Family of Sets

Family of setsWikipedia

Contents

1 Family of sets 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Special types of set family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Almost disjoint sets 32.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Other meanings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Antimatroid 53.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Paths and basic words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Convex geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5 Join-distributive lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6 Supersolvable antimatroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.7 Join operation and convex dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.8 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Class (set theory) 124.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Classes in formal set theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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5 Combinatorial design 145.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Fundamental combinatorial designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 A wide assortment of other combinatorial designs . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Delta-ring 236.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7 Disjoint sets 247.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.3 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.4 Disjoint unions and partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8 Dynkin system 288.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9 Field of sets 309.1 Fields of sets in the representation theory of Boolean algebras . . . . . . . . . . . . . . . . . . . . 30

9.1.1 Stone representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.1.2 Separative and compact elds of sets: towards Stone duality . . . . . . . . . . . . . . . . . 30

9.2 Fields of sets with additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.2.1 Sigma algebras and measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.2.2 Topological elds of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319.2.3 Preorder elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.2.4 Complex algebras and elds of sets on relational structures . . . . . . . . . . . . . . . . . . 32

9.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

10 Finite character 3410.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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10.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

11 Finite intersection property 3511.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.5 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

12 Hypergraph 3712.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3812.2 Bipartite graph model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.3 Acyclicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.4 Isomorphism and equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

12.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.5 Symmetric hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.6 Transversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.7 Incidence matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.8 Hypergraph coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.9 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.10Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.11Hypergraph drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.12Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

13 Indexed family 4713.1 Mathematical statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

13.2.1 Index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.2.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

13.3 Functions, sets and families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.5 Operations on families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.6 Subfamily . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.7 Usage in category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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14.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.1.1 Independent sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.1.2 Bases and circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.1.3 Rank functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.1.4 Closure operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.1.5 Flats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.1.6 Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

14.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.2.1 Uniform matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.2.2 Matroids from linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.2.3 Matroids from graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5514.2.4 Matroids from eld extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

14.3 Basic constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.3.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.3.2 Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.3.3 Sums and unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

14.4 Additional terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5714.5 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

14.5.1 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5714.5.2 Matroid partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5814.5.3 Matroid intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5814.5.4 Matroid software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

14.6 Polynomial invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5814.6.1 Characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5814.6.2 Tutte polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

14.7 Innite matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.8 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.9 Researchers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6114.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6114.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6114.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6314.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

15 Noncrossing partition 6515.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.2 Lattice structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.3 Role in free probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

16 Ordinal number 6916.1 Ordinals extend the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7016.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

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16.2.1 Well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7216.2.2 Denition of an ordinal as an equivalence class . . . . . . . . . . . . . . . . . . . . . . . 7216.2.3 Von Neumann denition of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7216.2.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

16.3 Transnite sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7316.4 Transnite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

16.4.1 What is transnite induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7316.4.2 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7416.4.3 Successor and limit ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7416.4.4 Indexing classes of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7416.4.5 Closed unbounded sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

16.5 Arithmetic of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7516.6 Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

16.6.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.6.2 Conality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

16.7 Some large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.8 Topology and ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.9 Downward closed sets of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

17 Pi system 7917.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7917.2 Relationship to -Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

17.2.1 The - Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8017.3 -Systems in Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

17.3.1 Equality in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.3.2 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

17.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8217.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8217.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

18 Ring of sets 8318.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8318.2 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8418.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8418.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

19 Russells paradox 8519.1 Informal presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

vi CONTENTS

19.2 Formal presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8519.3 Set-theoretic responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8619.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8619.5 Applied versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8819.6 Applications and related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

19.6.1 Russell-like paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8819.7 Related paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8919.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8919.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8919.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9019.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

20 Set (mathematics) 9220.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9320.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9320.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

20.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9520.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

20.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

20.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9720.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9820.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9820.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

20.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10120.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10120.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10220.10De Morgans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10220.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10320.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10320.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10320.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

21 Set theory 10421.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10521.2 Basic concepts and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10621.3 Some ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10721.4 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10721.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10821.6 Areas of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

