(d) 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. Væ(P(x) → Q(x))ii. Jx(P(x) → Q(x))iii. Vr(P(x) A Q(x))iv. 3x(P(x) A -Q(x))

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Answered: (d) Let P(x) be the statement " x² > 1"…

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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1. Construct a truth table for each of the following compound propositions and tell whether it is tautology, contradiction or contingency. a. (pq) → (~q-p) b. (p^q)^(-qV p)
7. Determine the truth values of the following universal propositions. If you believe the propo- sition is false, provide a counterexample from the domain. (a) VæP(x), where P(r) = "a l", and the domain of a is integers. (d) Væ(P(a) → Q(a)), where P(r) = "x < 4", Q(x) = "æ < 2", and the domain of r is real numbers.
1. State the value of x after the staement if P(x) then x := 1 is executed, where P(x) is the statement "x>1," if the value of x when this statement is reached is x=0 2. Let Q(x) be the statement " x+1 > 2x." If the domain consists pf the integers, what are these truth values? ∀xQ(x) 3. Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. ∀x(x^2+2≥1) 4. suppose that the domain of the propositional function P(x) consists of the integers -2,-1,0,1, and 2. Write out the proposition using disjunctions, conjunctions, and negations. ∀xP(x) 5. Translate this statement into a logical expressions using predicates, quantifiers, and lofical connectives. Something is not in the correct place.
5. If A = {1,3,5,7,9,11,13,15,17,19}, consider following statements. If they are statements find their truth values and if they are propositional function find their truth set. |(i) (Vx e A)(3y e A)(x+ y < 28) (ii) (Vx e A)(vy e A)x+y<28) (iii) (Vy e A)x+y <28) (iv) (3y e A)x+y < 28)
What is the Truth value of the Expression ∀ (x)P(x) in each of the following interpretation? P(x) is the property that X is yellow, and the domain of interpretation is the collection of all buttercups. P(x) is the property that x is yellow, and the domain of interpretation is the collection of all flowers. P(x) is the property that x is plant, and the domain of interpretation is the collection of all flowers. P(x) is the property that x is either positive or negative , and the domain of interpretation consists of the integers.
Determine the truth value of the following statements if the universe of discourse of each variable is the set of real numbers. Enter your answer as T or F. 1. Vx⇒y (x² = y) 2. Vx y (x = y²) 3. xVy (xy = 0) 4.3x (x² = -1) 5. Vx‡y (x + y = 1) 6. Vxy ((x+y= 2)^(2x - y = 1)) 7.3x (x² = 2) 8. Vx ‡ 0³y (xy: = 9. Vxzy (z = 10. xy ((x + 2y = 2) ^ (2x + 4y = 5)) 11. xVy 0 (xy = 1) 12. x³y (x + y‡y+x) T T T F LL T F T T TI F F F F : 1) x+y 2 ·)
2. Construct the truth table for the formula (A → B) → [((B→C) A-C) → -AJ.
d. What are the truth sets of the predicates P(x), Q(x), and R(x), where the domain is the set of integers and P(x) is “|x| = 4, " а. b. Q(x) is “x² = 16," С. R(x) is "|x| = x"
2. Determine the truth value of the statement 3y Vx(xy = 1) if the domain for the variable consists of a) The positive real numbers b) Nonzero real numbers c) Nonzero integers

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(d) 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 truth values for the following: i. Væ(P(x) → Q(x)) ii. Jæ(P(x) → Q(x)) iii. Væ(P(x) A Q(x)) iv. 3x(P(x) ¬Q(x))