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].

Let R = {(a, b): 2 divides a – b}. Show that the relation R transitive

Question 21 (OR 2nd Question) - CBSE Class 12 Sample Paper for 2020 Boards - Part 2
Question 21 (OR 2nd Question) - CBSE Class 12 Sample Paper for 2020 Boards - Part 3

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/

 

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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 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.