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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Solution

Here, set A is the set of all books in the library of a college.and 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} $\in R$
$\Rightarrow x$ and y have the same number of pages.
$\Rightarrow y$ and x have the same number of pages
$\Rightarrow (y, x) \in R$ . So, R is symmetric. Now, let (x,y) $\in R$ and $(y, z) \in R .$
$\Rightarrow x$ and y and have the same number of pages and y and z have the same number of pages.
$\Rightarrow x$ and z have the same number of pages $\Rightarrow (x, z) \in R$ .
Therefore, R is transitive. Hence, R is an equivalence relation.

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(v) Relation R in the set A of human beings in a town at a particular time given by
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