The relation R in the set Z of integers given by $\mathrm{R}=\{(a, b): a-b$ is divisible by 5$\}$ is

A relation ' $R$ ' is called equivalence relation, if it is reflexive, symmetric and transitive together. $\mathrm{R}=\{(a, a): a-a$ is divisible by 5$\}$. It is reflexive relation. Again, $\mathrm{R}=\{(a, b): a-b$ is divisible by 5$\}$ also $(b-a)$ is divisible by 5 . So it is symmetric relation. Again, $\mathrm{R}=(a, c): a-c$ is divisible by 5 , then $b-c$ is also divisible by 5 , so it is transitive. Hence ' $R$ ' is equivalence relation.