Concept# Equivalence class

Summary

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), 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.
Formally, given a set S and an equivalence relation ,\sim, on S, the 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 /{\s

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