Commutativity:
Let $\text{a},\text{b}\in\text{Q}_0,$ then $\Rightarrow\text{a }^*\text{ b}=\frac{\text{ab}}{2}=\frac{\text{ba}}{2}=\text{a }^*\text{ b}$ $\Rightarrow\text{a }^*\text{ b}=\text{b }^*\text{ a}$ Thus, * is commutative on Q0.Associativity:
Let $\text{a},\text{b},\text{c}\in\text{Q}_0,$ then $\Rightarrow(\text{a }^*\text{ b})\ ^*\ \text{c}=\frac{\text{ab}}{2}\ ....(1)$ and, $\text{a }^*\ (\text{b }^*\text{ c})=\text{a }^*\ \frac{\text{bc}}{2}=\frac{\text{abc}}{4}\ ....(2)$ From (1) & (2) $(\text{a }^*\text{ b})\ ^*\ \text{c}=\text{a }^*\ (\text{b }^*\text{ c})$ ⇒ * is accosiative on Q0.Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.