Find the identity element for the binary operation on set Q of rational numbers defined as follows:(i)a * b = {a^2} + {b^2}(ii) a * b = {(a - b)^2}(iii) a * b = a{b^2}

We are tasked with finding the identity element for a binary operation on the set of rational numbers Q. An identity element "e" for a binary operation * on a set S satisfies the following condition for all elements a in S:

a * e = e * a = a
Let's go through each of the three operations one by one and find the identity element for each.

(i) a * b = a² + b²
In this case, the binary operation is defined as:

a * b = a² + b²
We need to find the identity element "e" such that:

a * e = e * a = a
Substituting into the operation:

a * e = a² + e²

We want this to equal "a", so:

a² + e² = a

Rearranging this equation:

e² = a - a²

For e to be the identity element, the equation must hold for all values of a. However, notice that for different values of a, the right-hand side of the equation (a - a²) changes. This means that there is no single value of e that satisfies the equation for all values of a.

Thus, there is no identity element for this operation.

(ii) a * b = (a - b)²
In this case, the binary operation is defined as:

a * b = (a - b)²
We need to find the identity element "e" such that:

a * e = e * a = a
Substituting into the operation:

a * e = (a - e)²

We want this to equal "a", so:

(a - e)² = a

Taking the square root of both sides:

a - e = ±√a

Solving for e:

e = a - ±√a

For e to be the identity element, the equation must hold for all values of a. However, for different values of a, the right-hand side of the equation (a - ±√a) changes. This means there is no single value of e that satisfies the equation for all values of a.

Thus, there is no identity element for this operation either.

(iii) a * b = a * b²
In this case, the binary operation is defined as:

a * b = a * b²
We need to find the identity element "e" such that:

a * e = e * a = a
Substituting into the operation:

a * e = a * e²

We want this to equal "a", so:

a * e² = a

Dividing both sides by "a" (assuming a ≠ 0):

e² = 1

Solving for e:

e = ±1

Thus, the identity elements are e = 1 and e = -1. Both satisfy the equation for all a, so either of these could be the identity element.

Final Answer:
For operation (i), there is no identity element.
For operation (ii), there is no identity element.
For operation (iii), the identity elements are e = 1 and e = -1.