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:
A binary operation * on a set is a function * : A X A → A. We denote * (a, b) as a * b.
Here the function*:
For the ‘ * ’ to be commutative, a * b = b * a must be true for all a, b Q0. Let’s check.
⇒ a * b = b * a (as shown by 1 and 2)
Hence ‘ * ’ is commutative on Q0
For the ‘ * ’ to be associative, a * (b * c) = (a * b) * c must hold for every a, b, c ∈ Q0.
⇒ 3. = 4.
Hence ‘ * ’ is associative on Q0
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, a * e = a (a ∈Q0)
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.
Here the function*
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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