Question

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

Answer 1

b is the inverse of a if and only if a*b = b*a = identity.

Note that for this group, identity = -1 since a*(-1) = a - 1 + 1 = a ∀a ∈ Z.

Fix a ∈ Z.

Now,

a*b = -1

⇔ a + b + 1 = -1

⇔ b = - a - 2.

Also note that this group is abelian since a*b = b*a here.

Therefore, a^(-1) = - a - 2.

Answer 2

The inverse of a is -(a+2).

Step 1: Find the inverse of a.

Given- a×b = a+b+1

This expression will only satisfy when the value of b will be $a^{-1}$ (inverse of a). so,

Let identity Element = I

a×I = I×a = a

a = a + I + 1

I = -1

Inverse of a is $a^{-1}$

a×$a^{-1}$ = I

a + $a^{-1}$ + 1 = -1

$a^{-1}$ = -( a+2 )