Equivalence relation

In mathematics, an equivalence relation, denoted by an infix "~", is a binary relation on a set X that is reflexive, symmetric, and transitive. For all elements a, b, and c of X, the following must hold in order for '~' to be an equivalence relation:

• Reflexivity: a ~ a
• Symmetry: if a ~ b then b ~ a
• Transitivity: if a ~ b and b ~ c then a ~ c

Equivalence relations often group together objects that are similar in some sense. The mathematical notion of equivalence relation should not be confused with the logical notion of identity, denoted by infix '='. If a=b then a~b, but the converse is not necessarily true.

A set X together with an equivalence relation on X is called a setoid.

From order to equivalence via symmetry

Symmetry is the property setting equivalence relations apart from the order relations ubiquitous in mathematics. An equivalence relation is a:

• Reflexive and symmetric strict partial order;
• Symmetric preorder;
• Partial order whose antisymmetry property has been upgraded to full symmetry.

Hence equivalence relations can also be seen as the culmination of a hierarch of partial orders.

Equivalence class, quotient set, partition

Let X be a nonempty set with typical members a and b. Some definitions:

• The set of all a and b for which a~b holds make up an equivalence class of X by '~'. Let [a] =: {x∈X : x~a} denote the equivalence class to which a belongs. Then all members of X equivalent to each other are also members of the same equivalence class: ∀a,b∈X (a~b ↔ [a]=[b]).
• The set of all possible equivalence classes of X by '~', denoted X/~ =: {[x] : x∈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.
• The projection of '~' is the function π: X→X/~, defined as π(x) = [x], maps members of X into their respective equivalence classes by '~'.
• Given any function f, the equivalence kernel of f is the equivalence relation, denoted Ef, such that xEfy ↔ f(x)=f(y).
• A partition of X is a set P of subsets of X, such that every member of X is a member of a unique member of P. Each member of P is a cell of the partition. Moreover, the members of P are pairwise disjoint and their union is X.

The "Fundamental Theorem of Equivalence Relations" (Wallace 1998: 31, Th. 8; Dummit and Foote 2004: 3, Prop. 2):

• An equivalence relation '~' 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 member 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 member of X belongs to a unique equivalence class of X by '~'.

A theorem relating to 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 = g(π). If f is a surjection and a~b ↔ f(a)=f(b), then g is a bijection. (Birkhoff and Mac Lane 1999: 35, Th. 19)

Now let X be finite 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, the nth Bell number B_n:

$B_{n}=\sum _{k=0}^{\infty }{\frac {k^{n}}{ek!}}.$

Generating equivalence relations

• An equivalence relation ~ on X is the equivalence kernel of its surjective projection π: X → X/~. (Birkhoff and Mac Lane 1999: 33 Th. 18). 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 two 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 if:

a~b iff there exist elements x₁, x₂,...,x_n in X such that x₁ = a, x_n = b, and such that (x_i, x_{i +1}) or (x_{i +1}, x_i) is in R for every i = 1,...,n -1.

Note that the generated equivalence relation generated in this manner can be trivial. For instance, the equivalence relation '~' generated by:

• The binary relation ≤ has exactly one equivalence class, X itself, because x~y for all x and y;
• An antisymmetric relation 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 '~' 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.

• Let G be a group and H a subgroup of G. Define 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.

Algebraic characterizations

Transformation groups

It is very well known that lattice theory captures the mathematical structure of order relations. It is much less known that transformation groups (some authors prefer permutation groups) and their orbits capture the mathematical structure of equivalence relations. See also Lucas (1973: §31).

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 (Van Fraassen 1989: §10.3):

• '~' 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. (Wallace 1998: 202, Th. 6; Dummit and Foote 2004: 114, Prop. 2).

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.

‡Proof (adapted from Van Fraassen 1989: 246). 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:

• G is closed under composition. The composition of any two members of G exists, because the domain and codomain of any member of G is A. Moreover, the composition of bijections is bijective (Wallace 1998: 22, Th. 6);
• Existence of identity. The identity function, I(x)=x, is an obvious member of G;
• Existence of inverse. Every bijective function g has an inverse g^-1, such that gg^-1 = I;
• Composition associates. f(gh) = (fg)h. This holds for all functions over all domains (Wallace 1998: 24, Th. 7).

Let f and g be any two members 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. QED.

Category theory

The composition of morphisms central to category theory, denoted by concatenation, generalizes the composition of functions central to transformation groups. The two defining axioms of category theory assert that the composition of morphisms associates, and that the left and right identity morphisms exist. If all morphisms in a category were to have "inverses," the category would resemble a transformation group, whose close relation to equivalence relations has just been explained. A morphism f can be said to have an inverse when f is an automorphism, i.e., the domain and codomain of f are identical, and there exists a morphism g such that fg = gf = identity morphism. Hence the category-theoretic concept nearest to an equivalence relation is a category whose morphisms are all automorphisms.

Equivalence relations and mathematical logic

Some first order properties an equivalence relation may or may not have include:

• The number of equivalence classes is finite or infinite;
• If finite, the number of equivalence classes is exactly n, n a natural number;
• All equivalence classes have infinite cardinality;
• All equivalence classes have size exactly n, n a natural number.

