i. We are given with the set Z which is the set of integers.

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 *: Z X Z → Z is given by a * b = a + b – 4
For the ‘ * ’ to be commutative, a * b = b * a must be true for all a, b belong to Z. Let’s check.

1. a * b = a + b – 4

2. b * a = b + a – 4

a * b = b * a (as shown by 1 and 2)

Hence ‘ * ’ is commutative.

For the ‘ * ’ to be associative, a * (b * c) = (a * b) * c must hold for every a, b, c ∈ Z.

3. a * (b * c) = a * (b + c – 4) = a + (b + c – 4) – 4 = a + b + c – 8

4. (a * b) * c = (a + b – 4) * c = (a + b – 4) + c – 4 = a + b + c – 8

a * (b * c) = (a * b) * c

Hence ‘ * ’ is associative.

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

Let e be the identity element of Z.

Therefore, a * e = a
⇒ a + e – 4 = a
⇒ e – 4 = 0
⇒ e = 4

iii. Given a binary operation with the identity element e in A, an element a Z is said to be invertible with respect to the operation, if there exists an element b in Z 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 Z be the invertible elements in Z of a, here a Z.

a * b = e (We know the identity element from previous)
⇒ a + b – 4 = 4
⇒ b = 8 – a