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Check commutativity and associativity
Given:
\( a * b = \frac{ab}{4}, \quad a,b \in \mathbb{Q} \)
Commutativity:
\( a * b = \frac{ab}{4} = \frac{ba}{4} = b * a \)
✔ Operation is commutative
Associativity:
LHS:
\( (a*b)*c = \left(\frac{ab}{4}\right)*c = \frac{\frac{ab}{4} \cdot c}{4} = \frac{abc}{16} \)
RHS:
\( a*(b*c) = a*\left(\frac{bc}{4}\right) = \frac{a \cdot \frac{bc}{4}}{4} = \frac{abc}{16} \)
Thus:
\( (a*b)*c = a*(b*c) \)
✔ Operation is associative
Conclusion:
✔ The operation is both commutative and associative on \( \mathbb{Q} \).