On Q, the set of all rational numbers, a binary operation * is defined by `a*b=(a b)/5` for all `a , b in Qdot` Find the identity element for * in Q. Also, prove that every non-zero element of Q is invertible.

Explanation

Here, `a ∗ b = ab/5`

An identity element `e` in a relation is an element such that `a ∗ e = e ∗ a = a` .

So, in the given relation, `a ∗ e = a e/5 = a ⇒ a e = 5 a ⇒ e = 5 `

So, identity element is `5` for the given relation.

Now, for any element `x in Q`, If `a ∗ x = e`, then `x` is inverse of `a`.

Here, `a ∗ x = 5 ⇒ ax/5 = 5`

`x = 25/a `

So, inverse of given relation is `25/a` where `a in Q.
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