Equivalence class

In mathematics, when the elements of some set ${\displaystyle S}$ have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set ${\displaystyle S}$ into equivalence classes. These equivalence classes are constructed so that elements ${\displaystyle a}$ and ${\displaystyle b}$ belong to the same equivalence class if, and only if, they are equivalent.

Formally, given a set ${\displaystyle S}$ and an equivalence relation ${\displaystyle \,\sim \,}$ on ${\displaystyle S,}$ the equivalence class of an element ${\displaystyle a}$ in ${\displaystyle S,}$ denoted by ${\displaystyle [a],}$[1] is the set[2]

${\displaystyle \{x\in S:x\sim a\}}$

of elements which are equivalent to ${\displaystyle a.}$ It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of ${\displaystyle S.}$ This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of ${\displaystyle S}$ by ${\displaystyle \,\sim \,,}$ and is denoted by ${\displaystyle S/\sim .}$

When the set ${\displaystyle S}$ has some structure (such as a group operation or a topology) and the equivalence relation ${\displaystyle \,\sim \,}$ is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.