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.

Answer 2

Answer:

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Step-by-step explanation: