Question

Show that the relation $R$ in the set $A$ of all the books in a library of a college, given by $R=\left\{\right(x,y):xandyhavesamenumberofpages\}$ is an equivalence relation.

Answer 1

Set $A$ is the set of all books in the library of a college.
$R=\left\{\right(x,y):\text{x and y have the same number of pages}\}$
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 numbers of pages.
$\Rightarrow (x,z)\in R$
$\therefore R$ is transitive.
Hence, $R$ is an equivalence relation.