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
UNIVERSITY OF EDUCATION, WINNEBA
INSTITUTE FOR DISTANCE AND e-LEARNING
4-YR BACHELOR OF SCIENCE IN MATHEMATICS EDUCATION
TAKE HOME END OF SECOND SEMESTER EXAMINATION – ONLINE, 2019/2020 ACADEMIC YEAR
MATD472 ABSTRACT ALGEBRA
LEVEL: 300
INSTRUCTIONS: Attempt ALL questions and submit via UEW/LMS PLATFORM
(a) If
is the set of all residents of the United States, 
the set of all Canadian citizens, and 
the set of all women in the world, describe the sets [5Marks]
(i) 
(ii) 
(iii) 
(iv) 
(b) If 
, prove that 
for any set . [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 
is a set closed under an associative operation such that, given 
, there is an 
such that 
, and given 
, there is a 
such that 
. Show that is a group. [7 marks]
(ii) List the cyclic subgroup of 
generated by 
under +. [3 marks]
(e) (i) Let 
be the symmetric group of degree 3. Find all the subgroups of . [5 marks]
(ii) List the cosets of 
in 
. Is the factor group 
a cyclic? [5 marks]
Let
, and let 
.
(a) Find all left cosets and all right cosets of 
, and verify that is a normal subgroup of G. [20 marks]
(b) Show that 
has order 4, but is not cyclic. [10 marks]
(a) State the Chinese remainder theorem. [10 marks]
(b) Solve the system of congruences 
and 
[20 marks]
Note
As a reminder, you can present your response in EITHER:
Microsoft Word form. That is, Use MS Word in typing your responses and make sure you type all mathematical entities with the Equation feature of Word. Save the file as your name underscore index number (e.g. George Bondzie_2010110004).
OR
Write your responses legibly on foolscap sheets, number the pages, scanned the pages, and save them into a single PDF format and rename it as your name underscore index number (e.g. George Bondzie_2010110004).
In all circumstances, include the following: (i) Full Name (ii) Index Number (iii) Course Code (iv) Course Title, (v) Programme (vi) Level and (vii) Study Center. (iii) - (vi) can be gotten from this document.
WED. 5TH AUGUST, 2020 Page 2 of 2
