Let 'o' be a binary operation on the set Q_0 of all non-zero rati… - Vidyadip

Question

Let $'o'$ be a binary operation on the set $Q_0$ of all non-zero rational numbers defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{2} $ for all $\text{a},\text{b}\in\text{Q}_0.$
Find the identity element in $Q_0.$

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Answer

We have,
$\text{a }^*\text{ b}=\frac{\text{ab}}{2}$ for all $\text{a},\text{b}\in\text{Q}_0$
Let $\text{e}\in\text{Q}_0$ be the identity element with respect to $*.$
By identity property, we have,
$a * e = e * a = a$ for all $\text{a}\in\text{Q}_0$
$\Rightarrow\frac{\text{ae}}{2}=\text{a}\Rightarrow\text{e}=2$
Thus the required identity element is $2.$

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