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Contents

1 Binary relation 11.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Direct limit 112.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Algebraic objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Direct limit over a direct system in a category . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Related constructions and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Limit (mathematics) 143.1 Limit of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Limit of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Limit as standard part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Convergence and xed point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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3.5 Topological net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Linked set 184.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Preorder 195.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.5 Number of preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.6 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Reexive relation 236.1 Related terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.3 Number of reexive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.4 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 Sequence 277.1 Examples and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7.1.1 Important examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.1.2 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.1.3 Specifying a sequence by recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.2 Formal denition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2.2 Finite and innite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.2.3 Increasing and decreasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.2.4 Bounded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.2.5 Other types of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.3 Limits and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.3.1 Denition of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.3.2 Applications and important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.3.3 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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7.4 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.5 Use in other elds of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.5.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.5.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.5.3 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.5.4 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.5.5 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.5.6 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.5.7 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.6 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.7 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.8 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8 Transitive relation 408.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

8.3.1 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.3.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.3.3 Properties that require transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

8.4 Counting transitive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.6.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.6.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 43

8.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Chapter 1

Binary relation

Relation (mathematics)" redirects here. For a more general notion of relation, see nitary 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", anddivides 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 dened 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 onZZZ 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 Russells paradox.

1.1 Formal denition

A binary relation R is usually dened 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 dened 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 specied or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

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2 CHAPTER 1. BINARY RELATION

1.1.1 Is a relation more than its graph?According to the denition above, two relations with identical graphs but dierent domains or dierent codomainsare considered dierent. For example, ifG = f(1; 2); (1; 3); (2; 7)g , 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 dened as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then dened as the set of all x such that there exists at least oney such that (x; y) 2 R , the range of R is dened as the set of all y such that there exists at least one x such that(x; y) 2 R , and the eld of R is the union of its domain and its range.[2][3][4]A special case of this dierence in points of view applies to the notion of function. Many authors insist on distin-guishing between a functions 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 rst components. This dierence in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivityor being ontoas 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 denitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodenitions usually matters only in very formal contexts, like category theory.

1.1.2 ExampleExample: 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 rst 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 RJ means that the ball is owned by John.Two dierent relations could have the same graph. For example: the relation

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

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

1.2 Special types of binary relationsSome important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be dierent 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-denite[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 dierentfrom the denition 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 DifunctionalLess commonly encountered is the notion of difunctional (or regular) relation, dened as a relation R such thatR=RR1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element xX to a set xR = { yY| xRy }.[11] This set is sometimes called the successor neighborhood of x in R; one can dene 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 dene 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 justied 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 setIf 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:

reexive: for all x in X it holds that xRx. For example, greater than or equal to () is a reexive relation butgreater than (>) is not.

irreexive (or strict): for all x in X it holds that not xRx. For example, > is an irreexive relation, but is not. coreexive: for all x and y in X it holds that if xRy then x = y. An example of a coreexive 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 reexive and coreexive relation.

The previous 3 alternatives are far from being exhaustive; e.g. the red relation y=x2 from theabove picture is neither irreexive, nor coreexive, nor reexive, 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 denition 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 irreexive.[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 irreexive 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 denition for total is dierent 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 reexive 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 denition 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 reexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reexive. A relation that is only symmetric and transitive (without necessarily beingreexive) is called a partial equivalence relation.A relation that is reexive, 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 relationsIf 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, dened as R S = { (x, y) | (x, y) R or (x, y) S }. For example, is the union of >and =.

Intersection: R S X Y, dened 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), dened 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, dened 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 ().

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

Reexive closure: R =, dened as R = = { (x, x) | x X } R or the smallest reexive relation over X containingR. This can be proven to be equal to the intersection of all reexive relations containing R.

Reexive reduction: R , dened as R = R \ { (x, x) | x X } or the largest irreexive relation over Xcontained in R.

Transitive closure: R +, dened 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 , dened as a minimal relation having the same transitive closure as R. Reexive transitive closure: R *, dened as R * = (R +) =, the smallest preorder containing R. Reexive transitive symmetric closure: R , dened as the smallest equivalence relation over X containingR.

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

The complement S is dened 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 reexive relation is irreexive 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 RestrictionThe 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 reexive, irreexive, 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 systemsVarious 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 nitary relations (and in practice also niteand 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 classesCertain mathematical relations, such as equal to, member of, and subset of, cannot be understood to be binaryrelations as dened 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 specic 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 dened 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 MorseKelley 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 modication 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 denition one can for instance dene 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 irreexive relations is the same as that of reexive relations. The number of strict partial orders (irreexive 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 ane spaces) is in bijection with isomorphy

dependency relation, a nite, symmetric, reexive relation. independency relation, a symmetric, irreexive relation which is the complement of some dependency relation.

1.8 See also Conuence (term rewriting) Hasse diagram Incidence structure Logic of relatives Order theory Triadic relation

1.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 13301331. 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. 299300. 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. xxi. ISBN978-1-4020-6164-6.

[7] Kilp, Knauer and Mikhalev: p. 3. The same four denitions 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. 1921. 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. 2122. ISBN 978-3-89675-629-9.

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

[9] Ms, Stephan (2007), Reasoning on Spatial Semantic Integrity Constraints, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australiia, September 1923, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285302, 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 Bchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 3537. ISBN 978-1-4613-8853-1.

[16] M. E. Mller (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] Flaka, V.; Jeek, 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: 3033..

[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 andGraphs, 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

Direct limit

In mathematics, a direct limit (also called inductive limit) is a colimit of a directed family of objects. We willrst give the denition for algebraic structures like groups and modules, and then the general denition, which can beused in any category.

