Let A and B be finite sets. (a) Find and example of A and B for which |A U B|=

3.1.1 Describe the following sets in roster notation: that is, list their elements.

3.2.11 For the following proof sketch, determine the result that is being proved, and then turn the sketch into a formal proof.

Exercise 2 Over the course of the parts of this exercise you will show that multiplication of cosets in Z[i]/Z is not well-defined. (a) Let a, a, b, VEZ. Prove that a +i and a' + i represent the same coset in Z[i]/Z; that is, prove that (a + i) + Z= (a' + i) + Z (b) Prove that band & represent the same coset in Z[i]/Z; that is, prove that b+z=b'+Z (c) Determine the conditions under which (a +i)b and (a' + i)b' represent the same coset in Z[i]/Z. That is, determine what additionally must be assumed in order to ensure that (a + i)b+Z= (a' + i)b' +Z. Be sure to prove your condition is the correct one. (d) Find specific values of a, a, b, b' € Z for which (a+i)b+Z‡ (a¹ + i)b +Z.

Exercise 1 Let Z[i] = {a+bi | a,b€ Z}, and define the operation + in Z[i] as ordinary addition (as in C). Note that Z= { a+0i | a € Z } is naturally a subset of Z[i]. (a) Prove that (Z, +) is a normal subgroup of (Z[i], +). (b) Prove that the quotient group Z[i]/Z (with operation +) is an abelian group. (c) Let (5) denote the set (5) = { 5a +5bi | a,b € Z}. Prove that (5) is a normal subgroup of Z[i]. (d) Prove that Z[i]/(5) is an abelian group (with operation +).

Exercise 3 In this exercise you will show that multiplication of cosets in Z[i]/(5) is well-defined.

2. (7 points) Find god(216,315) and express it in the form 216n +315m for integers n, m. ged(216, 315) =