The given binary operation is a*b = a+b+2
In order to find the inverse of the relation, we have to find the identity element first.
Let that identity element be e then
a*e = a
From que.
a*e = a+e+2
So, from the above two relations we have
a+e+2 = a
or, e+2 = 0
∴ e = -2
Hence the identity element is -2 for this binary operation.
Now let a’ be the inverse of this relation
Then as per the definition of the inverse element
a*a’ = e
∴ a+a’+2 = -2
∴ a’ = -4 -a
And for 4 ,i.e. a = 4
a’ = -4 – 4
∴ a’ = -8
Thus the inverse element of 4 is -8 for the given binary operation.