$\begin{array}{1 1} \text{Reflexive Only} \\ \text{Symmetric Only} \\ \text{Transitive Only} \\ \text{Reflexive, Symmetric and Transitive} \end{array} $

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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.

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- A relation R in a set A is called reflexive. if $(a,a) \in R\;for\; all\; a\in A$
- A relation R in a set A is called symmetric. if $(a_1,a_2) \in R\;\Rightarrow \; (a_2,a_1)\in R \;$ for $\;a_1,a_2 \in A$
- A relation R in a set A is called transitive. if $(a_1,a_2) \in\; R$ and $(a_2,a_3)\in R \Rightarrow \;(a_1,a_3)\in R\; $for all $\; a_1,a_2,a_3 \in A$

Let $A=$Set of all book in library in college. Given $R=$ {$(x,y):x$ and $y$ have same number of pages}:

If $x=y$, such that $(x,x) \in R$, then $x$ and $y$ have the same number of pages. Hence, $R$ is reflexive.

For $R$ to be symmetric, $(x,y) \in R \; \Rightarrow \; (y,x) \in R$

If $x$ has same number of pages as $y$ then $y$ also has same number pages as $x$. Therefore, $R$ is symmetric

For $R$ to be transitive, if $(x,y),(y,z) \in R \; \Rightarrow \; (x,z) \in R$

If $x$ and $y$ have same number of pages and $y$ and $z$ have same number of pages Then $x$ and $z$ also have same number of pages. Hence $R$ is transitive

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