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Example 44 - Let R = {(a, b): f(a) = f(b)}. Examine equivalence - Examples


  1. Chapter 1 Class 12 Relation and Functions
  2. Serial order wise
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Example 44 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(a) 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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