21.6.1 Combinatorial set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

CONTENTS vii

21.6.2 Descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10921.6.3 Fuzzy set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10921.6.4 Inner model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10921.6.5 Large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11021.6.6 Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11021.6.7 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11021.6.8 Cardinal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11021.6.9 Set-theoretic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

21.7 Objections to set theory as a foundation for mathematics . . . . . . . . . . . . . . . . . . . . . . . 11121.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11121.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11121.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11221.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

22 Sigma-algebra 11322.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

22.1.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11322.1.2 Limits of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11422.1.3 Sub -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

22.2 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11522.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11522.2.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11522.2.3 Combining -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11522.2.4 -algebras for subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11622.2.5 Relation to -ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11622.2.6 Typographic note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

22.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11722.3.1 Simple set-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11722.3.2 Stopping time sigma-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

22.4 -algebras generated by families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11722.4.1 -algebra generated by an arbitrary family . . . . . . . . . . . . . . . . . . . . . . . . . . 11722.4.2 -algebra generated by a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11722.4.3 Borel and Lebesgue -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11822.4.4 Product -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11822.4.5 -algebra generated by cylinder sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11822.4.6 -algebra generated by random variable or vector . . . . . . . . . . . . . . . . . . . . . . 11922.4.7 -algebra generated by a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . 119

22.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11922.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12022.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

23 Sigma-ring 121

viii CONTENTS

23.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12123.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12123.3 Similar concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12123.4 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12123.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12223.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

24 Subset 12324.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12424.2 and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12424.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12424.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12524.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12524.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12524.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

25 Transversal (combinatorics) 12725.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12725.2 Systems of distinct representatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12825.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12825.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12825.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12825.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12825.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

26 Universal set 13026.1 Reasons for nonexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

26.1.1 Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13026.1.2 Cantors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

26.2 Theories of universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13026.2.1 Restricted comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13126.2.2 Universal objects that are not sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

26.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13126.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13126.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13226.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 133

26.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13326.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13626.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Chapter 1

Family of sets

In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family ofsubsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.The term collection is used here because, in some contexts, a family of sets may be allowed to contain repeatedcopies of any given member,[1][2][3] and in other contexts it may form a proper class rather than a set.

1.1 Examples The power set P(S) is a family of sets over S. The k-subsets S(k) of a set S form a family of sets. Let S = {a,b,c,1,2}, an example of a family of sets over S (in the multiset sense) is given by F = {A1, A2, A3,A4} where A1 = {a,b,c}, A2 = {1,2}, A3 = {1,2} and A4 = {a,b,1}.

The class Ord of all ordinal numbers is a large family of sets; that is, it is not itself a set but instead a properclass.

1.2 Special types of set family A Sperner family is a family of sets in which none of the sets contains any of the others. Sperners theorembounds the maximum size of a Sperner family.

A Helly family is a family of sets such that any minimal subfamily with empty intersection has bounded size.Hellys theorem states that convex sets in Euclidean spaces of bounded dimension form Helly families.

1.3 Properties Any family of subsets of S is itself a subset of the power set P(S) if it has no repeated members. Any family of sets without repetitions is a subclass of the proper class V of all sets (the universe). Halls marriage theorem, due to Philip Hall gives necessary and sucient conditions for a nite family ofnon-empty sets (repetitions allowed) to have a system of distinct representatives.

1.4 Related conceptsCertain types of objects from other areas ofmathematics are equivalent to families of sets, in that they can be describedpurely as a collection of sets of objects of some type:

1

2 CHAPTER 1. FAMILY OF SETS

A hypergraph, also called a set system, is formed by a set of vertices together with another set of hyperedges,each of which may be an arbitrary set. The hyperedges of a hypergraph form a family of sets, and any familyof sets can be interpreted as a hypergraph that has the union of the sets as its vertices.

An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shapeformed by unions of line segments, triangles, tetrahedra, and higher-dimensional simplices, joined face to face.In an abstract simplicial complex, each simplex is represented simply as the set of its vertices. Any family ofnite sets without repetitions in which the subsets of any set in the family also belong to the family forms anabstract simplicial complex.

