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Question
Write the identity element for the binary operation * defined on the set R of all real numbers by the rule
\[a * b = \frac{3ab}{7} \text{ for all a, b} \in R .\] ?
Solution
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 *.
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