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Given:
\( a * b = \frac{3ab}{5}, \quad a,b \in \mathbb{Q}_0 \)
Commutativity:
\( a*b = \frac{3ab}{5} = \frac{3ba}{5} = b*a \)
✔ Commutative
Associativity:
LHS:
\( (a*b)*c = \left(\frac{3ab}{5}\right)*c = \frac{3\cdot \frac{3ab}{5} \cdot c}{5} = \frac{9abc}{25} \)
RHS:
\( a*(b*c) = a*\left(\frac{3bc}{5}\right) = \frac{3a \cdot \frac{3bc}{5}}{5} = \frac{9abc}{25} \)
✔ Associative
Identity Element:
Let identity be \( e \), then:
\( a * e = a \Rightarrow \frac{3ae}{5} = a \)
Simplify:
\( \frac{3e}{5} = 1 \Rightarrow e = \frac{5}{3} \)
Verification:
\( a * \frac{5}{3} = \frac{3a \cdot \frac{5}{3}}{5} = a \)
✔ Identity = \( \frac{5}{3} \)
Conclusion:
✔ Commutative & Associative
✔ Identity element = \( \frac{5}{3} \)