Let P(x) be the statement x is a student, Q(x) be the statement x is a professor, and R(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 P(x), Q(x), R(x, y), quantifiers, and logical operators.

Is this the right way to express these following statements.

- Some student has asked every professor a question.

xy(P(x) (Q(y) R(x, y))

- There is a professor who has been asked a question by exactly one student.

xy[(P(x) Q(y) R(x, y) z((z x) R(z, y)]