# Let n be a fixed positive integer. Define a relation R on Z as follows:

Question:

Let $n$ be a fixed positive integer. Define a relation $R$ on $Z$ as follows:

$(a, b) \in R \Leftrightarrow a-b$ is divisible by $n$.

Show that R is an equivalence relation on Z.

Solution:

We observe the following properties of R. Then,
Reflexivity:

Let $a \in N$

Here,

$a-a=0=0 \times n$

$\Rightarrow a-a$ is divisible by $n$

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

$\Rightarrow(a, a) \in R$ for all $a \in Z$

So, $R$ is reflexive on $Z$.

Symmetry

Let $(a, b) \in R$

Here,

$a-b$ is divisible by $n$

$\Rightarrow a-b=n p$ for some $p \in Z$

$\Rightarrow b-a=n(-p)$

$\Rightarrow b-a$ is divisible by $n$               $[p \in Z \Rightarrow-p \in Z]$

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

So, $R$ is symmetric on $Z$.

Transitivity:

Let $(a, b)$ and $(b, c) \in R$

Here, $a-b$ is divisible by $n$ and $b-c$ is divisible by $n$.

$\Rightarrow a-b=n p$ for some $p \in Z$

and $b-c=n q$ for some $q \in Z$

Adding the above two, we get

$a-b+b-c=n p+n q$

$\Rightarrow a-c=n(p+q)$

Here, $p+q \in Z$

$\Rightarrow(a, c) \in R$ for all $a, c \in Z$

So, $R$ is transitive on $Z$.

Hence, R is an equivalence relation on Z.