Let P(x) be the statement x has pet a unicorn, and let S(x, y) be the statement x and y have shared a pizza, where the domain for the variables x and y consists of all students in the class. Express the following statements in predicate logic.

(a) Alice has shared a pizza with everyone except Carol.

(b) Everyone in the class who has pet a unicorn has shared a pizza with at least one other student in the class.

(c) There are two students in the class who have not shared a pizza with anyone.

(d) There are two students in the class who between them have shared a pizza with everyone else in the class.

(e) If everyone in the class shares a pizza with someone, then no one pets a unicorn.