We are given with the set Q₀ which is the set of non - zero rational numbers.

A general binary operation is nothing but association of any pair of elements a, b from an arbitrary set X to another element of X. This gives rise to a general definition as follows:

i. A binary operation * on a set is a function*:

Here the function o:

For the ‘o’ to be commutative, aob = boa must be true for all a, b Q₀. Let’s check.

⇒ a * b = b * a (as shown by 1 and 2)

Hence ‘o’ is commutative on Q₀

For the ‘o’ to be associative, ao(boc) = (aob)oc must hold for every a, b, c ∈ Q₀.

Hence ‘o’ is associative on Q₀

ii. Identity Element: Given a binary operation*: A X A → A, an element e ∈A, if it exists, is called an identity of the operation*, if a*e = a = e*a ∀ a ∈A.

Let e be the identity element of Q₀.

Therefore, aoe = a (a ∈Q_0)

iii. Given a binary operation with the identity element e in A, an element a A is said to be invertible with respect to the operation, if there exists an element b in A such that a * b = e = b * a and b is called the inverse of a and is denoted by a^–1.

Let us proceed with the solution.

Let b Q₀ be the invertible elements in Q₀ of a, where a Q₀.

∴a * b = e (We know the identity element from previous)