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Last updated at Jan. 30, 2020 by Teachoo

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Example 45 Determine which of the following binary operations on the set R are associative and which are commutative. (a) a * b = 1 ∀ a, b ∈ R Check commutative * is commutative if a * b = b * a Since a * b = b * a ∀ a, b ∈ R * is commutative a * b = 1 Check associative * is associative if a (a * b) * c = a * (b * c) Since (a * b) * c = a * (b * c) ∀ a, b, c ∈ R * is an associative binary operation Check associative * is associative if a (a * b) * c = a * (b * c) Since (a * b) * c = a * (b * c) ∀ a, b, c ∈ R * is an associative binary operation a * (b * c) = a * 1 = 1 Example 45 Determine which of the following binary operations on the set R are associative and which are commutative. (b) a ∗ b = ((a + b))/2 ∀ a, b ∈ R Check commutative * is commutative if a * b = b * a Since a * b = b * a ∀ a, b ∈ R * is commutative a * b = (𝑎 + 𝑏)/2 b * a = (𝑏 + 𝑎)/2 = (𝑎 + 𝑏)/2 Check associative * is associative if (a * b) * c = a * (b * c) Since (a * b) * c ≠ a * (b * c) ∀ a, b, c ∈ R * is not an associative binary operation (a * b)* c = ((𝑎 + 𝑏)/2) * c = ((𝑎 + 𝑏)/2 +𝑐)/2 = ((𝑎 + 𝑏 + 2𝑐)/2)/2 = (𝑎 + 𝑏 + 2𝑐)/4 a * (b * c) = a * ((𝑏 + 𝑐)/2) = (𝑎 + (𝑏 + 𝑐)/2)/2 = ((2𝑎 + 𝑏 + 𝑐)/2)/2 = (2𝑎 + 𝑏 + 𝑐)/4

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Chapter 1 Class 12 Relation and Functions

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.