Equivalence class

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Given a set X and an equivalence relation ~ over X an equivalence class is a subset of X of the form

{ x in X | x ~ a }

where a is an element in X. This equivalence class is usually denoted as [a]. Because of the properties of an equivalence relation it holds that a in [a] and that all equivalence classes will be either equal or disjoint. It follows that the set of all equivalence classes of X will form a partition of X. Conversely every partition of X also defines an equivalence relation over X.

It also follows from the properties of an equivalence relation that

x ~ y if and only if [x] = [y].

The notion of equivalence classes is useful for constructing sets out of already constructed ones. The set of all equivalence classes in X given an equivalence relation ~ is usually denoted as X / ~ and called the quotient set. In cases where X has some additional structure preserved under ~, the quotient naturally becomes an object of the same type; the map that sends x to [x] is then a homomorphism.

Examples:

  • The rational numbers can be constructed as the set of equivalence classes of pairs of integers (a,b) where the equivalence relation is defined by
(a,b) ~ (c,d) if and only if ad = bc.
  • The real numbers can be constructed as the set of equivalence classes on the set of Cauchy sequences of rational numbers, where the equivalence relation is defined by
(xn)n=1...infinity ~ (yn)n=1...infinity if and only if (xn - yn) -> 0 as n -> infinity

See also:

-- rational numbers -- multiplicatively closed set -- real numbers -- homotopy theory --