Mathematical relations dehFrom Wikipedia, the free encyclopedia

Contents

1 Binary relation 11.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Demonic composition 112.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Dense order 123.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Dependence relation 134.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Dependency relation 14

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5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6 Directed set 166.1 Equivalent definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.3 Contrast with semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.4 Directed subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7 Equality (mathematics) 197.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2 Types of equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.2.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2.2 Equalities as predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2.4 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2.5 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.3 Logical formalizations of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.4 Logical formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.5 Some basic logical properties of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.6 Relation with equivalence and isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

8 Equipollence (geometry) 238.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

9 Equivalence class 259.1 Notation and formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.4 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.6 Quotient space in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

10 Equivalence relation 30

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10.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.5 Well-definedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

10.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

10.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.10Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

10.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

10.11Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3510.12Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11 Euclidean relation 3911.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.2 Relation to transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

12 Exceptional isomorphism 4012.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

12.1.1 Finite simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.1.2 Groups of Lie type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.1.3 Alternating groups and symmetric groups . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.1.4 Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.1.5 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.1.6 Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12.2 Lie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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12.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

13 Fiber (mathematics) 4513.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

13.1.1 Fiber in naive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.1.2 Fiber in algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

13.2 Terminological variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4513.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

14 Finitary relation 4714.1 Informal introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.2 Relations with a small number of “places” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.3 Formal definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

15 Foundational relation 5115.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5115.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

16 Hypostatic abstraction 5216.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.4 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 54

16.4.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5416.4.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.4.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Chapter 1

Binary relation

“Relation (mathematics)" redirects here. For a more general notion of relation, see finitary relation. For a morecombinatorial viewpoint, see theory of relations. For other uses, see Relation § Mathematics.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subsetof A × B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and theprime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and“divides” in arithmetic, "is congruent to" in geometry, “is adjacent to” in graph theory, “is orthogonal to” in linearalgebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations arealso heavily used in computer science.A binary relation is the special case n = 2 of an n-ary relation R ⊆ A1 × … × An, that is, a set of n-tuples where thejth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation onZ×Z×Z is “lies between ... and ...”, containing e.g. the triples (5,2,8), (5,8,2), and (−4,9,−7).In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. Thisextension is needed for, among other things, modeling the concepts of “is an element of” or “is a subset of” in settheory, without running into logical inconsistencies such as Russell’s paradox.

1.1 Formal definition

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) ∈ G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X × Y for the set of pairs of G.The order of the elements in each pair ofG is important: if a ≠ b, then aRb and bRa can be true or false, independentlyof each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.A relation as defined by the triple (X, Y, G) is sometimes referred to as a correspondence instead.[1] In this case therelation from X to Y is the subset G of X × Y, and “from X to Y" must always be either specified or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

1

2 CHAPTER 1. BINARY RELATION

1.1.1 Is a relation more than its graph?

According to the definition above, two relations with identical graphs but different domains or different codomainsare considered different. For example, ifG = {(1, 2), (1, 3), (2, 7)} , then (Z,Z, G) , (R,N, G) , and (N,R, G) arethree distinct relations, where Z is the set of integers and R is the set of real numbers.Especially in set theory, binary relations are often defined as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least oney such that (x, y) ∈ R , the range of R is defined as the set of all y such that there exists at least one x such that(x, y) ∈ R , and the field of R is the union of its domain and its range.[2][3][4]

A special case of this difference in points of view applies to the notion of function. Many authors insist on distin-guishing between a function’s codomain and its range. Thus, a single “rule,” like mapping every real number x tox2, can lead to distinct functions f : R → R and f : R → R+ , depending on whether the images under thatrule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets ofordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivity—or being onto—as a property of functions, while the latter sees itas a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation,and the definitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodefinitions usually matters only in very formal contexts, like category theory.

1.1.2 Example

Example: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian, Venus}. Supposethat John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing.Then the binary relation “is owned by” is given as

R = ({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).

Thus the first element of R is the set of objects, the second is the set of persons, and the last element is a set of orderedpairs of the form (object, owner).The pair (ball, John), denoted by ₐ RJₒ means that the ball is owned by John.Two different relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball, John), (doll, Mary), (car, Venus)})

is different from the previous one as everyone is an owner. But the graphs of the two relations are the same.Nevertheless, R is usually identified or even defined as G(R) and “an ordered pair (x, y) ∈ G(R)" is usually denoted as"(x, y) ∈ R".

1.2 Special types of binary relations

Some important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be different sets, some authors call such binary relations heterogeneous.[5][6]

Uniqueness properties:

• injective (also called left-unique[7]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. Forexample, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = −5and z = +5 to y = 25.

• functional (also called univalent[8] or right-unique[7] or right-definite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=−5 and z=+5.

1.2. SPECIAL TYPES OF BINARY RELATIONS 3

Example relations between real numbers. Red: y=x2. Green: y=2x+20.

• one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties:

• left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is differentfrom the definition of total in the next section. Both relations in the picture are left-total. The relation x=y2,obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = −14 to any real number y.

• surjective (also called right-total[7] or onto): for all y in Y there exists an x in X such that xRy. The greenrelation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = −14.

Uniqueness and totality properties:

4 CHAPTER 1. BINARY RELATION

• A function: a relation that is functional and left-total. Both the green and the red relation are functions.

• An injective function: a relation that is injective, functional, and left-total.

• A surjective function or surjection: a relation that is functional, left-total, and right-total.

• A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known asone-to-one correspondence.[10] The green relation is bijective, but the red is not.

1.2.1 Difunctional

Less commonly encountered is the notion of difunctional (or regular) relation, defined as a relation R such thatR=RR−1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element x∈X to a set xR = { y∈Y| xRy }.[11] This set is sometimes called the successor neighborhood of x in R; one can define the predecessorneighborhood analogously.[12] Synonymous terms for these notions are afterset and respectively foreset.[5]

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1R and x2R have anon-empty intersection, then these two sets coincide; formally x1R ∩ x2R ≠ ∅ implies x1R = x2R.[11]

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If oneconsiders a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalencerelation), then it is also difunctional.[13] The converse of this latter statement also doesn't hold.A characterization of difunctional relations, which also explains their name, is to consider two functions f: A → Cand g: B→ C and then define the following set which generalizes the kernel of a single function as joint kernel: ker(f,g) = { (a, b) ∈ A × B | f(a) = g(b) }. Every difunctional relation R ⊆ A × B arises as the joint kernel of two functionsf: A→ C and g: B → C for some set C.[14]

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]Other authors however use the term “rectangular” to denote any heterogeneous relation whatsoever.[6]

1.3 Relations over a set

If X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.[16] In computerscience, such a relation is also called a homogeneous (binary) relation.[16][17][6] Some types of endorelations arewidely studied in graph theory, where they are known as simple directed graphs permitting loops.The set of all binary relations Rel(X) on a set X is the set 2X × X which is a Boolean algebra augmented with theinvolution of mapping of a relation to its inverse relation. For the theoretical explanation see Relation algebra.Some important properties of a binary relation R over a set X are:

• reflexive: for all x in X it holds that xRx. For example, “greater than or equal to” (≥) is a reflexive relation but“greater than” (>) is not.

• irreflexive (or strict): for all x in X it holds that not xRx. For example, > is an irreflexive relation, but ≥ is not.

• coreflexive: for all x and y in X it holds that if xRy then x = y. An example of a coreflexive relation is therelation on integers in which each odd number is related to itself and there are no other relations. The equalityrelation is the only example of a both reflexive and coreflexive relation.

The previous 3 alternatives are far from being exhaustive; e.g. the red relation y=x2 from theabove picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair(0,0), and (2,4), but not (2,2), respectively.

• symmetric: for all x and y in X it holds that if xRy then yRx. “Is a blood relative of” is a symmetric relation,because x is a blood relative of y if and only if y is a blood relative of x.

1.4. OPERATIONS ON BINARY RELATIONS 5

• antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, ≥ is anti-symmetric (so is >, butonly because the condition in the definition is always false).[18]

• asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is bothanti-symmetric and irreflexive.[19] For example, > is asymmetric, but ≥ is not.

• transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. A transitive relation is irreflexive if andonly if it is asymmetric.[20] For example, “is ancestor of” is transitive, while “is parent of” is not.

• total: for all x and y in X it holds that xRy or yRx (or both). This definition for total is different from left totalin the previous section. For example, ≥ is a total relation.

• trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomousrelation, while the relation “divides” on natural numbers is not.[21]

• Right Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz.

• Left Euclidean: for all x, y and z in X it holds that if yRx and zRx, then yRz.

• Euclidean: An Euclidean relation is both left and right Euclidean. Equality is a Euclidean relation because ifx=y and x=z, then y=z.

• serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. Butit is not a serial relation on the positive integers, because there is no y in the positive integers (i.e. the naturalnumbers) such that 1>y.[22] However, "is less than" is a serial relation on the positive integers, the rationalnumbers and the real numbers. Every reflexive relation is serial: for a given x, choose y=x. A serial relation canbe equivalently characterized as every element having a non-empty successor neighborhood (see the previoussection for the definition of this notion). Similarly an inverse serial relation is a relation in which every elementhas non-empty predecessor neighborhood.[12]

• set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relationson proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse> is not.

A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reflexive. A relation that is only symmetric and transitive (without necessarily beingreflexive) is called a partial equivalence relation.A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total is calleda total order, simple order, linear order, or a chain.[23] A linear order where every nonempty subset has a least elementis called a well-order.

1.4 Operations on binary relations

If R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y :

• Union: R ∪ S ⊆ X × Y, defined as R ∪ S = { (x, y) | (x, y) ∈ R or (x, y) ∈ S }. For example, ≥ is the union of >and =.

• Intersection: R ∩ S ⊆ X × Y, defined as R ∩ S = { (x, y) | (x, y) ∈ R and (x, y) ∈ S }.

