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# quiz 3 - 1(a ∀ xP x ⇔ ¬∃ x ¬ P x True(b ∃ xP x...

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CSE 260 QUIZ-3– Predicate Logic- ANSWERS (20 minutes) NAME: 1. (4 points) Convert the following with nested quantifiers into an equiv- alent compound proposition using only and and no quantifiers. x y P ( x, y ) where P ( x, y ) is a propositional function and the universe of discourse for x and y: { 1 , 2 } . x ( P ( x, 1) P ( x, 2)) ( P (1 , 1) P (1 , 2)) ( P (2 , 1) P (2 , 2)) or, y P (1 , y ) ∧ ∃ y P (2 , y ) ( P (1 , 1) P (1 , 2)) ( P (2 , 1) P (2 , 2)) 2. (8 points) Determine the truth value of each of these statements if the universe of discourse for all variables consists of all integers. (a) n ( n 2 0) True (b) n ( n 2 = 2) False (c) n m ( m + n = 0) False (d) m n ( m + n = 0) True (4 points) Complete the following equivalences by replacing ? marked below with an approriate quantifiers. ! x p ( x ) ⇔ ∃ x ( p ( x ) ? (( x 6 = y ) → ¬ p ( y ))) ! x p ( x ) ⇔ ∃ x ( p ( x ) y (( x 6 = y ) → ¬ p ( y ))) 3. (10 points) Answer the following true/false.

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Unformatted text preview: 1 (a) ∀ xP ( x ) ⇔ ¬∃ x ¬ P ( x ) True (b) ∃ xP ( x ) ⇔ ¬∀ xP ( x ) False 4. (10 points) Translate each of the following statements into logical ex-pressions using predicates, quanti±ers, and logical connectives. predicates: C(x): x is a CSE 260 student L(x): x loves music Universe of discourse for the variable x is all students. (a) Every student loves music ∀ x L ( x ) (b) No student loves music ∀ x ¬ L ( x ) (c) Some students love music ∃ x L ( x ) (d) Every CSE 260 student loves music. ∀ x ( C ( x ) → L ( x )) (e) Some CSE 260 students love music. ∃ x ( C ( x ) ∧ L ( x )) 2...
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quiz 3 - 1(a ∀ xP x ⇔ ¬∃ x ¬ P x True(b ∃ xP x...

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