List the cyclic subgroup of Z generated by -1 under +.

1.      (a) If A  is the set of all residents of the United States, B  the set of all Canadian citizens, and C  the set of all women in the world, describe the sets                  [5Marks]

(i) A∩B∩C

(ii) A-B

(iii) A-C

(iv) C-A

(b) If A⊂B , prove that (A⋃C)⊂(B⋃C)  for any set C .                                    [5 Marks]

(c) Show that if 80% of all DESA’s have gone to high school and 70% of all DESA’s read a daily newspaper, then at least 50% of DESA’s have both gone to high school and read a daily newspaper.                                                                           [10 Marks]

(d) (i) Suppose that G  is a set closed under an associative operation such that, given a,y∈G , there is an x∈G  such that ax=y , and given a,w∈G , there is a u∈G  such that au=w . Show that G  is a group.                                                                         [7 marks]

(ii) List the cyclic subgroup of Z  generated by -1  under +.                                [3 marks]

(e) (i) Let S3  be the symmetric group of degree 3. Find all the subgroups of S3 .            [5 marks]

(ii) List the cosets of 7  in Z16× . Is the factor group Z16×/7 a cyclic?          [5 marks]

2.      Let G=D8 , and let N={e,a2,a4,a6} .

(a) Find all left cosets and all right cosets of N , and verify that N  is a normal subgroup of G.                                                                     [20 marks]

(b) Show that G/N  has order 4, but is not cyclic.                                    [10 marks]

3.      (a) State the Chinese remainder theorem.                                                 [10 marks]

(b) Solve the system of congruences x≡7 (mod 21)  and x≡3 (mod 8)   [20 marks]

The post List the cyclic subgroup of Z generated by -1 under +. appeared first on Scholar Writers.

I absolutely LOVE this essay writing service. This is perhaps the tenth time I am ordering from them, and they have not failed me not once! My research paper was of excellent quality, as always. You can order essays, discussion, article critique, coursework, projects, case study, term papers, research papers, reaction paper, movie review, research proposal, capstone project, speech/presentation, book report/review, annotated bibliography, and more.

STUCK with your assignments? Hire Someone to Write Your papers. 100% plagiarism-free work Guarantee!