2.1 Formal denition

2.1.1 Algebraic objects

See also: Directed set and Filtered category

In this section objects are understood to be sets with a given algebraic structure such as groups, rings, modules (over axed ring), algebras (over a xed eld), etc. With this in mind, homomorphisms are understood in the correspondingsetting (group homomorphisms, etc.).Start with the denition of a direct system of objects and homomorphisms. Let hI;i be a directed set. LetfAi : i 2 Ig be a family of objects indexed by I and fij : Ai ! Aj be a homomorphism for all i j with thefollowing properties:

1. fii is the identity of Ai , and

2. fik = fjk fij for all i j k .

Then the pair hAi; fiji is called a direct system over I .The underlying set of the direct limit, A , of the direct system hAi; fiji is dened as the disjoint union of the Ai 'smodulo a certain equivalence relation :

lim!Ai =Gi

Ai

:

Here, if xi 2 Ai and xj 2 Aj , xi xj if there is some k 2 I such that fik(xi) = fjk(xj) . Heuristically, twoelements in the disjoint union are equivalent if and only if they eventually become equal in the direct system. Anequivalent formulation that highlights the duality to the inverse limit is that an element is equivalent to all its imagesunder the maps of the directed system, i.e. xi fik(xi) .One naturally obtains from this denition canonical morphisms i : Ai ! A sending each element to its equivalenceclass. The algebraic operations on A are dened via these maps in the obvious manner.An important property is that taking direct limits in the category of modules is an exact functor.

11

12 CHAPTER 2. DIRECT LIMIT

2.1.2 Direct limit over a direct system in a categoryThe direct limit can be dened in an arbitrary category C by means of a universal property. Let hXi; fiji be a directsystem of objects and morphisms in C (same denition as above). The direct limit of this system is an objectX in Ctogether with morphisms i : Xi ! X satisfying i = j fij . The pair hX;ii must be universal in the sensethat for any other such pair hY; ii there exists a unique morphism u : X ! Y making the diagram

commute for all i, j. The direct limit is often denoted

X = lim!Xiwith the direct system hXi; fiji being understood.Unlike for algebraic objects, the direct limit may not exist in an arbitrary category. If it does, however, it is unique ina strong sense: given another direct limit X there exists a unique isomorphism X X commuting with the canonicalmorphisms.We note that a direct system in a category C admits an alternative description in terms of functors. Any directed posethI;i can be considered as a small category I where the morphisms consist of arrows i ! j if and only if i j .A direct system is then just a covariant functor I ! C . In this case a direct limit is a colimit.

2.2 Examples A collection of subsets Mi of a set M can be partially ordered by inclusion. If the collection is directed, itsdirect limit is the unionSMi .

Let I be any directed set with a greatest element m. The direct limit of any corresponding direct system isisomorphic to Xm and the canonical morphism m: Xm X is an isomorphism.

Let p be a prime number. Consider the direct system composed of the groups Z/pnZ and the homomorphismsZ/pnZ Z/pn+1Z induced by multiplication by p. The direct limit of this system consists of all the roots ofunity of order some power of p, and is called the Prfer group Z(p).

Let F be a C-valued sheaf on a topological space X. Fix a point x in X. The open neighborhoods of x form adirected poset ordered by inclusion (U V if and only if U contains V). The corresponding direct system is(F(U), rU,V) where r is the restriction map. The direct limit of this system is called the stalk of F at x, denotedFx. For each neighborhood U of x, the canonical morphism F(U) Fx associates to a section s of F over Uan element sx of the stalk Fx called the germ of s at x.

Direct limits in the category of topological spaces are given by placing the nal topology on the underlyingset-theoretic direct limit.

Direct limits are linked to inverse limits via

Hom(lim!Xi; Y ) = lim Hom(Xi; Y ):

2.3. RELATED CONSTRUCTIONS AND GENERALIZATIONS 13

Consider a sequence {An, n} where An is a C*-algebra and n : An An is a *-homomorphism. TheC*-analog of the direct limit construction gives a C*-algebra satisfying the universal property above.

2.3 Related constructions and generalizationsThe categorical dual of the direct limit is called the inverse limit (or projective limit). More general concepts arethe limits and colimits of category theory. The terminology is somewhat confusing: direct limits are colimits whileinverse limits are limits.

2.4 See also Inverse, or projective limit

2.5 References Bourbaki, Nicolas (1968), Elements of mathematics. Theory of sets, Translated from the French, Paris: Her-mann, MR 0237342.

Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (2nded.), Springer-Verlag.

Chapter 3

Limit (mathematics)

This is an overview of the idea of a limit in mathematics. For specic uses of a limit, see Limit of a sequence andLimit of a function.

In mathematics, a limit is the value that a function or sequence approaches as the input or index approaches somevalue.[1] Limits are essential to calculus (and mathematical analysis in general) and are used to dene continuity,derivatives, and integrals.The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closelyrelated to limit and direct limit in category theory.In formulas, a limit is usually written as

limn!c f(n) = L

and is read as the limit of f of n as n approaches c equals L". Here lim indicates limit, and the fact that functionf(n) approaches the limit L as n approaches c is represented by the right arrow (), as in

f(n)! L :

3.1 Limit of a functionMain article: Limit of a function

Suppose f is a real-valued function and c is a real number. The expression

limx!c f(x) = L

means that f(x) can be made to be as close to L as desired by making x suciently close to c. In that case, the aboveequation can be read as the limit of f of x, as x approaches c, is L".Augustin-Louis Cauchy in 1821,[2] followed by Karl Weierstrass, formalized the denition of the limit of a functionas the above denition, which became known as the (, )-denition of limit in the 19th century. The denition uses (the lowercase Greek letter epsilon) to represent any small positive number, so that "f(x) becomes arbitrarily closeto L" means that f(x) eventually lies in the interval (L , L + ), which can also be written using the absolute valuesign as |f(x) L| < .[2] The phrase as x approaches c" then indicates that we refer to values of x whose distancefrom c is less than some positive number (the lower case Greek letter delta)that is, values of x within either (c , c) or (c, c + ), which can be expressed with 0 < |x c| < . The rst inequality means that the distance betweenx and c is greater than 0 and that x c, while the second indicates that x is within distance of c.[2]

14

3.2. LIMIT OF A SEQUENCE 15

Note that the above denition of a limit is true even if f(c) L. Indeed, the function f need not even be dened at c.For example, if

f(x) =x2 1x 1

then f(1) is not dened (see division by zero), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches2:Thus, f(x) can be made arbitrarily close to the limit of 2 just by making x suciently close to 1.In other words, limx!1 x

21x1 = 2

This can also be calculated algebraically, as x21x1 =(x+1)(x1)

x1 = x+ 1 for all real numbers x 1.