An incidence structure consists of a set of points, a set of lines, and an (arbitrary) binary relation, called theincidence relation, specifying which points belong to which lines. An incidence structure can be specied bya family of sets (even if two distinct lines contain the same set of points), the sets of points belonging to eachline, and any family of sets can be interpreted as an incidence structure in this way.

A binary block code consists of a set of codewords, each of which is a string of 0s and 1s, all the same length.When each pair of codewords has large Hamming distance, it can be used as an error-correcting code. A blockcode can also be described as a family of sets, by describing each codeword as the set of positions at which itcontains a 1.

1.5 See also Indexed family Class (set theory) Combinatorial design Russells paradox (or Set of sets that do not contain themselves)

1.6 Notes[1] Brualdi 2010, pg. 322

[2] Roberts & Tesman 2009, pg. 692

[3] Biggs 1985, pg. 89

1.7 References Biggs, Norman L. (1985), Discrete Mathematics, Oxford: Clarendon Press, ISBN 0-19-853252-0 Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Upper Saddle River, NJ: Prentice Hall, ISBN0-13-602040-2

Roberts, Fred S.; Tesman, Barry (2009), Applied Combinatorics (2nd ed.), Boca Raton: CRC Press, ISBN978-1-4200-9982-9

Chapter 2

Almost disjoint sets

In mathematics, two sets are almost disjoint [1][2] if their intersection is small in some sense; dierent denitions ofsmall will result in dierent denitions of almost disjoint.

2.1 DenitionThemost common choice is to take small to mean nite. In this case, two sets are almost disjoint if their intersectionis nite, i.e. if

jA \Bj

4 CHAPTER 2. ALMOST DISJOINT SETS

2.2 Other meaningsSometimes almost disjoint is used in some other sense, or in the sense of measure theory or topological category.Here are some alternative denitions of almost disjoint that are sometimes used (similar denitions apply to innitecollections):

Let be any cardinal number. Then two sets A and B are almost disjoint if the cardinality of their intersectionis less than , i.e. if

jA \Bj < :

The case of = 1 is simply the denition of disjoint sets; the case of

= @0

is simply the denition of almost disjoint given above, where the intersection of A and B is nite.

Let m be a complete measure on a measure space X. Then two subsets A and B of X are almost disjoint if theirintersection is a null-set, i.e. if

m(A \B) = 0:

Let X be a topological space. Then two subsets A and B of X are almost disjoint if their intersection is meagrein X.

2.3 References[1] Kunen, K. (1980), Set Theory; an introduction to independence proofs, North Holland, p. 47

[2] Jech, R. (2006) Set Theory (the third millennium edition, revised and expanded)", Springer, p. 118

Chapter 3

Antimatroid

{a,b}

{a,b,c}

{a,c} {b,c}

{a} {c}

abcaba

acacbccacabcbcba

{a} {c}

{a,b} {b,c}

{a,b,c,d}

abcd

acbd

cabd

cbad

{a,b,c,d}

Three views of an antimatroid: an inclusion ordering on its family of feasible sets, a formal language, and the corresponding pathposet.

In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by includingelements one at a time, and in which an element, once available for inclusion, remains available until it is included.Antimatroids are commonly axiomatized in two equivalent ways, either as a set system modeling the possible statesof such a process, or as a formal language modeling the dierent sequences in which elements may be included.Dilworth (1940) was the rst to study antimatroids, using yet another axiomatization based on lattice theory, andthey have been frequently rediscovered in other contexts;[1] see Korte et al. (1991) for a comprehensive survey ofantimatroid theory with many additional references.The axioms dening antimatroids as set systems are very similar to those ofmatroids, but whereasmatroids are denedby an exchange axiom (e.g., the basis exchange, or independent set exchange axioms), antimatroids are dened insteadby an anti-exchange axiom, from which their name derives. Antimatroids can be viewed as a special case of greedoidsand of semimodular lattices, and as a generalization of partial orders and of distributive lattices. Antimatroids areequivalent, by complementation, to convex geometries, a combinatorial abstraction of convex sets in geometry.

5

6 CHAPTER 3. ANTIMATROID

Antimatroids have been applied to model precedence constraints in scheduling problems, potential event sequencesin simulations, task planning in articial intelligence, and the states of knowledge of human learners.

