A relation R on set A ={ x:-10 ≤ x ≤ 10, x ∈ Z } is defined as R ={( x, y ):( x - y ) is divisible by 5 }. Show that R is an equivalence relation.…

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. $$[5]=\{-10,-5,0,5,10\} $$