Determine whether each of the following operations define a binary ope

1.	Determine whether each of the following operations define a binary operation on the given set or not: ‘O’ on Z defined by a O b = ab for all a,b Z.
Answer» 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 = 1/4 ⇒ 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.*

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

Answer»

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 = 1/4

⇒ 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.

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