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.

R $= \{(x, y) :$ x and y have same number of pages$\}$

(i) Reflexive
Books x and x have same number of pages.

Therefore, $(x,\;x) \in R$

Hence R is reflexive.

(ii) Symmetric

If $(x,\;y) \in R,$ i.e. Books x and y have same number of pages.
$\Rightarrow$ Books y and x have same number of pages.

$\Rightarrow$ $(y,\;x) \in R$
Therefore, R is symmetric.

(iii) Transitive

$If\;\;(x,\;y) \in R\;\;and\;\;(y,\;z) \in R$

$\Rightarrow$ Books x and y have same number of pages and books y and z have same number of pages.

$\Rightarrow$ Books x and z have same number of pages.

$\Rightarrow$ $(x,\;z) \in R\;\;\;\therefore \;\;R$ is transitive.

Hence, R is an equivalence relation.