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Equivalence relation
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In mathematics, an equivalence relation is, loosely, a binary relation on a set that specifies how to split up (i.e. partition) the set into subsets such that every element of the larger set is in exactly one of the subsets. Any two elements of the larger set are then considered "equivalent" with respect to the equivalence relation if and only if they are also elements of the same subset.
ough various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to equivalence relation R, the most common are "a ~ b" and "a = b", which are used when R is the obvious relation being referenced, and variations of "a ~R b", "a =R b", or "aRb".
A be a set and ~ be a binary relation on A.
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In mathematics, an equivalence relation is, loosely, a binary relation on a set that specifies how to split up (i.e. partition) the set into subsets such that every element of the larger set is in exactly one of the subsets. Any two elements of the larger set are then considered "equivalent" with respect to the equivalence relation if and only if they are also elements of the same subset.
Notation
Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to equivalence relation R, the most common are "a ~ b" and "a = b", which are used when R is the obvious relation being referenced, and variations of "a ~R b", "a =R b", or "aRb".
Definition
Let A be a set and ~ be a binary relation on A. ~ is called an equivalence relation if and only if for all , all the following holds true:
The equivalence class of a under ~, denoted [a], is defined as . A together with ~ is called a setoid.
Examples
Equivalence relations
A ubiquitous equivalence relation is the equality ("=") relation between elements of any set. Other examples include:
- "Has the same birthday as" on the set of all people, given naive set theory.
- "Is similar to" or "congruent to" on the set of all triangles.
- "Is congruent to modulo n" on the integers.
- "Has the same image under a function" on the elements of the domain of the function.
- Logical equivalence of logical sentences.
- "Is isomorphic to" on models of a set of sentences.
- In some axiomatic set theories other than the canonical ZFC (e.g., New Foundations and related theories):
- Let a, b, c, d be natural numbers, and let (a, b) and (c, d) be ordered pairs of such numbers. Then the equivalence classes under the relation (a, b) ~ (c, d) are the:
- Integers if a + d = b + c;
- Positive rational numbers if ad = bc. To obtain all the rational numbers, simply let a through d range over the integers just defined.
- Let (rn) and (sn) be any two Cauchy sequences of rational numbers. The real numbers are the equivalence classes of the relation (rn) ~ (sn), if the sequence (rn − sn) has limit 0.
- Green's relations are five equivalence relations on the elements of a semigroup.
- "Is parallel to" on the set of subspaces of an affine space.
- The binary relation of thermal equilibrium on the set of thermodynamic systems. The zeroth law of thermodynamics says that thermal equilibrium is a Euclidean relation. Thermal equilibrium is also trivially reflexive. For a proof that a relation that is both Euclidean and reflexive is also an equivalence relation, see here.
Relations that are not equivalences
- 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 partial order.
- The relation "has a common factor greater than 1 with" between natural numbers greater than 1, is reflexive and symmetric, but not transitive. (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).
- The empty relation R on a non-empty set X (i.e. aRb is never true) is vacuously symmetric and transitive, but not reflexive. (If X is also empty then R is reflexive.)
- 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.
- The relation "is a sibling of" on the set of all human beings is not an equivalence relation. Although siblinghood is both symmetric (if A is a sibling of B, then B is a sibling of A)and transitive (if A is a sibling of B and C is a sibling of B, then A is a sibling of C), it is not reflexive (A cannot be a sibling of A). This illustrates an important subtlety: a relation can be transitive and symmetric and still be non-reflexive.
- The concept of parallelism in ordered geometry is not symmetric and is, therefore, not an equivalence relation.
- An equivalence relation on a set is never an equivalence relation on a proper superset of that set. For example R = is an equivalence relation on but not on or on the natural numbers. The problem is that reflexivity fails because (4,4) is not a member.
Connection to other relations
- 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.
- A partial order replaces symmetry with antisymmetry and is thus reflexive, antisymmetric, and transitive. Equality is the only relation that is both an equivalence relation and a partial order.
- A strict partial order is irreflexive, transitive, and asymmetric.
