Equivalence class

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. Congruence is an example of an equivalence relation. The leftmost two triangles are congruent, while the third and fourth triangles are not congruent to any other triangle shown here. Thus, the first two triangles are in the same equivalence class, while the third and fourth triangles are each in their own equivalence class.

Formally, given a set $S$ and an equivalence relation $\,\sim \,$ on $S,$ the equivalence class of an element $a$ in $S,$ denoted by $[a],$ is the set

$\{x\in S:x\sim a\}$ 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 $\,\sim \,,$ and is denoted by $S/\sim .$ When the set $S$ has some structure (such as a group operation or a topology) and the equivalence relation $\,\sim \,$ 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.