📺 Watch Video Explanation:
Determine whether the operation is a binary operation or not
Given: An operation \( * \) on \( \mathbb{R} \) defined by
\( a * b = a + 4b^2 \quad \forall \, a, b \in \mathbb{R} \)
Concept:
A binary operation must satisfy the closure property, meaning the result must belong to the same set.
Solution:
Let \( a, b \in \mathbb{R} \).
\( a * b = a + 4b^2 \)
Since:
- \( b^2 \in \mathbb{R} \)
- \( 4b^2 \in \mathbb{R} \)
- Sum of real numbers is a real number
\( a + 4b^2 \in \mathbb{R} \)
Conclusion:
The set \( \mathbb{R} \) is closed under this operation.
✔ Therefore, the operation is a binary operation on \( \mathbb{R} \).