Question

Write the identity element for the binary operation * on the set R₀ of all non-zero real numbers by the rule \[a * b = \frac{ab}{2}\] for all a, b ∈ R₀.

Answer 1

Let e be the identity element in R₀ with respect to * such that

\[a * e = a = e * a, \forall a \in R_0 \]
\[a * e = a \text{ and } e * a = a, \forall a \in R_0 \]
\[\text{Then} , \]
\[\frac{ae}{2} = a \text{ and }\frac{ea}{2} = a, \forall a \in R_0 \]
\[ae = 2a, \forall a \in R_0 \]
\[a\left( e - 2 \right) = 0, \forall a \in R_0 \]
\[e - 2 = 0, \forall a \in R_0 (\because a\neq0)\]
\[e = 2 \in R_0\]

Thus, 2 is the identity element in R₀ with respect to *.