Now since x + 1 is continuous in x at 1, we can now plug in 1 for x, thus limx!1 x21x1 = 1 + 1 = 2 .

In addition to limits at nite values, functions can also have limits at innity. For example, consider

f(x) =2x 1x

f(100) = 1.9900 f(1000) = 1.9990 f(10000) = 1.99990

As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 asone could wish just by picking x suciently large. In this case, the limit of f(x) as x approaches innity is 2. Inmathematical notation,

limx!1

2x 1x

= 2:

3.2 Limit of a sequenceMain article: Limit of a sequence

Consider the following sequence: 1.79, 1.799, 1.7999,... It can be observed that the numbers are approaching 1.8,the limit of the sequence.Formally, suppose a1, a2, ... is a sequence of real numbers. It can be stated that the real number L is the limit of thissequence, namely:

limn!1 an = L

which is read as

The limit of an as n approaches innity equals L"

to mean

For every real number > 0, there exists a natural number N such that for all n > N, we have |an L| 0), we can nd a natural number N such that all terms(aN+1, aN+2, ...) are further closer to L (within of L). [1] This is often written more compactly using symbols. Forinstance,

for all > 0, there exists a natural number N such that L < an < L+ for all n N.

In even more compact notation

8 > 0; 9N 2 N s.t. 8n N; jan Lj < :

The dierence in the denitions of convergence for (one-sided) sequences in complex analysis and metric spaces isthat the absolute value |an L| is interpreted as the distance in the complex plane (

pzz ), and the distance under

the appropriate metric, respectively.

7.3.2 Applications and important results

Important results for convergence and limits of (one-sided) sequences of real numbers include the following. Theseequalities are all true at least when both sides exist. For a discussion of when the existence of the limit on one sideimplies the existence of the other see a real analysis text such as can be found in the references.[1][5]

The limit of a sequence is unique.

limn!1(an bn) = limn!1 an limn!1 bn limn!1 can = c limn!1 an limn!1(anbn) = (limn!1 an)(limn!1 bn)

limn!1 anbn = limn!1 anlimn!1 bn provided limn!1 bn 6= 0

limn!1 apn = [limn!1 an]p

If an bn for all n greater than some N, then limn!1 an limn!1 bn .

(Squeeze Theorem) If an cn bn for all n >N, and limn!1 an = limn!1 bn = L , then limn!1 cn = L.

If a sequence is bounded and monotonic then it is convergent.

A sequence is convergent if and only if every subsequence is convergent.

34 CHAPTER 7. SEQUENCE

The plot of a Cauchy sequence (X), shown in blue, asX versus n. Visually, we see that the sequence appears to be converging to thelimit zero as the terms in the sequence become closer together as n increases. In the real numbers every Cauchy sequence convergesto some limit.

7.3.3 Cauchy sequencesMain article: Cauchy sequenceA Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion ofa Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. Oneparticularly important result in real analysis is Cauchy characterization of convergence for sequences:

In the real numbers, a sequence is convergent if and only if it is Cauchy.

In contrast, in the rational numbers, e.g. the sequence dened by x1 = 1 and xn = xn + 2/xn/2 is Cauchy, but has norational limit, cf. here.

7.4 SeriesMain article: Series (mathematics)

A series is, informally speaking, the sum of the terms of a sequence. That is, adding the rst N terms of a (one-sided)sequence forms the Nth term of another sequence, called a series. Thus the N series of the sequence (a) results inanother sequence (SN) given by:

S1 = a1

S2 = a1 + a2

S3 = a1 + a2 + a3

... ...SN = a1 + a2 + a3 +

... ...

We can also write the nth term of the series as

7.5. USE IN OTHER FIELDS OF MATHEMATICS 35

SN =NXn=1

an:

Then the concepts used to talk about sequences, such as convergence, carry over to series (the sequence of partialsums) and the properties can be characterized as properties of the underlying sequences (such as (an) in the lastexample). The limit, if it exists, of an innite series (the series created from an innite sequence) is written as

limN!1

SN =

1Xn=1

an:

7.5 Use in other elds of mathematics

7.5.1 TopologySequence play an important role in topology, especially in the study of metric spaces. For instance:

A metric space is compact exactly when it is sequentially compact. A function from ametric space to another metric space is continuous exactly when it takes convergent sequencesto convergent sequences.

A metric space is a connected space if, whenever the space is partitioned into two sets, one of the two setscontains a sequence converging to a point in the other set.

A topological space is separable exactly when there is a dense sequence of points.

Sequences can be generalized to nets or lters. These generalizations allow one to extend some of the above theoremsto spaces without metrics.

Product topology

A product space of a sequence of topological spaces is the cartesian product of the spaces equipped with a naturaltopology called the product topology.More formally, given a sequence of spaces fXig , dene X such that

X :=Yi2I

Xi;

is the set of sequences fxig where each xi is an element of Xi . Let the canonical projections be written as pi :X Xi. Then the product topology on X is dened to be the coarsest topology (i.e. the topology with the fewestopen sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonotopology.

7.5.2 AnalysisIn analysis, when talking about sequences, one will generally consider sequences of the form

(x1; x2; x3; : : : ) or (x0; x1; x2; : : : )

which is to say, innite sequences of elements indexed by natural numbers.