If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relationover X and Z: (see main article composition of relations)

• Composition: S ∘ R, also denoted R ; S (or more ambiguously R ∘ S), defined as S ∘ R = { (x, z) | there existsy ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S }. The order of R and S in the notation S ∘ R, used here agrees withthe standard notational order for composition of functions. For example, the composition “is mother of” ∘ “isparent of” yields “is maternal grandparent of”, while the composition “is parent of” ∘ “is mother of” yields “isgrandmother of”.

6 CHAPTER 1. BINARY RELATION

A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a subset of S, that is, if x R yalways implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is containedin ≥.If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

• Inverse or converse: R −1, defined as R −1 = { (y, x) | (x, y) ∈ R }. A binary relation over a set is equal to itsinverse if and only if it is symmetric. See also duality (order theory). For example, “is less than” (<) is theinverse of “is greater than” (>).

If R is a binary relation over X, then each of the following is a binary relation over X:

• Reflexive closure: R =, defined as R = = { (x, x) | x ∈ X } ∪ R or the smallest reflexive relation over X containingR. This can be proven to be equal to the intersection of all reflexive relations containing R.

• Reflexive reduction: R ≠, defined as R ≠ = R \ { (x, x) | x ∈ X } or the largest irreflexive relation over Xcontained in R.

• Transitive closure: R +, defined as the smallest transitive relation over X containing R. This can be seen to beequal to the intersection of all transitive relations containing R.

• Transitive reduction: R −, defined as a minimal relation having the same transitive closure as R.

• Reflexive transitive closure: R *, defined as R * = (R +) =, the smallest preorder containing R.

• Reflexive transitive symmetric closure: R ≡, defined as the smallest equivalence relation over X containingR.

1.4.1 Complement

If R is a binary relation over X and Y, then the following too:

• The complement S is defined as x S y if not x R y. For example, on real numbers, ≤ is the complement of >.

The complement of the inverse is the inverse of the complement.If X = Y, the complement has the following properties:

• If a relation is symmetric, the complement is too.

• The complement of a reflexive relation is irreflexive and vice versa.

• The complement of a strict weak order is a total preorder and vice versa.

The complement of the inverse has these same properties.

1.4.2 Restriction

The restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x andy are in S.If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in generalnot equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother ofthe woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, thetransitive closure of “is parent of” is “is ancestor of"; its restriction to females does relate a woman with her paternalgrandmother.

1.5. SETS VERSUS CLASSES 7

Also, the various concepts of completeness (not to be confused with being “total”) do not carry over to restrictions.For example, on the set of real numbers a property of the relation "≤" is that every non-empty subset S of R with anupper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers thissupremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to theset of rational numbers.The left-restriction (right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain(codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.

1.4.3 Algebras, categories, and rewriting systems

Various operations on binary endorelations can be treated as giving rise to an algebraic structure, known as relationalgebra. It should not be confused with relational algebra which deals in finitary relations (and in practice also finiteand many-sorted).For heterogenous binary relations, a category of relations arises.[6]

Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstractrewriting system.

1.5 Sets versus classes

Certain mathematical “relations”, such as “equal to”, “member of”, and “subset of”, cannot be understood to be binaryrelations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems ofaxiomatic set theory. For example, if we try to model the general concept of “equality” as a binary relation =, wemust take the domain and codomain to be the “class of all sets”, which is not a set in the usual set theory.In most mathematical contexts, references to the relations of equality, membership and subset are harmless becausethey can be understood implicitly to be restricted to some set in the context. The usual work-around to this problemis to select a “large enough” set A, that contains all the objects of interest, and work with the restriction =A instead of=. Similarly, the “subset of” relation ⊆ needs to be restricted to have domain and codomain P(A) (the power set ofa specific set A): the resulting set relation can be denoted ⊆A. Also, the “member of” relation needs to be restrictedto have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown thatassuming ∈ to be defined on all sets leads to a contradiction in naive set theory.Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory,and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership,and subset are binary relations without special comment. (A minor modification needs to be made to the concept ofthe ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course onecan identify the function with its graph in this context.)[24] With this definition one can for instance define a functionrelation between every set and its power set.

1.6 The number of binary relations

The number of distinct binary relations on an n-element set is 2n2 (sequence A002416 in OEIS):Notes:

• The number of irreflexive relations is the same as that of reflexive relations.

• The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.

• The number of strict weak orders is the same as that of total preorders.

• The total orders are the partial orders that are also total preorders. The number of preorders that are neithera partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders,minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.

• the number of equivalence relations is the number of partitions, which is the Bell number.

8 CHAPTER 1. BINARY RELATION

The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its owncomplement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse com-plement).

1.7 Examples of common binary relations

• order relations, including strict orders:

• greater than• greater than or equal to• less than• less than or equal to• divides (evenly)• is a subset of

• equivalence relations:

• equality• is parallel to (for affine spaces)• is in bijection with• isomorphy

• dependency relation, a finite, symmetric, reflexive relation.

• independency relation, a symmetric, irreflexive relation which is the complement of some dependency relation.

1.8 See also

• Confluence (term rewriting)

• Hasse diagram

• Incidence structure

• Logic of relatives

• Order theory

• Triadic relation

1.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 1330–1331. ISBN 0-262-59020-4.

[2] Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN0-486-61630-4.

[3] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.

[4] Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979].Basic Set Theory. Dover. ISBN 0-486-42079-5.

[5] Christodoulos A. Floudas; PanosM. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science&BusinessMedia. pp. 299–300. ISBN 978-0-387-74758-3.

1.10. REFERENCES 9

[6] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN978-1-4020-6164-6.

[7] Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:

• Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook.Springer Science & Business Media. p. 506. ISBN 978-3-540-67995-0.

• Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 19–21. ISBN 978-0-13-460643-9.

• Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the HighLevel Petri Net Calculus. Herbert Utz Verlag. pp. 21–22. ISBN 978-3-89675-629-9.

[8] Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5

[9] Mäs, Stephan (2007), “Reasoning on Spatial Semantic Integrity Constraints”, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australiia, September 19–23, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285–302, doi:10.1007/978-3-540-74788-8_18

[10] Note that the use of “correspondence” here is narrower than as general synonym for binary relation.

[11] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science &Business Media. p. 200. ISBN 978-3-211-82971-4.

[12] Yao, Y. (2004). “Semantics of Fuzzy Sets in Rough Set Theory”. Transactions on Rough Sets II. Lecture Notes in ComputerScience 3135. p. 297. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.

[13] William Craig (2006). Semigroups Underlying First-order Logic. American Mathematical Soc. p. 72. ISBN 978-0-8218-6588-0.

[14] Gumm, H. P.; Zarrad, M. (2014). “Coalgebraic Simulations and Congruences”. Coalgebraic Methods in Computer Science.Lecture Notes in Computer Science 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[15] Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.

[16] M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.

[17] Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. SpringerScience & Business Media. p. 496. ISBN 978-3-540-67995-0.

[18] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006),ATransition to AdvancedMathematics (6th ed.), Brooks/Cole,p. 160, ISBN 0-534-39900-2

[19] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[20] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics – Physics Charles University. p. 1. Lemma 1.1 (iv). This source refers to asymmetric relations as “strictlyantisymmetric”.

[21] Since neither 5 divides 3, nor 3 divides 5, nor 3=5.

[22] Yao, Y.Y.; Wong, S.K.M. (1995). “Generalization of rough sets using relationships between attribute values” (PDF).Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..

[23] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

[24] Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p.3. ISBN 0-8218-1041-3.

1.10 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories: with Applications to Wreath Products and

Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

10 CHAPTER 1. BINARY RELATION

1.11 External links• Hazewinkel, Michiel, ed. (2001), “Binary relation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 2

Demonic composition

In mathematics, demonic composition is an operation on binary relations that is somewhat comparable to ordinarycomposition of relations but is robust to refinement of the relations into (partial) functions or injective relations.Unlike ordinary composition of relations, demonic composition is not associative.

2.1 Definition

SupposeR is a binary relation betweenX andY and S is a relation betweenY and Z. Their right demonic compositionR ;→ S is a relation between X and Z. Its graph is defined as

{(x, z) | x (S ◦R) z ∧ ∀y ∈ Y (xR y ⇒ y S z)}.

Conversely, their left demonic composition R ;← S is defined by

{(x, z) | x (S ◦R) z ∧ ∀y ∈ Y (y S z ⇒ xR y)}.

2.2 References• Backhouse, Roland; van der Woude, Jaap (1993), “Demonic operators and monotype factors”, Mathematical

Structures in Computer Science 3 (4): 417–433, doi:10.1017/S096012950000030X, MR 1249420.

11

Chapter 3

Dense order

In mathematics, a partial order < on a set X is said to be dense if, for all x and y in X for which x < y, there is a z inX such that x < z < y.

3.1 Example

The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers. Onthe other hand, the ordinary ordering on the integers is not dense.

3.2 Generalizations

Any binary relation R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y areR-related. Formally:

∀x ∀y xRy ⇒ (∃z xRz ∧ zRy).

Every reflexive relation is dense. A strict partial order < is a dense order iff < is a dense relation.

3.3 See also• Dense set

• Dense-in-itself

• Kripke semantics

3.4 References• David Harel, Dexter Kozen, Jerzy Tiuryn, Dynamic logic, MIT Press, 2000, ISBN 0-262-08289-6, p. 6ff

12

Chapter 4

Dependence relation

Not to be confused with Dependency relation, which is a binary relation that is symmetric and reflexive.

In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.LetX be a set. A (binary) relation ◁ between an element a ofX and a subset S ofX is called a dependence relation,written a ◁ S , if it satisfies the following properties:

• if a ∈ S , then a ◁ S ;

• if a ◁ S , then there is a finite subset S0 of S , such that a ◁ S0 ;

• if T is a subset of X such that b ∈ S implies b ◁ T , then a ◁ S implies a ◁ T ;

• if a ◁ S but a ̸◁S − {b} for some b ∈ S , then b ◁ (S − {b}) ∪ {a} .

