Proofs: Construct proofs to show that the following symbolic arguments are valid.  Commas mark the breaks between premises, ‘∴’ precedes the conclusion. You may use any rule, but you must use CP Z → (∼Y → X), Z → ∼Y ∴ Z → X

Suppose P, Q, R, S, C are five natural language statements that form an argument with premises P, Q, R, S and conclusion C. Suppose also that the argument is valid. Which of the following would not on its own be an explanation for why the argument is valid? R couldn't be true. P, Q, and S are jointly impossible. Necessarily, if P and R are true, so is C. C couldn't be false. P, Q, R, and C are jointly impossible.

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.

Proofs: Construct proofs to show that the following symbolic arguments are valid. Commas mark the breaks between premises, ‘∴’ precedes the conclusion. You may use the following rules: MP, MT, DS, HS, CD, Simp, Conj, Add ∼N • ∼M, ∼P → N, ∼N → Z, (Z • ∼∼P) → K ∴ K • ∼M

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).

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))

Provide a real-world example of when you would use each by creating a pseudocode representation of the programming problem.For example: if it is raining then             take an umbrella endif if it is raining then           take an umbrella else          bring sunscreen endif Also, answer the following: What does the "then" in the clause stand for? What does the "else" stand for?