Equivalence relation

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This article is about the mathematical concept. Equivalence relation_sentence_0

For the patent doctrine, see Doctrine of equivalents. Equivalence relation_sentence_1

"Equivalency" redirects here. Equivalence relation_sentence_2

For other uses, see Equivalence. Equivalence relation_sentence_3

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. Equivalence relation_sentence_4

The relation "is equal to" is the canonical example of an equivalence relation. Equivalence relation_sentence_5

Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Equivalence relation_sentence_6

Two elements of the given set are equivalent to each other, if and only if they belong to the same equivalence class. Equivalence relation_sentence_7

Notation Equivalence relation_section_0

Definition Equivalence relation_section_1

A binary relation ~ on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. Equivalence relation_sentence_8

That is, for all a, b and c in X: Equivalence relation_sentence_9

Equivalence relation_unordered_list_0

  • a ~ a. (Reflexivity)Equivalence relation_item_0_0
  • a ~ b if and only if b ~ a. (Symmetry)Equivalence relation_item_0_1
  • if a ~ b and b ~ c, then a ~ c. (Transitivity)Equivalence relation_item_0_2

Examples Equivalence relation_section_2

Simple example Equivalence relation_section_3

Equivalence relations Equivalence relation_section_4

The following relations are all equivalence relations: Equivalence relation_sentence_10

Relations that are not equivalences Equivalence relation_section_5

Equivalence relation_unordered_list_1

  • The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 does not imply that 5 ≥ 7. It is, however, a total order.Equivalence relation_item_1_3
  • The relation "has a common factor greater than 1 with" between natural numbers greater than 1, is reflexive and symmetric, but not transitive. For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1.Equivalence relation_item_1_4
  • The empty relation R (defined so that aRb is never true) on a non-empty set X is vacuously symmetric and transitive, but not reflexive. (If X is also empty then R is reflexive.)Equivalence relation_item_1_5
  • The relation "is approximately equal to" between real numbers, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes can accumulate to become a big change. However, if the approximation is defined asymptotically, for example by saying that two functions f and g are approximately equal near some point if the limit of f − g is 0 at that point, then this defines an equivalence relation.Equivalence relation_item_1_6

Connections to other relations Equivalence relation_section_6

Equivalence relation_unordered_list_2

  • A partial order is a relation that is reflexive, antisymmetric, and transitive.Equivalence relation_item_2_7
  • Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted for one another, a facility that is not available for equivalence related variables. The equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class.Equivalence relation_item_2_8
  • A strict partial order is irreflexive, transitive, and asymmetric.Equivalence relation_item_2_9
  • A partial equivalence relation is transitive and symmetric. Such a relation is reflexive if and only if it is serial, that is, if ∀a∃b a ~ b. Therefore, an equivalence relation may be alternatively defined as a symmetric, transitive, and serial relation.Equivalence relation_item_2_10
  • A ternary equivalence relation is a ternary analogue to the usual (binary) equivalence relation.Equivalence relation_item_2_11
  • A reflexive and symmetric relation is a dependency relation (if finite), and a tolerance relation if infinite.Equivalence relation_item_2_12
  • A preorder is reflexive and transitive.Equivalence relation_item_2_13
  • A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraic structure, and which respects the additional structure. In general, congruence relations play the role of kernels of homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many important cases, congruence relations have an alternative representation as substructures of the structure on which they are defined (e.g., the congruence relations on groups correspond to the normal subgroups).Equivalence relation_item_2_14
  • Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in classical mathematics (as opposed to constructive mathematics), since it is equivalent to the law of excluded middle.Equivalence relation_item_2_15

Well-definedness under an equivalence relation Equivalence relation_section_7

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 ~. Equivalence relation_sentence_11

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 morphism for ~, a class invariant under ~, or simply invariant under ~. Equivalence relation_sentence_12

This occurs, e.g. in the character theory of finite groups. Equivalence relation_sentence_13

