Algebra und Diskrete Mathematik by Dietlinde Lau PDF

By Dietlinde Lau

ISBN-10: 3540203974

ISBN-13: 9783540203971

ISBN-10: 3540203982

ISBN-13: 9783540203988

ISBN-10: 3540723641

ISBN-13: 9783540723646

ISBN-10: 3540725539

ISBN-13: 9783540725534

Band 1 Grundbegriffe der Mathematik, Algebraische Strukturen 1, Lineare Algebra und Analytische Geometrie, Numerische Algebra. Band 2 Lineare Optimierung, Graphen und Algorithmen, Algebraische Strukturen und Allgemeine Algebra mit Anwendungen

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Angenommen, die Menge (0, 1) ist abz¨ahlbar. a11 a12 a13 a14 . . a21 a22 a23 a24 . . a31 a32 a33 a34 . . . (∀i, j ∈ N : aij ∈ {0, 1, 2, . . an1 an2 an3 . , n ∈ N. Wenn wir zeigen k¨ alle x ∈ (0, 1) angeordnet haben) mindestens eine reelle Zahl y ∈ (0, 1) in der Aufz¨ ahlung nicht enthalten ist, h¨ atten wir einen Widerspruch zur Annahme und unsere Behauptung w¨ are bewiesen. B. y1 y2 y3 y4 . . , wobei yi := 0 1 falls aii = 0, (i ∈ N) sonst. Die Zahl y ist von 0 verschieden, da die Zahlen 0, a0000...

5 (mit Definition) Sei A eine Menge mit einer ¨außeren Verupfung ◦, die mit kn¨ upfung ∧K (K gewisse Menge) und einer inneren Verkn¨ ¨ einer Aquivalenzrelation R auf A vertr¨aglich sind. , die Definition der Verkn¨ upfung ◦ bzw. ∧K auf A/R ist unabh¨angig von der konkreten Wahl der sogenannten Vertreter a, b der ¨ Aquivalenzklassen [a], [b]. Beweis. (1): Seien a, a ∈ [a] und b, b ∈ [b]. Wir haben [a ◦ b] = [a ◦ b ] zu zeigen. Aus a, a ∈ [a] und b, b ∈ [b] folgt (a, a ) ∈ R und (b, b ) ∈ R. 1, (2) [a ◦ b] = [a ◦ b ] gilt.

Es gilt dann: Eine beliebige Boolesche Funktion ist genau dann mittels Superposition (Ineinandereinsetzen von Funktionen in Funktionen, Umordnen der Variablen, Identifizieren von Variablen) aus Elementen einer Menge A von Booleschen Funktionen erzeugbar, wenn zu jeder der 5 Relationen R0 R1 R2 R3 R4 := {0}, := {1}, := {(0, 1), (1, 0)}, := {(0, 0), (0, 1), (1, 1)}, := {(a, b, c, d) ∈ {0, 1}4 | a + b = c + d (mod 2)} = {(0, 0, 0, 0), (0, 0, 1, 1), (0, 1, 0, 1), (0, 1, 1, 0), (1, 1, 1, 1), (1, 1, 0, 0), (1, 0, 1, 0), (1, 0, 0, 1)} 11 in A eine Funktion existiert, die diese Relation nicht bewahrt.

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Algebra und Diskrete Mathematik by Dietlinde Lau

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