Write the following English statements using the following predicates, any needed logical operators, and quantifiers if needed. Assume the domain of x is all students and the domain of y is all states in US. B(x, y): Person x has been to state y. L(x, y): Person x likes state y. a. No one has been to every state in US. d. Alice has been to every state that Bob has been to. e. There is no state that both Alice and Bob have been to.

Convert each of the following argu- ments into formal statements, e.g., define sentences existentially and/or universally quantified statements. Then determine which rules of logic have been applied and explain whether or not they have been ap- plied correctly. if you do the homework you will pass the final. Somebody did not pass the final. Therefore somebody did not not do the homework. If you do the homework you will pass the final. Curly did not do the homework. Therefore Curly will not pass the final. If you don’t do the homework, you won’t pass the final. Moe did the homework. Therefore Moe passed the final.

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.

Using the predicate symbols shown and the appropriate quantifiers, write each English language statement as a predicate wff. (The domain is the whole world.) B(x): x is a bee. F(x): x is a flower. L(x, y): x loves y All bees love all flowers which one of these answer choices a. (∀x)(B(x) → [(∃y)(F(y) ∧ L(x, y)] b. (∀x)(B(x) → [(∀y)(F(y) ∧ L(x, y)] c. (∀x)(B(x) → [(∀y)(F(y) → L(x, y)] d. (∀x)(B(x) ∧ [(∃y)(F(y) → L(x, y)]

Let p, q, and r be the propositions p: You started a new video game this week. q: You finished all its quests.r: You didn’t win the platinum trophy.Express each of these propositions as an English sentence.¬p → ¬q(p ∧ q) → (¬r)((¬p) → (¬q)) ∨ (q → (¬r))(p ∧ q) ∨ (¬q ∧ r)

The teams is a sports competition are divided into six groups with five teams in each group. Each team plays the other teams in its group once only. Write a Prolog preciate schedule(S) that is true if S is a valid schedule. A schedule lists the fixtures for each day of the competition. In a fixture (A,B), team A is the home team and team B is the away team. A schedule is valid if: • there are no more than three matches on any day, • each team has the same number of home and away fixtures in the whole schedule, and • each team has at least four rest days between fixtures. Ideally the schedule S should be as short as possible, and the predicate schedule should be able to generate multiple valid schedules.

Define the following predicates: • A(x) : x is on the board of directors." • E(x) : x earns more than $100, 000. • W(x) : x works more than 60 hours per week. The domain is the set of all employees of a certain company. Match each English statement with its equivalent logical expression. ? v 1. Everyone who works more than 60 hours per week earns more than $100,000. A. 3x(E(x) ^ ¬W(x)) B. Vx(W(x) → E(x)) C. Ex(¬A(x) → E(x)) D. Vx(W(x) ^ E(x)) E. 3x(¬A(x) ^ E(x)) F. 3x(A(x) ^ ¬E(x)^ W(x)) G. Ex(A(x) V ¬E(x) V W(x)) H. Vx(A(x) → E(x)) I. Vx(A(x) ^ E(x)) 2 v 2. Someone on the board of directors does not earn more that $100.000 and works more than 60 hours per week. ? v 3. Every member of the board of directors earns more than $100,000. 2. v 4. There is someone who is not on the board of directors and earns more than $100,000. ? v 5. There is someone who earns more than $100,000, but does not work more than 60 hours per week.

For each of the following statements about regular expressions α, β and languages A, B, state whether they are true or false. Provide a one-sentence justification for each answer.

Exhibit the structure of the following statements by transforming them into a first-order formula,indicating the interpretation of predicates and the domain. For example the statement There isno greatest integer can be transformed into ¬∃x, ∀y, P(x, y) where P(x, y) is the predicate x ≥ y,the domain being the integers.Do not try to prove them (it may not be possible!).1. Every integer can be written as the sum of 2 squares.2. Every positive real number has a square root.3. The cosine function has zeroes.4. The cosine function has at least two distinct zeroes.5. There is a neutral element1for multiplication in real numbers.6. Every odd square can be written as the sum of three odd numbers.

Using the predicate symbols shown and appropriate quantifiers, write each English language statement as a predicate wff. (The domain is the whole world.) D(x): x is a day S(x): x is sunny R(x): x is rainy M: Monday T: Tuesday 1.All animals are dogs.  3.Some animals are not cats. 4.Every animal that is a dog is not a cat. 5.Some animals are dogs and cats. 6.No animal is both a dog and a cat. 7.An animal is always a dog only if it is a cat. 8.No animals are dogs. 9.Meg is a cat; all animals are cats. 10.Meg and Alex are both cats. 11.If one animal is a cat, then every animal will be a cat.