Given that ‘Ο’ is an operation that is valid in the Integers ‘Z’ and it is defined as given:

⇒ aΟb = 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 integers it gives Integers as a result of the operation,

⇒ aΟb∈Z ...... (1)

Let us values of a = 2 and b = – 2 on substituting in the R.H.S side we get,

⇒ a^b = 2 ^{– 2}

⇒

⇒ a^b∉Z ...... (2)

From (2), we can see that a^b doesn’t give only Integers as a result. So, this cannot be stated as a binary function.

∴ The operation ‘Ο’ does not define a binary function on Z.