Aside on equivalence relations

A relation on a set $ S$ is a subset $ R$ of $ S\times S$. By a slight abuse of notation, let us write, for any $ s,t\in S$, $ sRt$ if and only if $ (s,t)\in R$. An equivalence relation is a relation satisfying (for equivalence relations, we write $ s\sim t$ instead of $ sRt$)

Example 1.7.1  

The equivalence class of $ s\in S$ is

$\displaystyle [s]=\{t\in S\ \vert\ t\sim s\}. $

If $ [s]$ is an equivalence class of $ S$ then we call $ s$ (or any other element of $ [s]$) a representative of $ [s]$ in $ S$.

David Joyner 2007-09-03