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Check commutativity and associativity
Given:
\( a \circ b = \frac{ab}{2}, \quad a,b \in \mathbb{Q} \)
Commutativity:
\( a \circ b = \frac{ab}{2} = \frac{ba}{2} = b \circ a \)
✔ Operation is commutative
Associativity:
LHS:
\( (a \circ b)\circ c = \left(\frac{ab}{2}\right)\circ c = \frac{\frac{ab}{2} \cdot c}{2} = \frac{abc}{4} \)
RHS:
\( a \circ (b \circ c) = a \circ \left(\frac{bc}{2}\right) = \frac{a \cdot \frac{bc}{2}}{2} = \frac{abc}{4} \)
Thus:
\( (a \circ b)\circ c = a \circ (b \circ c) \)
✔ Operation is associative
Conclusion:
✔ The operation is both commutative and associative on \( \mathbb{Q} \).