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 April 16, 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.

Next: Question 33 (Choice 2) Important → Go Ad-free

Made by

Davneet Singh

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, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Class 6

Class 7

Class 8

Class 9

Class 10

Class 11

Class 12

Courses

Interesting

Popular