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Let `f : X->Y`be a function. Define a relation R in X given by `R = {(a , b): f(a) = f(b)}`. Examine if R is an equivalence relation. |
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Answer» `R = {(a,b): f(a) = f(b)}` As `f(a) = f(a)` `:. (a,a) in R` `:. R` is reflexive. If `f(a) = f(b)` Then, `f(b) = f(a)` Thus, `(b,a) in R` So, if `(a,b) in R`, then `(b,a) in R` `:. R` is ymmetric. If `(a,b) in R`, Then, `f(a) = f(b)`->(1) If `(b,c) in R`, Then, `f(b) = f(c)`->(2) From (1) and (2), `f(a) = f(c)` `:. (a,c) in R` `:. R` is transitive. As `R` is reflexive, summetric and transitive, `R` is an equivalence relation. |
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