Binary operation: A binary operation * on a set A is a function *: A × A → A. We denote *(a, b) by a * b. There are different types of binary operation Commutative Associative Distributive Identity Inverse We have * as the binary operation on the set of integers defined by a * b = |a - b |-1 We know that Commutative binary operation a * b = b * a Here L.H.S = a * b = |a - b |-1 R.H.S = b * a = |b- a| -1 = |a - b |-1 L.H.S = R.H.S ∴ a * b = b * a ∴ The operation * is commutative. We know that Associative bibary oeration (a*b)*c = a*(b*c) Here L.H.S = (a*b)*c = |a-b|-1 * c R.H.S = a*(b*c) = a*|b-c| - 1 ∴ L.H.S ≠ R.H.S ∴ (a*b)*c ≠ a*(b*c) ∴ The operation * is not associative. We know that Distributive Binary operation a*(b 0 c) = (a*b) 0 (a*c) (Left distributive over '0') (b 0 c) * a = (b*a 0 (c*a) (Right distributive over '0') Here, a*b = |a-b| - 1 Left distributive over '0' L.H.S = a*(b 0 c) =|a - (b 0 c)| - 1| R.H.S =(a*b) 0 (a*c) = (|a - b| -1|) 0 (|a - c| -a|) L.H.S ≠ R.H.S Similarly, in Right distributive over '0', L.H.S ≠ R.H.S Therefore, the operation * is not distributive. Hence, The binary operation * defined on the set of integers such that a*b = |a –b| -1 is commutative. For example : ∵ (2 × 3) × 4 = {|2 - 3| - 1} × 4 = {1 - 1} × 4 = 0 × 4 = |0 - 4| - 1 = 4 - 1 = 3 2 × (3 × 4) = 2 × {|3 - 4| - 1} = 2 × 0 = |2 - 0| - 1 = 2 - 1 = 1 Therefore, it is not Associative and Distributive.