Find which of the operations given above has identity.
(i) An element e ϵ Q will be the identity element for the operation * if
a * e = a = e * a, 
a ϵ Q.
a * b = a – b
This operation is not commutative,
Therefore, it does not have identity element.
(ii) An element e ϵ Q will be the identity element for the operation * if
a * e = a = e * a, 
a ϵ Q.
a * b = a2 + b2
If a * e = a, then a2 + e2 = a.
For a = -2, (-2)4 + e2 ≠ -2.
Therefore, there is no identity element.
(iii) An element e ϵ Q will be the identity element for the operation * if
a * e = a = e * a, 
a ϵ Q.
Now, a * b = a + ab
This is not commutative.
Therefore, there is no identity element.
(iv) An element e ϵ Q will be the identity element for the operation * if
a * e = a = e * a, 
a ϵ Q.
a * b = (a – b)2
If a * e = a, then (a – e)2 = a.
A square is always positive, thus for a = -2, (-2 –e)2 ≠ -2.
Therefore, there is no identity element.
(v) An element e ϵ Q will be the identity element for the operation * if
a * e = a = e * a, 
a ϵ Q.
a * b = 
If a * e = a, then 
Therefore, e =4 is the identity element.
a * 4 =4 * a = 
.
(vi) An element e ϵ Q will be the identity element for the operation * if
a * e = a = e * a, 
a ϵ Q.
Now, a * b = ab2
This operation is not commutative,
Therefore, there is not have identity element.