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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