1.

Let Q_(0) be the set of all nonzero reational numbers Let * be a binary operation on Q_(0) defined by a*b =(ab)/(4) for all a,b in Q_(0) (i) Show that * is commutative and associative (ii) Find the identity element in Q_(0) (iii) Find the inverse of an element a in Q_(0)

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Solution :(i) `forall a,b,c in Q_(0)` we have
`a*b=(AB)/(4)=(ba)/(4)=b*a`
And `(a*b)*C =(ab)/(4)*c=((ab)/(4).c)/(4)=(ABV)c/(16)`
Also `a*(b*c)=a*(bc)/(4)=(a(bc)/(4))/(4)=a(bc)/(16)`
But (ab) c = (bc) Hence (a*b)*c=a*(b*c)
(ii) Let e be the identitiy elemn NAD let a in `Q_(0)`Then
`A*e=a rarr (ae)/(4) rarr e=4`
(iii) Let a in `Q_(0)` and let its inverse be b then
`a*b=e rarr (ab)/(4) rarrb=(16)/(a) in Q_(0)`
Thus each a in `Q_(0) has (16)/(a)` as its inverse


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