Question

A binary operation * is defined on the set R of all real numbers by the rule

$a*b=\sqrt{a^2+b^2}\mathrm{for}\mathrm{all}a,b\in R.$

Write the identity element for * on R.

Answer 1

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

$$\begin{gathered} a*e=a=e*a,\forall a\in R \\ a*e=a\text{and}e*a=a,\forall a\in R \\ \mathrm{Then}, \\ \sqrt{a^2+e^2}=a\text{and}\sqrt{e^2+a^2}=a,\forall a\in R \\ \Rightarrow \sqrt{a^2+e}=a\text{and}\sqrt{e+a^2}=a,\forall a\in R\left[{\mathit{\text{∵ e}}}^{\mathit{2}}\mathit{\text{=}}\mathit{\text{e}}\right] \\ \Rightarrow a^2+e=a^2\text{and}\mathit{\text{e}}\text{+}{\mathit{\text{a}}}^2=a^2,\forall a\in R \\ \Rightarrow e=0\in R,\forall a\in R \end{gathered}$$

Thus, 0 is the identity element in R with respect to *.