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Quotient map

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Quotient map

This article is about equivalency in mathematics; for equivalency in music see equivalence class (music).

In mathematics, given a set Template:Mvar and an equivalence relation ~ on Template:Mvar, the equivalence class of an element Template:Mvar in Template:Mvar is the subset of all elements in Template:Mvar which are equivalent to Template:Mvar. It follows from the definition of the equivalence relations that the equivalence classes form a . The quotient set of Template:Mvar by ~ is the set of the equivalence classes. It is denoted as X / ~.

When Template:Mvar is equipped with some structure, and the equivalence relation is defined in relation with this structure, the quotient set often inherits some related structure. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogenous spaces, quotient rings, quotient monoids, and the quotient category.

Notation and formal definition

An equivalence relation is a binary relation ~ satisfying three properties:

  • For every element Template:Mvar in Template:Mvar, a ~ a (reflexivity),
  • For every two elements Template:Mvar and Template:Mvar in Template:Mvar, if a ~ b, then b ~ a (symmetry)
  • For every three elements Template:Mvar, Template:Mvar, and Template:Mvar in Template:Mvar, if a ~ b and b ~ c, then a ~ c (transitivity).

The equivalence class of an element Template:Mvar is denoted [a] and may be defined as the set

[a] = \{ x \in X \mid a \sim x \}

of elements that are related to Template:Mvar by ~. The alternative notation [a]R can be used to denote the equivalence class of the element Template:Mvar specifically with respect to the equivalence relation Template:Mvar. This is said to be the Template:Mvar-equivalence class of Template:Mvar.

The set of all equivalence classes in Template:Mvar given an equivalence relation ~ is denoted as X/~ and called the quotient set of Template:Mvar by ~. Each equivalence relation has a canonical projection map that maps each element to its equivalence class, the surjective function π from Template:Mvar to X/~ given by π(x) = [x].

Analogy with division

This operation can be thought of as the act of dividing the input set by the equivalence relation, hence both the name "quotient", and the notation, which are both reminiscent of division. One way in which the quotient set resembles division is that if Template:Mvar is finite and the equivalence classes are all equinumerous, then the number of equivalence classes in X/~ can be calculated by dividing the number of elements in Template:Mvar by the number of elements in each equivalence class. The quotient set X/~ may be thought of as the set Template:Mvar with all the equivalent points identified.

Examples

  • If Template:Mvar 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 (cardinality of X/~ would be the number of all car colors)
  • Consider the modulo 2 equivalence relation on the set Z of integers: x ~ y if and only if their difference xy is an even number. This relation gives rise to exactly two equivalence classes: one class consisting of all even numbers, and the other consisting of all odd numbers. Under this relation [7], [9], and [1] all represent the same element of Z/~.
  • Let Template:Mvar be the set of ordered pairs of integers (a,b) with Template:Mvar not zero, and define an equivalence relation ~ on Template:Mvar according to which (a,b) ~ (c,d) if and only if ad = bc. Then the equivalence class of the pair (a,b) can be identified with the rational number a/b, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers. The same construction can be generalized to the field of fractions of any integral domain.

Properties

Every element Template:Mvar of Template:Mvar is a member of the equivalence class [x]. Every two equivalence classes [x] and [y] are either equal or disjoint. Therefore, the set of all equivalence classes of Template:Mvar forms a partition of Template:Mvar: every element of Template:Mvar belongs to one and only one equivalence class. Conversely every partition of Template:Mvar comes from an equivalence relation in this way, according to which x ~ y if and only if Template:Mvar and Template:Mvar belong to the same set of the partition.

It follows from the properties of an equivalence relation that

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

In other words, if ~ is an equivalence relation on a set X, and Template:Mvar and Template:Mvar are two elements of Template:Mvar, then these statements are equivalent:

  • x \sim y
  • [x] = [y]
  • [x] \cap [y] \ne \emptyset.

Invariants

If ~ is an equivalence relation on Template:Mvar, and P(x) is a property of elements of Template:Mvar such that whenever x ~ y, P(x) is true if P(y) is true, then the property Template:Mvar is said to be an invariant of ~, or well-defined under the relation ~.

A frequent particular case occurs when Template:Mvar is a function from Template:Mvar to another set Template:Mvar; if f(x1) = f(x2) whenever x1 ~ x2, then Template:Mvar is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in the character theory of finite groups. Some authors use "compatible with ~" or just "respects ~" instead of "invariant under ~".

Any function f : XY itself defines an equivalence relation on Template:Mvar according to which x1 ~ x2 if and only if f(x1) = f(x2). The equivalence class of Template:Mvar is the set of all elements in Template:Mvar 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 Template:Mvar.

More generally, a function may map equivalent arguments (under an equivalence relation ~X on Template:Mvar) to equivalent values (under an equivalence relation ~Y on Template:Mvar). Such a function is known as a morphism from ~X to ~Y.

See also

nl:Equivalentierelatie#Equivalentieklasse

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