Given a dependence relation ◁ onX , a subset S ofX is said to be independent if a ̸◁S−{a} for all a ∈ S. If S ⊆ T, then S is said to span T if t ◁ S for every t ∈ T. S is said to be a basis of X if S is independent and S spans X.

Remark. If X is a non-empty set with a dependence relation ◁ , then X always has a basis with respect to ◁.Furthermore, any two bases ofX have the same cardinality.

4.1 Examples• Let V be a vector space over a field F. The relation ◁ , defined by υ ◁ S if υ is in the subspace spanned by S ,is a dependence relation. This is equivalent to the definition of linear dependence.

• LetK be a field extension of F.Define ◁ by α◁S if α is algebraic over F (S). Then ◁ is a dependence relation.This is equivalent to the definition of algebraic dependence.

4.2 See also• matroid

This article incorporates material from Dependence relation on PlanetMath, which is licensed under the Creative Com-mons Attribution/Share-Alike License.

13

Chapter 5

Dependency relation

For other uses, see Dependency (disambiguation).Not to be confused with Dependence relation, which is a generalization of the concept of linear dependence amongmembers of a vector space.

In mathematics and computer science, a dependency relation is a binary relation that is finite, symmetric, andreflexive; i.e. a finite tolerance relation. That is, it is a finite set of ordered pairs D , such that

• If (a, b) ∈ D then (b, a) ∈ D (symmetric)• If a is an element of the set on which the relation is defined, then (a, a) ∈ D (reflexive)

In general, dependency relations are not transitive; thus, they generalize the notion of an equivalence relation bydiscarding transitivity.Let Σ denote the alphabet of all the letters of D . Then the independency induced by D is the binary relation I

I = Σ× Σ \D

That is, the independency is the set of all ordered pairs that are not in D . The independency is symmetric andirreflexive.The pairs (Σ, D) and (Σ, I) , or the triple (Σ, D, I) (with I induced by D ) are sometimes called the concurrentalphabet or the reliance alphabet.The pairs of letters in an independency relation induce an equivalence relation on the free monoid of all possiblestrings of finite length. The elements of the equivalence classes induced by the independency are called traces, andare studied in trace theory.

5.1 Examples

Consider the alphabet Σ = {a, b, c} . A possible dependency relation is

D = {a, b} × {a, b} ∪ {a, c} × {a, c}= {a, b}2 ∪ {a, c}2

= {(a, b), (b, a), (a, c), (c, a), (a, a), (b, b), (c, c)}

The corresponding independency is

ID = {(b, c) , (c, b)}

Therefore, the letters b, c commute, or are independent of one another.

14

5.1. EXAMPLES 15

Aa

b

c

Chapter 6

Directed set

In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexiveand transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has anupper bound.[1] In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c.The notion defined above is sometimes called an upward directed set. A downward directed set is definedanalogously,[2] meaning when every doubleton is bounded below.[3] Some authors (and this article) assume that adirected set is directed upward, unless otherwise stated. Beware that other authors call a set directed if and only if itis directed both upward and downward.[4]

Directed sets are a generalization of nonempty totally ordered sets, that is, all totally ordered sets are directed sets(contrast partially ordered sets which need not be directed). Join semilattices (which are partially ordered sets) aredirected sets as well, but not conversely. Likewise, lattices are directed sets both upward and downward.In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limitused in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.

6.1 Equivalent definition

In addition to the definition above, there is an equivalent definition. A directed set is a set A with a preorder suchthat every finite subset of A has an upper bound. In this definition, the existence of an upper bound of the emptysubset implies that A is nonempty.

6.2 Examples

Examples of directed sets include:

• The set of natural numbers N with the ordinary order ≤ is a directed set (and so is every totally ordered set).

• Let D1 and D2 be directed sets. Then the Cartesian product set D1 × D2 can be made into a directed set bydefining (n1, n2) ≤ (m1, m2) if and only if n1 ≤ m1 and n2 ≤ m2. In analogy to the product order this is theproduct direction on the Cartesian product.

• It follows from previous example that the set N × N of pairs of natural numbers can be made into a directedset by defining (n0, n1) ≤ (m0, m1) if and only if n0 ≤ m0 and n1 ≤ m1.

• If x0 is a real number, we can turn the set R − {x0} into a directed set by writing a ≤ b if and only if|a − x0| ≥ |b − x0|. We then say that the reals have been directed towards x0. This is an example of a directedset that is not ordered (neither totally nor partially).

• A (trivial) example of a partially ordered set that is not directed is the set {a, b}, in which the only orderrelations are a ≤ a and b ≤ b. A less trivial example is like the previous example of the “reals directed towardsx0" but in which the ordering rule only applies to pairs of elements on the same side of x0.

16

6.3. CONTRAST WITH SEMILATTICES 17

• If T is a topological space and x0 is a point in T, we turn the set of all neighbourhoods of x0 into a directed setby writing U ≤ V if and only if U contains V.

• For every U: U ≤ U; since U contains itself.• For every U,V,W : if U ≤ V and V ≤ W, then U ≤ W; since if U contains V and V contains W then UcontainsW.

• For every U, V: there exists the set U ∩ V such that U ≤ U ∩ V and V ≤ U ∩ V; since both U and Vcontain U ∩ V.

• In a poset P, every lower closure of an element, i.e. every subset of the form {a| a in P, a ≤x} where x is afixed element from P, is directed.

6.3 Contrast with semilattices

Witness

Directed sets are a more general concept than (join) semilattices: every join semilattice is a directed set, as the joinor least upper bound of two elements is the desired c. The converse does not hold however, witness the directed set

18 CHAPTER 6. DIRECTED SET

{1000,0001,1101,1011,1111} ordered bitwise (e.g. 1000 ≤ 1011 holds, but 0001 ≤ 1000 does not, since in the lastbit 1 > 0), where {1000,0001} has three upper bounds but no least upper bound, cf. picture. (Also note that without1111, the set is not directed.)

6.4 Directed subsets

The order relation in a directed sets is not required to be antisymmetric, and therefore directed sets are not alwayspartial orders. However, the term directed set is also used frequently in the context of posets. In this setting, a subsetA of a partially ordered set (P,≤) is called a directed subset if it is a directed set according to the same partial order:in other words, it is not the empty set, and every pair of elements has an upper bound. Here the order relation on theelements of A is inherited from P; for this reason, reflexivity and transitivity need not be required explicitly.A directed subset of a poset is not required to be downward closed; a subset of a poset is directed if and only if itsdownward closure is an ideal. While the definition of a directed set is for an “upward-directed” set (every pair ofelements has an upper bound), it is also possible to define a downward-directed set in which every pair of elementshas a common lower bound. A subset of a poset is downward-directed if and only if its upper closure is a filter.Directed subsets are used in domain theory, which studies directed complete partial orders.[5] These are posets inwhich every upward-directed set is required to have a least upper bound. In this context, directed subsets againprovide a generalization of convergent sequences.

6.5 See also• Filtered category

• Centered set

• Linked set

6.6 Notes[1] Kelley, p. 65.

[2] Robert S. Borden (1988). A Course in Advanced Calculus. Courier Corporation. p. 20. ISBN 978-0-486-15038-3.

[3] Arlen Brown; Carl Pearcy (1995). An Introduction to Analysis. Springer. p. 13. ISBN 978-1-4612-0787-0.

[4] Siegfried Carl; Seppo Heikkilä (2010). Fixed Point Theory in Ordered Sets and Applications: From Differential and IntegralEquations to Game Theory. Springer. p. 77. ISBN 978-1-4419-7585-0.

[5] Gierz, p. 2.

6.7 References• J. L. Kelley (1955), General Topology.

• Gierz, Hofmann, Keimel, et al. (2003), Continuous Lattices and Domains, Cambridge University Press. ISBN0-521-80338-1.

Chapter 7

Equality (mathematics)

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions,asserting that the quantities have the same value or that the expressions represent the same mathematical object. Theequality between A and B is written A = B, and pronounced A equals B. The symbol "=" is called an "equals sign".

7.1 Etymology

The etymology of the word is from the Latin aequālis (“equal”, “like”, “comparable”, “similar”) from aequus (“equal”,“level”, “fair”, “just”).

7.2 Types of equalities

7.2.1 Identities

Main article: Identity (mathematics)

When A and Bmay be viewed as functions of some variables, then A = Bmeans that A and B define the same function.Such an equality of functions is sometimes called an identity. An example is (x + 1)2 = x2 + 2x + 1.

7.2.2 Equalities as predicates

When A and B are not fully specified or depend on some variables, equality is a proposition, which may be truefor some values and false for some other values. Equality is a binary relation, or, in other words, a two-argumentspredicate, which may produce a truth value (false or true) from its arguments. In computer programming, its com-putation from two expressions is known as comparison.

7.2.3 Congruences

In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties thatare considered. This is, in particular the case in geometry, where two geometric shapes are said equal when one maybe moved to coincide with the other. The word congruence is also used for this kind of equality.

7.2.4 Equations

An equation is the problem of finding values of some variables, called unknowns, for which the specified equalityis true. Equation may also refer to an equality relation that is satisfied only for the values of the variables that oneis interested on. For example x2 + y2 = 1 is the equation of the unit circle. There is no standard notation that

19

20 CHAPTER 7. EQUALITY (MATHEMATICS)

distinguishes an equation from an identity or other use of the equality relation: a reader has to guess an appropriateinterpretation from the semantic of expressions and the context.