The latter case with the function f can be expressed by a commutative triangle. Equivalence relation_sentence_14

See also invariant. Equivalence relation_sentence_15

Some authors use "compatible with ~" or just "respects ~" instead of "invariant under ~". Equivalence relation_sentence_16

More generally, a function may map equivalent arguments (under an equivalence relation ~A) to equivalent values (under an equivalence relation ~B). Equivalence relation_sentence_17

Such a function is known as a morphism from ~A to ~B. Equivalence relation_sentence_18

Equivalence class, quotient set, partition Equivalence relation_section_8

Equivalence class Equivalence relation_section_9

Main article: Equivalence class Equivalence relation_sentence_19

Quotient set Equivalence relation_section_10

Main article: Quotient set Equivalence relation_sentence_20

Projection Equivalence relation_section_11

Main article: Projection (relational algebra) Equivalence relation_sentence_21

Equivalence relation_description_list_3

  • Theorem on projections: Let the function f: X → B be such that a ~ b → f(a) = f(b). Then there is a unique function g : X/~ → B, such that f = gπ. If f is a surjection and a ~ b ↔ f(a) = f(b), then g is a bijection.Equivalence relation_item_3_16

Equivalence kernel Equivalence relation_section_12

Partition Equivalence relation_section_13

Main article: Partition of a set Equivalence relation_sentence_22

A partition of X is a set P of nonempty subsets of X, such that every element of X is an element of a single element of P. Each element of P is a cell of the partition. Equivalence relation_sentence_23

Moreover, the elements of P are pairwise disjoint and their union is X. Equivalence relation_sentence_24

Counting partitions Equivalence relation_section_14

Let X be a finite set with n elements. Equivalence relation_sentence_25

Since every equivalence relation over X corresponds to a partition of X, and vice versa, the number of equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number Bn: Equivalence relation_sentence_26

Fundamental theorem of equivalence relations Equivalence relation_section_15

A key result links equivalence relations and partitions: Equivalence relation_sentence_27

Equivalence relation_unordered_list_4

  • An equivalence relation ~ on a set X partitions X.Equivalence relation_item_4_17
  • Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.Equivalence relation_item_4_18

In both cases, the cells of the partition of X are the equivalence classes of X by ~. Equivalence relation_sentence_28

Since each element of X belongs to a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by ~, each element of X belongs to a unique equivalence class of X by ~. Equivalence relation_sentence_29

Thus there is a natural bijection between the set of all equivalence relations on X and the set of all partitions of X. Equivalence relation_sentence_30

Comparing equivalence relations Equivalence relation_section_16

See also: Partition of a set § Refinement of partitions Equivalence relation_sentence_31

If ~ and ≈ are two equivalence relations on the same set S, and a~b implies a≈b for all a,b ∈ S, then ≈ is said to be a coarser relation than ~, and ~ is a finer relation than ≈. Equivalence relation_sentence_32

Equivalently, Equivalence relation_sentence_33

Equivalence relation_unordered_list_5

  • ~ is finer than ≈ if every equivalence class of ~ is a subset of an equivalence class of ≈, and thus every equivalence class of ≈ is a union of equivalence classes of ~.Equivalence relation_item_5_19
  • ~ is finer than ≈ if the partition created by ~ is a refinement of the partition created by ≈.Equivalence relation_item_5_20

The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest. Equivalence relation_sentence_34

The relation "~ is finer than ≈" on the collection of all equivalence relations on a fixed set is itself a partial order relation, which makes the collection a geometric lattice. Equivalence relation_sentence_35