3.1 DenitionsAn antimatroid can be dened as a nite family F of sets, called feasible sets, with the following two properties:

The union of any two feasible sets is also feasible. That is, F is closed under unions.

If S is a nonempty feasible set, then there exists some x in S such that S \ {x} (the set formed by removing xfrom S) is also feasible. That is, F is an accessible set system.

Antimatroids also have an equivalent denition as a formal language, that is, as a set of strings dened from a nitealphabet of symbols. A language L dening an antimatroid must satisfy the following properties:

Every symbol of the alphabet occurs in at least one word of L.

Each word of L contains at most one copy of any symbol.

Every prex of a string in L is also in L.

If s and t are strings in L, and s contains at least one symbol that is not in t, then there is a symbol x in s suchthat tx is another string in L.

If L is an antimatroid dened as a formal language, then the sets of symbols in strings of L form an accessible union-closed set system. In the other direction, if F is an accessible union-closed set system, and L is the language of stringss with the property that the set of symbols in each prex of s is feasible, then L denes an antimatroid. Thus, thesetwo denitions lead to mathematically equivalent classes of objects.[2]

3.2 Examples A chain antimatroid has as its formal language the prexes of a single word, and as its feasible sets the setsof symbols in these prexes. For instance the chain antimatroid dened by the word abcd has as its formallanguage the strings {, a, ab, abc, abcd"} and as its feasible sets the sets , {a}, {a,b}, {a,b,c}, and{a,b,c,d}.[3]

A poset antimatroid has as its feasible sets the lower sets of a nite partially ordered set. By Birkhos rep-resentation theorem for distributive lattices, the feasible sets in a poset antimatroid (ordered by set inclusion)form a distributive lattice, and any distributive lattice can be formed in this way. Thus, antimatroids can beseen as generalizations of distributive lattices. A chain antimatroid is the special case of a poset antimatroidfor a total order.[3]

A shelling sequence of a nite set U of points in the Euclidean plane or a higher-dimensional Euclidean spaceis an ordering on the points such that, for each point p, there is a line (in the Euclidean plane, or a hyperplanein a Euclidean space) that separates p from all later points in the sequence. Equivalently, p must be a vertexof the convex hull of it and all later points. The partial shelling sequences of a point set form an antimatroid,called a shelling antimatroid. The feasible sets of the shelling antimatroid are the intersections of U with thecomplement of a convex set.[3]

A perfect elimination ordering of a chordal graph is an ordering of its vertices such that, for each vertex v,the neighbors of v that occur later than v in the ordering form a clique. The prexes of perfect eliminationorderings of a chordal graph form an antimatroid.[3]

3.3. PATHS AND BASIC WORDS 7

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A shelling sequence of a planar point set. The line segments show edges of the convex hulls after some of the points have beenremoved.

3.3 Paths and basic words

In the set theoretic axiomatization of an antimatroid there are certain special sets called paths that determine thewhole antimatroid, in the sense that the sets of the antimatroid are exactly the unions of paths. If S is any feasible setof the antimatroid, an element x that can be removed from S to form another feasible set is called an endpoint of S,and a feasible set that has only one endpoint is called a path of the antimatroid. The family of paths can be partiallyordered by set inclusion, forming the path poset of the antimatroid.For every feasible set S in the antimatroid, and every element x of S, one may nd a path subset of S for which xis an endpoint: to do so, remove one at a time elements other than x until no such removal leaves a feasible subset.Therefore, each feasible set in an antimatroid is the union of its path subsets. If S is not a path, each subset in thisunion is a proper subset of S. But, if S is itself a path with endpoint x, each proper subset of S that belongs to theantimatroid excludes x. Therefore, the paths of an antimatroid are exactly the sets that do not equal the unions oftheir proper subsets in the antimatroid. Equivalently, a given family of sets P forms the set of paths of an antimatroidif and only if, for each S in P, the union of subsets of S in P has one fewer element than S itself. If so, F itself is thefamily of unions of subsets of P.In the formal language formalization of an antimatroid wemay also identify a subset of words that determine the wholelanguage, the basic words. The longest strings in L are called basic words; each basic word forms a permutation ofthe whole alphabet. For instance, the basic words of a poset antimatroid are the linear extensions of the given partialorder. If B is the set of basic words, L can be dened from B as the set of prexes of words in B. It is often convenientto dene antimatroids from basic words in this way, but it is not straightforward to write an axiomatic denition of


antimatroids in terms of their basic words.

