Find which of the operations given above has identity.

Question:

Find which of the operations given above has identity.

Solution:

An element ∈ Q will be the identity element for the operation * if

* e = a = e * aa ∈ Q.

We are given

$a^{*} b=\frac{a b}{4}$

$\Rightarrow a^{*} e=a$

$\Rightarrow \frac{a e}{4}=a$

$\Rightarrow e=4$

Similarly, it can be checked for $e^{*} a=a$, we get $e=4$

Thus, $e=4$ is the identity.

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