7.2.5 Equivalence relations

Main article: Equivalence relation

Viewed as a relation, equality is the archetype of the more general concept of an equivalence relation on a set:those binary relations that are reflexive, symmetric, and transitive. The identity relation is an equivalence relation.Conversely, let R be an equivalence relation, and let us denote by xR the equivalence class of x, consisting of allelements z such that x R z. Then the relation x R y is equivalent with the equality xR = yR. It follows that equalityis the smallest equivalence relation on any set S, in the sense that it is the relation that has the smallest equivalenceclasses (every class is reduced to a single element).

7.3 Logical formalizations of equality

There are several formalizations of the notion of equality in mathematical logic, usually by means of axioms, such asthe first few Peano axioms, or the axiom of extensionality in ZF set theory.For example, Azriel Lévy gives as the five axioms for equality, first the three properties of an equivalence relation,and these two:

x = y ∧ x ∈ z⇒ y ∈ z, andx = y ∧ z ∈ x ⇒ z ∈ y.[1]

These extra two conditions allow substitution of equal quantities into complex expressions.There are also some logic systems that do not have any notion of equality. This reflects the undecidability of theequality of two real numbers defined by formulas involving the integers, the basic arithmetic operations, the logarithmand the exponential function. In other words, there cannot exist any algorithm for deciding such an equality.

7.4 Logical formulations

Equality is always defined such that things that are equal have all and only the same properties. Some people defineequality as congruence. Often equality is just defined as identity.A stronger sense of equality is obtained if some form of Leibniz’s law is added as an axiom; the assertion of this axiomrules out “bare particulars”—things that have all and only the same properties but are not equal to each other—whichare possible in some logical formalisms. The axiom states that two things are equal if they have all and only the sameproperties. Formally:

Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).

In this law, the connective “if and only if” can be weakened to “if"; the modified law is equivalent to the original.Instead of considering Leibniz’s law as an axiom, it can also be taken as the definition of equality. The property ofbeing an equivalence relation, as well as the properties given below, can then be proved: they become theorems. Ifa=b, then a can replace b and b can replace a.

7.5 Some basic logical properties of equality

The substitution property states:

• For any quantities a and b and any expression F(x), if a = b, then F(a) = F(b) (if both sides make sense, i.e.are well-formed).

7.6. RELATION WITH EQUIVALENCE AND ISOMORPHISM 21

In first-order logic, this is a schema, since we can't quantify over expressions like F (which would be a functionalpredicate).Some specific examples of this are:

• For any real numbers a, b, and c, if a = b, then a + c = b + c (here F(x) is x + c);

• For any real numbers a, b, and c, if a = b, then a − c = b − c (here F(x) is x − c);

• For any real numbers a, b, and c, if a = b, then ac = bc (here F(x) is xc);

• For any real numbers a, b, and c, if a = b and c is not zero, then a/c = b/c (here F(x) is x/c).

The reflexive property states:

For any quantity a, a = a.

This property is generally used in mathematical proofs as an intermediate step.The symmetric property states:

• For any quantities a and b, if a = b, then b = a.

The transitive property states:

• For any quantities a, b, and c, if a = b and b = c, then a = c.

The binary relation "is approximately equal" between real numbers or other things, even if more precisely defined,is not transitive (it may seem so at first sight, but many small differences can add up to something big). However,equality almost everywhere is transitive.Although the symmetric and transitive properties are often seen as fundamental, they can be proved, if the substitutionand reflexive properties are assumed instead.

7.6 Relation with equivalence and isomorphism

See also: Equivalence relation and Isomorphism

In some contexts, equality is sharply distinguished from equivalence or isomorphism.[2] For example, one may distin-guish fractions from rational numbers, the latter being equivalence classes of fractions: the fractions 1/2 and 2/4 aredistinct as fractions, as different strings of symbols, but they “represent” the same rational number, the same pointon a number line. This distinction gives rise to the notion of a quotient set.Similarly, the sets

{A,B,C} and {1, 2, 3}

are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of threeelements, and thus isomorphic, meaning that there is a bijection between them, for example

A 7→ 1,B 7→ 2,C 7→ 3.

However, there are other choices of isomorphism, such as

A 7→ 3,B 7→ 2,C 7→ 1,

and these sets cannot be identified without making such a choice – any statement that identifies them “dependson choice of identification”. This distinction, between equality and isomorphism, is of fundamental importance incategory theory, and is one motivation for the development of category theory.

22 CHAPTER 7. EQUALITY (MATHEMATICS)

7.7 See also• Equals sign

• Inequality

• Logical equality

• Extensionality

7.8 References[1] Azriel Lévy (1979) Basic Set Theory, page 358, Springer-Verlag

[2] (Mazur 2007)

• Mazur, Barry (12 June 2007),When is one thing equal to some other thing? (PDF)

• Mac Lane, Saunders; Garrett Birkhoff (1967). Algebra. American Mathematical Society.

Chapter 8

Equipollence (geometry)

In Euclidean geometry, equipollence is a binary relation between directed line segments. A line segment AB frompoint A to point B has the opposite direction to line segment BA. Two directed line segments are equipollent whenthey have the same length and direction.The concept of equipollent line segments was advanced by Giusto Bellavitis in 1835. Subsequently the term vectorwas adopted for a class of equipollent line segments. Bellavitis’s use of the idea of a relation to compare differentbut similar objects has become a common mathematical technique, particularly in the use of equivalence relations.Bellavitis used a special notation for the equipollence of segments AB and CD:

AB ≏ CD.

The following passages, translated by Michael J. Crowe, show the anticipation that Bellavitis had of vector concepts:

Equipollences continue to hold when one substitutes for the lines in them, other lines which are respec-tively equipollent to them, however they may be situated in space. From this it can be understood howany number and any kind of lines may be summed, and that in whatever order these lines are taken, thesame equipollent-sum will be obtained...

In equipollences, just as in equations, a line may be transferred from one side to the other, provided thatthe sign is changed...

Thus oppositely directed segments are negatives of each other: AB +BA ≏ 0.

The equipollence AB ≏ n.CD, where n stands for a positive number, indicates that AB is both parallelto and has the same direction as CD, and that their lengths have the relation expressed by AB = n.CD .

8.1 References

• Giusto Bellavitis (1835) “Saggio di applicasioni di un nuovo metodo di Geometria Analitica (Calculo delleequipollenze)", Annali delle Scienze del Regno Lombardo-Veneto, Padova 5: 244–59.

• Giusto Bellavitis (1854) Sposizione del Metodo della Equipollenze, link from Google Books.

• Michael J. Crowe (1967) A History of Vector Analysis, “Giusto Bellavitis and His Calculus of Equipollences”,pp 52–4, University of Notre Dame Press.

• Lena L. Severance (1930) The Theory of Equipollences; Method of Analytical Geometry of Sig. Bellavitis,link from HathiTrust.

23

24 CHAPTER 8. EQUIPOLLENCE (GEOMETRY)

8.2 External links• Axiomatic definition of equipollence

Chapter 9

Equivalence class

This article is about equivalency in mathematics. For equivalency in music, see equivalence class (music).In mathematics, when a set has an equivalence relation defined on its elements, there is a natural grouping of

Congruence is an example of an equivalence relation. The two triangles on the left are congruent, while the third and fourth trianglesare not congruent to any other triangle. Thus, the first two triangles are in the same equivalence class, while the third and fourthtriangles are in their own equivalence class.

elements that are related to one another, forming what are called equivalence classes. Notationally, given a set Xand an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X whichare equivalent to a. It follows from the definition of the equivalence relations that the equivalence classes form apartition of X. The set of equivalence classes is sometimes called the quotient set or the quotient space of X by ~and is denoted by X / ~.When X has some structure, and the equivalence relation is defined with some connection to this structure, thequotient set often inherits some related structure. Examples include quotient spaces in linear algebra, quotient spacesin topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.

9.1 Notation and formal definition

An equivalence relation is a binary relation ~ satisfying three properties:[1]

• For every element a in X, a ~ a (reflexivity),

• For every two elements a and b in X, if a ~ b, then b ~ a (symmetry)

• For every three elements a, b, and c in X, if a ~ b and b ~ c, then a ~ c (transitivity).

25

26 CHAPTER 9. EQUIVALENCE CLASS

The equivalence class of an element a is denoted [a] and is defined as the set

[a] = {x ∈ X | a ∼ x}

of elements that are related to a by ~. An alternative notation [a]R can be used to denote the equivalence class of theelement a, specifically with respect to the equivalence relation R. This is said to be the R-equivalence class of a.The set of all equivalence classes in Xwith respect to an equivalence relationR is denoted asX/R and called XmoduloR (or the quotient set of X by R).[2] The surjective map x 7→ [x] from X onto X/R, which maps each element to itsequivalence class, is called the canonical surjection or the canonical projection map.When an element is chosen (often implicitly) in each equivalence class, this defines an injective map called a section. Ifthis section is denoted by s, one has [s(c)] = c for every equivalence class c. The element s(c) is called a representativeof c. Any element of a class may be chosen as a representative of the class, by choosing the section appropriately.Sometimes, there is a section that is more “natural” than the other ones. In this case, the representatives are calledcanonical representatives. For example, in modular arithmetic, consider the equivalence relation on the integersdefined by a ~ b if a − b is a multiple of a given integer n, called the modulus. Each class contains a unique non-negative integer smaller than n, and these integers are the canonical representatives. The class and its representativeare more or less identified, as is witnessed by the fact that the notation a mod n may denote either the class or itscanonical representative (which is the remainder of the division of a by n).

9.2 Examples• If X is the set of all cars, and ~ is the equivalence relation “has the same color as.” then one particular equivalenceclass consists of all green cars. X/~ could be naturally identified with the set of all car colors (cardinality ofX/~ would be the number of all car colors).

