I've been looking around and found questions related to deriving partitions from equivalence relations; however I was wondering if there is a method to finding an equivalence relation from a given partition.
For example the partition $\{\{1, ..., 9\},\{10, ..., 99\},\{100, ..., 999\}, ...\}$ of the natural numbers (not counting 0). The best I can come up with (by guessing) for an equivalence relation is $x{R}y$ iff [$x$ and $y$ have the same number of digits], but that doesn't seem very mathematical. Otherwise my second best guess is something to do with logarithms. (Full disclosure, this is a homework problem.)
So I was wondering, is there a method to derive an equivalence relation based on the equivalence classes set by the partition?
The same number of decimal digits sounds good to me.
To make it harder to understand, we could write $x\sim y$ if $\lfloor \log_{10} x\rfloor=\lfloor \log_{10} y\rfloor$.