Q. 2 A5.0( 2 Votes )

# 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 the values a = 1 and b = 2

⇒ a – b = 1 – 2

⇒ a – b = – 1∉Z ^{+}

∴ The operation * does not define a binary operation on Z ^{+}

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