• Let X be the set of all rectangles in a plane, and ~ the equivalence relation “has the same area as”. For eachpositive real number A there will be an equivalence class of all the rectangles that have area A.[3]

• Consider the modulo 2 equivalence relation on the set Z of integers: x ~ y if and only if their difference x − yis an even number. This relation gives rise to exactly two equivalence classes: one class consisting of all evennumbers, and the other consisting of all odd numbers. Under this relation [7], [9], and [1] all represent thesame element of Z/~.[4]

• Let X be the set of ordered pairs of integers (a,b) with b not zero, and define an equivalence relation ~ on Xaccording to which (a,b) ~ (c,d) if and only if ad = bc. Then the equivalence class of the pair (a,b) can beidentified with the rational number a/b, and this equivalence relation and its equivalence classes can be used togive a formal definition of the set of rational numbers.[5] The same construction can be generalized to the fieldof fractions of any integral domain.

• If X consists of all the lines in, say the Euclidean plane, and L ~M means that L andM are parallel lines, thenthe set of lines that are parallel to each other form an equivalence class as long as a line is considered parallelto itself. In this situation, each equivalence class determines a point at infinity.

9.3 Properties

Every element x of X is a member of the equivalence class [x]. Every two equivalence classes [x] and [y] are eitherequal or disjoint. Therefore, the set of all equivalence classes of X forms a partition of X: every element of X belongsto one and only one equivalence class.[6] Conversely every partition of X comes from an equivalence relation in thisway, according to which x ~ y if and only if x and y belong to the same set of the partition.[7]

It follows from the properties of an equivalence relation that

x ~ y if and only if [x] = [y].

In other words, if ~ is an equivalence relation on a set X, and x and y are two elements of X, then these statementsare equivalent:

9.4. GRAPHICAL REPRESENTATION 27

• x ∼ y

• [x] = [y]

• [x] ∩ [y] ̸= ∅.

9.4 Graphical representation

Any binary relation can be represented by a directed graph and symmetric ones, such as equivalence relations, byundirected graphs. If ~ is an equivalence relation on a set X, let the vertices of the graph be the elements of X andjoin vertices s and t if and only if s ~ t. The equivalence classes are represented in this graph by the maximal cliquesforming the connected components of the graph.[8]

9.5 Invariants

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 ifP(y) is true, then the property P is said to be an invariant of ~, or well-defined under the relation ~.A frequent particular case occurs when f is a function from X to another set Y; if f(x1) = f(x2) whenever x1 ~ x2,then f is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of finite groups. Some authors use “compatible with ~" or just “respects ~" instead of “invariantunder ~".Any function f : X → Y itself defines an equivalence relation on X according to which x1 ~ x2 if and only if f(x1)= f(x2). The equivalence class of x is the set of all elements in X which get mapped to f(x), i.e. the class [x] is theinverse image of f(x). This equivalence relation is known as the kernel of f.More generally, a function may map equivalent arguments (under an equivalence relation ~X on X) to equivalentvalues (under an equivalence relation ~Y on Y). Such a function is known as a morphism from ~X to ~Y .

9.6 Quotient space in topology

In topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relationon a topological space using the original space’s topology to create the topology on the set of equivalence classes.In abstract algebra, congruence relations on the underlying set of an algebra allow the algebra to induce an algebraon the equivalence classes of the relation, called a quotient algebra. In linear algebra, a quotient space is a vectorspace formed by taking a quotient group where the quotient homomorphism is a linear map. By extension, in abstractalgebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotientalgebra. However, the use of the term for the more general cases can as often be by analogy with the orbits of a groupaction.The orbits of a group action on a set may be called the quotient space of the action on the set, particularly when theorbits of the group action are the right cosets of a subgroup of a group, which arise from the action of the subgroupon the group by left translations, or respectively the left cosets as orbits under right translation.A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the sensesof topology, abstract algebra, and group actions simultaneously.Although the term can be used for any equivalence relation’s set of equivalence classes, possibly with further structure,the intent of using the term is generally to compare that type of equivalence relation on a setX either to an equivalencerelation that induces some structure on the set of equivalence classes from a structure of the same kind on X, or to theorbits of a group action. Both the sense of a structure preserved by an equivalence relation and the study of invariantsunder group actions lead to the definition of invariants of equivalence relations given above.

9.7 See also• Equivalence partitioning, a method for devising test sets in software testing based on dividing the possible

28 CHAPTER 9. EQUIVALENCE CLASS

program inputs into equivalence classes according to the behavior of the program on those inputs

• Homogeneous space, the quotient space of Lie groups.

• Transversal (combinatorics)

9.8 Notes[1] Devlin 2004, p. 122

[2] Wolf 1998, p. 178

[3] Avelsgaard 1989, p. 127

[4] Devlin 2004, p. 123

[5] Maddox 2002, pp. 77–78

[6] Maddox 2002, p.74, Thm. 2.5.15

[7] Avelsgaard 1989, p.132, Thm. 3.16

[8] Devlin 2004, p. 123

9.9 References• Avelsgaard, Carol (1989), Foundations for Advanced Mathematics, Scott Foresman, ISBN 0-673-38152-8

• Devlin, Keith (2004), Sets, Functions, and Logic: An Introduction to Abstract Mathematics (3rd ed.), Chapman& Hall/ CRC Press, ISBN 978-1-58488-449-1

• Maddox, Randall B. (2002), Mathematical Thinking and Writing, Harcourt/ Academic Press, ISBN 0-12-464976-9

• Morash, Ronald P. (1987), Bridge to Abstract Mathematics, Random House, ISBN 0-394-35429-X

• Wolf, Robert S. (1998), Proof, Logic and Conjecture: A Mathematician’s Toolbox, Freeman, ISBN 978-0-7167-3050-7

9.10 Further reading

This material is basic and can be found in any text dealing with the fundamentals of proof technique, such as any ofthe following:

• Sundstrom (2003), Mathematical Reasoning: Writing and Proof, Prentice-Hall

• Smith; Eggen; St.Andre (2006), A Transition to Advanced Mathematics (6th Ed.), Thomson (Brooks/Cole)

• Schumacher, Carol (1996), Chapter Zero: Fundamental Notions of Abstract Mathematics, Addison-Wesley,ISBN 0-201-82653-4

• O'Leary (2003), The Structure of Proof: With Logic and Set Theory, Prentice-Hall

• Lay (2001), Analysis with an introduction to proof, Prentice Hall

• Gilbert; Vanstone (2005), An Introduction to Mathematical Thinking, Pearson Prentice-Hall

• Fletcher; Patty, Foundations of Higher Mathematics, PWS-Kent

• Iglewicz; Stoyle, An Introduction to Mathematical Reasoning, MacMillan

• D'Angelo; West (2000), Mathematical Thinking: Problem Solving and Proofs, Prentice Hall

9.10. FURTHER READING 29

• Cupillari, The Nuts and Bolts of Proofs, Wadsworth

• Bond, Introduction to Abstract Mathematics, Brooks/Cole

• Barnier; Feldman (2000), Introduction to Advanced Mathematics, Prentice Hall

• Ash, A Primer of Abstract Mathematics, MAA

Chapter 10

Equivalence relation

This article is about the mathematical concept. For the patent doctrine, see Doctrine of equivalents.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 amember of one and only one cell of the partition. The intersection of any two different cells is empty; the union ofall the cells equals the original set. These cells are formally called equivalence classes.

10.1 Notation

Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalentwith respect to an equivalence relation R, the most common are "a ~ b" and "a ≡ b", which are used when R is theobvious relation being referenced, and variations of "a ~R b", "a ≡R b", or "aRb" otherwise.

10.2 Definition

A given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric andtransitive. Equivalently, for all a, b and c in X:

• a ~ a. (Reflexivity)

• if a ~ b then b ~ a. (Symmetry)

• if a ~ b and b ~ c then a ~ c. (Transitivity)

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

10.3 Examples

10.3.1 Simple example

Let the set {a, b, c} have the equivalence relation {(a, a), (b, b), (c, c), (b, c), (c, b)} . The following sets are equivalenceclasses of this relation:[a] = {a}, [b] = [c] = {b, c} .The set of all equivalence classes for this relation is {{a}, {b, c}} .

30

10.4. CONNECTIONS TO OTHER RELATIONS 31

10.3.2 Equivalence relations

The following are all equivalence relations:

• “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.

10.3.3 Relations that are not equivalences

• The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 doesnot 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 reflexiveand symmetric, but not transitive. (Example: The natural numbers 2 and 6 have a common factor greater than1, 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, butnot 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 equiv-alence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes canaccumulate to become a big change. However, if the approximation is defined asymptotically, for example bysaying 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.

10.4 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 thatis reflexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted forone another, a facility that is not available for equivalence related variables. The equivalence classes of anequivalence 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. Transitive and symmetric imply reflexive if and onlyif for all a ∈ X, there exists a b ∈ X such that a ~ b.

• 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 algebraicstructure, and which respects the additional structure. In general, congruence relations play the role of kernelsof homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many importantcases congruence relations have an alternative representation as substructures of the structure on which theyare defined. E.g. the congruence relations on groups correspond to the normal subgroups.

32 CHAPTER 10. EQUIVALENCE RELATION

10.5 Well-definedness under an equivalence relation

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 ifP(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) thenf is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of finite groups. The latter case with the function f can be expressed by a commutative triangle. Seealso 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.

10.6 Equivalence class, quotient set, partition

Let a, b ∈ X . Some definitions:

10.6.1 Equivalence class

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 equivalenceclass of X by ~. Let [a] := {x ∈ X | a ∼ x} denote the equivalence class to which a belongs. All elements of Xequivalent to each other are also elements of the same equivalence class.

10.6.2 Quotient set

Main article: Quotient set

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 spacefor the details.

10.6.3 Projection

Main article: Projection (relational algebra)

The projection of ~ is the function π : X → X/∼ defined by π(x) = [x] which maps elements of X into theirrespective equivalence classes by ~.

Theorem on projections:[1] Let the function f: X → B be such that a ~ b→ f(a) = f(b). Then there is aunique function g : X/~ → B, such that f = gπ. If f is a surjection and a ~ b ↔ f(a) = f(b), then g is abijection.

