Answer :

(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.


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