Global InformationEquivalence class information
In mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set into equivalence classes. These equivalence classes are constructed so that elements and belong to the same equivalence class if, and only if, they are equivalent.
Formally, given a set and an equivalence relation on the equivalence class of an element in is denoted or, equivalently, to emphasize its equivalence relation The definition of equivalence relations implies that the equivalence classes form a partition of meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space of by and is denoted by
![{\displaystyle [a]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea82bc70a8e322f13a3c4e5b9d5d69e8ef097ad8)
![{\displaystyle [a]_{\sim }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3906ba39d6194f63cd682c68521fe0b61f18f284)
When the set 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.