Equivalence relation Totally Explained

Everything about Equivalence Relation totally explained

In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being "equivalent" in some way. Let a, b, and c be arbitrary elements of some set X. Then "a ~ b" or "a ≡ b" denotes that a is equivalent to b. An equivalence relation "~" is reflexive, symmetric, and transitive. In other words, the following must hold for "~" to be an equivalence relation on X:

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

The equivalence class a under "~", denoted [a], is the subset of X whose elements b are such that a~b. X together with "~" is called a setoid.

Examples of 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 (for example, New Foundations and related theories):

Similarity on the universe of well-orderings gives rise to equivalence classes that are the ordinal numbers.
Equinumerosity on the universe of:
- Finite sets gives rise to equivalence classes which are the natural numbers.
- Infinite sets gives rise to equivalence classes which are the transfinite cardinal numbers.
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.
Let (r_n) and (s_n) be any two Cauchy sequences of rational numbers. The real numbers are the equivalence classes of the relation (r_n) ~ (s_n), if the sequence (r_n − s_n) 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.

Examples of relations that are not equivalences

The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 doesn't 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. (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 don't have a common factor greater than 1).

The empty relation R on a non-empty set X (for example 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 or other things, even if more precisely defined, isn't an equivalence relation, because although reflexive and symmetric, it isn't transitive, since multiple small changes can accumulate to become a big change.

The relation "is a sibling of" on the set of all human beings isn't an equivalence relation. Although siblinghood is symmetric (if A is a sibling of B, then B is a sibling of A) it's neither reflexive (no one is a sibling of himself), nor transitive (since if A is a sibling of B, then B is a sibling of A, but A isn't a sibling of A). Instead of being transitive, siblinghood is "almost transitive", meaning that if A ~ B, and B ~ C, and A ≠ C, then A ~ C.

The concept of parallelism in ordered geometry isn't 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 = .

Generating equivalence relations

Given any set X, there's an equivalence relation over the set 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 X², and these equivalence classes partition X².

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

Let r be any sort of relation on X. Then r ∪ r⁻¹ is a symmetric relation. The transitive closure s of r ∪ r⁻¹ assures that s is transitive and reflexive. Moreover, s is the "smallest" equivalence relation containing r, and r/s partially orders X/s.

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

Modular lattices

The possible equivalence relations on any set X, when ordered by set inclusion, form a modular 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.

Group theory

It is very well known that lattice theory captures the mathematical structure of order relations. It is less known that transformation groups (some authors prefer permutation groups) and their orbits shed light on the mathematical structure of equivalence relations. 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.
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.
   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.
   For more on group theory and equivalence relations, see Lucas (1973: §31).
   ‡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 elements of G exists, because the domain and codomain of any element of G is A. Moreover, the composition of bijections is bijective (Wallace 1998: 22, Th. 6);

Existence of identity element. The identity function, I(x)=x, is an obvious element of G;

Existence of inverse function. 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 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

Relation with category theory and with 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.
   A morphism f can be said to have an inverse when f is an isomorphism, for example, there exists a morphism g such that fg and gf are the approrpiate identity morphisms. Hence the category-theoretic concept nearest to an equivalence relation is a (small) category whose morphisms are all isomorphisms. This is just the concept of groupoid.
   In a groupoid G, two objects x,y are 'equivalent' if there's an element g of the groupoid from x to y. There may be many such g, and they can be regarded as different `proofs' that x is equivalent to y.
   Regarding an equivalence relation as a special case of a groupoid has many implications: one is that whereas we don't have a notion of `free equivalence relation' we do have a notion of free groupoid on a directed graph. Thus we can talk of a `presentation of an equivalence relation', meaning a presentation of the corresponding groupoid. The other advantage is that it views bundles of groups, group actions, sets, and equivalence relations, as special cases of the same notion, that of groupoid, and so allows analogies between these theories and concepts.
   This also applies in many other contexts where `quotienting', and so the appropriate equivalence relations, often called congruences are important. This leads to the notion of internal groupoid in a category. For this, see the book `Galois theories' cited below.

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:

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's 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.

square

Hence an equivalence relation is a relation that's 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.Further Information

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