Question

Let * be the binary operation defined on Q. Find which of the following binary operations are commutative

(i) a * b = a – b ∀ a, b ∈ Q 

(ii) a * b = a² + b² ∀ a, b ∈ Q

(iii) a * b = a + ab ∀ a, b ∈ Q 

(iv) a * b = (a – b)² ∀ a, b ∈ Q

Answer 1

Given that * is a binary operation defined on Q.

(i) a * b = a – b, ∀ a, b ∈Q and b * a = b – a

So, a * b ≠ b * a

Thus, * is not commutative.

(ii) a * b = a² + b²

b * a = b² + a²

Thus, * is commutative.

(iii) a * b = a + ab

b * a = b + ab

So clearly, a + ab ≠ b + ab

Thus, * is not commutative.

(iv) a * b = (a – b)², ∀ a, b ∈Q

b * a = (b –a)²

Since, (a – b)² = (b – a)²

Thus, * is commutative.