Question

Let $n$ be a fixed positive integer. Define a relation $R$ on the set of integers $Z, a R b \Leftrightarrow n \mid(a-b)$. Then prove that $R$ is equivalence relation

Answer 1

$\begin{array}{ll} a R b \Leftrightarrow n \mid(a-b) & a, b \in Z \\n \in I^{+}\end{array}$

(i) $\mathrm{aRa} \Leftrightarrow \mathrm{n} \mid(\mathrm{a}-\mathrm{a}) \Rightarrow$ so $\mathrm{R}$ is reflexive

(ii) $a R a \Leftrightarrow n|(a-b)=n|(b-a) \Rightarrow R$ is symmetric

(iii) $a R b \Leftrightarrow n \mid(a-b)$ and $n \mid(b-c)$

$\Rightarrow n|(a-b)+(b-c) \Leftrightarrow n|(a-c) \Rightarrow R \text { is transitive }$