3.4 Convex geometriesSee also: Convex set, Convex geometry and Closure operator

If F is the set system dening an antimatroid, with U equal to the union of the sets in F, then the family of sets

G = fU n S j S 2 Fg

complementary to the sets in F is sometimes called a convex geometry, and the sets in G are called convex sets. Forinstance, in a shelling antimatroid, the convex sets are intersections of U with convex subsets of the Euclidean spaceinto which U is embedded.Complementarily to the properties of set systems that dene antimatroids, the set system dening a convex geometryshould be closed under intersections, and for any set S in G that is not equal to U there must be an element x not in Sthat can be added to S to form another set in G.A convex geometry can also be dened in terms of a closure operator that maps any subset of U to its minimalclosed superset. To be a closure operator, should have the following properties:

() = : the closure of the empty set is empty. Any set S is a subset of (S). If S is a subset of T, then (S) must be a subset of (T). For any set S, (S) = ((S)).

The family of closed sets resulting from a closure operation of this type is necessarily closed under intersections. Theclosure operators that dene convex geometries also satisfy an additional anti-exchange axiom:

If neither y nor z belong to (S), but z belongs to (S {y}), then y does not belong to (S {z}).

A closure operation satisfying this axiom is called an anti-exchange closure. If S is a closed set in an anti-exchangeclosure, then the anti-exchange axiom determines a partial order on the elements not belonging to S, where x y inthe partial order when x belongs to (S {y}). If x is a minimal element of this partial order, then S {x} is closed.That is, the family of closed sets of an anti-exchange closure has the property that for any set other than the universalset there is an element x that can be added to it to produce another closed set. This property is complementary tothe accessibility property of antimatroids, and the fact that intersections of closed sets are closed is complementaryto the property that unions of feasible sets in an antimatroid are feasible. Therefore, the complements of the closedsets of any anti-exchange closure form an antimatroid.[4]

3.5 Join-distributive latticesAny two sets in an antimatroid have a unique least upper bound (their union) and a unique greatest lower bound(the union of the sets in the antimatroid that are contained in both of them). Therefore, the sets of an antimatroid,partially ordered by set inclusion, form a lattice. Various important features of an antimatroid can be interpreted inlattice-theoretic terms; for instance the paths of an antimatroid are the join-irreducible elements of the correspondinglattice, and the basic words of the antimatroid correspond to maximal chains in the lattice. The lattices that arise fromantimatroids in this way generalize the nite distributive lattices, and can be characterized in several dierent ways.

The description originally considered by Dilworth (1940) concerns meet-irreducible elements of the lattice.For each element x of an antimatroid, there exists a unique maximal feasible set Sx that does not contain x (Sxis the union of all feasible sets not containing x). Sx is meet-irreducible, meaning that it is not the meet of any

3.6. SUPERSOLVABLE ANTIMATROIDS 9

two larger lattice elements: any larger feasible set, and any intersection of larger feasible sets, contains x and sodoes not equal Sx. Any element of any lattice can be decomposed as a meet of meet-irreducible sets, often inmultiple ways, but in the lattice corresponding to an antimatroid each element T has a unique minimal familyof meet-irreducible sets Sx whose meet is T ; this family consists of the sets Sx such that T {x} belongs to theantimatroid. That is, the lattice has unique meet-irreducible decompositions.

A second characterization concerns the intervals in the lattice, the sublattices dened by a pair of lattice elementsx y and consisting of all lattice elements z with x z y. An interval is atomistic if every element in it is thejoin of atoms (the minimal elements above the bottom element x), and it is Boolean if it is isomorphic to thelattice of all subsets of a nite set. For an antimatroid, every interval that is atomistic is also boolean.

