New PDF release: Algebra und Diskrete Mathematik

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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Dann heißt die Menge F ✷G := {(a, c) ∈ A × C | ∃b ∈ B : (a, b) ∈ F ∧ (b, c) ∈ G} Verkettung (Produkt, Hintereinanderausf¨ uhrung) von F und G. ) Seien F = {(0, 0), (0, 1), (2, 0), (2, 3)} und G = {(0, 2), (1, 1)}. Dann gilt F −1 = {(0, 0), (1, 0), (0, 2), (3, 2)} und F ✷G = {(0, 2), (0, 1), (2, 2)}. ) F¨ ur F = {(x, x2 ) | x ∈ R} und G = {(x, sin(x)) | x ∈ R} gilt F ✷G = {(x, sin(x2 )) | x ∈ R}. 1 Seien F ⊆ A × B, G ⊆ B × C und H ⊆ C × D Korrespondenzen. Dann gilt: (1) F ✷(G✷H) = (F ✷G)✷H; (2) (F ✷G)−1 = G−1 ✷F −1 .

6 Boolesche Funktionen und Pr¨ adikate 39 Mittels f (x1 , x2 , . . , xn ) := A ist dann eine n-stellige Boolesche Funktion definierbar, wobei obige Vorschrift bedeutet, daß bei konkreter Belegung der xi (i = 1, 2, . . , n) mit 0 oder 1 der zugeh¨ orige Funktionswert sich anhand der Formel A errechnet. Man beachte, daß jedem Term unendlich viele Boolesche Funktionen zugeordnet werden k¨ onnen. Außerdem k¨ onnen verschiedene Terme gleiche Funktionen beschreiben. Interessiert sind wir nat¨ urlich nur an einfachen Termen.

F¨ ur n = 2 gibt es 16 verschiedene Boolesche Funktionen, von denen 5 (∧, ∨, +, =⇒, ⇐⇒) in der folgende Tabelle definiert sind. Dabei sei ◦(x, y) := x ◦ y f¨ ur ◦ ∈ {∧, ∨, +, =⇒, ⇐⇒} vereinbart. x 0 0 1 1 y x ∧ y x ∨ y x + y x =⇒ y x ⇐⇒ y 0 0 0 0 1 1 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 1 . Die Erkl¨ arung f¨ ur die verwendeten Symbole findet man weiter unten. Anstelle von x∧y schreiben wir auch x·y oder kurz xy. Allgemein lassen sich Booleschen Funktionen f (n) durch Tabellen der Form x1 x2 0 0 0 0 ..

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Algebra und Diskrete Mathematik by Dietlinde Lau


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