33. Rewrite each of these statements so that negations ap-pear only within predicates (that is, so that no negationis outside a quantifier or an expression involving logicalconnectives).x(x, y)a) xy(x, y)c)x(P(x, y) v Q(x, y))d) (x³y-P(x, y)e)x(yz(x, y, z) z(x, y, z)b)x(x, y))

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Answered: 33. Rewrite each of these statements so…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. Using logical equivalence, prove that (p^q)⇒(q⇒p) is a tautology. In other words, show that (pa)(a⇒p)=T You have to give clear reasons for each step!
Essentials of DISCRETE MATHEMATICS
31. Express the negations of each of these statements so that all negation symbols immediately precede predicates. a) Vx3yz(x, y, z) b) Vx³yP(x,y) v \xyQ(x, y) c) Vx³y(P(x, y) 3zR(x, y, z)) d) Vx³y(P(x,y) → Q(x, y))
Express this in predicate logic: Each email address has exactly one email box. M(x): x is an email address B(x,y): x has an email box y
30. Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). (a)¬у³x²(x, y) (b) -VryP(x, y) (c) y(Q(y) AV-R(x, y)) (d)¬Jy(R(x, y) V VxS(x, y)) (e)y (VxzT(x, y, z) VrVzU (x, y, z)) 7
1. Find the contrapositive, converse and negation of the following statement. (Va € N) (Vb ≤ Z) (a² − b² > 0 ⇒ (a ≤ b) V (a + b ≤ N)) Contrapositive: • Converse: • Negation:
1. Show the validity of the logical equivalences. Derive them step by step using known logical rules: a) (p^q) →r=p→ (-qvr) b) (p→r)^(q→r) = (pvq) → r
23. Translate in two ways each of these statements into logi- cal expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. a) Someone in your class can speak Hindi. b) Everyone in your class is friendly. c) There is a person in your class who was not born in California. d) A student in your class has been in a movie. e) No student in your class has taken a course in logic programming.
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II. Define the predicate r(x) with the universe of discourse R by r(x) : ((Ve c) →x = 0. 1. Find the truth values of the following. (a) r(-1) (b) r(0) (c) r(1) 2. Write ~ r(x) such that there should be no negation before quantifiers nor before a logical connective.
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33. Rewrite each of these statements so that negations ap- pear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). b)x(x, y) a) xy(x, y) c)x(P(x, y) v Q(x, y)) d) (x³y-P(x, y)x(x, y)) e) xyz(x, y, z)=(x, y, z))