Thirdly, the lattices arising from antimatroids are semimodular lattices, lattices that satisfy the upper semimod-ular law that for any two elements x and y, if y covers x y then x y covers x. Translating this conditioninto the sets of an antimatroid, if a set Y has only one element not belonging to X then that one element maybe added to X to form another set in the antimatroid. Additionally, the lattice of an antimatroid has the meet-semidistributive property: for all lattice elements x, y, and z, if x y and x z are both equal then they alsoequal x (y z). A semimodular and meet-semidistributive lattice is called a join-distributive lattice.

These three characterizations are equivalent: any lattice with unique meet-irreducible decompositions has booleanatomistic intervals and is join-distributive, any lattice with boolean atomistic intervals has unique meet-irreducibledecompositions and is join-distributive, and any join-distributive lattice has unique meet-irreducible decompositionsand boolean atomistic intervals.[5] Thus, wemay refer to a lattice with any of these three properties as join-distributive.Any antimatroid gives rise to a nite join-distributive lattice, and any nite join-distributive lattice comes from anantimatroid in this way.[6] Another equivalent characterization of nite join-distributive lattices is that they are graded(any two maximal chains have the same length), and the length of a maximal chain equals the number of meet-irreducible elements of the lattice.[7] The antimatroid representing a nite join-distributive lattice can be recoveredfrom the lattice: the elements of the antimatroid can be taken to be the meet-irreducible elements of the lattice, andthe feasible set corresponding to any element x of the lattice consists of the set of meet-irreducible elements y suchthat y is not greater than or equal to x in the lattice.This representation of any nite join-distributive lattice as an accessible family of sets closed under unions (that is, asan antimatroid) may be viewed as an analogue of Birkhos representation theorem under which any nite distributivelattice has a representation as a family of sets closed under unions and intersections.

3.6 Supersolvable antimatroidsMotivated by a problem of dening partial orders on the elements of a Coxeter group, Armstrong (2007) studied an-timatroids which are also supersolvable lattices. A supersolvable antimatroid is dened by a totally ordered collectionof elements, and a family of sets of these elements. The family must include the empty set. Additionally, it musthave the property that if two sets A and B belong to the family, the set-theoretic dierence B \ A is nonempty, and xis the smallest element of B \ A, then A {x} also belongs to the family. As Armstrong observes, any family of setsof this type forms an antimatroid. Armstrong also provides a lattice-theoretic characterization of the antimatroidsthat this construction can form.

3.7 Join operation and convex dimensionIf A and B are two antimatroids, both described as a family of sets, and if the maximal sets in A and B are equal, wecan form another antimatroid, the join of A and B, as follows:

A _B = fS [ T j S 2 A ^ T 2 Bg:

Note that this is a dierent operation than the join considered in the lattice-theoretic characterizations of antimatroids:it combines two antimatroids to form another antimatroid, rather than combining two sets in an antimatroid to formanother set. The family of all antimatroids that have a given maximal set forms a semilattice with this join operation.


Joins are closely related to a closure operation that maps formal languages to antimatroids, where the closure of alanguage L is the intersection of all antimatroids containing L as a sublanguage. This closure has as its feasible setsthe unions of prexes of strings in L. In terms of this closure operation, the join is the closure of the union of thelanguages of A and B.Every antimatroid can be represented as a join of a family of chain antimatroids, or equivalently as the closure ofa set of basic words; the convex dimension of an antimatroid A is the minimum number of chain antimatroids (orequivalently the minimum number of basic words) in such a representation. If F is a family of chain antimatroidswhose basic words all belong to A, then F generates A if and only if the feasible sets of F include all paths of A. Thepaths of A belonging to a single chain antimatroid must form a chain in the path poset of A, so the convex dimensionof an antimatroid equals the minimum number of chains needed to cover the path poset, which by Dilworths theoremequals the width of the path poset.[8]

If one has a representation of an antimatroid as the closure of a set of d basic words, then this representation canbe used to map the feasible sets of the antimatroid into d-dimensional Euclidean space: assign one coordinate perbasic word w, and make the coordinate value of a feasible set S be the length of the longest prex of w that is asubset of S. With this embedding, S is a subset of T if and only if the coordinates for S are all less than or equal tothe corresponding coordinates of T. Therefore, the order dimension of the inclusion ordering of the feasible sets isat most equal to the convex dimension of the antimatroid.[9] However, in general these two dimensions may be verydierent: there exist antimatroids with order dimension three but with arbitrarily large convex dimension.

