Equivalence class
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 is in [a] and that any two equivalence classes are either equal or disjoint. It follows that the set of all equivalence classes of X forms a partition of X: every element of X belongs to one and only one equivalence class. Conversely every partition of X also defines an equivalence relation over X.
It also follows from the properties of an equivalence relation that
- a ~ b if and only if [a] = [b].
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 of X by ~. In cases where X has some additional structure preserved under ~, the quotient becomes an object of the same type in a natural fashion; the map that sends a to [a] is then a homomorphism.
Examples:
- If X s (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
- Every group can be partitioned into equivalence classes called conjugacy classes.
- The homotopy class of a continuous map f is the equivalence class of all maps homotopic to f.
See also:
-- rational numbers -- multiplicatively closed set -- real numbers -- homotopy theory -- up to