In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent.
Formally, given a set S and an equivalence relation ~ on S, the equivalence class of an element a in S, denoted by
[
a
]
{\displaystyle [a]}
, is the set
{
x
∈
S
∣
x
∼
a
}
{\displaystyle \{x\in S\mid x\sim a\}}
of elements which are equivalent to a. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S. This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S by ~, and is denoted by S / ~.
When the set S has some structure (such as a group operation or a topology) and the equivalence relation ~ 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.