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
- The [homotopy class]? of a [continous map]? f is the equivalence class of all maps homotopic? to f.
See also:
--
rational numbers -- [multiplicatively closed set]
? --
real numbers -- [homotopy theory]
? --