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Example 21 You are here

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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Example 21 Let f : X → Y be a function. Define a relation R in X given by R = {(a, b): f(a) = f(b)}. Examine whether R is an equivalence relation or not. Equivalence relation are Relations which are reflexive, transitive and symmetric. R = {(a, b): f(a) = f(b)} Check reflexive Since f (a) = f (a), ∴ (a, a) ∈ R, Hence, R is reflexive. Check symmetric If f (a) = f (b), then f (b) = f (a) Hence, (b, a) ∈ R So, if (a, b) ∈ R , then (b, a) ∈ R. ∴ R is symmetric. Check transitive If (a, b) ∈ R ∴ f(a) = f(b) Also if, (b, c) ∈ R ∴ f(b) = f(c) From (1) & (2) f(a) = f(c) ∴ (a, c) ∈ R, ∴ If (a, b) ∈ R & (b, c) ∈ R , then (a, c) ∈ R ∴ R is transitive. Hence, R is an equivalence relation.

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Davneet Singh

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.