Q. 55.0( 3 Votes )

# Define an operati

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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