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

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Solution

An element $e \in Q$ will be the identity element for the operation $$ if
$a b = a b^{2}$
If $a * e = a \Rightarrow a e^{2} = a$
$\Rightarrow e^{2} = 1 \Rightarrow e = \pm 1$
But identity is unique. Hence this operation has no identity.

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Let ∗ be a binary operation on the set Q of rational number as follows: (vi)a∗b=ab2 Show that none of the operations has an identity.

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

An element $e \in Q$ will be the identity element for the operation $$ if
$a b = a b^{2}$
If $a * e = a \Rightarrow a e^{2} = a$
$\Rightarrow e^{2} = 1 \Rightarrow e = \pm 1$
But identity is unique. Hence this operation has no identity.