Let R be the relation in the set Z of integers given by

R = {(a, b): 2 divides a – b}. Show that the relation R transitive? Write the equivalence class [0].

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Note : - This question is a part of Example 5 of NCERT – Chapter 1 Class 12

Check the answer here ttps://www.teachoo.com/3959/673/Example-5---R---(a--b)---2-divides-a-b--is-equivalence-relation/category/Examples/

 

  1. Class 12
  2. Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Transcript

Question 21 (OR 2nd Question) Let R be the relation in the set Z of integers given by R ={(a, b): 2 divides a – b}. Show that the relation R transitive? Write the equivalence class [0]. R = {(a, b) : 2 divides a – b} Check transitive If 2 divides (a – b) , & 2 divides (b – c) , So, 2 divides (a – b) + (b – c) also So, 2 divides (a – c) ∴ If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R Therefore, R is transitive. Rough 2 divides 8 & 2 divides 12 2 divides 8 + 12 = 20 also Now, Equivalance Class [0] This means that one element is 0, we need to find other elements which satisfy R R = {(a, b) : 2 divides a – b} So, (a, 0) ∈ R where a ∈ Z Because given that R is in set Z, so both a and b belong to set Z Now, If (a, 0) ∈ R 2 divides a – 0 i.e. 2 divides a So, possible values of a are 0, ±2, ± 4, ± 6, ….. i.e. all even numbers and 0 We use plus minus sign because 2 can divide 2 and –2 So, Equivalence Class [0] = {0, ±2, ± 4, ± 6, …..} Note: 1/2 marks will be deducted if you don’t write 0 or plus minus in the set

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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.