Determine whether or not each definition * given below gives a binary

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1.	Determine whether or not each definition * given below gives a binary operation. In the event that * is not a binary operation give justification of this. On Z+ , defined * by a*b = ab Here, Z+ denotes the set of all non – negative integers.
Answer» Given that ‘’ is an operation that is valid in the Positive integers ‘Z ⁺ ’ and it is defined as given: ⇒ ab = ab, where a,b∈Z⁺ , Since a∈Z⁺ and b∈Z⁺ , According to the problem it is given that on applying the operation ‘’ for two given positive integers it gives a Positive integer as a result of the operation, ⇒ ab∈Z ^{+ –} ...... (1) We know that p q>0 if p>0 and q>0. ⇒ ab>0∈N ⇒ ab∈Z ⁺ *∴ The operation defines a binary operation on Z ⁺.**

Determine whether or not each definition * given below gives a binary operation. In the event that * is not a binary operation give justification of this. On Z+ , defined * by a*b = ab Here, Z+ denotes the set of all non – negative integers.

Answer»

Given that ‘*’ is an operation that is valid in the Positive integers ‘Z ⁺ ’ and it is defined as given: ⇒ a*b = ab, where a,b∈Z⁺ ,

Since a∈Z⁺ and b∈Z⁺ ,

According to the problem it is given that on applying the operation ‘*’ for two given positive integers it gives a Positive integer as a result of the operation,

⇒ a*b∈Z ^{+ –} ...... (1)

We know that p q>0 if p>0 and q>0.

⇒ ab>0∈N

⇒ ab∈Z ⁺

∴ The operation * defines a binary operation on Z ⁺.

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