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.

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 $  $(y, x) \in R$

$\therefore R$ 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.