5. Given a predicate logic sentence:G: if (for some x) q(x, a, f(x,a)) then (for all y) p (y, f(b, y), b)Show that the sentence is satisfiable. 4. Given a propositional logic sentence:E: (P and (Q or R)) or (if (P and (Q or R)) then (P and Q))Find the total substitution result◄(P←(P and Q)}.

“Since you have posted multiple questions, we will provide the solution only to the first question…

Answered: 4. Given a propositional logic…

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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* A propositional variable p is a formula. (It is assumed that there are infinitary many such variables p, q, r, p1, p2, ...) * If A is a formula, then (~ A) is also a formula. * If A, B are formulas, then (A A B), (A v B), (A → B), (A = B) are formulas, respectively. *if A is a formula, then •A and OA are formulas, respectively. * We use - A, AB to mean ¬A, A → B, respectively since we use the text book by Girle [1] (See below). - Exercices. Give examples of formulas and explain why they satisfy this definition.
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.
2. Define a model in which all of the following propositions are true. ∃x(Gx∧Hx) ∀yHy→∃z¬F z F k↔∀x(Gx∨F x)
a 1. Consider the open sentence P(a, b) : > 1., where the domain of a is A = {-4,3,5} and the b domain of b is B = {-3,2,4}. a a) State the quantified statement R: Va € A, 3b € B, > 1. in words. b b) State -R in symbols. c) State - R in words. d) Determine, with justification, whether R is true.
2. Which is the main connective of the following sentence: -(F G)→ H A. - B.. C. → D. () 3. Which is the main connective of the following sentence: {[-(A-B) v C]~D} v E A. - B. → C. V D. →
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!
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
For the second picture I only need numbers #11 and #13 to be done For each proof, you must include (i.e., write) the premises in that proof. I do not want to see any proofs without premises. Do not use any transformation rules (e.g. contraposition) in your proof other than DN. Only use the eight inference rules. YOU CANNOT USE CONDITIONAL PROOF (CP), INDIRECT PROOF (IP), OR ASSUMED PREMISES (AP).
4. Let P be "It is humid" and let Q be "sum of 9 and 11 is 21". Give a simple verbal sentence which describes each of the following statements: (a) ¬Q, (b) P V Q, (c) (¬P) ^ Q, (d) (¬P) V (–¬Q).
Any time you see A, B, or C in any of the quiz questions, those statement letters are true. Any time you see X, Y, or Z in any of the quiz questions, those statement letters are false. Not all letters, (A, B, C, X, Y, Z), will be in all statements. (-Av-C)v (X A) True False

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4. Given a propositional logic sentence: E: (P and (Q or R)) or (if (P and (Q or R)) then (P and Q)) Find the total substitution result◄(P←(P and Q)}.