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: 
What do you know about monotonous functions? Is this function (strictly) monotonous, and if yes, does this tell you something about injectivity?
**Hint** Let $a=x^2 \geq 0$ and $b=y^2 \geq 0$. The equation reduces to $$2a+a^2=2b+b^2$$