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Determine whether the operation is a binary operation or not
Given: An operation \( * \) on \( \mathbb{Q} \) defined by
\( a * b = \frac{a – 1}{b + 1} \quad \forall \, a, b \in \mathbb{Q} \)
Concept:
A binary operation must be defined for all elements of the set and must satisfy the closure property.
Solution:
Let \( a, b \in \mathbb{Q} \).
\( a * b = \frac{a – 1}{b + 1} \)
For this expression to be defined, the denominator must not be zero:
\( b + 1 \neq 0 \Rightarrow b \neq -1 \)
But \( -1 \in \mathbb{Q} \), so for \( b = -1 \), the operation is not defined.
Conclusion:
Since the operation is not defined for all elements of \( \mathbb{Q} \), it is not a binary operation.
❌ Therefore, the operation is NOT a binary operation on \( \mathbb{Q} \).