Define an operation * on the set of all rational numbers □ as follows:

r*s = r + s –(r × s),

for any two rational numbers r, s. Answer the following with justification:

(i) Is ℚ closed under the operation *?

(ii) Is * an associative operation of ℚ ?

(iii) Is * a commutative operation of ℚ?

(iv) What is a * 1 for any a in ℚ?

(v) Find two integers a ≠ 0 and b ≠ 0 such that a * b = 0.

(i) Yes, Q is closed under the operation * as addition, multiplication and subtraction are always closed.

(ii) yes * is an associative operation of Q because as per associative law while adding rational numbers, they can be grouped in any order.

(iii) yes * is a commutative operation of Q as two rational numbers can be added in any order.

(iv) For any a * 1 in Q the value of ‘a’ would be a itself as 1 has the multiplicative identity property.

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

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

2 * 2 = 2 + 2 – ( 2× 2)

RHS 4 – 4

= 0 = LHS