Quotient Space

up:: Equivalence Relation

Given a set $X$ and an Equivalence Relation $\sim$, the set of the Equivalence Classes

$$X/\sim =\{[x]\mid x\in X\}$$

is called its quotient space.

Note that each equivalence class can be seen as different elements from each other, since All different equivalence classes are disjoint.

Examples of quotient spaces

${\mathbb{Z}}_2=\{0,1\}$ can be seen as $\mathbb{Z}/\sim$ with

$$m\sim n⟺m\mathrm{mod}2=n\mathrm{mod}2$$

Thus, $0:=[0]=\{n\in \mathbb{Z}\mid n\mathrm{mod}2=0\}$ and $1:=[1]=\{n\in \mathbb{Z}\mid n\mathrm{mod}2=1\}$.