Example 21 - Chapter 1 Class 12 Relation and Functions
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
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 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