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

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 = \|a – b\| 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 = \|a – b\|, 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) Let us take a = 2 and b = 2, ⇒ \|a – b\| = \|2 – 2\| ⇒ \|a – b\| = \|0\| ⇒ \|a – b\| = 0∉Z ⁺ *∴ The operation does not define a binary function 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 = |a – b| 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 = |a – b|, 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)

Let us take a = 2 and b = 2,

⇒ |a – b| = |2 – 2|

⇒ |a – b| = |0|

⇒ |a – b| = 0∉Z ⁺

∴ The operation * does not define a binary function on Z⁺ .

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