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If * is a binary operation on Q where Q is set of rational numbers such that a * b = ab/2 ∀ a, b ∈ Q then find the identity element of Q with respect to operation * ? 

If * is a binary operation on Q where Q is set of rational numbers such that a * b = ab/2 ∀ a, b ∈ Q then find the identity element of Q with respect to operation * ?  Correct Answer 2

Concept:

Let * be a binary operation on a non-empty set S. If there exists an element e in S such that a * e = e * a = a ∀ a ∈ S. Then the element e is said to be an identity element of S with respect to *.

Calculation:

Given: * is a binary operation on Q where Q is set of rational numbers such that a * b = ab/2 ∀ a, b ∈ Q.

Let e be the identity element of Q with respect to *.

As we know that if e is an identity element of a non-empty set S with respect to a binary operation * then a * e = e * a = a ∀ a ∈ S.

Let a ∈ Q and because e is the identity element of Q with respect to given operation *.

.e a * e = a =  e * a ∀ a ∈ Q.

According to the definition of *, we have

⇒ a * e = ae/2 = a

⇒ e = 2 ∈ Q

Hence, 2 is the identity element of Q with respect to given operation *

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