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.
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 + .