10.6.4 Equivalence kernel

The equivalence kernel of a function f is the equivalence relation ~ defined by x ∼ y ⇐⇒ f(x) = f(y) . Theequivalence kernel of an injection is the identity relation.

10.6.5 Partition

Main article: Partition of a set

10.7. FUNDAMENTAL THEOREM OF EQUIVALENCE RELATIONS 33

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 elementof P. Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their unionis 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, andvice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which isthe nth Bell number Bn:

Bn =1

e

∞∑k=0

kn

k!,

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

10.7 Fundamental theorem of equivalence relations

A key result links equivalence relations and partitions:[2][3][4]

• 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 belongsto 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 setof all possible equivalence relations on X and the set of all partitions of X.

10.8 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 acoarser 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 equivalenceclass 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 allpairs 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 orderrelation.

10.9 Generating equivalence relations

• Given any set X, there is an equivalence relation over the set [X→X] of all possible functions X→X. Two suchfunctions are deemed equivalent when their respective sets of fixpoints have the same cardinality, correspondingto 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].

34 CHAPTER 10. EQUIVALENCE RELATION

• An equivalence relation ~ on X is the equivalence kernel of its surjective projection π : X→ X/~.[5] Conversely,any surjection between sets determines a partition on its domain, the set of preimages of singletons in thecodomain. Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are threeequivalent ways of specifying the same thing.

• 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 anybinary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containingR. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x1, x2, ..., xn in Xsuch 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 equivalencerelation ~ 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 ofpaper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder soas to glue together its two open ends, resulting in a torus.

10.10 Algebraic structure

Much of mathematics is grounded in the study of equivalences, and order relations. Lattice theory captures themathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics asorder relations, the algebraic structure of equivalences is not as well known as that of orders. The former structuredraws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.

10.10.1 Group theory

Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalencerelations 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. Hencepermutation groups (also known as transformation groups) and the related notion of orbit shed light on the mathe-matical structure of equivalence relations.Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G denotethe set of bijective functions over A that preserve the partition structure of A: ∀x ∈ A ∀g ∈ G (g(x) ∈ [x]). Then thefollowing three connected theorems hold:[6]

• ~ partitions A into equivalence classes. (This is the Fundamental Theorem of Equivalence Relations,mentionedabove);

• Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the parti-tion‡;

• Given a transformation groupG overA, there exists an equivalence relation ~ over A, whose equivalence classesare the orbits of G.[7][8]

In sum, given an equivalence relation ~ over A, there exists a transformation group G over A whose orbits are theequivalence classes of A under ~.This transformation group characterisation of equivalence relations differs fundamentally from the way lattices char-acterize order relations. The arguments of the lattice theory operations meet and join are elements of some universe

10.11. EQUIVALENCE RELATIONS AND MATHEMATICAL LOGIC 35

A. Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a setof 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 cosetsof H in G. Interchanging a and b yields the left cosets.‡Proof.[9] Let function composition interpret group multiplication, and function inverse interpret group inverse. ThenG is a group under composition, meaning that ∀x ∈ A ∀g ∈ G ([g(x)] = [x]), because G satisfies the following fourconditions:

• G is closed under composition. The composition of any two elements of G exists, because the domain andcodomain of any element of G is A. Moreover, the composition of bijections is bijective;[10]

• Existence of identity function. 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.[11]

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 compositionpreserves the partitioning of A. □Related thinking can be found in Rosen (2008: chpt. 10).

10.10.2 Categories and groupoids

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing thisequivalence relation as follows. The objects are the elements of G, and for any two elements x and y of G, there existsa 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 graphdoes. Thus it is meaningful to speak of a “presentation of an equivalence relation,” i.e., a presentation of thecorresponding groupoid;

• Bundles of groups, group actions, sets, and equivalence relations can be regarded as special cases of the notionof groupoid, a point of view that suggests a number of analogies;

• In many contexts “quotienting,” and hence the appropriate equivalence relations often called congruences, areimportant. This leads to the notion of an internal groupoid in a category.[12]

10.10.3 Lattices

The possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called ConX 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 toits kernel ker f. Likewise, ker(ker) is an equivalence relation on X^X.

10.11 Equivalence relations and mathematical logic

Equivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation withexactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical forany 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 relationsnot satisfying the given property while satisfying all the other properties. Hence the three defining properties ofequivalence relations can be proved mutually independent by the following three examples:

36 CHAPTER 10. EQUIVALENCE RELATION

• 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.

10.12 Euclidean relations

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 relationwith”). By “relation” is meant a binary relation, in which aRb is generally distinct from bRa. An Euclidean relationthus comes in two forms:

(aRc ∧ bRc) → aRb (Left-Euclidean relation)(cRa ∧ cRb) → aRb (Right-Euclidean relation)

The following theorem connects Euclidean relations and equivalence relations:

Theorem If a relation is (left or right) Euclidean and reflexive, it is also symmetric and transitive.

Proof for a left-Euclidean relation

(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. □

with an analogous proof for a right-Euclidean relation. Hence an equivalence relation is a relation that is Euclideanand reflexive. The Elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed thereflexivity of equality too obvious to warrant explicit mention.

10.13 See also

• Partition of a set

• Equivalence class

• Up to

• Conjugacy class

• Topological conjugacy

10.14. NOTES 37

10.14 Notes[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] Karel Hrbacek & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, pages 29–32, Marcel Dekker

[5] Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 33, Th. 18. Chelsea.

[6] Rosen (2008), pp. 243-45. Less clear is §10.3 of Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press.

[7] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 202, Th. 6.

[8] Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. John Wiley & Sons: 114, Prop. 2.

[9] Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press: 246.

[10] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 22, Th. 6.

[11] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 24, Th. 7.

[12] Borceux, F. and Janelidze, G., 2001. Galois theories, Cambridge University Press, ISBN 0-521-80309-8

10.15 References• 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., Symmetriesin Physics: Philosophical Reflections. Cambridge Univ. Press: 422-433.

• Robert Dilworth and Crawley, Peter, 1973. Algebraic Theory of Lattices. Prentice Hall. Chpt. 12 discusseshow 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: Axiomsdefining equivalence, pp 48–50, John Wiley & Sons.

10.16 External links• Hazewinkel, Michiel, ed. (2001), “Equivalence relation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Bogomolny, A., "Equivalence Relationship" cut-the-knot. Accessed 1 September 2009

• Equivalence relation at PlanetMath

• Binary matrices representing equivalence relations at OEIS.

38 CHAPTER 10. EQUIVALENCE RELATION

Logical matrices of the 52 equivalence relations on a 5-element set (Colored fields, including those in light gray, stand for ones; whitefields for zeros.)

Chapter 11

Euclidean relation

In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity thatformalizes Euclid's “Common Notion 1” in The Elements: things which equal the same thing also equal one another.

11.1 Definition

A binary relation R on a set X is Euclidean (sometimes called right Euclidean) if it satisfies the following: for everya, b, c in X, if a is related to b and c, then b is related to c.[1]

To write this in predicate logic:

∀a, b, c ∈ X (aR b ∧ aR c → bR c).

Dually, a relation R on X is left Euclidean if for every a, b, c in X, if b is related to a and c is related to a, then b isrelated to c:

∀a, b, c ∈ X (bR a ∧ cR a → bR c).

11.2 Relation to transitivity

The property of being Euclidean is different from transitivity: both the Euclidean property and transitivity infer arelation between b and c from relations between a and b and between a and c, but with different argument orderingsin the relations. However, if a relation is symmetric, then the argument orders do not matter; thus a symmetric relationwith any one of these three properties (transitive, right Euclidean, left Euclidean) must have all three.[1]

If a relation is Euclidean and reflexive, then it must also be symmetric and hence transitive (following the previousparagraph), and so it must be an equivalence relation. Consequently, equivalence relations are exactly the reflexiveEuclidean relations.[1]

11.3 References[1] Fagin, Ronald (2003), Reasoning About Knowledge, MIT Press, p. 60, ISBN 978-0-262-56200-3.

39

Chapter 12

Exceptional isomorphism

In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism betweenmembers ai and bj of two families (usually infinite) of mathematical objects, that is not an example of a pattern ofsuch isomorphisms.[note 1] These coincidences are at times considered a matter of trivia,[1] but in other respects theycan give rise to other phenomena, notably exceptional objects.[1] In the below, coincidences are listed in all placesthey occur.

12.1 Groups

12.1.1 Finite simple groups

The exceptional isomorphisms between the series of finite simple groups mostly involve projective special lineargroups and alternating groups, and are:[1]

• L2(4) ∼= L2(5) ∼= A5, the smallest non-abelian simple group (order 60);

• L2(7) ∼= L3(2), the second-smallest non-abelian simple group (order 168) – PSL(2,7);

• L2(9) ∼= A6,

• L4(2) ∼= A8,

• PSU4(2) ∼= PSp4(3), between a projective special orthogonal group and a projective symplectic group.

12.1.2 Groups of Lie type

In addition to the aforementioned, there are some isomorphisms involving SL, PSL, GL, PGL, and the natural mapsbetween these. For example, the groups over F5 have a number of exceptional isomorphisms:

• PSL(2, 5) ∼= A5∼= I, the alternating group on five elements, or equivalently the icosahedral group;

• PGL(2, 5) ∼= S5, the symmetric group on five elements;

• SL(2, 5) ∼= 2 · A5∼= 2I, the double cover of the alternating group A5, or equivalently the binary icosahedral

group.

12.1.3 Alternating groups and symmetric groups

There are coincidences between alternating groups and small groups of Lie type:

• L2(4) ∼= L2(5) ∼= A5,

40

12.1. GROUPS 41

The compound of five tetrahedra expresses the exceptional isomorphism between the icosahedral group and the alternating group onfive letters.

• L2(9) ∼= Sp4(2)′ ∼= A6,

• Sp4(2) ∼= S6,

• L4(2) ∼= O6(+, 2)′ ∼= A8,

• O6(+, 2) ∼= S8.

