# Thread: Equivalence Class Representatives

1. ## Equivalence Class Representatives

Okay, so this question seemed easy to me earlier, but now i'm getting paranoid about it.

Problem:

Define a relation $\sim$ on the set $\mathbb{N}$ of natural numbers by

$a \sim b$ iff $a = b \cdot 10^k$ for some $k \in \mathbb{Z}$

(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 $S$ be a set which we have defined an equivalence relation on. The set of equivalence class representatives is the subset of $S$ containing precisely one element from each equivalence class.

so, for example. lets say i have the set $\mathbb{Z}$, and my relation on this set is

for $a,b \in \mathbb{Z}$, $a \sim b$ iff $a$ and $b$ have the same parity.

clearly this partitions the integers into the set of even integers and the set of odd integers. so if $e$ is any even integer, its equivalence class is $[e] = \{ 0,~ \pm 2,~ \pm 4, \cdots \}$ and if $o$ is any odd integer, its equivalence class is given by $[o] = \{ \pm 1,~ \pm 3,~ \pm 5, \cdots \}$

for the set of equivalence class representatives, we would simply choose any one even number and any one odd number, so $\{ 0,1 \}$ suffices.

What I have tried:

I realize for any $a \in \mathbb{N}$, $[a] = \{a \cdot 10^k | k \in \mathbb{N} \}$

but it gets more complicated. if $a$ 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 $\mathbb{N}$, 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?

2. i think $\{ n ~|~ n \in \mathbb{N} \mbox{ and } n \mbox{ is not a multiple of } 10 \}$ works.

does it?