31. Express the negations of each of these statements so thatall 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))

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Answered: 31. Express the negations of each of…

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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C. Let Q(x,y):x² + y° = x² - y°. If the domain for both variables consists of all integers, what are the truth values? 11. Q(1,1) 12. Q(2,0) 13. 3xQ(x,2) 14. VyQ(1,y) 15. EX3YQ(x, y) 16. Vx3yQ(x, y) 17. 3YVXQ(x,y) 18. VyaxQ(x,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
1. Let R+ := {x € R|x > 0}. Write negations for the following statements and determine if the original statement or its negation is true. a) 3x € + Vy € R+ (xy < y). b) Vx € R+ y ≤ R+ (xy < y).
2. Write negations of the following conditional statements: - (p→q) = p^~q (a) VneZ,ifn divisible by 2 then n is divisible by 2. (b) VxeR, if x(x+1)> 0 then x >0 or x<-1
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3. Write the contrapositive of the conditional statements: contrapositive(p →q)=-q→p (a) VneZifn° divisible by 2 then n is divisible by 2. (b) VxeR, if x(x+1)>0 then x>0 or x<-1
2. Let p, q, and r be as in Exercise 1. Translate the follow- ing into English sentences. (a) (p ^ q) → r (c) p (q vr) (e) (pvq) ^r (b) (pr)→ 9 (d) (p → (q vr))

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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) xy(P(x, y) ZR(x, y, z)) d) Vxy(P(x,y) → Q(x, y))