Let $R$ be an equivalence relation defined on $\Bbb{N}^2$, where $$(x,y)R(w,z)\iff2(x-w)=z-y.$$ Find equivalence classes and quotient set.
First, we can express $2(x-w)=z-y$ as $$2x+y=2w+z.$$ So, for example $[(1,1)]=\{(x,y)\in\Bbb{N}^2\mid2x+y=3\}=\{(1,1)\}$.
Also, $[(1,2)]=\{(x,y)\in\Bbb{N}^2\mid2x+y=4\}=\{(1,2)\}$.
In addition, $[(1,3)]=\{(x,y)\in\Bbb{N}^2\mid2x+y=5\}=\{(1,3),(2,1)\}$.
$[(1,4)]=\{(x,y)\in\Bbb{N}^2\mid2x+y=6\}=\{(1,4),(2,2)\}$.
$[(1,5)]=\{(x,y)\in\Bbb{N}^2\mid2x+y=7\}=\{(1,5),(2,3),(3,1)\}$.
$[(1,6)]=\{(x,y)\in\Bbb{N}^2\mid2x+y=8\}=\{(1,6),(2,4),(3,2)\}$.
We can guess that $$[(a,b)]=\{(x,y)\in\Bbb{N}^2\mid2x+y=2a+b,\;a\in\Bbb{N},b=2-a\}\cup\{(1,1),(1,2)\}.$$ I don't think it has a nice expression. I am not sure how to find the quotient set.
Any help?