Abstract Algebra (MATS2103)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2016
UNIVERSITY OF BAMENDA
FACULTY OF SCIENCE
END OF SEMESTER EXAMINATION
DEPARTMENT: Mathematics and computer Science INSTRUCTOR: Fomboh nee Nforba
MONTH: March COURSE CODE AND NUMBER: MATS2103
YEAR: 2016 COURSE TITLE: Abstract algebra
Date: CREDIT VALUE: six credits
TIME ALLOWED: 3 Hours
1. 25 marks.
(i) if A, B and C are sets, such that A⊑C, prove that
(a) A∩(B-C)=⊘, (b) A-B=A∩(C-B) (4+3=7 marks)
(ii) Prove by mathematical induction that
∑
( + 1)
=
(
+ 1
)(
+ 2
)
(3 + 5) (8 marks)
(iii) Using congruence, find the remainder when 4
554
is divided by 7. (5 marks)
(iv) Prove that a≡ ( 15) ⟺a≡b(mod 3) and a ≡b(mod 5). (5 marks)
2. 18 marks
(i) Define the g.c.d., d, of two integers a and b. (lmark)
(ii)Given that d is the g.c.d. of a and b, prove that c = ax + by, x, y ∈ ℤ if and only if d|c.
(6marks)
(iii) Find d, the g.c.d. of 7200 and 3132 and express it in the form 7200p + 3132g = d; p, q ∈ ℤ (6marks)
Hence,
(a) find the particular solution and all integer solutions of the equation 7200p + 3132q = 72 (2marks)
(b) determine whether or not the equation 7200p + 3132q = 72 has positive solutions. (3marks)
3. 19 marks.
(i) Consider the function f:A→B. When is f said to be bijective? (1 mark)
(ii) the function f: ]1, ∞
→
1, ∞ is defined by f(x)=
!"
!#
. Prove that f is bijective. ( 8 marks)
(iii) Let f: A→B and g: B→C be mappings. Prove that
(a) if f is injective and g is injective, then gof is injective (4 marks)
(b) if X and Y are subsets of B. then f
-1
(X/Y)=f
-1
(X)/f
01
(Y) (6 marks)
4. 13 marks.
Let R be a binary relation defined on a set A.
(i) When is R said to be an equivalence relation? (1 mark)
(ii) Use your definition in (a) to determine whether the relation R, defined on ℤ by “xRy⟺ x+y is even” is
an equivalence relation.
(iii) Prove that if R is an equivalence relation, then aRb⟺[a]=[b]. ( 5 marks)
GOOD LUCK.
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