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
Solution:
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.