These can all be explained in a systematic way by using linear algebra (and the action of Sn on affine n -space) todefine the isomorphism going from the right side to the left side. (The above isomorphisms for A8 and S8 are linkedvia the exceptional isomorphism SL4/µ2

∼= SO6 .) There are also some coincidences with symmetries of regularpolyhedra: the alternating group A5 agrees with the icosahedral group (itself an exceptional object), and the doublecover of the alternating group A5 is the binary icosahedral group.

12.1.4 Cyclic groups

Cyclic groups of small order especially arise in various ways, for instance:

• C2∼= {±1} ∼= O(1) ∼= Spin(1) ∼= Z∗ , the last being the group of units of the integers

42 CHAPTER 12. EXCEPTIONAL ISOMORPHISM

12.1.5 Spheres

The spheres S0, S1, and S3 admit group structures, which arise in various ways:

• S0 ∼= O(1) ,

• S1 ∼= SO(2) ∼= U(1) ∼= Spin(2) ,

• S3 ∼= Spin(3) ∼= SU(2) ∼= Sp(1) .

12.1.6 Coxeter groups

B2 C2≅ ≅

D3A3 ≅

≅

≅

E4A4 ≅

E5D5 ≅

≅

The exceptional isomorphisms of connected Dynkin diagrams.

12.2. LIE THEORY 43

There are some exceptional isomorphisms of Coxeter diagrams, yielding isomorphisms of the corresponding Coxetergroups and of polytopes realizing the symmetries. These are:

• A2 = I2(2) (2-simplex is regular 3-gon/triangle);

• BC2 = I2(4) (2-cube (square) = 2-cross-polytope (diamond) = regular 4-gon)

• A3 = D3 (3-simplex (tetrahedron) is 3-demihypercube (demicube), as per diagram)

• A1 = B1 = C1 (= D1?)

• D2 = A1 × A1

• A4 = E4

• D5 = E5

Closely related ones occur in Lie theory for Dynkin diagrams.

12.2 Lie theory

In low dimensions, there are isomorphisms among the classical Lie algebras and classical Lie groups called acciden-tal isomorphisms. For instance, there are isomorphisms between low-dimensional spin groups and certain classicalLie groups, due to low-dimensional isomorphisms between the root systems of the different families of simple Liealgebras, visible as isomorphisms of the corresponding Dynkin diagrams:

• Trivially, A0 = B0 = C0 = D0

• A1 = B1 = C1 , or sl2 ∼= so3 ∼= sp1

• B2 = C2, or so5 ∼= sp2

• D2 = A1 × A1, or so4 ∼= sl2 ⊕ sl2 ; note that these are disconnected, but part of the D-series

• A3 = D3 sl4 ∼= so6

• A4 = E4; the E-series usually starts at 6, but can be started at 4, yielding isomorphisms

• D5 = E5

Spin(1) = O(1)Spin(2) = U(1) = SO(2)Spin(3) = Sp(1) = SU(2)Spin(4) = Sp(1) × Sp(1)Spin(5) = Sp(2)Spin(6) = SU(4)

12.3 See also• Exceptional object

• Mathematical coincidence, for numerical coincidences

12.4 References[1] Because these series of objects are presented differently, they are not identical objects (do not have identical descriptions),

but turn out to describe the same object, hence one refers to this as an isomorphism, not an equality (identity).

44 CHAPTER 12. EXCEPTIONAL ISOMORPHISM

12.5 References[1] Wilson, Robert A. (2009), “Chapter 1: Introduction”, The finite simple groups, Graduate Texts in Mathematics 251 251,

Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792, 2007preprint; Chapter doi:10.1007/978-1-84800-988-2_1.

Chapter 13

Fiber (mathematics)

In mathematics, the term fiber (or fibre in British English) can have two meanings, depending on the context:

1. In naive set theory, the fiber of the element y in the set Y under a map f : X → Y is the inverse image of thesingleton {y} under f.

2. In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because,in general, not every point is closed.

13.1 Definitions

13.1.1 Fiber in naive set theory

Let f : X → Y be a map. The fiber of an element y ∈ Y , commonly denoted by f−1(y) , is defined as

f−1({y}) = {x ∈ X | f(x) = y}.

In various applications, this is also called:

• the inverse image of {y} under the map f

• the preimage of {y} under the map f

• the level set of the function f at the point y.

The term level set is only used if f maps into the real numbers and so y is simply a number. If f is a continuousfunction and if y is in the image of f, then the level set of y under f is a curve in 2D, a surface in 3D, and moregenerally a hypersurface of dimension d-1.

13.1.2 Fiber in algebraic geometry

In algebraic geometry, if f : X → Y is a morphism of schemes, the fiber of a point p in Y is the fibered productX ×Y Spec k(p) where k(p) is the residue field at p.

13.2 Terminological variance

The recommended practice is to use the terms fiber, inverse image, preimage, and level set as follows:

• the fiber of the element y under the map f

45

46 CHAPTER 13. FIBER (MATHEMATICS)

• the inverse image of the set {y} under the map f

• the preimage of the set {y} under the map f

• the level set of the function f at the point y.

By abuse of language, the following terminology is sometimes used but should be avoided:

• the fiber of the map f at the element y• the inverse image of the map f at the element y• the preimage of the map f at the element y• the level set of the point y under the map f.

13.3 See also• Fibration

• Fiber bundle

• Fiber product

• Image (category theory)

• Image (mathematics)

• Inverse relation

• Kernel (mathematics)

• Level set

• Preimage

• Relation

• Zero set

Chapter 14

Finitary relation

This article is about the set-theoretic notion of relation. For the common case, see binary relation.For other uses, see Relation (disambiguation).

In mathematics, a finitary relation has a finite number of “places”. In set theory and logic, a relation is a propertythat assigns truth values to k -tuples of individuals. Typically, the property describes a possible connection betweenthe components of a k -tuple. For a given set of k -tuples, a truth value is assigned to each k -tuple according towhether the property does or does not hold.An example of a ternary relation (i.e., between three individuals) is: "X was introduced to Y byZ ", where (X,Y, Z)is a 3-tuple of persons; for example, "Beatrice Wood was introduced to Henri-Pierre Roché by Marcel Duchamp" istrue, while "Karl Marx was introduced to Friedrich Engels by Queen Victoria" is false.

14.1 Informal introduction

Relation is formally defined in the next section. In this section we introduce the concept of a relation with a familiareveryday example. Consider the relation involving three roles that people might play, expressed in a statement of theform "X thinks that Y likes Z ". The facts of a concrete situation could be organized in a table like the following:Each row of the table records a fact or makes an assertion of the form "X thinks that Y likes Z ". For instance, thefirst row says, in effect, “Alice thinks that Bob likes Denise”. The table represents a relation S over the set P of peopleunder discussion:

P = {Alice, Bob, Charles, Denise}.

The data of the table are equivalent to the following set of ordered triples:

S = {(Alice, Bob, Denise), (Charles, Alice, Bob), (Charles, Charles, Alice), (Denise, Denise, Denise)}.

By a slight abuse of notation, it is usual to write S(Alice, Bob, Denise) to say the same thing as the first row ofthe table. The relation S is a ternary relation, since there are three items involved in each row. The relation itselfis a mathematical object defined in terms of concepts from set theory (i.e., the relation is a subset of the Cartesianproduct on {Person X, Person Y, Person Z}), that carries all of the information from the table in one neat package.Mathematically, then, a relation is simply an “ordered set”.The table for relation S is an extremely simple example of a relational database. The theoretical aspects of databasesare the specialty of one branch of computer science, while their practical impacts have become all too familiar in oureveryday lives. Computer scientists, logicians, and mathematicians, however, tend to see different things when theylook at these concrete examples and samples of the more general concept of a relation.For one thing, databases are designed to deal with empirical data, and experience is always finite, whereasmathematicsat the very least concerns itself with potential infinity. This difference in perspective brings up a number of ideas thatmay be usefully introduced at this point, if by no means covered in depth.

47

48 CHAPTER 14. FINITARY RELATION

14.2 Relations with a small number of “places”

The variable k giving the number of "places" in the relation, 3 for the above example, is a non-negative integer,called the relation’s arity, adicity, or dimension. A relation with k places is variously called a k -ary, a k -adic, ora k -dimensional relation. Relations with a finite number of places are called finite-place or finitary relations. Itis possible to generalize the concept to include infinitary relations between infinitudes of individuals, for exampleinfinite sequences; however, in this article only finitary relations are discussed, which will from now on simply becalled relations.Since there is only one 0-tuple, the so-called empty tuple ( ), there are only two zero-place relations: the one thatalways holds, and the one that never holds. They are sometimes useful for constructing the base case of an inductionargument. One-place relations are called unary relations. For instance, any set (such as the collection of Nobellaureates) can be viewed as a collection of individuals having some property (such as that of having been awardedthe Nobel prize). Two-place relations are called binary relations or, in the past, dyadic relations. Binary relations arevery common, given the ubiquity of relations such as:

• Equality and inequality, denoted by signs such as ' = ' and ' < ' in statements like ' 5 < 12 ';

• Being a divisor of, denoted by the sign ' | ' in statements like ' 13 | 143 ';

• Set membership, denoted by the sign ' ∈ ' in statements like ' 1 ∈ N '.

A k -ary relation is a straightforward generalization of a binary relation.

14.3 Formal definitionsWhen two objects, qualities, classes, or attributes, viewed together by the mind, are seen under some

connexion, that connexion is called a relation.—Augustus De Morgan[1]

The simpler of the two definitions of k-place relations encountered in mathematics is:Definition 1. A relation L over the sets X1, …, Xk is a subset of their Cartesian product, written L ⊆ X1 ×… × Xk.Relations are classified according to the number of sets in the defining Cartesian product, in other words, accordingto the number of terms following L. Hence:

• Lu denotes a unary relation or property;• Luv or uLv denote a binary relation;• Luvw denotes a ternary relation;• Luvwx denotes a quaternary relation.

