Can you possibility give us a reference on this?

Is there a textbook or lecture notes?

Frankly, I do not follow the construction.

PS

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?