In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:

The notion of equivalence classes is useful for constructing sets out of already constructed ones. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" 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 ~. This operation can be thought of (very informally indeed) 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 X is finite and the equivalence classes are all equinumerous, then the order of X/~ is the quotient of the order of X by the order of an equivalence class. In the field of Mathematics, two sets A and B are equinumerous if they have the same Cardinality, i The quotient set is to be thought of as the set X with all the equivalent points identified.

For any equivalence relation, there is a canonical projection map π from X to X/~ given by π(x) = [x]. This map is always surjective. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In cases where X has some additional structure, one considers equivalence relations which preserve that structure. Then one says that that structure is well-defined, and the quotient set inherits the structure to become an object of the same category in a natural fashion; the map that sends a to [a] is then an epimorphism in that category. In Mathematics, the term well-defined is used to specify that a certain concept or object (a function, a property, a relation, etc In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics and related technical fields the term map or mapping is often a Synonym for function. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which See congruence relation. See Congruence (geometry for the term as used in elementary geometry

The alternative notation [a]_R can be used to denote that we mean the equivalence class of the element a specifically with respect to the equivalence relation R. This is said to be the R-equivalence class of a.

Examples

• 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. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the parity of an object states whether it is even or odd 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. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions

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 Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function ↔ 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). In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage This equivalence relation is known as the kernel of f. In Mathematics, the kernel of a function f may be taken to be either the Equivalence relation on the function's domain
• 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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of The equivalence classes are known as right cosets of H in G; one of them is H itself. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH They all have the same number of elements (or cardinality in the case of an infinite H). In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness If H is a normal subgroup, then the set of all cosets is itself a group in a natural way. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup.
• Every group can be partitioned into equivalence classes called conjugacy classes. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class
• The homotopy class of a continuous map f is the equivalence class of all maps homotopic to f. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output
• 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. Natural language processing ( NLP) is a subfield of Artificial intelligence and Computational linguistics. 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.

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. In Mathematics, two sets are said to be disjoint if they have no element in common 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. In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " 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 ~. In Mathematics, the term well-defined is used to specify that a certain concept or object (a function, a property, a relation, etc 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. In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance.

Some authors use "compatible with ~" or just "respects ~" instead of "invariant under ~".

See also

• First Isomorphism Theorem
• In music see octave equivalency, transpositional equivalency, inversional equivalency, enharmonic equivalency. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural Music is an Art form in which the medium is Sound organized in Time. In Music, an octave ( is the the use of which is "common in most musical systems In Music transposition refers to the process of moving a collection of notes ( pitches) up or down in pitch by a constant interval. In Music theory, the word inversion has several meanings There are inverted chords, inverted melodies, inverted intervals, and In modern Music and notation, an enharmonic equivalent is a Note ( enharmonic tone) interval ( enharmonic interval) or Musical set theory takes advantage of all of these, to varying degrees, while other theories take more or less advantage of a selection. Musical set theory provides concepts for categorizing musical objects and describing their relationships
• In computing a form of testing is based on equivalence partitions, which are based on equivalence classes. Equivalence partitioning is a Software testing related technique with the goal To reduce the number of test cases to a necessary minimum

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic