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Let $R$ be a relation on two countable sets $A$ and $B$, where $R\subset A\times B$, with the following properties:

  1. $\forall a\in A$ the set $\{b\in B: (a,b)\in R\}$ is finite.
  2. For any finite set $A_0\subset A$: $$|\{b\in B : \exists a_0\in A_0, \text{such that} \ \ (a_0,b)\in R \}|\leq n|A_0|$$ where $n\in\mathbb{N}$.

I need to show then, that there exist $n$ disjoint sets, $B_1,B_2,\dots, B_n$, where $B_i\subset B\ \ \ \ \ \ \forall \ \ 1 \leq i\leq n$, and there exists $n$ one to one and onto functions $f_1,f_2,\dots, f_n$ such that $f_i:B_i\to A \ \ \ \ \ \ \ \forall \ \ 1 \leq i\leq n $?

Thank you for your help.


marked as duplicate by Asaf Karagila, Cameron Buie, Arthur, mrs, Omnomnomnom Sep 3 '13 at 14:51

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  • $\begingroup$ There was no answer there and many wrote that it was very unclear, so i hope i made it more clear... but i'm stuck... any ideas? $\endgroup$ – Daniella Sep 3 '13 at 14:44
  • $\begingroup$ Don't post duplicate questions. $\endgroup$ – Asaf Karagila Sep 3 '13 at 14:49