Determine whether each of the following operations define a binary operation on the given set or not: ‘*’ on N defined by a * b = a + b – 2 for all a,b N.

Question

Philoid · Accepted Answer

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

⇒ a*b = a + b – 2, 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)

Let us take the values of a = 1 and b = 1, substituting in the R.H.S side we get,

⇒ a + b – 2 = 1 + 1 – 2

⇒ a + b – 2 = 0∉N ...... (2)

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

∴ The operation ‘*’ does not define a binary operation on N.

Determine whether each of the following operations define a binary operation on the given set or not:‘*’ on N defined by a * b = a + b – 2 for all a,b N.

Determine whether each of the following operations define a binary operation on the given set or not:
‘’ on N defined by a b = a + b – 2 for all a,b N.