An equivalence relation with exactly 2 infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number. Equivalence relations are not all that difficult or interesting, but may be a source of easy examples or counterexamples.

Examples of equivalence relations

The obvious example, one writ large over all mathematics and human reasoning, is the equality ("=") relation between numbers, sets, etc. Other examples include the relation:

• "Has the same birthday as" on the set of all human beings.
• "Is in thermal equilibrium with" on physical objects.
• "Is similar to" or "congruent to" on the set of all triangles.
• Logical equivalence of statements in logic.
• "Is isomorphic to" on models of a set of sentences.
• "Has the same image under a function" on the elements of the domain of the function.
• Similarity on the set of well-orderings. The resulting equivalence classes are the ordinal numbers.
• In some axiomatic set theories other than the canonical ZFC, equinumerosity on the universe of:
  • Finite sets gives rise to equivalence classes which are the natural numbers, the set of which is denoted N.
  • Infinite sets gives rise to equivalence classes which are the transfinite cardinal numbers.
• Let a,b,c,d∈N, and (a,b) and (c,d) be ordered pairs. If (a,b)~(c,d) iff a+d = b+c, the equivalence classes under ~ are the integers. If (a,b)~(c,d) iff ad = bc, the equivalence classes are the rational numbers.
• Let {r_i} and {s_i} be any two Cauchy sequences of rational numbers. If {r_i}~{s_i} iff limi→∞{r_i} = limi→∞{s_i} (limits taken in the sense of Cauchy), the equivalence classes under ~ are the irrational numbers.
• "Is congruent to modulo n" on the integers.
• Green's relations are five equivalence relations on the elements of a semigroup.

Examples of relations that are not equivalences

• The relation "≥" between real numbers is reflexive and transitive, but not symmetric. E.g. 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. (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 (except when X is also empty).
• The relation "is approximately equal to" between real numbers or other things, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not transitive. The reason is empirical and pragmatic: multiple small changes can cumulate to a big change.
• The relation "is friends with" on the set of human beings is always symmetric and never reflexive. Transitivity is an empirical matter.
• The relation "is a sibling of" on the set of all human beings is not an equivalence relation, a fact worth pondering. Upon first inspection, one might think that equivalence fails here simply because siblinghood is not reflexive, but this would be incorrect. While this relation is symmetric (if A is a sibling of B, then B is a sibling of A), it is neither transitive nor reflexive. The failure of reflexivity is evident; no one is a sibling to himself. Moreover, if the relation were transitive it would have to be reflexive. Let '~' denote siblinghood and assume it transitive. If A~B. then B~A (by symmetry) and A~A (by transitivity). But it is agreed that siblinghood cannot be reflexive; hence it cannot be transitive as well. Instead, siblinghood is "almost transitive", meaning that if A~B, and B~C, and A≠C, then A~C.
• Relations that are transitive, symmetric, but irreflexive are very rare (e.g., the empty relation). The previous example illustrates why. More generally, if a relation R is symmetric and transitive, then for any x, y in the field of R, if xRy, then yRx (by symmetry), and xRx (by transitivity). Hence if there exist an x and yin the field of R, such that y≠x and xRy both come out true, the reflexivity of R follows and R must be an equivalence relation. R can be transitive, symmetric but irreflexive iff x is an "island," namely for an x that is not related to anything at all.

Euclid anticipated equivalence

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

Things which equal the same thing also equal one another.

Nowadays, a binary relation satisfying Common Notion 1 is called Euclidean. Euclid regrettably did not mention symmetry or reflexivity. Nevertheless, if a relation is Euclidian and reflexive, it is also symmetric and transitive. The proof follows:

• (aRc ∧ bRc) → aRb [a/c] = (aRa ∧ bRa) → aRb [reflexive; erase T∧] = bRa → aRb.
• (aRc ∧ bRc) → aRb [a/b; b/a; c/b] = (bRb ∧ aRb) → bRa [reflexive; erase T∧] = aRb → bRa. This line and the preceding one imply that R is symmetric.
• (aRc ∧ bRc) → aRb [c/b; b/c] = (aRb ∧ cRb) → aRc [symmetry] = (aRb ∧ bRc) → aRc. Hence R is transitive.

Hence an equivalence relation is a relation that is Euclidean and reflexive. Euclid probably would have deemed reflexivity too obvious to warrant explicit mention. If this is granted, a charitable reading of his Common Notion 1 emerges, one crediting him with being the first to conceive of equivalence relations and their importance in deductive systems.

See also

• Directed set
• Equivalence
• group action
• Partial equivalence relation
• Partial order
• Partition of a set
• Permutation group
• Total order

References

• Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 2nd ed. Chelsea.
• Castellani, E., 2003, "Symmetry and equivalence" in K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections. Cambridge University Press: 422-433.
• Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. John Wiley & Sons.
• John Randolph Lucas, 1973. A Treatise on Time and Space. London: Methuen. Section 31.
• Bas Van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press.
• Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag.

External links

• Equivalence Relations.