3.8 EnumerationThe number of possible antimatroids on a set of elements grows rapidly with the number of elements in the set. Forsets of one, two, three, etc. elements, the number of distinct antimatroids is

1, 3, 22, 485, 59386, 133059751, ... (sequence A119770 in OEIS).

3.9 ApplicationsBoth the precedence and release time constraints in the standard notation for theoretic scheduling problems maybe modeled by antimatroids. Boyd & Faigle (1990) use antimatroids to generalize a greedy algorithm of EugeneLawler for optimally solving single-processor scheduling problems with precedence constraints in which the goal isto minimize the maximum penalty incurred by the late scheduling of a task.Glasserman & Yao (1994) use antimatroids to model the ordering of events in discrete event simulation systems.Parmar (2003) uses antimatroids to model progress towards a goal in articial intelligence planning problems.In mathematical psychology, antimatroids have been used to describe feasible states of knowledge of a human learner.Each element of the antimatroid represents a concept that is to be understood by the learner, or a class of problems thathe or she might be able to solve correctly, and the sets of elements that form the antimatroid represent possible sets ofconcepts that could be understood by a single person. The axioms dening an antimatroid may be phrased informallyas stating that learning one concept can never prevent the learner from learning another concept, and that any feasiblestate of knowledge can be reached by learning a single concept at a time. The task of a knowledge assessment systemis to infer the set of concepts known by a given learner by analyzing his or her responses to a small and well-chosenset of problems. In this context antimatroids have also been called learning spaces and well-graded knowledgespaces.[10]

3.10 Notes[1] Two early references are Edelman (1980) and Jamison (1980); Jamison was the rst to use the term antimatroid.

Monjardet (1985) surveys the history of rediscovery of antimatroids.

[2] Korte et al., Theorem 1.4.

3.11. REFERENCES 11

[3] Gordon (1997) describes several results related to antimatroids of this type, but these antimatroids were mentioned earliere.g. by Korte et al. Chandran et al. (2003) use the connection to antimatroids as part of an algorithm for eciently listingall perfect elimination orderings of a given chordal graph.

[4] Korte et al., Theorem 1.1.[5] Adaricheva, Gorbunov & Tumanov (2003), Theorems 1.7 and 1.9; Armstrong (2007), Theorem 2.7.[6] Edelman (1980), Theorem 3.3; Armstrong (2007), Theorem 2.8.[7] Monjardet (1985) credits a dual form of this characterization to several papers from the 1960s by S. P. Avann.[8] Edelman & Saks (1988); Korte et al., Theorem 6.9.[9] Korte et al., Corollary 6.10.[10] Doignon & Falmagne (1999).

3.11 References Adaricheva, K. V.; Gorbunov, V. A.; Tumanov, V. I. (2003), Join-semidistributive lattices and convex ge-ometries, Advances in Mathematics 173 (1): 149, doi:10.1016/S0001-8708(02)00011-7.

Armstrong, Drew (2007), The sorting order on a Coxeter group, arXiv:0712.1047. Birkho, Garrett; Bennett, M.K. (1985), The convexity lattice of a poset,Order 2 (3): 223242, doi:10.1007/BF00333128. Bjrner, Anders; Ziegler, Gnter M. (1992), 8 Introduction to greedoids, in White, Neil, Matroid Appli-cations, Encyclopedia of Mathematics and its Applications 40, Cambridge: Cambridge University Press, pp.284357, doi:10.1017/CBO9780511662041.009, ISBN 0-521-38165-7, MR 1165537

Boyd, E. Andrew; Faigle, Ulrich (1990), An algorithmic characterization of antimatroids, Discrete AppliedMathematics 28 (3): 197205, doi:10.1016/0166-218X(90)90002-T.

Chandran, L. S.; Ibarra, L.; Ruskey, F.; Sawada, J. (2003), Generating and characterizing the perfect elimina-tion orderings of a chordal graph (PDF), Theoretical Computer Science 307 (2): 303317, doi:10.1016/S0304-3975(03)00221-4.

