Question

Determine whether the operation ′∗′ on N defined by a ∗ b = \[{{a}^{b}}\]
for all a, b ∈ N is a binary operation or not :

Answer 1

Hint: Just follow the definition of the binary operation.

 Complete step-by-step answer:

As per information provided in the question, it is given that ‘∗’ is an operation that applies to the Natural Numbers ‘N’ and it is defined as given: a∗ b = \[{{a}^{b}}\]
where a, b ∈ N.

Now we have both ‘a’ and ‘b’ as a Natural number.

According to the question, it is given that on applying the operation ‘∗’ for two

given natural numbers, as a result, we get a natural number.

⇒a ∗ b ∈ N ..........eq(i)

We also know that \[{{p}^{q}}\] > 0 if p > 0 and q > 0.

So, we can state that,

⇒ \[{{a}^{b}}\] > 0

⇒\[{{a}^{b}}\] ∈ N ..........eq(ii)

Operation is defined on a ∗ b = \[{{a}^{b}}\]

Here, we have a*b as an L.H.S part and \[{{a}^{b}}\] as the R.H.S part

From eq(i) and eq(ii) we can see that both L.H.S and R.H.S have only Natural

numbers as a result.

Thus we can clearly state that ‘∗’ is a Binary Operation on ‘N’.

Note: For this type of question, one must remember the basic definition of the binary operation. That is, a binary operation is a calculation that combines two elements (which is also called operands) to produce another element.

Also, a binary operation on elements of the set is such an operation whose domain and codomain remains the same.