Page No 25:
Question 9:
Let * be a binary operation on the set Q of rational numbers as follows:
(i) a * b = a − b (ii) a * b = a2 + b2
(iii) a * b = a + ab (iv) a * b = (a − b)2
(v) 
(vi) a * b = ab2
Find which of the binary operations are commutative and which are associative.
a ∈ N.
But this relation is not true for any a ∈ N.
Thus, the operation * does not have any identity in N.
(i) On Q, the operation * is defined as a * b = a − b.
It can be observed that:
and 
∴
; where
Thus, the operation * is not commutative.
It can also be observed that:
Thus, the operation * is not associative.
(ii) On Q, the operation * is defined as a * b = a2 + b2.
For a, b ∈ Q, we have:
∴a * b = b * a
Thus, the operation * is commutative.
Thus, the operation * is not associative.
(iii) On Q, the operation * is defined as a * b = a + ab.
It can be observed that:
Thus, the operation * is not commutative.
It can also be observed that:
Thus, the operation * is not associative.
(iv) On Q, the operation * is defined by a * b = (a − b)2.
For a, b ∈ Q, we have:
a * b = (a − b)2
b * a = (b − a)2 = [− (a − b)]2 = (a − b)2
∴ a * b = b * a
Thus, the operation * is commutative.
It can be observed that:
Thus, the operation * is not associative.
(v) On Q, the operation * is defined as 
For a, b ∈ Q, we have:
∴ a * b = b * a
Thus, the operation * is commutative.
For a, b, c ∈ Q, we have:
∴(a * b) * c = a * (b * c)
Thus, the operation * is associative.
(vi) On Q, the operation * is defined as a * b = ab2
It can be observed that:
Thus, the operation * is not commutative.
It can also be observed that:
Thus, the operation * is not associative.
Hence, the operations defined in (ii), (iv), (v) are commutative and the operation defined in (v) is associative.
