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Set Theory and Algebra MCQ - Algebra

6:  

The set of integers Z with the binary operation "*" defined as a*b =a +b+ 1 for a, b ∈ Z, is a group. The identity element of this group is

A.

0

B.

1

C.

-1

D.

12

 
 

Option: C

Explanation :

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7:  

 In the group (G, .), the value of (a- 1 b)- 1 is

A.

ab-1

B.

b- 1a

C.

a-1b

D.

ba-1

 
 

Option: B

Explanation :

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8:  

114. The identity element in the group
G={[x x]}:x €R,x#0} with respect to matrix
     {[x x]}
multiplication is

A.

(1/0 0/1)

B.

(-1/0 0/-1)

C.

((1/2)/(1/2) (1/2)/(1/2))

D.

(1/1 1/1)

 
 

Option: C

Explanation :

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9:  

If (G, .) is a group, such that (ab)2 = a2 b2 ∀ a, b ∈ G, then G is a/an

A.

commutative semi group

B.

abelian group

C.

non-abelian group

D.

none of these

 
 

Option: B

Explanation :

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10:  

 (Z,*) is a group with a*b = a+b+1   ∀ a, b ∈Z. The inverse of a is

A.

0

B.

-2

C.

a-2

D.

-a-2

 
 

Option: D

Explanation :

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sachin kushare said: (6:15pm on Monday 25th February 2013)
we have a*e=a....(Identity property)so, a e 1=aso, e=-1we have a*b=-1....(Inverse property)so, a b 1=-1 b=-2-aso correct option (D)
Simran said: (4:49pm on Tuesday 27th June 2017)
Inverse : let a.(-a-2) = a (-a-2) 1= -1 = etherefore a inverse = -a-2 for all a belongs to Z

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