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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:- [a] = { x in X | x ~ a }
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 ~. 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.
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 an epimorphism. See congruence relation.
1 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 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.
- The rational numbers can be constructed as the set of equivalence classes of 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. Is this the origin of the term quotient set?
- Any function f : X → Y defines an equivalence relation on X by x ~ y iff f(x) = f(y). 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 groupIn mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste G G and a subgroupGroup theory In mathematics, given a group G under a binary operation , we say that some subset H of G is a subgroup of G if H also forms a group under the operation . More precisely, H is a subgroup of G if the restriction of to H is a group operation on H, we can define an equivalence relation on G by x ~ y iff xy -1 ∈ H. The equivalence classes are known as right cosetIn mathematics, if G is a group, H a subgroup of G and g an element of G then gH { gh : h an element of H } is a left coset of H in G and Hg { hg : h an element of H } is a right coset of H in G''. Some properties We have gH H if and only if g is an elemes of H in G. If H is a normal subgroupIn mathematics, a normal subgroup ''N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G the element g-1ng is still in N''. The statement N is a normal subgroup of G is written: :. Another way to put t, then the set of all cosets is itself a group in a natural way.
- Every group can be partitioned into equivalence classes called conjugacy classIn mathematics, the elements of any group may be partitioned into conjugacy classes members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a group's structure. Definition Suppose G is aes.
- The homotopyAlgebraic topology Homotopy theory In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. class of a continuousIn mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. If small changes in the input can produce a broken jump in the changes of the output, the function is said to be map f is the equivalence class of all maps homotopic to f.
- In natural language processingNatural Language Processing (NLP is a subfield of artificial intelligence and linguistics. It studies the problems inherent in the processing and manipulation of natural language, and, natural language understanding devoted to making computers "understand, 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 synonomous, and thus constitute one equivalence class. There are separate equivalence classes for GE shareholders and Jack Welch.
