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Determine whether the operation is a binary operation or not
Given: An operation \( O \) on \( \mathbb{Z} \) defined by
\( a \, O \, b = ab \quad \forall \, a, b \in \mathbb{Z} \)
Concept:
A binary operation on a set satisfies the closure property, meaning the result of the operation on any two elements of the set must also belong to the same set.
Solution:
Let \( a, b \in \mathbb{Z} \).
\( a \, O \, b = ab \)
We know that the product of any two integers is always an integer.
\( ab \in \mathbb{Z} \)
Conclusion:
Since the result is always an integer, the set \( \mathbb{Z} \) is closed under this operation.
✔ Therefore, the operation defines a binary operation on \( \mathbb{Z} \).