Determine whether each of the following operations define a binary operation on the given set or not: ‘O’ on N defined by a O b = a b + b a for all a,b N.

Question

Philoid · Accepted Answer

Given that ‘Ο’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:

⇒ aΟb = a^b + b^a, where a,b∈N

Since a∈N and b∈N,

According to the problem it is given that on applying the operation ‘Ο’ for two given natural numbers it gives a natural number as a result of the operation,

⇒ aΟb∈N ...... (1)

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

⇒ a^b>0 and b^a>0.

We also know that the sum of two natural numbers is a natural number.

⇒ a^b + b^a∈N

∴ The operation ‘Ο’ defines the binary operation on N.

Determine whether each of the following operations define a binary operation on the given set or not:‘O’ on N defined by a O b = ab + ba for all a,b N.

Determine whether each of the following operations define a binary operation on the given set or not:
‘O’ on N defined by a O b = a^b + b^a for all a,b N.