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

$$\begin{gathered} a*e=a=e*a,\forall a\in R_0 \\ a*e=a\text{and}e*a=a,\forall a\in R_0 \\ \mathrm{Then}, \\ \frac{ae}{2}=a\text{and}\frac{ea}{2}=a,\forall a\in R_0 \\ ae=2a,\forall a\in R_0 \\ a\left(e-2\right)=0,\forall a\in R_0 \\ e-2=0,\forall a\in R_0\text{(∵}\mathit{\text{a}}\text{≠0)} \\ e=2\in R_0 \end{gathered}$$

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