Problem II.6. Use the distributivity law to prove that (A ∩ B) ∪ (B ∩ C) ∪ (C ∩ A) = (A ∪ B) ∩ (B ∪ C) ∩ (C ∪ A). [Hint: There is a similar problem worked out in the book. See the proof Proposition 6.3.5 on page 72.]

2. Problem II.7. For subsets of a universal set U prove that B ⊆ Ac if and only if A ∩ B = ∅. By taking complements deduce that Ac ⊆ B if and only if A ∪ B = U. Deduce that B = Ac if and only if A ∩ B = ∅ and A ∪ B = U.

3. Problem II.8. Let X be a set. Given sets A, B ∈ P(X), their symmetric difference A∆B is defined by A∆B = (A − B) ∪ (B − A) = (A ∪ B) − (A ∩ B). Prove that (i) the symmetric difference is associative, (A∆B)∆C = A∆(B∆C) for all A, B, C ∈ P(X), (ii) there exists a unique set N ∈ P(X) such that A∆N = A for all A ∈ P(X) [Hint: Guess what N is!], (iii) for each A ∈ P(X), there exists a unique A0 ∈ P(X) such that A∆A0 = N, (iv) for each A, B ∈ P(X), there exists a unique set C such that A∆C = B.

4. Problem II.9. Using the notation in the previous problem, prove that for sets A, B, C, D ∈ P(X) A∆B = C∆D ⇔ A∆C = B∆D.

Problem II.15. Let X be a set. Given A ∈ P(X) define the characteristic function χA : X → {0, 1} by χA(x) = 0, if x ∈ X − A 1, if x ∈ A. Suppose that A and B are subsets of X. (i) Prove that the function x 7→ χA(x)χB(x) (multiplication of integers) is the characteristic function of the intersection A ∩ B. (ii) Find the subset C whose characteristic function is given by χC(x) = χA(x) + χB(x) − χA(x)χB(x)