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.