CLO1:Let the domain be Z and the propositional functions P(x), Q(x), R(x) and S(x) be:• P(x): "x is a positive number".• Q(x): "x is a negative number".• R(x): "x > 0".• Sx): "x is an even number".Translate the following English sentences into logic statements using quantifiers:1. All positive numbers are larger than 0.2. All numbers can be either positive or negative.3. Some numbers are both negative and even.4. No number can be both negative and positive at the same time.Use the following list of symbols (as needed) in your answer: v3-AV

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Answered: CLO1: Let the domain be Z and the…

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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3. Let S(x) be the statement “x is a student," P(x) be the statement “x is a professor," and Q(x, y) be the statement “x asked a question to y." If the domain of x and y consists of all people, express each of the following sentences in terms of S(x), P(x), Q(x,y), quantifiers, and logical operators. Example: Some student has asked every professor a question. Answer: ax(S(x) A Vy(P(y)→ Q(x, y))) a. Every professor has been asked a question by some student. b. There is a professor who has never been asked a question by any student. c. There is a student who has asked a question to exactly one professor. d. There are two different students who have asked each other a question.
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What do the following statements mean in ordinary language? Write out their negations using quantifiers. (a) (Vn € N) (x ‡ 2¹) (b) (a, b ≤ N) (π = a/b) (c) (Vb = B) (3a = A) (ƒ(a) = b), assume f: A → B (d) (Vn € N) (3m € N) (n² — n = 2m) (e) (3A CR) (Vx ≤ R) (Ja € A) (x > a) (f) (Vn ≤ N) (‡p ≤ N) ((p > n) and [(Vq € N) (q/p ⇒ (q = 1) or q = p)])
Consider the two following logical expressions: (1) (p → q) → r (2) p → (q → r) Let the first expression be X and the second expression be Y. With proof, determine which of the following (if any) is a tautology? (a) X → Y (b) Y → X
132 #1
Let T(x,y) = "x will take y", S(x) = "x is a CS student", H(y) = "y is a hard course" and P(y) = "y is an elective course" Assume the domain of x is all students and the domain of y is all courses. Select the negation of "Some CS students will not take all elective courses" O 1.All CS students will take some elective courses. O 2. All students will not take all elective courses. O 3. Some CS students will not take some elective courses. O 4. Some CS students will take all elective courses.
Proficiency #8. Which of the following are logically equivalent to the statement -[(Vx E U) (¬p(x) V q(x))] if U is the set of all parrots, p(x) is the statement "x is talkative", q(x) is the statement “x is green" (Choose all that apply): I. There is an parrot who is not green and is talkative. II. No talkative parrots are green. III. All green parrots are talkative. IV. Not all parrots are not green. V. At least one talkative parrot is not green. VI. Not all talkative parrots are green.
Let P (x) be the statement “ x2 > 1 ” and Q(x) be the statement “x+1 < 4”.The universe of discourse consists of all real numbers. What are the truthvalues for the following:i. ∀x(P (x) →Q(x))ii. ∃x(P (x) →Q(x))iii. ∀x(P (x) ∧Q(x))iv. ∃x(P (x) ∧¬Q(x))
2 Translating English Sentences into Formulas Consider the following two predicates: S(x) : x is a student in CSE015; M(x) : x plays a musical instrument. Let the domain be the set of all people. Using the above predicates, translate the following sentences into formulas using the appropriate quantifiers and operators. a) Not every student in CSE015 plays a musical instrument. b) A person is either a student in CSE015 or plays a musical instrument, but not both. c) There exists at least one student in CSE015 who does not play a musical instrument.
Which of the following is a logical equivalence in first-order logic? A. ¬(∀x)P(x) ≡ (∃x)¬P(x) B. (∀x)(∃y)P(x, y) ≡ (∃y)(∀x)P(x, y) C. (∀x)(P(x) → Q(x)) ≡ (∃x)(P(x) ∧ Q(x)) D. (∀x)(P(x) ∧ Q(x)) ≡ (∀x)(P(x) ∨ Q(x))
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

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