Question

Let $Z$ be the set of all integers and $R$ be the relation on $Z$ defined as $R=\{(a, b): a, b \in Z$, and $(a-b)$ is divisible by 5 \} Prove that $R$ is an equivalence relation.

Answer 1

SOLUTION : $\Rightarrow \mathrm{b}-\mathrm{a}$ is divisible by 5

$\Rightarrow \mathrm{b}-\mathrm{a} \in \mathrm{R}$

$\therefore \mathrm{R}$ is symmetric

Transitive : $(a, b) \in R$ and $(b, c) \in R$

$\Rightarrow a-b$ and $b-c$ are both divisible by 5

$\Rightarrow a-b+b-c$ is divisible by 5

$\Rightarrow \mathrm{a}-\mathrm{c}$ is divisible by 5

$\Rightarrow(a, c) \in R$

$\therefore \mathrm{R}$ is transitive

Since $R$ is reflexive, symmetric and transitive.

Hence, $R$ is an equivalence relation.