In one of my exercises we are shown that the relation "$\sim$" is defined as the following:
$$\langle n,m \rangle \sim \langle k,l \rangle \iff |(n\setminus m)| = |(k\setminus l)|$$
in which $|X|$ symbolizes the number of elements of the set. We need to:
(a) Show that "$\sim$" is a equivalence relation on $\omega\times\omega$.
(b) Find a "representative system" ("system of representatives") of this equivalence relation.
For part (a) it is straight forward, we show that it is reflexive, symmetric and transitive. But for part (b) I am stuck. So far I think I have an understanding of the definition for a representative, in terms of this exercise.(Please correct me if I am wrong):
I understand that if "$\sim$" is an equivalence relation on $\omega\times\omega$, then for $x \in \omega\times\omega$, the set
$[x]_{\sim} = \{ y \in \omega\times\omega : y\sim x\}$ is called an equivalence class of $x$. So every $y \in [x]_{\sim}$ (i.e $y\sim x$) is then called a "representative" of the equivalence class of $x$.
However I am lost on how to apply this definition to the question. Some guidance would be very much appreciated. Thank you in advance.
A representative system is simply some set that contains exactly each element from each equivalence class. There are many different possibilities for such a set; the exercise simply asks you to describe one of those possibilities, of your own choice.
Each equivalence class is described by a natural number, and contains all pairs $\langle n,m\rangle$ such that $n\setminus m$ has that many elements. (Since $n$ is always finite, $n\setminus m$ will always be finite too).
So what you need to do is simply, for each natural number $k$, to decide on one particular pair $\langle n_k,m_k\rangle$ where $n_k\setminus m_k$ has $k$ elements.