36 CHAPTER 7. SEQUENCE

It may be convenient to have the sequence start with an index dierent from 1 or 0. For example, the sequence denedby xn = 1/log(n) would be dened only for n 2. When talking about such innite sequences, it is usually sucient(and does not change much for most considerations) to assume that the members of the sequence are dened at leastfor all indices large enough, that is, greater than some given N.The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This typecan be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are oftenfunction spaces. Even more generally, one can study sequences with elements in some topological space.

Sequence spaces

Main article: Sequence space

A sequence space is a vector space whose elements are innite sequences of real or complex numbers. Equivalently, itis a function space whose elements are functions from the natural numbers to the eldK of real or complex numbers.The set of all such functions is naturally identied with the set of all possible innite sequences with elements in K,and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalarmultiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped witha norm, or at least the structure of a topological vector space.The most important sequences spaces in analysis are the p spaces, consisting of the p-power summable sequences,with the p-norm. These are special cases of Lp spaces for the counting measure on the set of natural numbers.Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectivelydenoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology of pointwiseconvergence, under which it becomes a special kind of Frchet space called FK-space.

7.5.3 Linear algebraSequences over a eld may also be viewed as vectors in a vector space. Specically, the set of F-valued sequences(where F is a eld) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers.

7.5.4 Abstract algebraAbstract algebra employs several types of sequences, including sequences of mathematical objects such as groups orrings.

Free monoid

Main article: Free monoid

If A is a set, the free monoid over A (denoted A*, also called Kleene star of A) is a monoid containing all the nitesequences (or strings) of zero or more elements of A, with the binary operation of concatenation. The free semigroupA+ is the subsemigroup of A* containing all elements except the empty sequence.

Exact sequences

Main article: Exact sequence

In the context of group theory, a sequence

G0f1! G1 f2! G2 f3! fn! Gn

of groups and group homomorphisms is called exact if the image (or range) of each homomorphism is equal to thekernel of the next:

7.6. TYPES 37

im(fk) = ker(fk+1)

Note that the sequence of groups and homomorphisms may be either nite or innite.A similar denition can be made for certain other algebraic structures. For example, one could have an exact sequenceof vector spaces and linear maps, or of modules and module homomorphisms.

Spectral sequences

Main article: Spectral sequence

In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groupsby taking successive approximations. Spectral sequences are a generalization of exact sequences, and since theirintroduction by Jean Leray (1946), they have become an important research tool, particularly in homotopy theory.

7.5.5 Set theoryAn ordinal-indexed sequence is a generalization of a sequence. If is a limit ordinal and X is a set, an -indexedsequence of elements of X is a function from to X. In this terminology an -indexed sequence is an ordinarysequence.

7.5.6 ComputingAutomata or nite statemachines can typically be thought of as directed graphs, with edges labeled using some specicalphabet, . Most familiar types of automata transition from state to state by reading input letters from , followingedges with matching labels; the ordered input for such an automaton forms a sequence called a word (or input word).The sequence of states encountered by the automaton when processing a word is called a run. A nondeterministicautomaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some inputletter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence ofsingle states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally usedto mean the latter.

7.5.7 StreamsInnite sequences of digits (or characters) drawn from a nite alphabet are of particular interest in theoretical com-puter science. They are often referred to simply as sequences or streams, as opposed to nite strings. Innite binarysequences, for instance, are innite sequences of bits (characters drawn from the alphabet {0, 1}). The set C = {0,1} of all innite, binary sequences is sometimes called the Cantor space.An innite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to1 if and only if the n th string (in shortlex order) is in the language. This representation is useful in the diagonalizationmethod for proofs.[6]

7.6 Types 1-sequence Arithmetic progression Cauchy sequence Farey sequence Fibonacci sequence

38 CHAPTER 7. SEQUENCE

Geometric progression

Look-and-say sequence

ThueMorse sequence

7.7 Related concepts List (computing)

Ordinal-indexed sequence

Recursion (computer science)

Tuple

Set theory

7.8 Operations Cauchy product

Limit of a sequence

7.9 See also Enumeration

Net (topology) (a generalization of sequences)

On-Line Encyclopedia of Integer Sequences

Permutation

Recurrence relation

Sequence space

Set (mathematics)

7.10 References[1] Gaughan, Edward. 1.1 Sequences and Convergence. Introduction to Analysis. AMS (2009). ISBN 0-8218-4787-2.

[2] Edward B. Sa & Arthur David Snider (2003). Chapter 2.1. Fundamentals of Complex Analysis. ISBN 01-390-7874-6.

[3] James R. Munkres. Chapters 1&2. Topology. ISBN 01-318-1629-2.

[4] Lando, Sergei K. 7.4 Multiplicative sequences. Lectures on generating functions. AMS. ISBN 0-8218-3481-9.

[5] Dawikins, Paul. Series and Sequences. Pauls Online Math Notes/Calc II (notes). Retrieved 18 December 2012.

[6] Oazer, Kemal. FORMAL LANGUAGES, AUTOMATAANDCOMPUTATION: DECIDABILITY (PDF). cmu.edu.Carnegie-Mellon University. Retrieved 24 April 2015.

7.11. EXTERNAL LINKS 39

7.11 External links The dictionary denition of sequence at Wiktionary Hazewinkel, Michiel, ed. (2001), Sequence, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

The On-Line Encyclopedia of Integer Sequences Journal of Integer Sequences (free) Sequence at PlanetMath.org.

Chapter 8

Transitive relation

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b,and b is in turn related to an element c, then a is also related to c. Transitivity is a key property of both partial orderrelations and equivalence relations.

8.1 Formal denitionIn terms of set theory, the transitive relation can be dened as:

8a; b; c 2 X : (aRb ^ bRc)) aRc

8.2 ExamplesFor example, is greater than, is at least as great as, and is equal to (equality) are transitive relations:

whenever A > B and B > C, then also A > Cwhenever A B and B C, then also A Cwhenever A = B and B = C, then also A = C.

