First off, I am quite open to changing the name of the question if anyone has suggestions, so that it might be more accessible and helpful to future mathonaughts.
I need to describe partitions for given equivalence relations.
The following 2 are straight forward but I feel are important to ensure I have a firm grasp of what's meant to be done.
For $x,y \in \mathbb R \ xRy \quad iff \quad x^2=y^2$
$\mathscr P = \{\overline n \in \mathbb R : y=x\}$
For $(x,y)R(u,v) \quad iff \quad x+v=y+u$
$\mathscr P = \{ \overline {(x,y)} \in \mathbb R : u-v = x-y => u=x \land v=y \}$
Am I right so far? The next thing has me somewhat stumped
For $n,m\in \mathbb Z$ nRm iff n and m have he same 10 digits. I've devised an answer as shown below, but I am not certain it is rigorous enough and would greatly appreciate any feedback you are willing to share.
Let $N_n = \{k_i: i \in \mathbb N, 0 < i \le 10, k_i = \frac {(n \pmod {10^i}) \ - (n\pmod {10^{i-1}})} {10^{i-1}}, \ 10^9 \le n < 10^{10} \}$. => This gives a set for all the digits in the number n.
Let $M_m$ = Same thing as above except for number m
Therefore $\mathscr P = \{ \overline n : 10^9 \le n < 10^{10}, \ 10^9 \le m < 10^{10}, \ s.t. M_m = N_n \}$ => if I'm not mistaken on notation, $\overline n$ would be the set of all numbers m which contain the same digits as n (albeit in a different order). My one concern is that some sets of $\mathscr P$ might have overlap, but and there is a major place where I could use input... If you hypothetically had $n_1 = 2531 \quad and \quad n_2 = 5321$ but if I'm not mistaken, in the partition, those would wind up being considered the same subset. If you could shed light and give me an idea of how close I am, it would be greatly appreciated.
The second one should be something like the collection of pairs with difference $x - y = a$ for all $a$, borrowing the notation of drhab:
$\mathscr P=\{\{\{x, x+a\}\mid x\in\mathbb R\}\mid a\in\mathbb R\}$
I'm not sure if it is easy to write an easy non-verbal notation for the third case if it is just homework, and it needs to be able to handle leading $0$'s.