12. (2 2 = 4 points) Suppose the domain of the propositional function P(x, y) consists of pairs x and y, where x is 2 or 4, and y is 1, 3, or 5. Write out these propositions using disjunctions and conjunctions.

a) yP(2, y)

b) xP(x, y)

13. (4 3 = 12 points) Express these propositions using quantifiers. Then, form the negation of the proposition so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase It is not the case that.) Make sure to identify the domain of discourse.

a) No dinosaur is a carnivore.

b) At least one CS course is theoretical.

c) Everyone is not happy.