Relations with more than four terms are usually referred to as k-ary or n-ary, for example, “a 5-ary relation”. A k-aryrelation is simply a set of k-tuples.The second definition makes use of an idiom that is common in mathematics, stipulating that “such and such is ann-tuple” in order to ensure that such and such a mathematical object is determined by the specification of n componentmathematical objects. In the case of a relation L over k sets, there are k + 1 things to specify, namely, the k sets plusa subset of their Cartesian product. In the idiom, this is expressed by saying that L is a (k + 1)-tuple.Definition 2. A relation L over the sets X1, …, Xk is a (k + 1)-tuple L = (X1, …, Xk, G(L)), where G(L) is a subsetof the Cartesian product X1 × … × Xk. G(L) is called the graph of L.Elements of a relation are more briefly denoted by using boldface characters, for example, the constant element a =(a1, …, ak) or the variable element x = (x1, …, xk).A statement of the form "a is in the relation L " or "a satisfies L " is taken to mean that a is in L under the firstdefinition and that a is in G(L) under the second definition.The following considerations apply under either definition:

14.4. HISTORY 49

• The sets Xj for j = 1 to k are called the domains of the relation. Under the first definition, the relation does notuniquely determine a given sequence of domains.

• If all of the domains Xj are the same set X, then it is simpler to refer to L as a k-ary relation over X.

• If any of the domains Xj is empty, then the defining Cartesian product is empty, and the only relation over sucha sequence of domains is the empty relation L = ∅ . Hence it is commonly stipulated that all of the domainsbe nonempty.

As a rule, whatever definition best fits the application at hand will be chosen for that purpose, and anything that fallsunder it will be called a relation for the duration of that discussion. If it becomes necessary to distinguish the twodefinitions, an entity satisfying the second definition may be called an embedded or included relation.If L is a relation over the domains X1, …, Xk, it is conventional to consider a sequence of terms called variables, x1,…, xk, that are said to range over the respective domains.Let a Boolean domain B be a two-element set, say, B = {0, 1}, whose elements can be interpreted as logical values,typically 0 = false and 1 = true. The characteristic function of the relation L, written ƒL or χ(L), is the Boolean-valuedfunction ƒL : X1 × … × Xk → B, defined in such a way that ƒL( x ) = 1 just in case the k-tuple x is in the relation L.Such a function can also be called an indicator function, particularly in probability and statistics, to avoid confusionwith the notion of a characteristic function in probability theory.It is conventional in appliedmathematics, computer science, and statistics to refer to a Boolean-valued function like ƒLas a k-place predicate. From the more abstract viewpoint of formal logic and model theory, the relation L constitutesa logical model or a relational structure that serves as one of many possible interpretations of some k-place predicatesymbol.Because relations arise in many scientific disciplines as well as in many branches of mathematics and logic, thereis considerable variation in terminology. This article treats a relation as the set-theoretic extension of a relationalconcept or term. A variant usage reserves the term “relation” to the corresponding logical entity, either the logicalcomprehension, which is the totality of intensions or abstract properties that all of the elements of the relation inextension have in common, or else the symbols that are taken to denote these elements and intensions. Further, somewriters of the latter persuasion introduce terms with more concrete connotations, like “relational structure”, for theset-theoretic extension of a given relational concept.

14.4 History

The logician Augustus De Morgan, in work published around 1860, was the first to articulate the notion of relationin anything like its present sense. He also stated the first formal results in the theory of relations (on De Morgan andrelations, see Merrill 1990). Charles Sanders Peirce restated and extended De Morgan’s results. Bertrand Russell(1938; 1st ed. 1903) was historically important, in that it brought together in one place many 19th century results onrelations, especially orders, by Peirce, Gottlob Frege, Georg Cantor, Richard Dedekind, and others. Russell and A.N. Whitehead made free use of these results in their Principia Mathematica.

14.5 Notes[1] De Morgan, A. (1858) “On the syllogism, part 3” in Heath, P., ed. (1966) On the syllogism and other logical writings.

Routledge. P. 119,

14.6 See also• Correspondence (mathematics)

• Functional relation

• Incidence structure

• Hypergraph

50 CHAPTER 14. FINITARY RELATION

• Logic of relatives

• Logical matrix

• Partial order

• Projection (set theory)

• Reflexive relation

• Relation algebra

• Sign relation

• Transitive relation

• Relational algebra

• Relational model

14.7 References• Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplificationof the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9,317–78, 1870. Reprinted, Collected Papers CP 3.45–149, Chronological Edition CE 2, 359–429.

• Ulam, S.M. and Bednarek, A.R. (1990), “On the Theory of Relational Structures and Schemata for ParallelComputation”, pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), Analogies Between Analogies: TheMathematical Reports of S.M. Ulam andHis Los Alamos Collaborators, University of California Press, Berkeley,CA.

14.8 Bibliography• Bourbaki, N. (1994) Elements of the History of Mathematics, John Meldrum, trans. Springer-Verlag.

• Carnap, Rudolf (1958) Introduction to Symbolic Logic with Applications. Dover Publications.

• Halmos, P.R. (1960) Naive Set Theory. Princeton NJ: D. Van Nostrand Company.

• Lawvere, F.W., and R. Rosebrugh (2003) Sets for Mathematics, Cambridge Univ. Press.

• Lucas, J. R. (1999) Conceptual Roots of Mathematics. Routledge.

• Maddux, R.D. (2006) Relation Algebras, vol. 150 in 'Studies in Logic and the Foundations of Mathematics’.Elsevier Science.

• Merrill, Dan D. (1990) Augustus De Morgan and the logic of relations. Kluwer.

• Peirce, C.S. (1984) Writings of Charles S. Peirce: A Chronological Edition, Volume 2, 1867-1871. PeirceEdition Project, eds. Indiana University Press.

• Russell, Bertrand (1903/1938) The Principles of Mathematics, 2nd ed. Cambridge Univ. Press.

• Suppes, Patrick (1960/1972) Axiomatic Set Theory. Dover Publications.

• Tarski, A. (1956/1983) Logic, Semantics, Metamathematics, Papers from 1923 to 1938, J.H. Woodger, trans.1st edition, Oxford University Press. 2nd edition, J. Corcoran, ed. Indianapolis IN: Hackett Publishing.

• Ulam, S.M. (1990) Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los AlamosCollaborators in A.R. Bednarek and Françoise Ulam, eds., University of California Press.

• R. Fraïssé, Theory of Relations (North Holland; 2000).

Chapter 15

Foundational relation

In set theory, a foundational relation on a set or proper class lets each nonempty subset admit a relational minimalelement.Formally, let (A, R) be a binary relation structure, where A is a class (set or proper class), and R is a binary relationdefined on A. Then (A, R) is a foundational relation if and only if any nonempty subset in A has a R-minimal element.In predicate logic,

(∀S)(S ⊆ A ∧ S ̸= ∅ ⇒ (∃x ∈ S)(S ∩R−1{x} = ∅)

), [1]

in which ∅ denotes the empty set, and R−1{x} denotes the class of the elements that precede x in the relation R. Thatis,

R−1{x} = {y|yRx}. [2]

Here x is an R-minimal element in the subset S, since none of its R-predecessors is in S.

15.1 See also• Binary relation

• Well-order

15.2 References[1] See Definition 6.21 in Zaring W.M., G. Takeuti (1971). Introduction to axiomatic set theory (2nd, rev. ed.). New York:

Springer-Verlag. ISBN 0387900241.

[2] See Theorem 6.19 and Definition 6.20 in Zaring W.M., G. Takeuti (1971). Introduction to axiomatic set theory (2nd, rev.ed.). New York: Springer-Verlag. ISBN 0387900241.

51

Chapter 16

Hypostatic abstraction

Hypostatic abstraction in mathematical logic, also known as hypostasis or subjectal abstraction, is a formal op-eration that transforms a predicate into a relation; for example “Honey is sweet” is transformed into “Honey possessessweetness”. The relation is created between the original subject and a new term that represents the property expressedby the original predicate.Hypostasis changes a propositional formula of the form X is Y to another one of the form X has the property of beingY or X has Y-ness. The logical functioning of the second object Y-ness consists solely in the truth-values of thosepropositions that have the corresponding concrete term Y as the predicate. The object of thought introduced in thisway may be called a hypostatic object and in some senses an abstract object and a formal object.The above definition is adapted from the one given by Charles Sanders Peirce (CP 4.235, “The Simplest Mathematics”(1902), in Collected Papers, CP 4.227–323). As Peirce describes it, the main point about the formal operation ofhypostatic abstraction, insofar as it operates on formal linguistic expressions, is that it converts an adjective or predicateinto an extra subject, thus increasing by one the number of “subject” slots -- called the arity or adicity -- of the mainpredicate.The transformation of “honey is sweet” into “honey possesses sweetness” can be viewed in several ways:

The grammatical trace of this hypostatic transformation is a process that extracts the adjective “sweet” from thepredicate “is sweet”, replacing it by a new, increased-arity predicate “possesses”, and as a by-product of the reaction,as it were, precipitating out the substantive “sweetness” as a second subject of the new predicate.The abstraction of hypostasis takes the concrete physical sense of “taste” found in “honey is sweet” and gives it formalmetaphysical characteristics in “honey has sweetness”.

52

16.1. SEE ALSO 53

16.1 See also

16.2 References• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6 (1931–1935), Charles Hartshorne and PaulWeiss, eds., vols. 7–8 (1958), Arthur W. Burks, ed., Harvard University Press, Cambridge, MA.

16.3 External links• J. Jay Zeman, Peirce on Abstraction

54 CHAPTER 16. HYPOSTATIC ABSTRACTION

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56 CHAPTER 16. HYPOSTATIC ABSTRACTION

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