Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.(i) On `Z^+`, define ∗ by `a ∗ b = a – b`(ii) On `Z^+`, define ∗ by `a ∗ b = ab`(iii) On `R`, define ∗ by `a ∗ b = ab^2` (iv) On `Z^+`, define ∗ by `a ∗ b = |a – b|`(v) On `Z^+`, define ∗ by `a ∗ b = a `

Determine whether or not each of the definition of given below gives a

1.	Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.(i) On `Z^+`, define ∗ by `a ∗ b = a – b`(ii) On `Z^+`, define ∗ by `a ∗ b = ab`(iii) On `R`, define ∗ by `a ∗ b = ab^2` (iv) On `Z^+`, define ∗ by `a ∗ b = \|a – b\|`(v) On `Z^+`, define ∗ by `a ∗ b = a `
Answer» (i) On `Z^+, ab = a-b` In this case, `a` and `b`, both are positive and `a` is less than `b`. Then, `a-b` will be negative and will not fall in the set of positive integers. `:.` This is not a binary operation. (ii) On `Z^+, ab = ab` In this case, `a` and `b`, both are positive and multiplication of two positive numbers is always positive number and fall into the set `Z^+`. `:.` This is a binary operation. (iii)On `Z^+, ab = ab^2` In this case, `a` and `b`, both are positive. So, `b^2` will also be positive and multiplication of two positive numbers is always positive number and fall into the set `Z^+`. `:.` This is a binary operation. (iv) On `Z^+, ab = \|a-b\|` In this case, `a` and `b`, both are positive and modulus of any number is always positive and fall into the set `Z^+`. `:.` This is a binary operation. (v) On `Z^+, a**b = a` In this case, `a` and `b`, both are positive. So, result of this operation will also positive and fall into the set `Z^+`. `:.` This is a binary operation.