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" where the domain of x and y consists of all people. Express the following sentences in terms of predicates, quantifiers, and logical operators 1. There is a professor who has never been asked a question by a student. There is a professor who has asked every other professor a question. Some student has never been asked a question by a professor a. b. c. 2. Write the following arguments in argument form and prove that they are valid using the rules of inference and the laws of propositional logic. a. If I eat spicy foods, then I have strange dreams. I have strange dreams if there is thunder while I sleep. I did not have strange dreams. Therefore, I did not eat spicy food and it did not thunder. b. There is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre