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)
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