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Determine whether or not each of the def...

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 `

Text Solution

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(i) On `Z^+, a**b = 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^+, a**b = 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.

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