Write the following English statements using the following predicates and any needed quantifier. The domain of all variables are all people in a university    S(x): x is a student F(x): is a faculty member A(x, y): x has asked y a question Some students have not asked any faculty member a question There are at least two students who have asked every faculty member a question There is a faculty member who has asked every other faculty member a question

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.

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.

(2) If ¬L is context free, then L must also be context free. True False

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.

Question 7 PS:Please choose the right answer only.

3. Let S(x) be the statement "x studied for the exam" and H(x, y) be the statement "x scored higher than y in the exam," where the domain of x and y consists of all students in this class. Which logical expression corresponds to the following statement? "Every student in this class who studied for the exam scored higher in the exam than every student who did not study for the exam." a. VxVy (H (x, y) → (S(x) ^ ¬S(y))) b. VxVy(S(x)^¬S(y) ^ H (x, y)) c. VxVy((S(x) ^ ¬S(y)) → H (x, y)) d. VxVy((S(x) → ¬S(y)) → H (x, y))