- A partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reflexive if and only if for all a?X exists b?X such that a~b.
- A dependency relation is reflexive and symmetric.
- A preorder is reflexive and transitive.
Equivalence class, quotient set, partition
Let X be a nonempty set, and let . Some definitions:
Equivalence class
The set of all a and b for which a ~ b holds make up an equivalence class of X by ~. Let denote the equivalence class to which a belongs. Then all elements of X equivalent to each other are also elements of the same equivalence class.
Quotient set
The set of all possible 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 : 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 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 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.
Fundamental Theorem of Equivalence Relations
- An equivalence relation ~ on a set X partitions X.
- Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.
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 from the set of all possible equivalence relations on X and the set of all partitions of X.
Counting possible 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 possible equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number Bn:
-
Generating equivalence relations
- Given any set X, there is an equivalence relation over the set [X?X] of all possible 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].
- An equivalence relation ~ on X is the equivalence kernel of its surjective projection p : 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.
- The intersection of any collection of equivalence relations over X (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;
- Any subset of the identity relation on X has equivalence classes that are the singletons of X.
- 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, b ? [0,1] ((a, 0) ~ (a, 1) ? (0, b) ~ (1, b)). Then the quotient space X/~ can be naturally identified 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.
Algebraic structure
Much of mathematics is grounded in the study of equivalences (the subject of this entry), and order relations. It is very well known that 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, sets closed under bijections preserving 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: ?x ? A ?g ? G (g(x) ? [x]). Then the following three connected theorems hold :
- ~ partitions A into equivalence classes. (This is the Fundamental Theorem of Equivalence Relations, mentioned above);
- Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition‡;
- Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G. .
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, A ? A.
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−1 ? H). 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.
‡Proof . Let function composition interpret group multiplication, and function inverse interpret group inverse. Then G is a group under composition, meaning that ?x ? A ?g ? G ([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.
Categories and groupoids
The composition of morphisms central to category theory, denoted here by concatenation, generalizes the composition of functions central to transformation groups. The axioms of category theory assert that the composition of morphisms associates, and that the left and right identity morphisms exist for any morphism.
If a morphism f has an inverse, f is an isomorphism, i.e., there exists a morphism g such that the compositions fg and gf equal the appropriate identity morphisms. Hence the category-theoretic concept nearest to an equivalence relation is a (small) category whose morphisms are all isomorphisms. Groupoid is another name for a small category of this nature.
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 elements x and y are "equivalent" if there is an element g of the groupoid from x to y. There may be many such g, each of which can be regarded as a distinct "proof" that x and y are equivalent.
The advantages of regarding an equivalence relation as a special case of a groupoid include:
- 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;
- 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;
- 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 .
Lattices
The possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. 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. 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 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 ?-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:
- Reflexive and transitive: The relation = on N. Or any preorder;
- Symmetric and transitive: The relation R on N, defined as aRb ? ab ? 0. Or any partial equivalence relation;
- 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.
Properties definable in first-order logic that an equivalence relation may or may not possess include:
- The number of equivalence classes is finite or infinite;
- The number of equivalence classes equals the (finite) natural number n;
- All equivalence classes have infinite cardinality;
- The number of elements in each equivalence class is the natural number n.
Euclid anticipated equivalence
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"). The following theorem connects Euclidean relations and equivalence relations:
Theorem. If a relation is Euclidean and reflexive, it is also symmetric and transitive.
Proof:
- (aRc ? bRc) ? aRb [a/c] = (aRa ? bRa) ? aRb [reflexive; erase T?] = bRa ? aRb. Hence R is symmetric.
- (aRc ? bRc) ? aRb [symmetry] = (aRc ? cRb) ? aRb. Hence R is transitive.
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. If this (and taking "equality" as an all-purpose abstract relation) is granted, a charitable reading of Common Notion 1 would credit Euclid with being the first to conceive of equivalence relations and their importance in deductive systems.
See also
External links
- Bogomolny, A., "" cut-the-knot. Accessed 7 December 2007
- at PlanetMath
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