Dietlinde Lau's Algebra und Diskrete Mathematik PDF

By Dietlinde Lau

ISBN-10: 3540723641

ISBN-13: 9783540723646

Algebra und Diskrete Mathematik gehören zu den wichtigsten mathematischen Grundlagen der Informatik. Dieses zweibändige Lehrbuch liegt jetzt in korrigierter zweiter Auflage vor und führt umfassend und lebendig in den Themenkomplex ein. Dabei ermöglichen ein klares Herausarbeiten von Lösungsalgorithmen, viele Beispiele, ausführliche Beweise und eine deutliche optische Unterscheidung des Kernstoffs von weiterführenden Informationen einen raschen Zugang zum Stoff. Die umfangreiche Sammlung von Übungsaufgaben erleichtert nicht nur eine aktive Erarbeitung des Inhalts, sondern zeigt auch die unterschiedlichsten Anwendungsmöglichkeiten auf.

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Example text

Setzt man diesen Prozeß fort, erh¨ alt man die abz¨ ahlbare Teilmenge {ai | i ∈ N} von B. A. sei A ⊆ C. 2 kann C\A weder endlich noch abz¨ ahlbar sein. Folglich ist C\A u ¨ berabz¨ahlbar und enth¨ alt nach (a) eine abz¨ ahlbare Teilmenge N . Sei M := (C\A)\N . Dann gilt C\A = N ∪ M und C = (A ∪ N ) ∪ M . Zwischen den abz¨ahlbaren Mengen N und A ∪ N gibt es eine bijektive Abbildung. Ordnet man außerdem jedem x ∈ M wieder x zu, hat man eine bijektive Abbildung zwischen C\A und C erhalten, womit C\A ∼ C gezeigt ist.

Weiter mit Definitionen Sei A eine Menge. Man sagt • A ist unendlich :⇐⇒ ∃B : B ⊂ A ∧ B ∼ A; • A ist endlich :⇐⇒ A ist nicht unendlich. Beispiel Die Unendlichkeit der Menge N0 ergibt sich aus der Bijektion f := {(n, 2 · n) | n ∈ N0 } von N0 auf die echte Teilmenge der geraden Zahlen von N0 . Eine erste Unterscheidung zwischen verschiedenen Unendlichkeiten“ liefern ” die Definitionen Sei A eine Menge. Dann heißt • A abz¨ ahlbar (unendlich) :⇐⇒ A ∼ N; • Au ahlbar :⇐⇒ A unendlich und nicht abz¨ahlbar.

A2 , b1 ) (a2 , b✟ (a2 , b3 ) ✟ (a✟ 2) 2 , b4 ) ✟✑ ✑✟✟ ✑ ✑ ✑ ✟✟✑ ✟✟ ✟ ✑ ✑ ✑ ✰ ✑ ✰ ✰ ✑ ✟ ✟ ✑ ... ✟ (a3 , b3 ) (a3 , b4 ) (a3 , b1 ) (a3 , b✟ 2) ✑✟ ✑ ✟ ✑ ✑ ✟ ✑ ✟ ✑ ✰ ✑ ✰ ✑ ✟ ... (a4 , b1 ) (a4 , b2 ) (a4 , b3 ) (a4 , b4 ) ✑ ✑✏✏ ✶ ✏ ✑ ✰ ✏ ✑ .. .. . . , unsere bijektive Abbildung von A1 × A2 auf N sieht wie folgt aus: (a1 , b1 ) → 1, (a1 , b2 ) → 2, (a2 , b1 ) → 3, . ). 57). 4 (a) Q ∼ N. (b) A abz¨ahlbar =⇒ {M ∈ P(A) | M ist endlich} abz¨ahlbar.

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Algebra und Diskrete Mathematik by Dietlinde Lau


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