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) (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:

and

∴ ; where

Thus, the operation * is not commutative.

It can also be observed that:

Thus, the operation * is not associative.

(ii) On Q, the operation * is defined as a * b = a² + b².

For a, b ∈ Q, we have:

∴a * b = b * a

Thus, the operation * is commutative.

It can be observed that:

Thus, ,the operation * is not associative.

(iii) On Q, the operation * is defined as a * b = a + ab.

It can be observed that:

Thus, the operation * is not commutative.

It can also be observed that:

Thus, the operation * is not associative.

(iv) On Q, the operation * is defined by a * b = (a − b)².

For a, b ∈ Q, we have:

a * b = (a − b)²

b * a = (b − a)² = [− (a − b)]² = (a − b)²

∴ a * b = b * a

Thus, the operation * is commutative.

It can be observed that:

Thus, the operation * is not associative.

(v) On Q, the operation * is defined as

For a, b ∈ Q, we have:

∴ a * b = b * a

Thus, the operation * is commutative.

For a, b, c ∈ Q, we have:

∴(a * b) * c = a * (b * c)

Thus, the operation * is associative.

(vi) On Q, the operation * is defined as a * b = ab²

It can be observed that:

Thus, the operation * is not commutative.

It can also 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.