Question

Prove that the set Z of all integers with binary operation * defined by a * b = a + b + 1, where a, b is in G is an abelian group.

Answer 1

Correct option is -1

a∗b=a+b+1 (a,b,z is a group)

at a=−1⇒a∗b=−1+b+1=b

at b=−1⇒a∗b=a−1+1=a

⇒a∗0=a+0+1

⇒ identity element is −1.

Answer 2

Answer:

HENCE PROVED

Step-by-step explanation:

Given,

   a*b=a+b+1

To prove,

  a*b=a+b+1 is in G is an abelian group.

solution,

•     In the set of integers are binary operators.

• The set of integers Z is a grojup under ordinary addition, In case of the identity 0 and the inverse of an element a is -a. In fact all these form abelian groups.

• let us consider  (a,b,z is a group),

              a*b=a+b+1,

• a= -1 ------ a*b = -1 +b+1 =b

• b= -1 --------a*b = -1 +a+1 = a

• a*0 =a+0+1

• identity element is  -1

                                          Hence proved