Equivalence relation/N times N/Jumps (2,0) and (3,3)/Example

Visualization of the example. The equivalence classes are colored differently.

We consider the lattice product set ${}M=\mathbb {N} \times \mathbb {N}$. We fix the jumps (think about a Meadow jumping mouse that can perform these jumps and no other jumps)

$\pm (2,0)\,\,{\text{ and }}\,\,\pm (3,3).$

We say that two points ${}P=(a,b),\,Q=(c,d)\in M$ are equivalent if it is possible, starting in ${}P$, to reach the point ${}Q$ with a sequence of such jumps. This defines an equivalence relation (the main reason is that for every jump also its negative jump is allowed). Typical questions are: How can we characterize equivalent points, how can we decide whether two points are equivalent or not?