# 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

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]

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