Question

Let * be a binary operation on the set Q of rational numbers as follows:

(i) a * b = a − b       (ii) a * b = a² + b²
(iii) a * b = a + ab     (iv) a * b = (a − b)²
(v) a*b=ab/4              (vi) a * b = ab²

Find which of the binary operations are commutative and which are associative.

Answer 1

(i) On Q, the operation * is defined as a * b = a − b. It can be observed that:

Thus, the operation * is not associative.

Hence, the operations defined in (ii), (iv), (v) are commutative and the operation defined in (v) is associative.