Lecture_20 - Lecture 20 Lecture 20 Examples Examples Fix a...

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Unformatted text preview: Lecture 20 Lecture 20 Examples Examples Fix a language over the following signature: P(­),Q(­),R(­,­),a,b Let M be the following interpretation: X = {0,1,2,3,4} PM = {0,1} QM = {1,3,4} RM = {(0,0),(0,1),(2,1),(2,4),(3,4),(4,0),(4,2)} aM = 1, bM = 3 Evaluate Evaluate ∃ x(¬(x≈ a)∧ P(x)) ∧∃ x(¬(x≈ a)∧ Q(x)) ∃ x(¬(x≈ a)∧ ∧ P(x) Q(x)) ∀x(R(x,x) → R(x,a)) ∀x(R(b,x) → ¬Q(x)) ∃ x∀y(R(x,y) → P(y)) ∀x∃ y(R(x,y) ∧ Q(y)) Find the set Find the set R(x,a) ∃ yR(x,y) ∃ z[R(x,z)∧ R(y,z)] ∀y[R(x,y) →(y ≈ a)] ∀y[R(x,y) →Q(y)] ∃ z[R(x,z)∧ Q(z)] ∧ ≈ y y Find a defining condition Find a defining condition {2,3,4} {0} {0,2,3,4} {1} {(4,2),(2,4),(1,3)} {(0,0,0),(1,1,1),(2,2,2),(3,3,3),(4,4,4)} Give counterexamples to the Give counterexamples to the following ∀x(P(x) ∨ Q(x)) ╞ ∀xP(x) ∨ xQ(x) ∀ ∃ xP(x) ∧∃ xQ(x) ╞ ∃ x(P(x) ∧ Q(x)) ∀x∃ yR(x,y) ╞ ∃ x∀yR(x,y) ∃ x∀yR(x,y) ╞ ∀x∃ yR(x,y) ...
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