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Question
A binary operation * is defined on the set R of all real numbers by the rule \[a * b = \sqrt{ a^2 + b^2} \text{for all a, b } \in R .\]
Write the identity element for * on R.
Solution
Let e be the identity element in R with respect to * such that
\[a * e = a = e * a, \forall a \in R\]
\[a * e = a \text{ and }e * a = a, \forall a \in R\]
\[Then, \]
\[\sqrt{a^2 + e^2} = a \text{ and }\sqrt{e^2 + a^2} = a, \forall a \in R\]
\[ \Rightarrow \sqrt{a^2 + e} = \text{a and}\sqrt{e + a^2} = a, \forall a \in R \left[ \because e^2 =e \right]\]
\[ \Rightarrow a^2 + e = a^2 \text{ and }e+ a^2 = a^2 , \forall a \in R\]
\[ \Rightarrow e = 0 \in R, \forall a \in R\]
Thus, 0 is the identity element in R with respect to *.
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