Question no. 3

True when the domain consists of all nonnegative integers. For all nonnegative integers x, x 0) d. 3x(x = 3) e. 3x(x3 = -1) (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. 3x(S(x) A Vy(Pty) Q(x,y))) 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