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Determine whether the operation is a binary operation or not
Given: An operation \( O \) on \( \mathbb{N} \) defined by
\( a \, O \, b = a^b + b^a \quad \forall \, a, b \in \mathbb{N} \)
Concept:
A binary operation on a set must satisfy the closure property, meaning the result must always belong to the same set.
Solution:
Let \( a, b \in \mathbb{N} \).
\( a \, O \, b = a^b + b^a \)
Since:
- \( a^b \in \mathbb{N} \)
- \( b^a \in \mathbb{N} \)
The sum of two natural numbers is also a natural number.
\( a^b + b^a \in \mathbb{N} \)
Conclusion:
The result always belongs to \( \mathbb{N} \), so the set is closed under this operation.
✔ Therefore, the operation is a binary operation on \( \mathbb{N} \).