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Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation. |
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Answer» Set `A` is the set of all books in the library of a college. `R = {x, y): x` and `y` have the same number of pages}. Now, R is reflexive since `(x, x) in R` as `x` and `x` has the same number of pages. Let `(x, y) in R => x` and `y` have the same number of pages. `=> y` and `x` have the same number of pages. `=> (y, x) in R`. `:. R` is symmetric. Now, let `(x, y) in R` and `(y, z) in R`. `=> x` and `y` and have the same number of pages and `y` and `z` have the same number of pages. It means `x` and `z` have the same number of pages. => `(x, z) in R`. `:. R` is transitive. As `R` is reflexive, summetric and transitive, `R` is an equivalence relation. |
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