A binary operation * is defined on the set R of all real numbers … - Vidyadip

Question

A binary operation $*$ is defined on the set $R$ of all real numbers by the rule $\text{a}\times\text{b}=\sqrt{\text{a}^2+\text{b}^2}\ \forall\text{ a, b}\in\text{R}$.
Write the identity element for $*$ on $R.$

✓

Answer

Let e be the identity element in $R$ with respect to $*$ such thata $* e = a = e * a, \forall\text{ a}\in\text{R}$
$a * e = a$ and $e * a = a, \forall\text{ a}\in\text{R}$
Then,
$\sqrt{\text{a}^2+\text{e}^2}=\text{a}$ and $\sqrt{\text{e}^2+\text{a}^2}=\text{a},\forall\text{ a}\in\text{R}$
Implies that $\sqrt{\text{a}^2+\text{e}}=\text{a}$ and $\sqrt{\text{e}+\text{a}^2}=\text{a},\forall\text{ a}\in\text{R} [ \because e^2 = e]$
Implies that $a^2 + e = a^2$ and $e + a^2 = a^2,$ $\forall\text{ a}\in\text{R}$
Implies that $\text{e}=0\in\text{R},\forall\text{ a}\in\text{R}$
Thus, $0$ is the identity element in $R$ with respect to $*.$

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