Logic/Math/(CS)

2. (15 pt., 3 pt. each) Determine the truth value of each of the following statements if the domain consists of all integers. If it is true, find a domain for which it is false. If it is false, find a domain for which it is true. Justify your answer. -2. True when the domain consists of all nonnegative integers. For all nonnegative integers x, x 0) d. 3x(x? = 3) . x(x3 -1) 3. (15 pt., 3 pt. each) Let S(x) be the statement x is a student," P(x) be the statement "x is a professor," and Q(x, y) be the statement x asked a question to y." If the domain of x and y consists of all people, express each of the following sentences in terms of S(x), P(x), Q(x,y), quantifiers, and logical operators. Example: Some student has asked every professor a question. Answer: x (S (x) V () > Q(x, ))) a. Every professor has been asked a question by some student. b. There is no professor who has been asked a question by every student. c. There is a professor who has never been asked a question by any student. d. There is a student who has asked a question to exactly one professor. e. There are two different students who have asked each other a question. Part II: Proving logical equivalence using laws of propositional logic (50 pt.) 4. (35 pt.) Use the laws of propositional logic to prove that the following compound propositions are logically equivalent. (5 pt.) (p q) v (p r) and p (q v r) a. (10 pt.) +q and (p q) v (--) b. . (10 pt.) (q) and p > q d. (10 pt.) -(v (q(r>p))) and - (q r)