# 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.

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