m is said to be related to n if m and n are integers and m − n is divisible by 13. Does this define an equivalence relation?

Solution:

We observe the following properties of relation R.

Let $R=\{(m, n): m, n \in Z: m-n$ is divisible by 13$\}$

Relexivity : Let $m$ be an arbitrary element of $Z$. Then,

$m \in R$

$\Rightarrow m-m=0=0 \times 13$

$\Rightarrow m-m$ is divisible by 13

$\Rightarrow(m, m)$ is reflexive on $Z$.

Symmetry: Let $(m, n) \in R$. Then,

$m-n$ is divisible by 13

$\Rightarrow m-n=13 p$

Here, $p \in Z$

$\Rightarrow n-m=13(-p)$

Here, $-p \in Z$

$\Rightarrow n-m$ is divisible by 13

$\Rightarrow(n, m) \in R$ for all $m, n \in Z$

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

Transitivity : Let $(m, n)$ and $(n, o) \in R$

$\Rightarrow m-n$ and $n-o$ are divisible by 13

$\Rightarrow m-n=13 p$ and $n-o=13 q$ for some $p, q \in Z$

Adding the above two, we get

$m-n+n-o=13 p+13 q$

$\Rightarrow m-o=13(p+q)$

Here, $p+q \in Z$

$\Rightarrow m-o$ is divisible by 13

$\Rightarrow(m, o) \in R$ for all $m, o \in Z$

Hence, R is an equivalence relation on Z.