, denn sie bedeutet, d~ in der Schreibweise der Ergebnisse von Operationen, die durch die Zeichen Ii und U symbolisiert sind, weder Exponenten noch Koefflzienten notwendig sind.

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