On the other hand, is the mother of is not a transitive relation, because if Alice is the mother of Brenda, and Brendais the mother of Claire, then Alice is not the mother of Claire. What is more, it is antitransitive: Alice can never bethe mother of Claire.Then again, in biology we often need to consider motherhood over an arbitrary number of generations: the relationis a matrilinear ancestor of. This is a transitive relation. More precisely, it is the transitive closure of the relationis the mother of.More examples of transitive relations:

is a subset of (set inclusion) divides (divisibility) implies (implication)

8.3 Properties

40

8.4. COUNTING TRANSITIVE RELATIONS 41

8.3.1 Closure properties

The converse of a transitive relation is always transitive: e.g. knowing that is a subset of is transitive and is asuperset of is its converse, we can conclude that the latter is transitive as well.The intersection of two transitive relations is always transitive: knowing that was born before and has the same rstname as are transitive, we can conclude that was born before and also has the same rst name as is also transitive.The union of two transitive relations is not always transitive. For instance was born before or has the same rst nameas is not generally a transitive relation.The complement of a transitive relation is not always transitive. For instance, while equal to is transitive, not equalto is only transitive on sets with at most one element.

8.3.2 Other properties

A transitive relation is asymmetric if and only if it is irreexive.[1]

8.3.3 Properties that require transitivity Preorder a reexive transitive relation partial order an antisymmetric preorder Total preorder a total preorder Equivalence relation a symmetric preorder Strict weak ordering a strict partial order in which incomparability is an equivalence relation Total ordering a total, antisymmetric transitive relation

8.4 Counting transitive relationsNo general formula that counts the number of transitive relations on a nite set (sequence A006905 in OEIS) isknown.[2] However, there is a formula for nding the number of relations that are simultaneously reexive, symmet-ric, and transitive in other words, equivalence relations (sequence A000110 in OEIS), those that are symmetricand transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and an-tisymmetric. Pfeier[3] has made some progress in this direction, expressing relations with combinations of theseproperties in terms of each other, but still calculating any one is dicult. See also.[4]

8.5 See also Transitive closure Transitive reduction Intransitivity Reexive relation Symmetric relation Quasitransitive relation Nontransitive dice Rational choice theory

42 CHAPTER 8. TRANSITIVE RELATION

8.6 Sources

8.6.1 References[1] Flaka, V.; Jeek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: School

of Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relationsas strictly antisymmetric.

[2] Steven R. Finch, Transitive relations, topologies and partial orders, 2003.

[3] Gtz Pfeier, "Counting Transitive Relations", Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

[4] Gunnar Brinkmann and Brendan D. McKay,Counting unlabelled topologies and transitive relations"

8.6.2 Bibliography Ralph P. Grimaldi, Discrete and Combinatorial Mathematics, ISBN 0-201-19912-2. Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

8.7 External links Hazewinkel, Michiel, ed. (2001), Transitivity, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Transitivity in Action at cut-the-knot

8.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 43

8.8 Text and image sources, contributors, and licenses8.8.1 Text

Binary relation Source: https://en.wikipedia.org/wiki/Binary_relation?oldid=669144544 Contributors: AxelBoldt, Bryan Derksen, Zun-dark, Tarquin, Jan Hidders, Roadrunner, Mjb, Tomo, Patrick, Xavic69, Michael Hardy, Wshun, Isomorphic, Dominus, Ixfd64, Takuya-Murata, Charles Matthews, Timwi, Dcoetzee, Jitse Niesen, Robbot, Chocolateboy, MathMartin, Tobias Bergemann, Giftlite, Fropu,Dratman, Jorge Stol, Jlr~enwiki, Andycjp, Quarl, Guanabot, Yuval madar, Slipstream, Paul August, Elwikipedista~enwiki, Shanes,EmilJ, Randall Holmes, Ardric47, Obradovic Goran, Eje211, Alansohn, Dallashan~enwiki, Keenan Pepper, PAR, Adrian.benko, OlegAlexandrov, Joriki, Linas, MFH, Dpv, Pigcatian, Penumbra2000, Fresheneesz, Chobot, YurikBot, Hairy Dude, Koeyahoo, Trovatore,Bota47, Arthur Rubin, Netrapt, SmackBot, Royalguard11, SEIBasaurus, Cybercobra, Jon Awbrey, Turms, Lambiam, Dbtfz, Mr Stephen,Mets501, Dreftymac, Happy-melon, Petr Matas, CRGreathouse, CBM, Yrodro, WillowW, Xantharius, Thijs!bot, Egrin, Rlupsa, JAnD-bot, MER-C, Magioladitis, Vanish2, David Eppstein, Robin S, Akurn, Adavidb, LajujKej, Owlgorithm, Djjrjr, Policron, DavidCBryant,Quux0r, VolkovBot, Boute, Vipinhari, Anonymous Dissident, PaulTanenbaum, Jackfork, Wykypydya, Dmcq, AlleborgoBot, AHMartin,Ocsenave, Sftd, Paradoctor, Henry Delforn (old), MiNombreDeGuerra, DuaneLAnderson, Anchor Link Bot, CBM2, Classicalecon,ClueBot, Snigbrook, Rhubbarb, Hans Adler, SilvonenBot, BYS2, Plmday, Addbot, LinkFA-Bot, Tide rolls, Jarble, Legobot, Luckas-bot,Yobot, Ht686rg90, Pcap, Labus, Nallimbot, Reindra, FredrikMeyer, AnomieBOT, Floquenbeam, Royote, Hahahaha4, Materialscientist,Belkovich, Citation bot, Racconish, Jellystones, Xqbot, Isheden, Geero, GhalyBot, Ernsts, Howard McCay, Constructive editor, Mark Re-nier, Mfwitten, RandomDSdevel, NearSetAccount, SpaceFlight89, Yunshui, Miracle Pen, Brambleclawx, RjwilmsiBot, Nomen4Omen,Chharvey, SporkBot, OnePt618, Sameer143, Socialservice, ResearchRave, ClueBot NG, Wcherowi, Frietjes, Helpful Pixie Bot, Ko-ertefa, ChrisGualtieri, YFdyh-bot, Dexbot, Makecat-bot, Lerutit, Jochen Burghardt, Jodosma, Karim132, Monkbot, Pratincola, ,Some1Redirects4You and Anonymous: 102

