# Definition Given a [[set]] $X$ and an [[Equivalence Relation]] $\sim$, then we have a partition of $X$ into *mutually disjoint* [[subset|subsets]] called *equivalence classes*. A class is denoted by $[a]$, and is composed of all elements in $X$ such that $x \sim a$, i.e., $ [a] = \left\{x \in X | x \sim a\right\} $ Note that $[a]$ is never empty since $a \sim a$ by the reflectivity property of [[Equivalence Relation|equivalence relations]]. By mutually disjoint we mean that either $[a] = [b]$ or $[a] \cap [b] = \emptyset$. The element $a$, or any element of $[a]$, is called a *representative* of the class. # Examples [[Conjugacy Class|Conjugacy classes]] are an example of an equivalence class where the set $X$ is actually a group.