4 -(1,5+1,5+2 marks) Explain why a) the groups z, and S, are not isomorphic b) the groups Z, x Z2 and Z, xZ, xZ, are no isomorphic; c) the function from ring R-a+b/2a,bEto ring S-abv3a,bE defined...
How many non-isomorphic unital rings are there of order 4? Question 3: How many non-isomorphic unital rings R4 are there of order 4? Hint: we can assume that the additive group of R4 can be either (74, +) or (Z2 X Z2, +). Thus the elements of R4 are one or the other of these groups, with a multiplication defined in some way. In the former case, 1 can be assumed to be the multiplicative identity. Why can't 2 be...
1- (2,5+2,5 mark) Consider in GL(2, Q), the subset (a=1 or a=-1),bez Prove that H, with multiplication, is a subgroup of GL(2,Q) a) Is the function b) an homomorphism of groups? Justify your answer 2 (3 marks) Let G be a group and a E G an element of order 12. Find the orders of each of the elements of (a) 3- (1+1,5 marks) Let G be a group such that any non-identity element has order 2. Prove that a)...
(8) Let TC C(R111) be defined by rar' t br + c)-(2a b)z? t (2b α-c) r + c b (a) Find M(T) :-M(T, B. B) where B-(z2, 1, 1} (b) Compute det(M(T). Is Tinvertible? e) If possible, write an explieit formmla for T (az2 b c). (8) Let TC C(R111) be defined by rar' t br + c)-(2a b)z? t (2b α-c) r + c b (a) Find M(T) :-M(T, B. B) where B-(z2, 1, 1} (b) Compute det(M(T)....
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x, y, z) = x2 + y2 + x2 = 2a?, (1) where a is a fixed real positive constant, and the point u = (0,a,a) on the surface f(x, y, z) = 2a. a) Find the gradient of f(x, y, z) at the point u. b) Calculate the normal derivative of f(x, y, 2) at u. c) Find the equation of the tangent plane...
I. a) (4 points) For a given function F(x, y, z) = xz + (y + z)(x + z) Draw the logic circuit diagram of the function: b) Using Boolean Algebra to simply the above function c) Use Demorgan's Theorem to find out the complement of the above function F(x,y,z)xz+ + 2)(x +z)
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....
8. let salle &]: xy, 2 e R} a). Prove that (5, +,-) is a ring, where t' and are the usual addition and multiplication of matrices. (You may assume standard properities of matrix Operations ) b). Let T be the set of matrices in 5 of the form { x so]. Prove that I is an ideal in the ring s. c). Let & be the function f: 5-71R, given by f[ 8 ] = 2 i prove that...
Question 2 The function r) is defined by 3(x + 1) 1 f12r + 71-4 1+4 () (111) 13 marks Show that f(x)= [4 marks) Find (3) Find the domain off- [2 marks] Given that g(x) = In(x + 1) Find the solution of r if fg(x) = 5, by leaving your answer in terms of e exponential function) [4 marks) (b) Sketch the graph of the function, In)-1-cosx). not by plotting points, but starting with the graph of a...
Part 1 Part 2 7.1.2. Let R be a commutative ring and a, b E R, and define The goal of this problem is to prove that (a, b) is an ideal of R (a) Explain how you know that 0 E (a, b b) What do two random elements of (a, b) look like? Explain why their sum must be in (c) For s E R and z E (a,b), explain why sz E (a, b). 7.2.1. In the...
solve parts b,d and f 2. Compute the integral of f over S where (a) f(ayz)xy+z.S is the region in the first octant with xy+ (b) f(xy.z)xxyz, S is the region defined in 2(a) (c) f(x,y.z) x + y2-xz, s is the region bounded by the x'y plane, the plane z (d) f(x,y,z) 2, and the cylinderx2 y z, s is the region in the first octant bounded by r2 + y2 + 2 4 (e) f(xy,z-2, s is the...