![cover image](https://wikiwandv2-19431.kxcdn.com/_next/image?url=https://upload.wikimedia.org/wikipedia/commons/thumb/1/12/Congruent_non-congruent_triangles.svg/640px-Congruent_non-congruent_triangles.svg.png&w=640&q=50)
Equivalence class
Mathematical concept / From Wikipedia, the free encyclopedia
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.
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/1/12/Congruent_non-congruent_triangles.svg/320px-Congruent_non-congruent_triangles.svg.png)
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
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.