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.”

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