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Determine whether the operation is a binary operation or not
Given: An operation \( * \) on \( \mathbb{R} \) defined by
\( a * b = ab^2 \quad \forall \, a, b \in \mathbb{R} \)
Concept:
A binary operation must satisfy the closure property, meaning the result must always belong to the same set.
Solution:
Let \( a, b \in \mathbb{R} \).
\( a * b = ab^2 \)
Since:
- \( b^2 \in \mathbb{R} \)
- The product of real numbers is a real number
\( ab^2 \in \mathbb{R} \)
Conclusion:
The set \( \mathbb{R} \) is closed under this operation.
✔ Therefore, the operation is a binary operation on \( \mathbb{R} \).