Question

Write the identity element for the binary operation * defined on the set R of all real numbers by the rule

\[a * b = \frac{3ab}{7} \text{ for all a, b} \in R .\] ?

Answer 1

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

\[a * e = a = e * a, \forall a \in R\]
\[a * e = a \text{ and }e * a = a, \forall a \in R\]
\[\text{ Then }, \]
\[\frac{3ae}{7} = a \text { and }\frac{3ea}{7} = a, \forall a \in R\]
\[e = \frac{7}{3} , \forall a \in R\]

Thus, \[\frac{7}{3}\] is the identity element in R with respect to *.