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Determine whether the operation is a binary operation or not
Given: An operation \( * \) on \( \mathbb{N} \) defined by
\( a * b = ab \quad \forall \, a, b \in \mathbb{N} \)
Concept:
A binary operation on a set is a rule that combines any two elements of the set and gives a result that is also in the same set (closure property).
Solution:
Let \( a, b \in \mathbb{N} \).
\( a * b = ab \)
The product of two natural numbers is always a natural number.
\( ab \in \mathbb{N} \)
Conclusion:
Since the result is always in \( \mathbb{N} \), the set is closed under this operation.
✔ Therefore, the operation is a binary operation on \( \mathbb{N} \).