finding a general formula for an equivalence class (discrete)

Question

I have the following question -

Consider the following equivalence relation $R$ on $\mathbb{Z} \times \mathbb{Z}$.

$(a,b)R(c,d) \leftrightarrow |a|+|b|=|c|+|d|$.

a.) Write out all elements of $[(1,2)]$.

b.) Find a formula for |[(x,y)]|.

I think I figured out part a, would it just be $[(1,2)]=\{(0,3),(3,0),(2,1),(-2,-1),(-1,-2),(1,2),(0,-3),(-3,0)\}$.

But I'm not sure how I would do part b and find a general formula for the equivalence classes. Suggestions?

Answer 1

$$[(x,y)]=\{(a,b)\in\mathbb{Z}^2\colon(a,b)R(x,y)\}=\{(a,b)\in\mathbb{Z}^2\colon|a|+|b|=|x|+|y|\}=\{(a,\pm(|x|+|y|-|a|)\colon a\in\mathbb{Z}\}$$