Problem 1. [20 points ] Let a and b be two integers. Prove that divisors(a) 1"] divisors{b) = divisor5(b) r1 divisor5(a - 5), via the following two steps. We will grade only the second part; the first part is very similar and thus we do not require you to submit it and will not grade it. Use the twocolumn proof format from Lab 6 and Lab- 7. a) [I] points neither required nor graded] Prove that diVi50r5[o} ['1 diViEDr5(b) g divisors(b] divisor5(a _ 5)] b} [ED points] Prove that divisors(b] divisors(o 5] Q divisors(a.] F1 divisors-[b] Problem 2. [20 points] Your domain of discourse is sets. Prove that P(A] ['1 13(3) 2 P(A F1 B], via the following two steps. We will grade only the second part; the first part is very similar and thus we do not require you to submit it and will not grade it. Use the twocolumn proof format from Lab 6 and Lab 7. a) [1] points neither required nor graded} Prove that PU!) n P[B) g P(A H B). Hint: First prove that if [C g A] A (C l'_: B) then C Q A n 3. Write that as a separate proof . Then use this result in your main proof. b} [20 points} Prove that P(A F1 B] Q P(A) 13(3). Hint: First prove that if (0 g A n B) then (G g A] A {C Q 3). Write that as a separate proof . Then use this result in your main proof. Problem 3. [20 points] a) [15 points] Prove that F = {{z,yj la: y E Z} on the set R is an equivalence relation. Use the two column proof format from Lab 6 and Lab 7. b} [5 points] Explain why (3' = {{$,y] I: - y E N} on the set R is not an equivalence relation. Problem 4. [20 points 1 a) [15 points ] Let R be a relation defined on E x Z where (a,b)R(c,d) if a g c and b g d. Prove that R is transitive using the twocolumn proof format. b} [5 points] Now let R be a relation defined on E X E where (a,b)R[c,d} if a. g c or b 5 d. Is R transitive? Justify your