Direct limit Source: https://en.wikipedia.org/wiki/Direct_limit?oldid=636684609Contributors: AxelBoldt, Magnus~enwiki, TakuyaMu-rata, Charles Matthews, Giftlite, Fropu, Rich Farmbrough, Gauge, Oleg Alexandrov, Marudubshinki, Salix alba, Masnevets, Shell Kin-ney, Prime Entelechy, SmackBot, RDBury, Turms, Tesseran, Mets501, P199, A. Pichler, CRGreathouse, Mct mht, Cydebot, Headbomb,RobHar, Jakob.scholbach, VolkovBot, Kyle the bot, Kkilger, DesolateReality, Rswarbrick, Beroal, Sandrobt, Marc van Leeuwen, Wik-Head, Addbot, Fryed-peach, Luckas-bot, Point-set topologist, Ringspectrum, Erik9bot, Sawomir Biay, Tiled, WikitanvirBot, ZroBot,Quasihuman, IkamusumeFan, ChrisGualtieri, Makecat-bot, Mark viking and Anonymous: 24

Limit (mathematics) Source: https://en.wikipedia.org/wiki/Limit_(mathematics)?oldid=662863699 Contributors: AxelBoldt, BryanDerksen, Zundark, The Anome, Tarquin, Andre Engels, JeLuF, Toby Bartels, Miguel~enwiki, Lir, Chas zzz brown, Michael Hardy,Wshun, Modster, Oliver Pereira, Cyp, Alvaro, Poor Yorick, Rossami, Nikai, Pizza Puzzle, Timwi, Dcoetzee, Dysprosia, Jitse Niesen,Phr, Taxman, Omegatron, PuzzletChung, Robbot, Fredrik, Tomchiukc, Troworld, Ojigiri~enwiki, Mushroom, Dbroadwell, Tobias Berge-mann, Marc Venot, Giftlite, Harp, MathKnight, Patrick-br, Gscshoyru, Trevor MacInnis, PhotoBox, Skal, Paul August, Sixpence, Spoon!,Bobo192, Nk, Haham hanuka, LutzL, Alansohn, ChrisUK, Matsw, Sligocki, Jaw959, Piranhaex, EmilSit, Colin Kimbrell, SteinbDJ,MIT Trekkie, Oleg Alexandrov, Sam Vimes, MONGO, Palica, Graham87, Qwertyus, Broccoli, Salix alba, Tawker, MapsMan, SLi,Windchaser, Privong, Natkuhn, JonathanFreed, Theshibboleth, Visor, DVdm, WriterHound, Algebraist, YurikBot, Red Slash, Bhny,Stephenb, Kjmathew, Closedmouth, Arthur Rubin, Kier07, Gesslein, Memodude, GrinBot~enwiki, A bit iy, SmackBot, InverseHyper-cube, Melchoir, Vald, Bluebot, Quinsareth, Oli Filth, Colonies Chris, E946, Jmlk17, Vina-iwbot~enwiki, EunuchOmerta, Tmchk, Argle-bargleIV, Jim.belk, Inkwell, A. Parrot, Loadmaster, Levineps, BranStark, BrainMagMo, JForget, Tobias087, CBM, FilipeS, Equendil,Doctormatt, Cydebot, Thijs!bot, King Bee, Rlupsa, Jojan, Paxinum, Danger, Rbb l181, JAnDbot, Thenub314, Nickvkalker, MSBOT,EulerGamma, Ciaccona, ANONYMOUS COWARD0xC0DE, JoergenB, DerHexer, MartinBot, Miraculousrandomness, Jonathan Hall,Padillah, CommonsDelinker, Ulfalizer, Ohms law, SJP, Potatoswatter, Fylwind, Alan R. Fisher, Gilb 4, Useight, Idioma-bot, PhilipTrueman, TXiKiBoT, Hqb, Anonymous Dissident, Sankalpdravid, Liko81, Plclark, Broadbot, UnitedStatesian, Bob f it, Popopp, Yk YkYk, Meters, Programmerq, Dmcq, Symane, Katzmik, Minestrone Soup, SieBot, Mmtrebuchet, AS, Berserkerus, MiNombreDeGuerra,Amahoney, Dolphin51, ClueBot, LizardJr8, Lartoven, 4-409r-0, CogitoErgoCogitoSum, Corkgkagj, SoxBot III, Hpmv, XLinkBot, Ne-penthes, Good Olfactory, Addbot, Melab-1, CUSENZA Mario, LinkFA-Bot, Jubeidono, PV=nRT, Zorrobot, Bimonte, LuK3, Luckas-bot, Tohd8BohaithuGh1, TaBOT-zerem, Estudiarme, Htyuiop, Mattia Luigi Nappi, Sivanov87, AnomieBOT, Gtz, Ipatrol, Henry Go-dric, Cristiano Ton, Materialscientist, ArthurBot, Ayda D, , RibotBOT, Uuo, Raulshc, LepiCane, Confront, LucienBOT,Erimaxbau, Andrew.clemens, Tkuvho, DrilBot, Pinethicket, Ebony Jackson, Eyrryds, MarcelB612, Galoa2804~enwiki, Duoduoduo,Science220, Alph Bot, Jowa fan, EmausBot, Unrealomega-1, Slightsmile, Slawekb, Joe Gazz84, ZroBot, Quondum, Chewings72,Anita5192, Sonicyouth86, ClueBot NG, Jack Greenmaven, CocuBot, Otaskrebche, Helpful Pixie Bot, Telanian183, Vagobot, Brad7777,Jfd34, DarafshBot, Dexbot, Kavigupta, Ugog Nizdast, Tbyrd89, MagicNut1994, Ob7, KasparBot and Anonymous: 248

