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Question 33 (Choice 1) - CBSE Class 12 Sample Paper for 2023 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at Dec. 13, 2024 by Teachoo

Define the relation R in the set 𝑁 × 𝑁 as follows:

For (a, b), (c, d) ∈ 𝑁 × 𝑁, (a, b) R (c, d) iff ad = bc. Prove that R is an equivalence relation in 𝑁 × 𝑁.

 

 

Transcript

Question 33 (Choice 1) Define the relation R in the set 𝑁 × 𝑁 as follows: For (a, b), (c, d) ∈ 𝑁 × 𝑁, (a, b) R (c, d) iff ad = bc. Prove that R is an equivalence relation in 𝑁 × 𝑁.If (a, b) R (c, d) , then ad = bc Check Reflexive If (a, b) R (a, b), then ab = ba Since, ab = ba Hence , R is reflexive. Check symmetric If (a, b) R (c, d) , then ad = bc Now, If (c, d) R (a, b) , then cb = da Since, ad = bc, da = cb ∴ cb = da So, if (a, b) R (c, d) , then (c, d) R (a, b) So, R is symmetric. Check transitive Putting (2) in (1) ad = bc ad = b(𝑑𝑒/𝑓) adf = bde af = be Hence (a, b) R (e, f) If (a, b) R (c, d) , then ad = bc If (c, d) R (e, f) , then cf = de c = 𝑑𝑒/𝑓 We need to prove that (a, b) R (e, f) , i.e. af = be So, if (a, b) R (c, d) & (c, d) R (e, f) , then (a, b) R (e, f) Thus R is transitive. Thus, R is an equivalence relation.
  1. Class 12
  2. Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

    3. CBSE Class 12 Sample Paper for 2023 Boards

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

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo