Abstract Algebra (MATS2103)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2015
University of Bamenda-Faculty of Science
Department of Mathematics, and Computer-Sciences
Abstract algebra Mats 2103
February 2015
Duration: 3 heures-
Exercice 1 (10 marks) Give the definitions of (a) a tautology (b) a relation between two sets (c) an ordered set
(d) a mapping (e) a one-to-one mapping (f) a binary operation (g) a group (h) a unitary ring
Exercice 2 (1+1+1=3 marks) Compute the truth tables for the following propositional formulas : (a) (p => p)
=> p (b) p => (p => p) (c) p V q => p ∧q
Exercice 3 (1+1+2+2=6 marks) Let E be a set and † be a relation on E.
1.
Give the definition of † to be a transitive relation.
2.
Give the definition of the transitive closure †
of †
.
3.
Prove that T
+
is indeed a transitive relation on E.
4.
Let E = {1,2,3} and †
= {(1,1), (1,2), (1,3), (2,3), (3,1)}. Give the transitive closure T
+
of †
.
Exercice 4 (1 x 12 = 12 marks) Determine if each of the following relations R on the integers Z, is (i) reflexive,
(ii) symmetric, (iii) antisymmetric, and (iv)transitive.
(a)
(x, y) e R if and only if x is a multiple of y
(b)
(x, y) e R if and only if x ≥ y
2
(c)
(x, y) € R if and only if x ≠y.
(d)
(x, y) e R if and only if xy ≥ 1.
(e)
(x, y) e R if and only ifx = y + 1 or x = y - 1.
Exercice 5 (1 x 20= 20 marks) For each of the following binary operations, state whether the binary operation is
associative, whether it is commutative, whether there is an identity element and, if there is an identity element,
which elements have inverses.
(1)
The binary operation ⨁ on Z defined by x ⨁ y = —xy for all x, y ∈ Z.
(2)
The binary operation * on IR defined by x * y = x + 2y for all x, y € IR.
(3)
The binary operation ⨂ on IR defined by x ⨂ y = x + y + xy for all x, y ∈ IR.
(4)
The binary operation o on IR defined by x o y = x for all x, y ∈ R
Exercice 6 (2+2+2+1 = 7 marks ) Give the addition and the multiplication table of Z
6
. Which elements of
Z
6
have multiplicative inverses? Is Z
6
a field?
Exercice 7 (5 marks ) Give the multiplicative table of S
3
the symetric group on three objects. Solve in S
3
the equation x o (1,2,3) = (1,3,2).
Exercice 8 (7 marks ) Prove that the set of all the matrices of the form
0 0
0
is a subring of the ring
M
2x2
(IR).
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