Equivalence class

For an equivalence relation ~ on A, the class [a] is {x∈A : x~a}

Equivalence class

Let $\sim$ be an equivalence relation on a set $A$ , and fix $a\in A$ . The equivalence class of $a$ is the subset

[a]=\{x\in A : x\sim a\}\subseteq A,

so $[a]$ is a subset of $A$ .

Distinct equivalence classes are disjoint, and the set of all classes forms a partition of $A$ .

Examples:

If $\sim$ is equality, then $[a]=\{a\}$ .
For congruence modulo $3$ on $\mathbb{Z}$ , the class of $1$ consists of all integers of the form $3k+1$ .
For $x\sim y$ iff $x-y\in\mathbb{Z}$ on $\mathbb{R}$ , each class is a translate of $\mathbb{Z}$ inside $\mathbb{R}$ .