Equivalence relation

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The 52 equivalence relations on a 5-element set depicted as 5x5 logical matrices (colored fields, including those in light gray, stand for ones; white fields for zeros.) The row and column indices of nonwhite cells are the related elements, while the different colors, other than light gray, indicate the equivalence classes (each light gray cell is its own equivalence class). Set partitions 5; matrices.svg
The 52 equivalence relations on a 5-element set depicted as 5×5 logical matrices (colored fields, including those in light gray, stand for ones; white fields for zeros.) The row and column indices of nonwhite cells are the related elements, while the different colors, other than light gray, indicate the equivalence classes (each light gray cell is its own equivalence class).

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:

Contents

As a consequence of the reflexive, symmetric, and transitive properties, any equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.

Notation

Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a ~ b" and "ab", which are used when R is implicit, and variations of "a ~Rb", "aRb", or "aRb" to specify R explicitly. Non-equivalence may be written "ab" or "".

Definition

A given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. That is, for all a, b and c in X:

X together with the relation ~ is called a setoid. The equivalence class of under ~, denoted , is defined as .

Examples

Simple example

Let the set have the equivalence relation . The following sets are equivalence classes of this relation:

.

The set of all equivalence classes for this relation is . This set is a partition of the set .

Equivalence relations

The following are all equivalence relations:

Relations that are not equivalences

Connections to other relations

Well-definedness under an equivalence relation

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

More generally, a function may map equivalent arguments (under an equivalence relation ~A) to equivalent values (under an equivalence relation ~B). Such a function is known as a morphism from ~A to ~B.

Equivalence class, quotient set, partition

Let . Some definitions:

Equivalence class

A subset Y of X such that a ~ b holds for all a and b in Y, and never for a in Y and b outside Y, is called an equivalence class of X by ~. Let denote the equivalence class to which a belongs. All elements of X equivalent to each other are also elements of the same equivalence class.

Quotient set

The set of all equivalence classes of X by ~, denoted , is the quotient set of X by ~. If X is a topological space, there is a natural way of transforming X/~ into a topological space; see quotient space for the details.

Projection

The projection of ~ is the function defined by which maps elements of X into their respective equivalence classes by ~.

Theorem on projections: [2] Let the function f: XB be such that a ~ bf(a) = f(b). Then there is a unique function g : X/~B, such that f = gπ. If f is a surjection and a ~ bf(a) = f(b), then g is a bijection.

Equivalence kernel

The equivalence kernel of a function f is the equivalence relation ~ defined by . The equivalence kernel of an injection is the identity relation.

Partition

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

Counting partitions

Let X be a finite set with n elements. 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:

    (Dobinski's formula).

Fundamental theorem of equivalence relations

A key result links equivalence relations and partitions: [3] [4] [5]

In both cases, the cells of the partition of X are the equivalence classes of X by ~. 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 ~. Thus there is a natural bijection between the set of all equivalence relations on X and the set of all partitions of X.

Comparing equivalence relations

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

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.

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. [6]

Generating equivalence relations

Given any binary relation on , the equivalence relation generated by is the intersection of the equivalence relations on that contain . (Since is an equivalence relation, the intersection is nontrivial.)

Algebraic structure

Much of mathematics is grounded in the study of equivalences, and order relations. Lattice theory captures the mathematical structure of order relations. 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. The former structure draws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.

Group theory

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. Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations. Hence permutation groups (also known as transformation groups) and the related notion of orbit shed light on the mathematical structure of equivalence relations.

Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀xAgG (g(x) ∈ [x]). Then the following three connected theorems hold: [8]

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 ~.

This transformation group characterisation of equivalence relations differs fundamentally from the way lattices characterize order relations. The arguments of the lattice theory operations meet and join are elements of some universe A. Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a set of bijections, AA.

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−1H). The equivalence classes of ~also called the orbits of the action of H on Gare the right cosets of H in G. Interchanging a and b yields the left cosets.

Related thinking can be found in Rosen (2008: chpt. 10).

Categories and groupoids

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. 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.

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

Lattices

The equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called ConX by convention. The canonical map ker: X^XConX, relates the monoid X^X of all functions on X and ConX. ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f: XX to its kernel kerf. Likewise, ker(ker) is an equivalence relation on X^X.

Equivalence relations and mathematical logic

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

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

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

Euclidean relations

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

Things which equal the same thing also equal one another.

Nowadays, the property described by Common Notion 1 is called Euclidean (replacing "equal" by "are in relation with"). By "relation" is meant a binary relation, in which aRb is generally distinct from bRa. A Euclidean relation thus comes in two forms:

(aRcbRc) → aRb (Left-Euclidean relation)
(cRacRb) → aRb (Right-Euclidean relation)

The following theorem connects Euclidean relations and equivalence relations:

Theorem
If a relation is (left or right) Euclidean and reflexive, it is also symmetric and transitive.
Proof for a left-Euclidean relation
(aRcbRc) → aRb [a/c] = (aRabRa) → aRb [reflexive; erase T∧] = bRaaRb. Hence R is symmetric.
(aRcbRc) → aRb [symmetry] = (aRccRb) → aRb. Hence R is transitive.

with an analogous proof for a right-Euclidean relation. Hence an equivalence relation is a relation that is Euclidean and reflexive. The Elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed the reflexivity of equality too obvious to warrant explicit mention.

See also

Notes

  1. If: Given a, let a~b hold using seriality, then b~a by symmetry, hence a~a by transitivity. Only if: Given a, choose b=a, then a~b by reflexivity.
  2. Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 35, Th. 19. Chelsea.
  3. Wallace, D. A. R., 1998. Groups, Rings and Fields. p. 31, Th. 8. Springer-Verlag.
  4. Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. p. 3, Prop. 2. John Wiley & Sons.
  5. Karel Hrbacek & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, pages 29–32, Marcel Dekker
  6. Birkhoff, Garrett (1995), Lattice Theory, Colloquium Publications, 25 (3rd ed.), American Mathematical Society, ISBN   9780821810255 . Sect. IV.9, Theorem 12, page 95
  7. Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 33, Th. 18. Chelsea.
  8. Rosen (2008), pp. 243–45. Less clear is §10.3 of Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press.
  9. Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press: 246.
  10. Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 22, Th. 6.
  11. Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 24, Th. 7.
  12. Proof. [9] Let function composition interpret group multiplication, and function inverse interpret group inverse. Then G is a group under composition, meaning that ∀xAgG ([g(x)] = [x]), because G satisfies the following four conditions: Let f and g be any two elements of G. By virtue of the definition of G, [g(f(x))] = [f(x)] and [f(x)] = [x], so that [g(f(x))] = [x]. Hence G is also a transformation group (and an automorphism group) because function composition preserves the partitioning of A.
  13. Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 202, Th. 6.
  14. Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. John Wiley & Sons: 114, Prop. 2.
  15. Borceux, F. and Janelidze, G., 2001. Galois theories, Cambridge University Press, ISBN   0-521-80309-8

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