Finding complete set of representatives

Question

For $x, y \in \mathbb N \times \mathbb N$ with $x=(a,b)$ and $y=(c,d)$. Define $ x \sim y$ if $a+d=b+c$ and show that $\{(n,0) : n\in \mathbb N \} \cup \{(0,m) : m \in \mathbb N, \ m\neq 0\}$ is a complete set of representatives for $(\mathbb N \times \mathbb N)/\sim$ (We assume $0 \in \mathbb N$)

I have written the equivalence class of an element $(a,b)$ with $[(a,b)]=\{(n,m) \in \mathbb N \times \mathbb N : n+b=m+a\}$

I can see that if we consider set of real numbers, an equivalence class is a line having slope 1 and $y-axis$ is a complete set of representatives. However I could not see how can I show the given set is a complete set of representatives.

In addition, what if we take equivalence relation as $|a|+|b|=|c|+|d|$ on $\mathbb R^2$? I am in we get the lines having slope 1 in first quadrant and third quadrant and having slope -1 in second quadrant and fourth quadrant. Is it true? How can I describe geometrically equivalence classes better and what is the complete set of representatives for this?

I appreciate any help.

Thank you

Answer 1

If $a+d=b+c$ we also have $a-b=c-d$ so that the equivalence class of $(a,b)$ is characterized by the number $a-b$. Do you see how to do it now?

In the second part of the question, the equivalence class of $(a,b)$ is characterized by $|a|+|b|$, so $\{(0,a)\mid a\geq0\}$ is a complete set of representatives.

Answer 2

$$(a,b) \sim\begin{cases}(a-b,0)&\text{ if }\;a\ge b,\\ (0,b-a) &\text{ if }\;a<b.\end{cases}$$ Note this is the starting point of the construction of $\mathbf Z$ from $\mathbf N$: $$\mathbf Z\overset{\text{def}}{=} (\mathbf N\times \mathbf N)/\sim$$

Note: This construction is quite similar to the construction of the field of fractions of $\mathbf Z$:

On $\mathbf Z\times\mathbf Z^*$, one defines the equivalence relation: $$(p,q)\sim(r,s)\quad (q,s\ne 0)\iff ps=qr$$ and $\:\mathbf Q\overset{\text{def}}{=} (\mathbf Z\times \mathbf Z^*)/\sim$.