Find the equivalence class on a equivalence relation

Question

$m R n \iff \exists k \in \Bbb Z\ :\ m^2 - n^2 = 2k$

• Determine the equivalence class of $5$

• Determine quotient set $\Bbb Z/R$

How do I do this?

Answer 1

$m$ is in the class of $5$ if and only if

$m^2=25+2k$ for some $k$

For example $7$ is in the class because

$7^2=49=25 +2 \cdot 7$

We can set $m=5+s$ to get that

$m^2=25+s^2+2s=25+2k$

so

$s^2+2s=2k$

if $m$ is odd then $s^2$ is even then you get

$k=\frac{s^2}{2}+s$

so each odd numbers greater of $5$ are in the class of $5$

Answer 2

$m R n\stackrel{\text{(why?)}}\iff 2\mid m-n\lor 2\mid m+n\stackrel{\text{(why?)}}\iff 2\mid m-n$. So the equivalence class of $5$ is the set of odd numbers. And the other class is the set of even numbers.