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Let $\mathrm{n}$ be a fixed positive integer. Define a relation $\mathrm{R}$ in $\mathrm{Z}$ as follows $\forall \mathrm{a}, \mathrm{b} \in \mathrm{Z}$, aRb if and only if $\mathrm{a}-\mathrm{b}$ is divisible by $\mathrm{n}$. Show that $\mathrm{R}$ is an equivalence relation.
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Given that, $\forall \mathrm{a}, \mathrm{b} \in \mathrm{Z}$, aRb if and only if $\mathrm{a}-\mathrm{b}$ is divisible by $n$.
Now,
I. Reflexive : $\mathrm{a} \mathrm{R} \mathrm{a} \Rightarrow(\mathrm{a}-\mathrm{a})$ is divisible by $\mathrm{n}$, which is true for any integer a as ' $O$ ' is divisible by $n$.
Hence, $R$ is reflexive.
II. Symmetric : $\mathrm{a}$ R b
$\Rightarrow \quad a-b$ is divisible by $n$.
$\Rightarrow-b+a$ is divisible by $n$.
$\Rightarrow-(b-a)$ is divisible by $n$.
$\Rightarrow \quad(b-a)$ is divisible by $n$.
$\Rightarrow \mathrm{bRa}$
Hence, $R$ is symmetric.
III. Transitive : Let aRb and bRc $\Rightarrow \quad(\mathrm{a}-\mathrm{b})$ is divisible by $\mathrm{n}$ and $(\mathrm{b}-\mathrm{c})$ is divisible by $n$
$\Rightarrow \quad(\mathrm{a}-\mathrm{b})+(\mathrm{b}-\mathrm{c})$ is divisibly by $\mathrm{n}$
$\Rightarrow \quad(\mathrm{a}-\mathrm{c})$ is divisible by $\mathrm{n}$
$\Rightarrow \quad \mathrm{aRc}$
Hence, $R$ is transitive
So, $\mathrm{R}$ is an equivalence relation.
Now,
I. Reflexive : $\mathrm{a} \mathrm{R} \mathrm{a} \Rightarrow(\mathrm{a}-\mathrm{a})$ is divisible by $\mathrm{n}$, which is true for any integer a as ' $O$ ' is divisible by $n$.
Hence, $R$ is reflexive.
II. Symmetric : $\mathrm{a}$ R b
$\Rightarrow \quad a-b$ is divisible by $n$.
$\Rightarrow-b+a$ is divisible by $n$.
$\Rightarrow-(b-a)$ is divisible by $n$.
$\Rightarrow \quad(b-a)$ is divisible by $n$.
$\Rightarrow \mathrm{bRa}$
Hence, $R$ is symmetric.
III. Transitive : Let aRb and bRc $\Rightarrow \quad(\mathrm{a}-\mathrm{b})$ is divisible by $\mathrm{n}$ and $(\mathrm{b}-\mathrm{c})$ is divisible by $n$
$\Rightarrow \quad(\mathrm{a}-\mathrm{b})+(\mathrm{b}-\mathrm{c})$ is divisibly by $\mathrm{n}$
$\Rightarrow \quad(\mathrm{a}-\mathrm{c})$ is divisible by $\mathrm{n}$
$\Rightarrow \quad \mathrm{aRc}$
Hence, $R$ is transitive
So, $\mathrm{R}$ is an equivalence relation.
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