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