Generating equivalence relations Equivalence relation_section_17

Equivalence relation_unordered_list_6

  • Given any set X, there is an equivalence relation over the set [X → X] of all functions X→X. Two such functions are deemed equivalent when their respective sets of fixpoints have the same cardinality, corresponding to cycles of length one in a permutation. Functions equivalent in this manner form an equivalence class on [X → X], and these equivalence classes partition [X → X].Equivalence relation_item_6_21
  • An equivalence relation ~ on X is the equivalence kernel of its surjective projection π : X → X/~. Conversely, any surjection between sets determines a partition on its domain, the set of preimages of singletons in the codomain. Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are three equivalent ways of specifying the same thing.Equivalence relation_item_6_22
  • The intersection of any collection of equivalence relations over X (binary relations viewed as a subset of X × X) is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given any binary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containing R. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x1, x2, ..., xn in X such that a = x1, b = xn, and (xi, xi+1) ∈ R or (xi+1, xi) ∈ R, i = 1, ..., n−1. Note that the equivalence relation generated in this manner can be trivial. For instance, the equivalence relation ~ generated by any total order on X has exactly one equivalence class, X itself, because x ~ y for all x and y. As another example, any subset of the identity relation on X has equivalence classes that are the singletons of X.Equivalence relation_item_6_23
  • Equivalence relations can construct new spaces by "gluing things together." Let X be the unit Cartesian square [0, 1] × [0, 1], and let ~ be the equivalence relation on X defined by (a, 0) ~ (a, 1) for all a ∈ [0, 1] and (0, b) ~ (1, b) for all b ∈ [0, 1]. Then the quotient space X/~ can be naturally identified (homeomorphism) with a torus: take a square piece of paper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder so as to glue together its two open ends, resulting in a torus.Equivalence relation_item_6_24

Algebraic structure Equivalence relation_section_18

Much of mathematics is grounded in the study of equivalences, and order relations. Equivalence relation_sentence_36

Lattice theory captures the mathematical structure of order relations. Equivalence relation_sentence_37

Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders. Equivalence relation_sentence_38

The former structure draws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids. Equivalence relation_sentence_39

Group theory Equivalence relation_section_19

Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalence relations are grounded in partitioned sets, which are sets closed under bijections that preserve partition structure. Equivalence relation_sentence_40

Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations. Equivalence relation_sentence_41

Hence permutation groups (also known as transformation groups) and the related notion of orbit shed light on the mathematical structure of equivalence relations. Equivalence relation_sentence_42

Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Equivalence relation_sentence_43

Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀x ∈ A ∀g ∈ G (g(x) ∈ [x]). Equivalence relation_sentence_44

Then the following three connected theorems hold: Equivalence relation_sentence_45

Equivalence relation_unordered_list_7

  • ~ partitions A into equivalence classes. (This is the Fundamental Theorem of Equivalence Relations, mentioned above);Equivalence relation_item_7_25
  • Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition;Equivalence relation_item_7_26

Equivalence relation_unordered_list_8

  • Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.Equivalence relation_item_8_27

In sum, given an equivalence relation ~ over A, there exists a transformation group G over A whose orbits are the equivalence classes of A under ~. Equivalence relation_sentence_46

This transformation group characterisation of equivalence relations differs fundamentally from the way lattices characterize order relations. Equivalence relation_sentence_47

The arguments of the lattice theory operations meet and join are elements of some universe A. Equivalence relation_sentence_48

Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a set of bijections, A → A. Equivalence relation_sentence_49

Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a ~ b ↔ (ab ∈ H). Equivalence relation_sentence_50

The equivalence classes of ~—also called the orbits of the action of H on G—are the right cosets of H in G. Interchanging a and b yields the left cosets. Equivalence relation_sentence_51

Related thinking can be found in Rosen (2008: chpt. Equivalence relation_sentence_52

10). Equivalence relation_sentence_53

Categories and groupoids Equivalence relation_section_20

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows. Equivalence relation_sentence_54

The objects are the elements of G, and for any two elements x and y of G, there exists a unique morphism from x to y if and only if x~y. Equivalence relation_sentence_55

The advantages of regarding an equivalence relation as a special case of a groupoid include: Equivalence relation_sentence_56

