Equivalence Class Question

Question

On the set $N\times N$ define $(m,n)\simeq(k,l)$ if $m+l=n+k$.

Draw a sketch of $N\times N$ that shows several equivalence classes. (hint: sketch points on graph paper).

I'm not quite sure how to sketch the equivalence classes. I know that the following is in the relation, but do I just graph the points? Also, how do I find out how many equivalence classes there are and how do I distinguish them? Are there $4$ different equivalence classes ($1$ for $m$, $1$ for $n$, $1$ for $k$, and $1$ for $l$)? I know that if my relation were $\mod p$, there would be $p$ equivalence classes.

Answer 1

Points on the same line belong to the same equivalence class and each different line represent a different equivalence class.

Answer 2

You can start out by simply using the points on a coordinate grid for $\mathbb{N}\times\mathbb{N}$. Every point $(m,n)$ represents the pair $(m,n)\in\mathbb{N}^2$. So far, so obvious. For the actual equivalence classes, you could ask yourself: what is it that characterizes every equivalence class? For instance, take the point $(3,2)$. We have $3+2=5$. Where do you find other points with $m+n=5?$ Try points nearby first and try to find some kind of pattern.

A different approach would be the following: I am going to assume you are familiar with the normal vector form of a line in $\mathbb{R}^2$.

Every point in your class fulfills $m+n=c$. Note that you could also write this as $(m,n)\cdot (1,1)=c$. Can you find a geometric interpretation of that? I have already mentioned normal vectors, so maybe that will help you to find an actual proof.