Download Algebra und Diskrete Mathematik by Dietlinde Lau PDF

By Dietlinde Lau

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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3 Relationen 17 {a, b} r ☞ ✎✛ ✒✻ ■ ❅ ✎ ✍ ☞❅✌ ☞ ✎ ❅ r ✌ ✍ ❅r✛ ✌ ✍✛ ■ ✒{b} {a}❅ ❅✎ ☞ ❅ ❅r✛ ✌ ✍ ∅ Verzichtet man auf diejenigen Pfeile, die sich aus der Transitivit¨at und Reflexivit¨ at ergeben und vereinbart man, die gr¨ oßeren“ Elemente oberhalb der ” kleineren“ Elemente anzuordnen, so gen¨ ugt zur Veranschaulichung der Re” lation R folgende Zeichnung: {a, b} s ❅ ❅ s ❅s {b} {a}❅ ❅ ❅ s ∅ Eine solche Darstellung einer reflexiven teilweisen Ordnung nennt man Hasse6 -Diagramm. B. zum Hasse-Diagramm 6s 4s ❅ ❅ ❅ ❅ ❅s ❅s s5 2❅ 3✟✟✟ ❅ ✟ ❅s ✟✟ 1 die Relation {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), (2, 6), (3, 6)} (= {(a, b) ∈ {1, 2, 3, 4, 5, 6}2 | a teilt b}).

3 F¨ ur beliebige ϕ, ψ ∈ F ORM gilt: (1) ¬∀xk ϕ ≡ ∃xk ¬ϕ, ¬∃xk ϕ ≡ ∀xk ¬ϕ; ur alle Q ∈ {∃, ∀} und (2) falls xk ∈ f r(ψ), dann (Qxk ϕ) ◦ ψ ≡ Qxk (ϕ ◦ ψ) f¨ ◦ ∈ {∧, ∨}; (3) (∀xk ϕ) ∧ (∀xk ψ) ≡ ∀xk (ϕ ∧ ψ), (∃xk ϕ) ∨ (∃xk ψ) ≡ ∃xk (ϕ ∨ ψ); (4) ∀xk (∀xl ϕ) ≡ ∀xl (∀xk ϕ), ∃xk (∃xl ϕ) ≡ ∃xl (∃xk ϕ). 7 Graphen 51 Beweis.

F ORM wird durch u : V ar −→ A erf¨ ullt, genau dann, wenn vA,u (ϕ) = 1. ur alle u : V ar −→ A ϕ ist wahr (oder g¨ ultig) in A genau dann, wenn vA,u (ϕ) = 1 f¨ gilt. Wir schreiben A |= ϕ f¨ ur ϕ ist wahr in A“. ” A |= ϕ steht f¨ ur ϕ ist nicht wahr (oder falsch) in A“. ” Falls A |= ϕ, heißt A ein Model der Formel ϕ. Gilt A |= σ f¨ ur alle σ ∈ Σ ⊆ F ORM , nennen wir A ein Model der Menge Σ. Offenbar ist eine geschlossene Formel aus F ORM entweder wahr oder falsch in A. Formeln α, β ∈ F ORM heißen ¨ aquivalent (Bezeichnung: α ≡ β) genau dann, wenn ur jede Struktur A und f¨ ur jede Abbildung u : V ar −→ A.

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