Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on N defined by a * b = ab for all a,b N.
Given that ‘*’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
⇒ a*b = ab, where a,b∈N
Since a∈N and b∈N,
According to the problem it is given that on applying the operation ‘*’ for two given natural numbers it gives a natural number as a result of the operation,
⇒ a*b∈N ...... (1)
We also know that pq>0 if p>0 and q>0.
So, we can state that,
⇒ ab>0
⇒ ab∈N ...... (2)
From (1) and (2) we can see that both L.H.S and R.H.S gave only Natural numbers as a result.
Thus we can clearly state that ‘*’ is a Binary Operation on ‘N’.