Let `f:X->Y` be a function. Define a relation `R` in `X` given by `R={ – InterviewSolution

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1.	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.
Answer» Here R satisfies the following properties : (i) Reflexivity Let `a in X.` Then `f(a) =f(a) rArr (a, a) in R` `:.` R is reflexive . (ii) Symmetry Let `(a,b ) in R` Then `(a,b) in R rArr f(a) =f(b) rArr f(b) =f(a) rArr (b,a) in R` `:.` R is symmetric. (iii) Transitivity Let `(a,b) in R " and " (b,c) in R .` Then `(a,b) in R , (b,c) in R` `rArr f(a) =f(b) " and " f(b) =f( c)` `rArr f(a) =f(c )` `rArr (a,c) in R` `:.` R is transitive. Hence R is an equivalence relation .

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