等价类

在数学中，给定一个集合 X 和在 X 上的一个等价关系 ~，则 X 中的一个元素 a 的等价类是在 X 中等价于 a 的所有元素的子集:

[a] = { x ∈ X | x ~ a }

等价类的概念有助于从已经构造了的集合构造集合。在 X 中的给定等价关系 ~ 的所有等价类的集合表示为 X / ~ 并叫做 X 除以 ~ 的商集。这种运算可以(实际上非常不正式的)被认为是输入集合除以等价关系的活动，所以名字“商”和这种记法都是模仿的除法。商集类似于除法的一个方面是如果 X 是有限的并且等价类都是等势的，则 X/~ 的序是 X 的序除以一个等价类的序的商。等价点商集要被认为是带有所有等价点都识别出来的集合 X。

对于任何等价关系，都有从 X 到 X/~ 的一个规范投影映射 π，给出为 π(x) = [x]。这个映射总是满射的。在 X 有某种额外结构的情况下，考虑保持这个结构的等价关系。接着称这个结构是良好定义的，而商集在自然方式下继承了这个结构而成为同一个范畴的对象；从 a 到 [a] 的映射则是在这个范畴内的满态射。参见同余关系。

例子

• If X is the set of all cars, and ~ is the equivalence relation "has the same color as", 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 set Z 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. Under this relation [7] [9] and [1] all represent the same element of Z / ~.
• The rational numbers can be constructed as the set of equivalence classes of ordered pairs of integers (a,b) with b not zero, where the equivalence relation is defined by

(a,b) ~ (c,d) if and only if ad = bc.

Here the equivalence class of the pair (a,b) can be identified with rational number a/b.

• Any function f : X → Y defines an equivalence relation on X by x₁ ~ x₂ if and only if f(x₁) = f(x₂). The equivalence class of x is the set of all elements in X which get mapped to f(x), i.e. the class [x] is the inverse image of f(x). This equivalence relation is known as the kernel of f.
• Given a group G and a subgroup H, we can define an equivalence relation on G by x ~ y if and only if xy ^-1 ∈ H. The equivalence classes are known as right cosets of H in G; one of them is H itself. They all have the same number of elements (or cardinality in the case of an infinite H). 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 homotopy class of a continuous map f is the equivalence class of all maps homotopic to f.
• In natural language processing, an equivalence class is a set of all references to a single person, place, thing, or event, either real or conceptual. For example, in the sentence "GE shareholders will vote for a successor to the company's outgoing CEO Jack Welch", GE and the company are synonymous, and thus constitute one equivalence class. There are separate equivalence classes for GE shareholders and Jack Welch.

性质

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].

If ~ is an equivalence relation on X, and P(x) is a property of elements of x, such that whenever x ~ y, P(x) is true if P(y) is true, then the property P is said to be well-defined or a class invariant under the relation ~. A frequent particular case occurs when f is a function from X to another set Y; if x₁ ~ x₂ implies f(x₁) = f(x₂) then f is said to be a class invariant under ~, or simply invariant under ~. This occurs, e.g. in the character theory of finite groups. The latter case with the function f can be expressed by a commutative triangle. See also invariant.

参见

• 等价关系