Answer :

For the operation r*s = r + s –(r × s)
(i)
 Let r = 1/2 and s = 1/4
Now ,




Which is also a rational number.
Hence Q is closed under the operation * .


(ii) Let three rational number be o,p,q
For the operation r*s = r + s –(r × s)
(o*p)*q = [ o + p -( o × p)]*q
            = [o + p - op]*q
            = [(o + p - op) + q - (( o + p - op) × q)]
            = [ o + p - op + q - oq - pq +opq ]    ..... (1)
Now,
o*(p*q) = o* [ p + q - (p × q ) ]
             = o* [ p + q - pq ]
            = [( o + (p + q - pq) - (o ×(p + q - pq))]
            = [ o + p + q - pq - op - oq + opq]  ..... (2)
From 1 and 2
(o*p)*q = o*(p*q)
Hence * is an associative operation.


(iii) Let two rational number be p,q.
For the operation r*s = r + s –(r × s)
p*q = p + q - ( p × q )
      = p + q - pq
Now,
q*p = q + p - ( q × p )
      = q + p - qp
Hence * is a commutative operation. 


(iv) For the operation r*s = r + s –(r × s)
a * 1 = a + 1 - ( a × 1)
        = a + 1 - a
        = 1
For any a  in Q a * 1 is 1.


(v) The value of two integers would be a = 2 and b = 2


For the operation r*s = r + s –(r × s)


a * b = 0 

a + b - ( a × b ) = 0   
a + b - ab = 0
a + b ( 1 - a) = 0
b ( 1- a) = -a
b = - a/(1 -  a)
Hence the value is a , - a/(1 -  a).

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