Linked set Source: https://en.wikipedia.org/wiki/Linked_set?oldid=604894240 Contributors: Michael Hardy, David Eppstein, Yaddie,Helpful Pixie Bot, Brad7777 and Anonymous: 1

Preorder Source: https://en.wikipedia.org/wiki/Preorder?oldid=611521876 Contributors: AxelBoldt, Patrick, Repton, Delirium, An-dres, Dysprosia, Greenrd, Big Bob the Finder, BenRG, Tobias Bergemann, Giftlite, Markus Krtzsch, Lethe, Fropu, Vadmium, De-fLog~enwiki, Zzo38, Jh51681, Barnaby dawson, Paul August, EmilJ, Msh210, Melaen, Joriki, Linas, Dionyziz, Mandarax, Salix alba,Cjoev, VKokielov, Mathbot, Jrtayloriv, YurikBot, Laurentius, Hairy Dude, WikidSmaht, Trovatore, Modify, Netrapt, Wasseralm, Smack-Bot, XudongGuan~enwiki, DCary, Jdthood, Mets501, PaulGS, Stotr~enwiki, Zero sharp, CRGreathouse, Michael A. White, DavidEppstein, Jwuthe2, PaulTanenbaum, SieBot, Thehotelambush, Functor salad, He7d3r, Sun Creator, Cenarium, 1ForTheMoney, Palnot,, Legobot, Luckas-bot, AnomieBOT, DannyAsher, Xqbot, VladimirReshetnikov, ComputScientist, BrideOfKripkenstein, Noted-grant, WikitanvirBot, Lclem, Dfabera, SporkBot, RichardMills65, Khazar2, Lerutit, Jochen Burghardt, Reatlas, Damonamc and Anony-mous: 26

Reexive relation Source: https://en.wikipedia.org/wiki/Reflexive_relation?oldid=645017228Contributors: AxelBoldt, DavidSJ, Patrick,Wshun, TakuyaMurata, Looxix~enwiki, William M. Connolley, Charles Matthews, Josh Cherry, MathMartin, Henrygb, Tobias Berge-mann, Giftlite, Jason Quinn, Gubbubu, Urhixidur, Ascnder, Paul August, BenjBot, Spayrard, Spoon!, Jet57, LavosBacons, Wtmitchell,Bookandcoee, Oleg Alexandrov, Joriki, Mel Etitis, LOL, MFH, Isnow, Audiovideo, Margosbot~enwiki, Fresheneesz, Chobot, Yurik-Bot, Laurentius, Maelin, Mathlaura, KarlHeg, Arthur Rubin, Isaac Dupree, Jdthood, Mhym, Ceosion, Mike Fikes, Fjbex, CRGreathouse,CBM, Gregbard, Farzaneh, Wikid77, JAnDbot, Policron, Joshua Issac, VolkovBot, Jackfork, Jamelan, Ocsenave, SieBot, Davidellerman,Henry Delforn (old), Hello71, Cuyaken, ClueBot, Ywanne, Da rulz07, SoxBot III, WikHead, Addbot, Download, Favonian, Luckas-bot, Yobot, Renato sr, Pkukiss, Galoubet, ArthurBot, Xqbot, Z0973, I dream of horses, RedBot, MastiBot, Gamewizard71, EmausBot,

44 CHAPTER 8. TRANSITIVE RELATION

Dmayank, DimitriC, ClueBot NG, Kasirbot, Joel B. Lewis, BG19bot, ChrisGualtieri, Eptied, Lerutit, Jochen Burghardt, Seanhalle andAnonymous: 37

