Show that the relation $R$ in the set $A$ of all the books in a library of a college,given by $R = \{(x, y) : x \text{ and } y \text{ have the same number of pages} \}$ is an equivalence relation.

(N/A) Set $A$ is the set of all books in the library of a college.
$R = \{(x, y) : x \text{ and } y \text{ have the same number of pages} \}$
$1.$ Reflexive: For any book $x \in A$,$x$ has the same number of pages as itself. Therefore,$(x, x) \in R$ for all $x \in A$. Thus,$R$ is reflexive.
$2.$ Symmetric: Let $(x, y) \in R$. This means $x$ and $y$ have the same number of pages. It follows that $y$ and $x$ also have the same number of pages. Therefore,$(y, x) \in R$. Thus,$R$ is symmetric.
$3.$ Transitive: Let $(x, y) \in R$ and $(y, z) \in R$. This means $x$ and $y$ have the same number of pages,and $y$ and $z$ have the same number of pages. Consequently,$x$ and $z$ must have the same number of pages. Therefore,$(x, z) \in R$. Thus,$R$ is transitive.
Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation.