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Given:
\( a \circ b = \frac{ab}{2}, \quad a,b \in \mathbb{Q}_0 \)
i. Commutativity:
\( a \circ b = \frac{ab}{2} = \frac{ba}{2} = b \circ a \)
✔ 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:
\( a \circ (b \circ c) = a \circ \left(\frac{bc}{2}\right) = \frac{a \cdot \frac{bc}{2}}{2} = \frac{abc}{4} \)
✔ Associative
ii. Identity Element:
\( a \circ e = a \Rightarrow \frac{ae}{2} = a \Rightarrow e = 2 \)
✔ Identity = 2
iii. Inverse Element:
Find \( b \) such that:
\( a \circ b = 2 \)
\( \frac{ab}{2} = 2 \Rightarrow ab = 4 \Rightarrow b = \frac{4}{a} \)
✔ Inverse of \( a \) = \( \frac{4}{a} \)
✔ Every element is invertible
Conclusion:
✔ Commutative & Associative
✔ Identity = 2
✔ Inverse of \( a \) = \( \frac{4}{a} \)