Sequence Source: https://en.wikipedia.org/wiki/Sequence?oldid=665140592 Contributors: AxelBoldt, Mav, Zundark, Tarquin, XJaM,Toby Bartels, Imran, Camembert, Youandme, Lir, Patrick, Michael Hardy, Ihcoyc, Poor Yorick, Nikai, EdH, Charles Matthews, Dys-prosia, Greenrd, Hyacinth, Zero0000, Sabbut, Garo, Robbot, Lowellian, MathMartin, Stewartadcock, Henrygb, Bkell, Tosha, Centrx,Giftlite, BenFrantzDale, Lupin, Herbee, Horatio, Edcolins, Vadmium, Leonard Vertighel, Manuel Anastcio, Alexf, Fudo, Melikamp,Sam Hocevar, Tsemii, Ross bencina, Jiy, TedPavlic, Paul August, JoeSmack, Elwikipedista~enwiki, Syp, Pjrich, Shanes, Jonathan Drain,Nk, Obradovic Goran, Haham hanuka, Zaraki~enwiki, Merope, Jumbuck, Reubot, Jet57, Olegalexandrov, Ringbang, Djsasso, Total-cynic, Oleg Alexandrov, Hoziron, Linas, Madmardigan53, MFH, Isnow, Graham87, Dpv, Mendaliv, Salix alba, Figs, VKokielov, Log-gie, Rsenington, RexNL, Pexatus, Fresheneesz, Kri, Ryvr, Chobot, Lightsup55, Krishnavedala, Wavelength, Michael Slone, Grubber,Arthur Rubin, JahJah, Pred, Finell, KHenriksson, Gelingvistoj, Chris the speller, Bluebot, Nbarth, Mcaruso, Suicidalhamster, SundarBot,Dreadstar, Fagstein, Just plain Bill, Xionbox, Dreftymac, Gco, CRGreathouse, CBM, Gregbard, Cydebot, Xantharius, Epbr123, KClier,Saber Cherry, Rlupsa, Marek69, Urdutext, Icep, Ste4k, Mutt Lunker, JAnDbot, Asnac, Coolhandscot, Martinkunev, VoABot II, Avjoska,JamesBWatson, Brusegadi, Minimiscience, Stdazi, DerHexer, J.delanoy, Trusilver, Suenm~enwiki, Ncmvocalist, Belovedfreak, Policron,JingaJenga, VolkovBot, ABF, AlnoktaBOT, Philip Trueman, Digby Tantrum, JhsBot, Isis4563, Wolfrock, Xiong Yingfei, Newbyguesses,SieBot, Scarian, Yintan, Xelgen, Outs, Paolo.dL, OKBot, Pagen HD,Wahrmund, Classicalecon, Atif.t2, Crambo0349, ClueBot, Justin WSmith, Fyyer, SuperHamster, Excirial, Estirabot, Jotterbot, Thingg, Downgrader, Aj00200, XLinkBot, Stickee, Rror, WikHead, Brent-smith101, Addbot, Non-dropframe, Kongr43gpen, Matj Grabovsk, Legobot, Luckas-bot, Yobot, Eric-Wester, 4th-otaku, AnomieBOT,Jim1138, Law, Materialscientist, E2eamon, ArthurBot, Ayda D, Xqbot, Omnipaedista, RibotBOT, Charvest, Shadowjams, Thehelpful-bot, Dan6hell66, Constructive editor, Mark Renier, Tal physdancer, SixPurpleFish, Pinethicket, BRUTE, SkyMachine, PiRSquared17,Roy McCoy, RjwilmsiBot, Tzfyr, EmausBot, John of Reading, GoingBatty, Wikipelli, K6ka, Brent Perreault, Nellandmice, Bethnim,Ida Shaw, Alpha Quadrant, KuduIO, D.Lazard, SporkBot, Wayne Slam, Donner60, Chewings72, ClueBot NG, Satellizer, Widr, MerlI-wBot, Helpful Pixie Bot, HMSSolent, Curb Chain, Calabe1992, Brad7777, Minsbot, Praxiphenes, EuroCarGT, Ven Seyranyan., Jegyao,DavyRalph, Graphium, Jochen Burghardt, Brirush, Mark viking, LoMaPh, Immonster, EricsonWillians, Emlynlee, Buscus 3, JackHoang,Some1Redirects4You and Anonymous: 209

Transitive relation Source: https://en.wikipedia.org/wiki/Transitive_relation?oldid=670393145Contributors: Zundark, Patrick,MichaelHardy, Rp, Looxix~enwiki, Andres, Charles Matthews, Dcoetzee, Jitse Niesen, Fredrik, MathMartin, Tobias Bergemann, Giftlite, Ben-FrantzDale, Gubbubu, Chowbok, Paul August, MyNameIsNotBob, Spoon!, Polluks, DanShearer, Woohookitty, Linas, LOL, Isnow,Palica, Jrmie Lumbroso~enwiki, Salix alba, Mathbot, Fresheneesz, YurikBot, Laurentius, Sasuke Sarutobi, 48v, Bota47, Arthur Ru-bin, MaratL, Wasseralm, JJL, SmackBot, InverseHypercube, Nbarth, Robma, Cybercobra, Jna runn, Lambiam, Coredesat, Lyon-sam, Cbuckley, CRGreathouse, Aggarwal kshitij, CBM, Thomasmeeks, Gogo Dodo, Tawkerbot4, AntiVandalBot, Mhaitham.shammaa,MER-C, .anacondabot, Magioladitis, Albmont, David Eppstein, Edward321, MartinBot, Extransit, Tomaz.slivnik, Policron, VolkovBot,AThomas203, Jamelan, Cnilep, SieBot, Paradoctor, Henry Delforn (old), Anchor Link Bot, ClueBot, Tomvanderweide, Sarbogard, Ot-tawahitech, Alexbot, Wikibojopayne, Pa68, SilvonenBot, Addbot, Luckas-bot, Yobot, Ptbotgourou, GrouchoBot, Undsoweiter, RedBot,Katovatzschyn, EmausBot, Slightsmile, IGeMiNix, ChuispastonBot, ClueBot NG, Pars99, Sourabh.khot, Justincheng12345-bot, Lerutit,Loraof and Anonymous: 60

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Binary relationFormal definitionIs a relation more than its graph?Example

Special types of binary relationsDifunctional

Relations over a setOperations on binary relationsComplementRestrictionAlgebras, categories, and rewriting systems

Sets versus classesThe number of binary relationsExamples of common binary relationsSee alsoNotesReferencesExternal links

Direct limitFormal definitionAlgebraic objects Direct limit over a direct system in a category

ExamplesRelated constructions and generalizationsSee alsoReferences

Limit (mathematics)Limit of a function Limit of a sequence Limit as standard partConvergence and fixed pointTopological net See also NotesExternal links

Linked setReferences

PreorderFormal definitionExamplesUsesConstructionsNumber of preordersIntervalSee alsoReferences

Reflexive relationRelated termsExamplesNumber of reflexive relationsPhilosophical logicSee alsoNotesReferencesExternal links

SequenceExamples and notation Important examplesIndexingSpecifying a sequence by recursion

Formal definition and basic properties Formal definitionFinite and infinite Increasing and decreasingBoundedOther types of sequences

Limits and convergenceDefinition of convergenceApplications and important resultsCauchy sequences

SeriesUse in other fields of mathematicsTopology AnalysisLinear algebra Abstract algebraSet theoryComputing Streams

TypesRelated conceptsOperationsSee alsoReferencesExternal links

Transitive relationFormal definition ExamplesProperties Closure properties Other properties Properties that require transitivity

Counting transitive relationsSee alsoSources References Bibliography

External linksText and image sources, contributors, and licensesTextImagesContent license