Let ∗ be a binary operation on the set Q of rational numbers as follows and Find which of the binary operations are commutative and which are associative.

1. a ∗ b = a – b
2. a ∗ b = a2 + b2

Arrangement:

(i) a ∗ b = a – b

a ∗ b = a – b = – (b – a) = – b * c ≠ b * a (Not commutative)

(a * b) * c = (a – b) * c = (a – (b – c) = a – b + c ≠ a * (b *c) (Not cooperative)

(ii) a ∗ b = a2 + b2

a ∗ b = a2 + b2 = b2 + a2 = b * a (activity is commutative) Check for cooperative:

(a * b) * c = (a2 + b2) * c2 = (a2 + b2) + c2 a * (b *c) = a * (b2 + c2 ) = a2 * (b2 + c2 )2 (a * b) * c ≠ a * (b *c) (Not cooperative)