equivalence class

In mathematics, if a set S has an equivalence relation defined over it, then an equivalence class for an element a in S is a subset of S that contains all of the elements that are equivalent to a. Any two elements belong to the same equivalence class if and only if they are equivalent. The equivalence class of an element a is denoted [a].

An equivalence relation satisfies the properties of symmetry, reflexivity, and transitivity.

For equivalent elements a, b, c ∈ S:

• a ~ a (Reflexivity)
• if a ~ b if and only if b ~ a (Symmetry) 
• if a ~ b and b ~ c, then a ~ c (Transitivity)

The equivalence class of an element a in set S is formally defined as:

[a] = {x ∈ S | a ~ x}

To explicitly indicate which equivalence relation defines the class, the equivalence class for a can alternatively be denoted [a]_R.

The set of all equivalence classes in set S with respect to an equivalence relation R is denoted S/R. This is pronounced “S modulo R.”