On the set Q⁺ of all positive rational numbers, define an operation * on Q⁺ by for all a, b ∈ Q⁺. Show that

(i) * is a binary operation on Q⁺,

(ii) * is commutative,

(iii) * is associative.

Find the identity element in Q⁺ for *. What is the inverse of a ∈ Q⁺?

(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 an associative binary operation.

For a binary operation *, e identity element exists if a*e = e*a = a.

(1)

(2)

using a*e = a

Either a = 0 or e = 2 as given a≠0, so e = 2.

For a binary operation * if e is identity element then it is invertible with respect to * if for an element b, a*b = e = b*a where b is called inverse of * and denoted by a^-1.

a*b = 2