i think works.
does it?
Okay, so this question seemed easy to me earlier, but now i'm getting paranoid about it.
Problem:
Define a relation on the set of natural numbers by
iff for some
(b) Give a complete set of equivalence class representatives
Things That Might Come In Handy:
I suppose whoever helps me with this will know what an equivalence class is, so i will tell you about equivalence class representatives.
let be a set which we have defined an equivalence relation on. The set of equivalence class representatives is the subset of containing precisely one element from each equivalence class.
so, for example. lets say i have the set , and my relation on this set is
for , iff and have the same parity.
clearly this partitions the integers into the set of even integers and the set of odd integers. so if is any even integer, its equivalence class is and if is any odd integer, its equivalence class is given by
for the set of equivalence class representatives, we would simply choose any one even number and any one odd number, so suffices.
What I have tried:
I realize for any ,
but it gets more complicated. if happens to be a multiple of 10, then for some of the k's in the above set, -k also works, but not all. secondly, what would i choose for the set of representatives. i was thinking about choosing , but this will cause repeats, as for example, 1, 10, 100, 1000 etc will be in there when all are in [1] and [10] and [100] etc. how would i define my set here?