Equivalence class

In mathematics, when the elements of some set ${\displaystyle S}$ have a notion of equivalence (formalized as an equivalence relation), 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 ${\displaystyle S}$ is denoted ${\displaystyle [a]}$ or, equivalently, ${\displaystyle [a]_{\sim }}$ to emphasize its equivalence relation ${\displaystyle \sim .}$ The definition of equivalence relations implies that the equivalence classes form a partition of ${\displaystyle S,}$ 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 ${\displaystyle S}$ 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.