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Determine whether the operation is a binary operation or not
Given: A set \( S = \{0,1,2,3,4,5\} \) and an operation \( +_6 \) defined by
\[
a +_6 b =
\begin{cases}
a + b, & \text{if } a + b < 6 \\
a + b - 6, & \text{if } a + b \geq 6
\end{cases}
\]
Concept:
A binary operation must satisfy the closure property, meaning the result must always belong to the same set.
Solution:
Take any \( a, b \in S \). Then \( a + b \) is between 0 and 10.
- If \( a + b < 6 \), then the result is clearly in \( S \).
- If \( a + b \geq 6 \), then \( a + b – 6 \) will lie between 0 and 4, which is also in \( S \).
Thus, in all cases:
\( a +_6 b \in S \)
Conclusion:
The set \( S \) is closed under this operation.
✔ Therefore, the operation \( +_6 \) is a binary operation on \( S \).