# Example 45 - Chapter 1 Class 12 Relation and Functions

Last updated at Jan. 30, 2020 by Teachoo

Last updated at Jan. 30, 2020 by Teachoo

Transcript

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 10 years. He provides courses for Maths and Science at Teachoo.