What is the complete set of representatives of an equivalence class?

Question

I have been researching the topic, but I can't find anything that explains specifically and in detail what it is. I just find a bunch of exercises about the topic.

Answer 1

When you have an equivalence relation defined on a set $S$, you partition the set into equivalence classes.

A representative of an equivalence class is any element of $S$ which belongs to that equivalence class. A complete set of representatives $R$ is a list of elements of $S$ such that you have a representative for each class. That is to say that any element of $S$ will be equivalent to exactly one element of $R$.

For example, if we take $S=\mathbb{Z}$ and define our equivalence as $a\sim b$ if $3|(a-b)$. This is the same as saying $a\sim b$ if $a\equiv b \mod{3}$. We can then take $R=\{0,1,2\}$ as given any integer $x$, either $x=3k$, $x=3k+1$ or $x=3k+2$ for some integer $k$ and hence $x$ will lie in the equivalence class of $0,1$ or $2$ respectively. Furthermore, these classes are all distinct so we have a complete set of representatives.

Finally, I just want to say that the set of representatives is by no means unique since we could have also taken $R$ to be $\{1,2,3\}$ or even $\{-1,6,52\}$.