Can you possibility give us a reference on this?
Is there a textbook or lecture notes?
Frankly, I do not follow the construction.
I have given thought to your posting. I had fallen to the sin of familiarity.
I know a similar approach: (a,b)~(c,d) iff (a-b)=(c-d).
But of course that is equivalent to the relation you posted.
Have you shown that that is an equivalence relation on NxN?
The equivalence class for [3,1] contains (4,2),(5,3),(9,7) etc.
The equivalence class for [1,3] contains (2,4),(3,5),(7,9) etc.
The equivalence class for [0,0] contains (2,2),(3,3),(7,7) etc.
Thus we can identify n with [a,b] iff a-b=n.
Thus we can identify -n with [b,a] iff a-b=n.
Thus we can identify with [0,0].
We define as .
Can you carry on from here?