Equivalence relation_unordered_list_9

  • Whereas the notion of "free equivalence relation" does not exist, that of a free groupoid on a directed graph does. Thus it is meaningful to speak of a "presentation of an equivalence relation," i.e., a presentation of the corresponding groupoid;Equivalence relation_item_9_28
  • Bundles of groups, group actions, sets, and equivalence relations can be regarded as special cases of the notion of groupoid, a point of view that suggests a number of analogies;Equivalence relation_item_9_29
  • In many contexts "quotienting," and hence the appropriate equivalence relations often called congruences, are important. This leads to the notion of an internal groupoid in a category.Equivalence relation_item_9_30

Lattices Equivalence relation_section_21

The equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. Equivalence relation_sentence_57

The canonical map ker: X^X → Con X, relates the monoid X^X of all functions on X and Con X. ker is surjective but not injective. Equivalence relation_sentence_58

Less formally, the equivalence relation ker on X, takes each function f: X→X to its kernel ker f. Likewise, ker(ker) is an equivalence relation on X^X. Equivalence relation_sentence_59

Equivalence relations and mathematical logic Equivalence relation_section_22

Equivalence relations are a ready source of examples or counterexamples. Equivalence relation_sentence_60

For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number. Equivalence relation_sentence_61

An implication of model theory is that the properties defining a relation can be proved independent of each other (and hence necessary parts of the definition) if and only if, for each property, examples can be found of relations not satisfying the given property while satisfying all the other properties. Equivalence relation_sentence_62

Hence the three defining properties of equivalence relations can be proved mutually independent by the following three examples: Equivalence relation_sentence_63

Equivalence relation_unordered_list_10

  • Reflexive and transitive: The relation ≤ on N. Or any preorder;Equivalence relation_item_10_31
  • Symmetric and transitive: The relation R on N, defined as aRb ↔ ab ≠ 0. Or any partial equivalence relation;Equivalence relation_item_10_32
  • Reflexive and symmetric: The relation R on Z, defined as aRb ↔ "a − b is divisible by at least one of 2 or 3." Or any dependency relation.Equivalence relation_item_10_33

Properties definable in first-order logic that an equivalence relation may or may not possess include: Equivalence relation_sentence_64

Equivalence relation_unordered_list_11

  • The number of equivalence classes is finite or infinite;Equivalence relation_item_11_34
  • The number of equivalence classes equals the (finite) natural number n;Equivalence relation_item_11_35
  • All equivalence classes have infinite cardinality;Equivalence relation_item_11_36
  • The number of elements in each equivalence class is the natural number n.Equivalence relation_item_11_37

Euclidean relations Equivalence relation_section_23

Euclid's The Elements includes the following "Common Notion 1": Equivalence relation_sentence_65

Equivalence relation_description_list_12

  • Things which equal the same thing also equal one another.Equivalence relation_item_12_38

Nowadays, the property described by Common Notion 1 is called Euclidean (replacing "equal" by "are in relation with"). Equivalence relation_sentence_66

By "relation" is meant a binary relation, in which aRb is generally distinct from bRa. Equivalence relation_sentence_67

A Euclidean relation thus comes in two forms: Equivalence relation_sentence_68

Equivalence relation_description_list_13

  • (aRc ∧ bRc) → aRb (Left-Euclidean relation)Equivalence relation_item_13_39
  • (cRa ∧ cRb) → aRb (Right-Euclidean relation)Equivalence relation_item_13_40

The following theorem connects Euclidean relations and equivalence relations: Equivalence relation_sentence_69

with an analogous proof for a right-Euclidean relation. Equivalence relation_sentence_70

Hence an equivalence relation is a relation that is Euclidean and reflexive. Equivalence relation_sentence_71

The Elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed the reflexivity of equality too obvious to warrant explicit mention. Equivalence relation_sentence_72

See also Equivalence relation_section_24

Equivalence relation_unordered_list_14

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Equivalence relation.