A relation R on set $\displaystyle \mathrm{A}=\{\mathrm{x}:-10 \leq \mathrm{x} \leq 10, \mathrm{x} \in \mathrm{Z}\}$ is defined as $\displaystyle \mathrm{R}=\{(\mathrm{x}, \mathrm{y}):(\mathrm{x}-\mathrm{y})$ is divisible by $\displaystyle 5 \}$. Show that R is an equivalence relation. Also, write the equivalence class [$\displaystyle 5$].

For reflexive relation To prove $\displaystyle (\mathrm{x}, \mathrm{x}) \in \mathrm{R}, \mathrm{x}-\mathrm{x}=0$ which is divisible by $\displaystyle 5$ $\displaystyle \therefore(\mathrm{x}, \mathrm{x}) \in \mathrm{R} \Rightarrow \mathrm{R}$ is reflexive For symmetric relation Let $\displaystyle (\mathrm{x}, \mathrm{y}) \in \mathrm{R} \Rightarrow \mathrm{x}-\mathrm{y}$ is divisible by $\displaystyle 5$\Rightarrow \mathrm{x}-\mathrm{y}=5 m \Rightarrow \mathrm{y}-\mathrm{x}=5(-m)$\displaystyle \Rightarrow \mathrm{y}-\mathrm{x}$ is divisible by $\displaystyle 5$ $\displaystyle \Rightarrow(\mathrm{y}, \mathrm{x}) \in \mathrm{R} \therefore \mathrm{R}$ is symmetric For transitive relation\begin{aligned} & \text { Let }(\mathrm{x}, \mathrm{y}) \in \mathrm{R} \text { and }(\mathrm{y}, \mathrm{z}) \in \mathrm{R} & \begin{array}{l} \mathrm{x}-\mathrm{y} \text { is divisible by } 5 \mathrm{y}-\mathrm{z} \text { is divisible by } 5 \Rightarrow \mathrm{x}-\mathrm{y}+\mathrm{y}-\mathrm{z}=5(m-n) \Rightarrow \mathrm{x}-\mathrm{z}=5(m-n) \therefore \mathrm{x}-\mathrm{z} \text { is divisible by } 5 \end{array} \end{aligned}$\displaystyle \Rightarrow(\mathrm{x}, \mathrm{z}) \in \mathrm{R} \therefore \mathrm{R}$ is transitive. $\displaystyle \mathrm{R}$ is an equivalence relation.[5]=\{-10,-5,0,5,10\} $$