Equivalence relation

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This article is about the mathematical concept. For the patent doctrine, see Doctrine of equivalents.
Logical matrices of the 52 equivalence relations on a 5-element set (Colored fields, including those in light gray, stand for ones; white fields for zeros.)

In mathematics, an equivalence relation is the relation that holds between two elements if and only if they are members of the same cell within a set that has been partitioned into cells such that every element of the set is a member of one and only one cell of the partition. The intersection of any two different cells is empty; the union of all the cells equals the original set. These cells are formally called equivalence classes.

Notation[edit]

Although various notations are used throughout 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 the obvious relation being referenced, and variations of "a ~R b", "aR b", or "aRb" otherwise.

Definition[edit]

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. Equivalently, for all a, b and c in X:

X together with the relation ~ is called a setoid. The equivalence class of a under ~, denoted [a], is defined as [a] = \{b\in X | a\sim b\}.

Examples[edit]

Simple example[edit]

Let the set \{a,b,c\} have the equivalence relation \{(a,a),(b,b),(c,c),(b,c),(c,b)\}. The following sets are equivalence classes of this relation:

[a]=\{a\}, ~~~~ [b]=[c]=\{b,c\}.

The set of all equivalence classes for this relation is \{\{a\},\{b,c\}\}.

Equivalence relations[edit]

The following are all equivalence relations:

Relations that are not equivalences[edit]

  • 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.

Connections to other relations[edit]

Well-definedness under an equivalence relation[edit]

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[edit]

Let X be a nonempty set, and let a,b\in X. Some definitions:

Equivalence class[edit]

Main article: 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 [a] := \{x \in X | a \sim x\} 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[edit]

Main article: Quotient set

The set of all possible equivalence classes of X by ~, denoted  X/\mathord{\sim} := \{[x] | x \in X\}, 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[edit]

The projection of ~ is the function \pi: X \to X/\mathord{\sim} defined by \pi(x) = [x] which maps elements of X into their respective equivalence classes by ~.

Theorem on projections:[1] 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[edit]

The equivalence kernel of a function f is the equivalence relation ~ defined by x\sim y \iff f(x) = f(y). The equivalence kernel of an injection is the identity relation.

Partition[edit]

Main article: Partition of a set

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 possible partitions[edit]

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:

B_n = \frac{1}{e}\sum_{k=0}^\infty \frac{k^n}{k!},

where the above is one of the ways to write the nth Bell number.

Fundamental theorem of equivalence relations[edit]

A key result links equivalence relations and partitions:[2][3]

  • 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.

Comparing equivalence relations[edit]

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,

  • ~ 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 equality equivalence relation is the finest equivalence relation on any set, while the trivial relation that makes all pairs of elements related 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.

Generating equivalence relations[edit]

  • Given any set X, there is an equivalence relation over the set [XX] of all possible functions XX. 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 [XX], and these equivalence classes partition [XX].
  • 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;
    • 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 (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.

Algebraic structure[edit]

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[edit]

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 and 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:[5]

  • ~ 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.[6][7]

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 G—are the right cosets of H in G. Interchanging a and b yields the left cosets.

Proof.[8] 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. \square

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

Categories and groupoids[edit]

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.[11]

Lattices[edit]

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^XCon 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: XX to its kernel ker f. Likewise, ker(ker) is an equivalence relation on X^X.

Equivalence relations and mathematical logic[edit]

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:

  • Reflexive and transitive: The relation ≤ on N. Or any preorder;
  • Symmetric and transitive: The relation R on N, defined as aRbab ≠ 0. Or any partial equivalence relation;
  • Reflexive and symmetric: The relation R on Z, defined as aRb ↔ "ab 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.

Euclidean relations[edit]

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:

  • (aRcbRc) → aRb [a/c] = (aRabRa) → aRb [reflexive; erase T∧] = bRaaRb. Hence R is symmetric.
  • (aRcbRc) → aRb [symmetry] = (aRccRb) → aRb. Hence R is transitive. \square

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[edit]

Notes[edit]

  1. ^ Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 35, Th. 19. Chelsea.
  2. ^ Wallace, D. A. R., 1998. Groups, Rings and Fields. p. 31, Th. 8. Springer-Verlag.
  3. ^ Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. p. 3, Prop. 2. John Wiley & Sons.
  4. ^ Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 33, Th. 18. Chelsea.
  5. ^ Rosen (2008), pp. 243-45. Less clear is §10.3 of Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press.
  6. ^ Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 202, Th. 6.
  7. ^ Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. John Wiley & Sons: 114, Prop. 2.
  8. ^ Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press: 246.
  9. ^ Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 22, Th. 6.
  10. ^ Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 24, Th. 7.
  11. ^ Borceux, F. and Janelidze, G., 2001. Galois theories, Cambridge University Press, ISBN 0-521-80309-8

References[edit]

  • Brown, Ronald, 2006. Topology and Groupoids. Booksurge LLC. ISBN 1-4196-2722-8.
  • Castellani, E., 2003, "Symmetry and equivalence" in Brading, Katherine, and E. Castellani, eds., Symmetries in Physics: Philosophical Reflections. Cambridge Univ. Press: 422-433.
  • Robert Dilworth and Crawley, Peter, 1973. Algebraic Theory of Lattices. Prentice Hall. Chpt. 12 discusses how equivalence relations arise in lattice theory.
  • Higgins, P.J., 1971. Categories and groupoids. Van Nostrand. Downloadable since 2005 as a TAC Reprint.
  • John Randolph Lucas, 1973. A Treatise on Time and Space. London: Methuen. Section 31.
  • Rosen, Joseph (2008) Symmetry Rules: How Science and Nature are Founded on Symmetry. Springer-Verlag. Mostly chpts. 9,10.
  • Raymond Wilder (1965) Introduction to the Foundations of Mathematics 2nd edition, Chapter 2-8: Axioms defining equivalence, pp 48–50, John Wiley & Sons.

External links[edit]