Question

Let 'o' be a binary operation on the set Q₀ of all non-zero rational numbers defined by \[a o b = \frac{ab}{2}, \text{ for all a, b } \in Q_0\] :

 Find the identity element in Q₀.

Answer 1

Let e be the identity element in Q_o with respect to * such that

\[a o e = a = e o a, \forall a \in Q_0 \] 
\[a o e = a \text{ and }e o a = a, \forall a \in Q_0 \] 
\[ \Rightarrow \frac{ae}{2} = a \text{ and }\frac{ea}{2} = a, \forall a \in Q_0 \] 
\[e = 2 \in Q_0 , \forall a \in Q_0\]

Thus, 2 is the identity element in Q_o with respect to o.