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.

 

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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.

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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.