Let Q⁺ be the set of all positive rational numbers.

(i) Show that the operation * on Q⁺ defined by is a binary operation.

(ii) Show that * is commutative.

(iii) Show that * is not associative.

(i) * is an operation as where a, b ∈ Q⁺. Let and b = 2 two integers.

So, * is a binary operation from .

(ii) For commutative binary operation, a*b = b*a.

Since a*b = b*a, hence * is a commutative binary operation.

(iii) For associative binary operation, a*(b*c) = (a*b) *c.

As a*(b*c) ≠(a*b) *c, hence * is not associative binary operation.