1. Let A, B and C be sets. Prove that A \ (B \ C) = (A \ B) U (A \ Cc) by:
(a) Using set-theoretic identities.
(b) Using set builder notation and showing that the two sets are defined by logically equivalent expressions.
(c) Arguing in words that the set on left-hand side of the equality is a subset of the one on the right-hand side, and vice-versa.
2. (a) Argue that the relation "logically implies" is transitive on the set of all statements. That is, if a, b and c are statements such that a ⇒ b and b ⇒ c, then a ⇒ c.
(b) Suppose a, b, c and d are statements such that a ⇒ b, b ⇒ c, c ⇒ d, and d ⇒ a. Argue that any two of these statements are logically equivalent.
(c) Use the method from part (b) to prove that the following statements are all (logically) equivalent:
a. A ⊆ B
b. A ∩ B = A
c. A U B = B
d. Bc ⊆ Ac
3. Let A, B, C and D be sets. Determine whether the following statements are true or false. Prove the statements which are true; note that Venn Diagrams are not proofs.
Give a counter example for each statement which is false.
(a) If A ∩ B = A ∩ C, then B = C.
(b) If A ⊕ B = A ⊕ C, then B = C.
(c) If A 
B and B ⊆ C, then A C.
(d) A \ (B U C) = (A \ B) U (A \ C).
4. Let A, B and C be sets. Prove that if A ⊆ B and C 
D, then AxC Bx D.
5. Let S be a set that contains at least two different elements. Let it be the relation on 2(8), the set of all subsets of S, defined by (X, Y) ∈ R it if and only if X ∩ Y = ø. Determine whether R is reflexive, symmetric, antisymmetric, or transitive.
6. Five mathematicians, A, B, C, D, E, are working in the department one Sunday. Mathematician E makes a pot of coffee to see her through the long afternoon of marking midterms. When she returns to the coffee room for her first cup, she discovers that half of the coffee has already been consumed. She knows no one else has been in the department that day, so she questions her four colleagues and receives the following responses:
A: "C did it." B: "Not me." C: "D did it." D: "C is lying."
If only one of these four statements is true, and only one of the four people drank the coffee, who is the coffee thief? Justify your answer