In set theory, an **equivalence class** is a set of all objects that are related to a specific object by some equivalence relation.

For example, if *R* is an equivalence relation on a set *S*, and *a* ∈ *S*, then the equivalence class [*a*] is the set of all elements of *S* that are *R*-related to *a*.

[*a*] = { *x*|*Rxa* }

The notation for equivalence classes varies among logic texts. On this site, the notation [*a*] is used to denote an equivalence class of the element *a*. If there are multiple relations being considered, a subscript of the specific relation may be added: [*a*]_{R}. Some texts will place a line over the element name instead, using the notation *a* to indicate an equivalence class.

Suppose that *S* is a set and *R* is an equivalence class on that set, such that:

*S*= { a, b, c, d, e }*R*= { <a, a>, <b, b>, <c, c>, <d, d>, <e, e>, <b, c>, <c, b>, <b, e>, <c, e> }

Then the following are true:

- [
*a*] = { a } - [
*b*] = { b, c } - [
*c*] = { b, c } - [
*e*] = { b, c, e }

(1) is true because the only element *R*-related to *a* (that is, the only *x* such that <*x*, *a*> ∈ *R*) is *a* itself. (2) and (3) are true because *b* and *c* are both *R*-related to themselves, and to each other, but nothing else is related to them. Similarly, in (4), *b* and *c* are both *R*-related to *e*, and *e* is *R*-related to itself.

With an equivalence relation on a set *S*, for every *x* ∈ *S*, *x* ∈ [*x*]. This is because equivalence relations are reflexive — that is, every element is related to itself, and is therefore a member of its own equivalence class.