1. In (Z x Z, o) ,the composition is defined by = (a, b) o (c, d) = (a -c, b - d) Where (a, b),(c, d) ∈ Z x Z. Is the composition is commutative and associative.
2. In (M2(R), *), the * is defined by A*B = (AB-BA)/2 .Is * is commutative and associative.
3. Let (G,*) is a group where * is defined by a*b = a o c o b for all a,b ∈ G.
Calculate the identity element of the group.
4. (P(X), *) is defined by A*B = A∩B for all A,B ∈ P(X).Check is it group or not?
Where P(X) is defined by the power set of X.
5. (P(X), *) is defined by A*B=A∪B for all A,B ∈ P(X).Check is it group or not?
Where P(X) is defined by the power set of X.
6. In a group G ,if a = a-1 for all a ∈ G, then show that G is a commutative group.
7. S is the set of 2 x 2 real matrices and binary operation * is defined on S by
A*B = (AB+BA) for all A,B ∈ S, then check that(S,*) is a semi group.
8. The set {1,2,3,5,7,8,9} under multiplication modulo 10 is not a group.Then justify
The options
a) It is not closed
b)2 does not have an inverse.
c) 3 has an inverse.
d) 8 does not have an inverse.
9. Prove that a group G is commutative if and only if (ab)-1 = a-1b-1 for all a,b ∈ G
10. Let G denote the set of all non zero real numbers. Define a binary operation * on G by
a*b = |a|b for all a,b ∈ G. Then is (G,*) a group?