Let p, q, and r be the following simple statements: p: Sydney is the capital of Australia.

Is (X and Y) or (Z and W) a tautology? Tautology means for any assignment, the formula evaluates to true. Yes O No

I will thumbs up if you answer all. please convert FOL to CNF  Convert each of the following English statements into a single first-order logic sentence using the following constants and predicates. •Constants: Apples, Oranges, Chess, Go, John, Mary. •Predicates: oLikes(x,y): person x likes food y oPlays(x,y): person x plays game y a.If a personlikesApples, then they play chess. b.If a person likesOranges, then they play Go. c.A person likesApples or Oranges, but not both. d.John likes apples. e.Mary does not like anything that John likes. and then: Convert each of the first-order logic sentences in Problem 2to conjunctive normal form (CNF). There is no need to show intermediate steps. Assign each clause number: C1, C2, etc.

Determine if the following statements are logically equivalent:p ∨ (q ∧ r),(p ∧ q) ∨ (p ∧ r)Variables: p,q,r

1. Let p, q, and r be the statementsp: I finish my assignment before 2:00 pm todayq: I go to the gym todayr: It is raining today   (a) Invent a compound statement that uses all three of the proposi-tions p, q, and r as well as whatever logical connectives you choose. (b) Write an English sentence that corresponds to the statement youconstructed in (a). (c) Make a truth table for the statement you constructed in (a).

Consider the following statement: Statement A: ∀ integers m and n, if 2m + n is odd then m and n are both odd. Disprove Statement A. That is, show that Statement A is false

2. Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. If true, give a brief explanation for why. If false, give a counterexample (i.e. a set of values x,y that make the statement false. You must also explain why they make the statement false.) a) yvx((y > x²) → y > x) b) Vy!x(√√ 0)) d) x(x < 0⇒x² ≥y) e) \xy(x² = y² ↔ y³ = x³)

For each of the following statements, determine whether they are true or false. If false, write their logical negation (distributing the ¬ across any expressions as necessary) and explain how this negation is true. In each of these, x and y are assumed to be integers. (a) (∀x)(∀y)((y > x) ⇒ (x = 0)) (b) (∃x)(∀y)((y > x) ⇒ (x = 0)) (c) (∀x)(∃y)((y > x) ⇒ (x = 0)) (d) (∀x)(((∀y)(y > x)) ⇒ (x = 0))

Which statement is NOT correct? O A. Every WFF is equivalent to a WFF in prenex normal form. B. Skolemization does not change the meaning of sentences. C. Skolemization generates equisatisfiable sentences. D. A formula in Skolem form does not have existentially quantified variables.

Consider the following true propositions:• p : The applicant has passed the learner permit test.• q : The applicant has passed the road test.• r : The applicant is allowed a driver’s license.For each of the following sentences write,symbolically, the compound proposition that correspondstothe given sentence in English asit is written (do not change the order or form of the expression).a) The applicant did not pass the road test but passed the learner permit test.b) If the applicant passes the learner permit test and the road test, then the applicant is allowed adriver’s license.c) Passing the learner permit test and the road test are necessary for being allowed a driver’slicense.d) The applicant passed either the learner permit test or the road test, but not both.e) It is not true that the applicant does not pass the road test and is allowed a driver’s license.