Equivalence relation
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,, and :
- ,
- if then , and
- if and then .
Notation
Various notations are used in the literature to denote that two elements and of a set are equivalent with respect to an equivalence relation ; the most common are "" and "", which are used when is implicit, and variations of "", "", or "" to specify explicitly. Non-equivalence may be written "" 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:- a ~ a.
- a ~ b if and only if b ~ a.
- if a ~ b and b ~ c then a ~ c.
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:- "Is equal to" on the set of numbers. For example, is equal to.
- "Has the same birthday as" on the set of all people.
- "Is similar to" on the set of all triangles.
- "Is 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.
- "Has the same absolute value" on the set of real numbers
- "Has the same cosine" on the set of all angles.
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 total order.
- The relation "has a common factor greater than 1 with" between natural numbers greater than 1, is reflexive and symmetric, but not transitive..
- The empty relation R on a non-empty set X is vacuously symmetric and transitive, but not 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.
Connections to other relations
- A partial order is a relation that is reflexive, antisymmetric, and transitive.
- 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.
- A strict partial order is irreflexive, transitive, and asymmetric.
- A partial equivalence relation is transitive and symmetric. Such a relation is reflexive if and only if it is serial, i.e. if ∀ a ∃ b a ~ b. Therefore, an equivalence relation may be alternatively defined as a symmetric, transitive, and serial relation.
- A reflexive and symmetric relation is a dependency relation, if finite, and a tolerance relation if infinite.
- A preorder is reflexive and transitive.
- 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.
- Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in classical mathematics, since it is equivalent to the law of excluded middle.
Well-definedness under an equivalence relation
A frequent particular case occurs when f is a function from X to another set Y; if x1 ~ x2 implies f = f 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 to equivalent values. 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 ~.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:Fundamental theorem of equivalence relations
A key result links equivalence relations and partitions:- An equivalence relation ~ on a set X partitions X.
- Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.
Comparing equivalence relations
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 ≈. Equivalently,- ~ 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 ~.
- ~ is finer than ≈ if the partition created by ~ is a refinement of the partition created by ≈.
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.
Generating equivalence relations
Given any binary relation on, the equivalence relation generated by is the intersection of the equivalence relations on that contain.- Given any set X, there is an equivalence relation over the set 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 , and these equivalence classes partition .
- 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.
- The intersection of any collection of equivalence relations over 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 ∈ R or ∈ 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 relations can construct new spaces by "gluing things together." Let X be the unit Cartesian square × , and let ~ be the equivalence relation on X defined by ~ for all a ∈ and ~ for all 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
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 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. Then the following three connected theorems hold:
- ~ partitions A into equivalence classes. ;
- 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.
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 ↔. 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.
Related thinking can be found in Rosen.
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:
- 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
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 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.
- The number of equivalence classes is finite or infinite;
- The number of equivalence classes equals the natural number n;
- All equivalence classes have infinite cardinality;
- The number of elements in each equivalence class is the natural number n.
Euclidean relations
Nowadays, the property described by Common Notion 1 is called Euclidean. By "relation" is meant a binary relation, in which aRb is generally distinct from bRa. A Euclidean relation thus comes in two forms:
The following theorem connects Euclidean relations and equivalence relations:
; Theorem
; Proof for a left-Euclidean relation:
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.