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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.
Solution
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.
