Let Takes(x, y) be the propositional function x takes course y, Teaches(x, y) be the propositional function x teaches course y, and Passes(x, y) be the propositional function x passes course y. The universe of discourse is the set of all living people and all courses (i.e., you do not have to check this in your expressions).

For example, if we need to say Adam takes and passes CS 110, we can write it as Takes(Adam, CS 110) Passes(Adam, CS 110)

Write each of the following propositions symbolically in one expression:

1. Peter takes CS 220 and CS 410, but not CS 680.

2. Bob passes every course that he takes except CS 220.

3. Francesca passes every course that is taught by Prof. Einstein.

4. There is a course that both Julia and Peter took, but both of them failed it.

Find out for each of the following propositions whether it is a tautology, a contradiction, or neither (a contingency). Prove your answer either by truth tables or by using rules of propositional logic.

1. [(p q) (q p)] (p q) Prove this using a truth table

2. (p q r) [(q r) (p q)] Prove this by applying rules of propositional logic.

Is the below argument valid? Prove your answer by replacing each proposition with a variable to obtain the form of the argument. Then prove that the form is valid or invalid.

If x is an irrational number, then 5x is an irrational number. 5x is an irrational number. Therefore x is an irrational number.

(See the Additional Exercise 1.11.3.a in your book to better understand how to solve such questions)