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
Given that, * be the binary operation defined on Q.
A binary operation ‘*’ is commutative if a*b = b*a ∀ a, b ∈ Q
(i) a * b = a – b ∀ a, b ∈Q
a * b = a – b = -b+a = -(b-a) = -(b*a)
∴ a*b ≠ b*a
Hence, ‘*’ is not commutative on Q.
(ii) a * b = a2 + b2 ∀ a, b ∈Q
a * b = a2 + b2
= b2 + a2 [∵ addition is commutative on Q ⇒ a+b = b+a ]
= b*a
∴ a*b = b*a
Hence, ‘*’ is commutative on Q.
(iii) a * b = a + ab ∀ a, b ∈ Q
a * b = a + ab and b*a = b + ab
⇒ a + ab ≠ b + ab
∴ a*b ≠ b*a
Hence, ‘*’ is not commutative on Q.
(iv) a * b = (a – b)2 ∀ a, b ∈Q
a * b = (a – b)2 = {-(b-a)}2
= (b-a)2
=b*a
∴ a*b = b*a
Hence, ‘*’ is commutative on Q.