# Equivalence class

In mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation) defined on them, 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 denoted by [1] is the set[2]

of elements which are equivalent to It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of This partition—the set of 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.