# Example 41 - Chapter 1 Class 12 Relation and Functions (Term 1)

Last updated at Aug. 11, 2021 by Teachoo

Last updated at Aug. 11, 2021 by Teachoo

Transcript

Example 41 If R1 and R2 are equivalence relations in a set A, show that R1 ∩ R2 is also an equivalence relation. R1 is an equivalence relation 1. R1 is symmetric (a, a) ∈ R1, for all a ∈ A. 2. R1 is reflexive If (a, b) ∈ R1 , then (b, a) ∈ R1 3. R1 is transitive If (a, b) ∈ R1 & (b, c) ∈ R1 , then (a, c) ∈ R1 R2 is an equivalence relation 1. R2 is symmetric (a, a) ∈ R2, for all a ∈ A. 2. R2 is reflexive If (a, b) ∈ R2 , then (b, a) ∈ R2 3. R2 is transitive If (a, b) ∈ R2 & (b, c) ∈ R2 , then (a, c) ∈ R2 We have to prove R1 ∩ R2 is equivalence relation Check reflexive For all a ∈ A (a, a) ∈ R1, & (a, a) ∈ R2 Hence, (a, a) ∈ both R1 & R2 Hence, (a, a) ∈ R1 ∩ R2 ∴ R1 ∩ R2 is reflexive. Check symmetric R1 is symmetric ,hence If (a, b) ∈ R1 , then (b, a) ∈ R1 R2 is symmetric, hence If (a, b) ∈ R2 , then (b, a) ∈ R2 From (1) and (2) If (a, b) ∈ R1 ∩ R2, then (b, a) ∈ R1 ∩ R2 Hence , R1 ∩ R2 is symmetric. Checking transitive R1 is transitive, Hence, if (a, b) ∈ R1 & (b, c) ∈ R1 , then (a, c) ∈ R1 R2 is transitive, Hence, if (a, b) ∈ R2 & (b, c) ∈ R2 , then (a, c) ∈ R2 From (3) & (4) If (a, b) ∈ R1 ∩ R2 and (b, c) ∈ R1 ∩ R2 , then (a, c) ∈ R1 ∩ R2, ∴ R1∩ R2 is transitive. Thus, R1 ∩ R2 is an equivalence relation.

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Chapter 1 Class 12 Relation and Functions (Term 1)

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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.