Equivalence class

Let $\sim$ be an equivalence relation on a set $X$. For $x\in X$, the equivalence class of $x$ is

Equivalence classes are the “blocks” determined by $\sim$; they are pairwise disjoint and their union is all of $X$. Quotient constructions replace elements by their classes.

Examples:

• If $\sim$ is congruence mod $3$ on $\mathbb{Z}$, then $[1]=\{\dots,-5,-2,1,4,7,\dots\}$.
• If $\sim$ is equality on $X$, then $[x]=\{x\}$ for each $x\in X$.
• On $\mathbb{R}$ with $x\sim y \iff x-y\in\mathbb{Q}$, the class of $0$ is $[0]=\mathbb{Q}$, and the class of $\sqrt{2}$ is $\sqrt{2}+\mathbb{Q}$.