# 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.

Toolbox:
• A relation R in a set A in an equivalance relation. if R is reflexive, symmetric and transitive.
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
Hence R is an equivalance relation.
edited Mar 8, 2013