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

-a -2

a - 2

-2

None of these

Given the operation a * b = a + b + 1:

Let's find the identity element 'e' such that a * e = a.

a * e = a + e + 1 = a

This implies e + 1 = 0, so e = -1.

Now, to find the inverse 'x' of 'a' such that a * x = e = -1:

a * x = a + x + 1 = -1

This implies a + x = -2.

x = -2 - a

x = -a - 2