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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 |
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Answer» 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. |
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