Consider the statement (formula)(3x)A(x) → A(z)where z is a new variable not free (not an “input variable”) in A(x).Find now a specific example of A(x) over the set N and choose a specificvalue of z EN so that (1) becomes false (meaning we cannot prove it,since proofs start from true axioms and preserve truth at every step).(1)

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Answered: the statement (3x)A(x) → A(z) where z…

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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Let W(x,y) mean that student x has visited website y, where the domain for x consists of all students in your school and the domain for y consists of all websites. Express each of these statements by a simple English sentence. ∃y∀z(y ≠ (David Belcher) ∧ (W(David Belcher, z) → W(y, z))) Everyone besides David Belcher has visited all the websites that David Belcher has visited. There is no person besides David Belcher who has visited all the websites that David Belcher has visited. There is a person who has visited all the websites that David Belcher has visited. There is a person who has not visited all of the websites that David Belcher has visited. There is a person besides David Belcher who has visited all the websites that David Belcher has visited.
e) Let A(x), C(x), S(x) and GP(x) be the statements "x is an animal," "x is a cat," "x is small," and "x is a good pet," respectively. Express each of these quantifications in English where the universe consists of all animals. i. Vx (C(x) → A(x)) ii. -3x(C(x) ^ ¬ S(x)) i. Vx (C(x) S(x) A A(x)) iv. Vx (S(x) A A(x)→ GP(x))
question e)
prove these deductions (a) x∩y=∅, (z⊂x), ∴(z∩y)=∅ (b) p⊂q, ∴(r∖q) = (r∖p)∖q
Show by example that these three statements are generally false:(a) A and AT have the same nullspace.(b) A and AT have the same free variables.(c) If R is the reduced form rref(A) then RT is rref(AT ).
(a) How many triples (x, y, z) of nonnegative integers satisfy x+y+z= 100? ww many triples (x, y, z) of nonnegative integers satisfy x+y+z= 100 with 0 ≤ x ≤ 2?
For each of these universally quantified statements, find a counterexample. If there are none, then state that no counterexample exists. The domain for all variables is all real numbers. (a)x=² (b) Vx((x+1)² = x² + 1) (c) \x\y((x≥y) → ((x5⁰ > y) v (y ≥ 0)))
Use the rules of inference to show that if 3x(M(x) AP(x)) and Vx(K (x) → ¬M(x)) are true, then 3x(P(x) ^ ¬K(x)) is also true, where the domains of all the quantifiers are the same.
1. (a) (b) (c) (d) Prove or disprove that, for any universal set U and predicates P and Q, [3x = U, P(x) ^ Q(x)] → [3r EU, P(x)) ^ (3x = U, Q(x))] Prove or disprove that, for any universal set U and predicates P and Q, [3r EU, P(x)) ^ (3xU, Q(x))] → [r U, P(x) ^ Q(x)] Prove or disprove that, for any universal set U and predicate P [3r € U, P(x)] → [Vr € U, P(x)] Prove or disprove that, for any universal set U and predicate P [VxU, P(x)] → [3r € U, P(x)]
True or False The following propositions are logically equivalent. ∀x∃y(P(x)→Q(y)) (∀xP(x))→(∃yQ(y)) With all steps
5. Use a table to express the values of each of these Boolean functions. а) F(x, у, 2) %3DХу b) F(x, y, 2) —х + уz с) F(x, у, 2) —ху+ (хуг) d) F(x, y, 2) — x(уz + у2)

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