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.

a. Every professor has been asked a question by some student.
b. There is a student who has never asked a question to a professor.
c. There is a professor who has been asked a question by exactly one student.
d. There are two different students who have asked each other a question