9. Suppose n : (Z4 ⓇZ3) + (Z2 Z3) is defined by n(a,b) = (a mod 2, 0). This is a homomorphism. (You can believe me about that, and you don't need to check.) Write down the elements of the kernel of n.
= a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by ¢(a) (a) (5 Pts) Prove that o is injective. Answer (b) (5 Pts) Prove that o is surjective onto its image. Answer
= a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by ¢(a) (a) (5 Pts) Prove that o is injective. Answer
(b) (5 Pts) Prove that o is surjective onto its image. Answer
g(p+1)/2 (a) Suppose 9 is a p rimitive root of an odd prime p. Prove that- (mod p)
g(p+1)/2 (a) Suppose 9 is a p rimitive root of an odd prime p. Prove that- (mod p)
Please solve all questions
1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) = 4.+ 12Z (a) Prove that o is a ring homomorphism. Note: You must first show that o is well-defined (b) Is o injective? explain (c) Is o surjective? explain 2. In Z, let I = (3) and J = (18). Show that the group I/J is isomorphic to the group Z6 but that the ring I/J is not ring-isomorphic to the ring Z6. 3....
quention for 8
iz) 23)1Dy ave 7. (10M) Prove that o: Z x Z Z given by (a, b) a+b homomorphism and find its kernel. Describe the set is a 8. (10M) Prove that there is no homomorphism from Zs x Z2 onto Z4 x Z 9.(10M) Let G be a order of the element gH in G/H must divide the order of g in G. finite group and let H be a normal subgroup of G. Prove that (16M)...
7. Prove or disprove: If we know that 2X +6=4 (mod 8), then X +3 = 2 (mod 8). 8. Prove or disprove: If we know that 2X+6 = 4 (mod 7), then X+3 = 2 (mod 7). 9. Let S be the set {311, 254, -172,45,2019, 111,3}. Find a subset T such that the sum of the elements in divisible by 7
Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(x)-3 E(Y)9 E(Z)-2 Var(X) = 36 par(r)=19 par(Z)-10 Compute the values of the expressions below E (32 +3) 5Y+ 2x Var (5-2)-
Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. Var (r)-30 Var (r)-36 Var (z) 23 Compute the values of the expressions below 2X + 32 Var (Z-4)-
Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(x)-5 ECY)4 E(Z)--8 Var (x)-39 Var (Y)-11 Var (z) 37 Compute the values of the expressions below. E(2 -2z) 35 30 Var (sz)-5-D
Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(x)-4 E(Y) 2 E(Z)-7 Var (x) -28 Var(Y)-3 Var (Z) -44 Compute the values of the expressions below. E(Y 1) 5Z + 4x Var (4Y-3)