Show that the relation R in the set A of all the books in a library of a college,

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 $(x, y) \in R \Rightarrow x$ and $y$ have the same number of pages.

$\Rightarrow y$ and $x$ have the same number of pages.

$\Rightarrow(y, x) \in R$

∴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.

$\Rightarrow(x, z) \in R$

∴R is transitive.

Hence, R is an equivalence relation.