Let $$ be a binary operation on the set Q of rational number as follows:
$(i) a b = a^{2} + b^{2}$
Show that none of the opeartion has identity.

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Solution

An element $e \in Q$ will be the identity element for the operation $$ if
$a e = a = e * a, \forall a \in Q a * b = a^{2} + b^{2}$
If $a * e = a$ , then $a^{2} + e^{2} = a$
For $a = - 2, (- 2)^{4} + e^{2} = 4 + e^{2} \neq - 2$
Hence, there is no identity element.

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Let ∗ be a binary operation on the set Q of rational number as follows: (i)a∗b=a2+b2 Show that none of the opeartion has identity.

An element e∈Q will be the identity element for the operation ∗ if a∗e=a=e∗a,∀a∈Qa∗b=a2+b2 If a∗e=a, then a2+e2=a For a=−2,(−2)4+e2=4+e2≠−2 Hence, there is no identity element.

Let $$ be a binary operation on the set Q of rational number as follows:
$(i) a b = a^{2} + b^{2}$
Show that none of the opeartion has identity.

An element $e \in Q$ will be the identity element for the operation $$ if
$a e = a = e * a, \forall a \in Q a * b = a^{2} + b^{2}$
If $a * e = a$ , then $a^{2} + e^{2} = a$
For $a = - 2, (- 2)^{4} + e^{2} = 4 + e^{2} \neq - 2$
Hence, there is no identity element.