Examples

Chapter 1 Class 12 Relation and Functions
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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.

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.