Question

If the group (z, ∗) of all integers, where a ∗ b = a + b + 1 for all a, b ∈ z, the inverse of -2 is
1. -2
2. 0
3. -4
4. 2

Answer 1

Correct Answer - Option 2 : 0

Given: a ∗ b = a + b + 1

Solution:

a ∗ b = a + b + 1

the identity element of b for any element a, b ∈ z

a ∗ b = a

a + b + 1 = a

b = -1

Hence identity element is -1

Hence e = -1

now for a inverse ,

a*a^-1 = a+a^-1+1 = e

a+a^-1+1 = -1

a-1 = -2 - a

when a = -2, a-1= 0

Ans: Option 2