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Determine whether the operation is a binary operation or not
Given: A set \( S = \{1,2,3,4,5\} \) and an operation \( \times_6 \) defined by
\( a \times_6 b = \text{remainder when } ab \text{ is divided by } 6 \)
Concept:
A binary operation on a set must satisfy the closure property, i.e., the result of the operation must always belong to the same set.
Solution:
Take elements \( a = 2 \), \( b = 3 \) from the set \( S \).
\( a \times_6 b = 2 \times 3 = 6 \)
Now, the remainder when 6 is divided by 6 is:
\( 6 \div 6 \Rightarrow \text{remainder} = 0 \)
But \( 0 \notin S \).
Conclusion:
Since the result is not always an element of \( S \), the set is not closed under this operation.
❌ Therefore, the operation is NOT a binary operation on \( S \).