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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]; it consists of precisely those elements of X which are equivalent to a.

Examples:

  • If X is the set of all cars, and ~ is the equivalence relation of "having the same color", then one particular equivalence class consists of all green cars. X / ~ could be naturally identified with the set of all car colors.
  • Consider the "modulo 2" equivalence relation on the the set of integers: x~y if and only if x-y is even. This relation gives rise to exactly two equivalence classes: [0] consisting of all even numbers, and [1] consisting of all odd numbers.
  • Given a group G and a subgroup H, we can define an equivalence relation on G by x ~ y iff xy -1H. The equivalence classes are known as right cosets of H in G. If H is a normal subgroup, then the set of all cosets is itself a group in a natural way.
  • Every group can be partitioned into equivalence classes called conjugacy classes.
  • 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.

Properties

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.

See also:

-- rational numbers -- multiplicatively closed set[?] -- homotopy theory[?] -- up to



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