Let `'o'` be a binary operation on the set `Q_0` of all non-zero rational numbers defined by `a\ o\ b=(a b)/2` , for all `a ,\ b in Q_0` . Show that `'o'` is both commutative and associate (ii) Find the identity element in `Q_0` (iii) Find the invertible elements of `Q_0` .

Explanation

i) * is commutative if `a * b=b * a` for all `a, b`.

`a * b=frac{a b}{2}` `b * a=frac{b a}{2}`

Thus `a * b=b * a`. Thus `{ }^{*}` is commutative.

*is associative if `a *(b * c)=(a * b) * c` for all `a, b, c`.

`a *(b * c)=a *(frac{b c}{2})=frac{a times frac{b c}{2}}{2}=frac{a b c}{4}` `(a * b) * c=(frac{a b}{2}) * c=frac{frac{a b}{2} times c}{2}=frac{a b c}{4}`

Thus `a *(b * c)=(a * b) * c`. Thus * is associative.



ii) Calculation If `{e}` is the identity element then `a * e=a`

`a * e=frac{a e}{2}=a`

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