Let $$ be a binary operation on the set Q of rational number as follows:
$(i i i) a b = a + a b$
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
$a b = a + a b$
If a $* e = a \Rightarrow a + a e = a \Rightarrow a e = 0 \Rightarrow e = 0, a \neq 0$
Also if
$\Rightarrow e * a = a \Rightarrow e + e a = a \Rightarrow e = \frac{a}{1 - a}, a \neq 1$
$∴ e = 0 = \frac{a}{1 - a}, a \neq 0$
But the 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:
$(i i i) a b = a + a b$
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