Dilworth, Robert P. (1940), Lattices with unique irreducible decompositions, Annals of Mathematics 41 (4):771777, doi:10.2307/1968857, JSTOR 1968857.

Doignon, Jean-Paul; Falmagne, Jean-Claude (1999), Knowledge Spaces, Springer-Verlag, ISBN 3-540-64501-2.

Edelman, Paul H. (1980), Meet-distributive lattices and the anti-exchange closure, Algebra Universalis 10(1): 290299, doi:10.1007/BF02482912.

Edelman, Paul H.; Saks, Michael E. (1988), Combinatorial representation and convex dimension of convexgeometries, Order 5 (1): 2332, doi:10.1007/BF00143895.

Glasserman, Paul; Yao, David D. (1994), Monotone Structure in Discrete Event Systems, Wiley Series in Prob-ability and Statistics, Wiley Interscience, ISBN 978-0-471-58041-6.

Gordon, Gary (1997), A invariant for greedoids and antimatroids, Electronic Journal of Combinatorics 4(1): Research Paper 13, MR 1445628.

Jamison, Robert (1980), Copoints in antimatroids, Proceedings of the Eleventh Southeastern Conference onCombinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1980), Vol. II, Con-gressus Numerantium 29, pp. 535544, MR 608454.

Korte, Bernhard; Lovsz, Lszl; Schrader, Rainer (1991), Greedoids, Springer-Verlag, pp. 1943, ISBN3-540-18190-3.

Monjardet, Bernard (1985), A use for frequently rediscovering a concept,Order 1 (4): 415417, doi:10.1007/BF00582748. Parmar, Aarati (2003), Some Mathematical Structures Underlying Ecient Planning, AAAI Spring Sympo-sium on Logical Formalization of Commonsense Reasoning (PDF).

Chapter 4

Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathe-matical objects) that can be unambiguously dened by a property that all its members share. The precise denitionof class depends on foundational context. In work on ZermeloFraenkel set theory, the notion of class is informal,whereas other set theories, such as Von NeumannBernaysGdel set theory, axiomatize the notion of proper class,e.g., as entities that are not members of another entity.A class that is not a set (informally in ZermeloFraenkel) is called a proper class, and a class that is a set is sometimescalled a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes inmany formal systems.Outside set theory, the word class is sometimes used synonymously with set. This usage dates from a histor-ical period where classes and sets were not distinguished as they are in modern set-theoretic terminology. Manydiscussions of classes in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguousconcept.

4.1 ExamplesThe collection of all algebraic objects of a given type will usually be a proper class. Examples include the class ofall groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objectsforms a proper class (or whose collection of morphisms forms a proper class) is called a large category.The surreal numbers are a proper class of objects that have the properties of a eld.Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets, theclass of all ordinal numbers, and the class of all cardinal numbers.One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers. This methodis used, for example, in the proof that there is no free complete lattice.

4.2 ParadoxesThe paradoxes of naive set theory can be explained in terms of the inconsistent assumption that all classes are sets.With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper. For example,Russells paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper.

4.3 Classes in formal set theoriesZF set theory does not formalize the notion of classes, so each formula with classes must be reduced syntactically toa formula without classes.[1] For example, one can reduce the formula A = fx j x = xg to 8x:(x 2 A $ x = x). Semantically, in a metalanguage, the classes can be described as equivalence classes of logical formulas: If A

12

4.4. REFERENCES 13

is a structure interpreting ZF, then the object language class builder expression fx j g is interpreted in A by thecollection of all the elements from the domain of A on which x: holds; thus, the class can be described as the setof all predicates equivalent to (including itself). In particular, one can identify the class of all sets with the setof all predicates equivalent to x=x.Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply toclasses. However, if an inaccessible cardinal is assumed, then the sets of smaller rank form a model of ZF (aGrothendieck universe), and its subsets can be thought of as classes.In ZF, the concept of a function can also be generalised to classes. A class function is not a function in the usualsense, since it is not a set; it is rather a formula (x; y) with the property that for any set x there is no more than oneset y such that the pair (x,y) satises . For example, the class function mapping each set to its successor may beexpressed as the formula y = x [ fxg . The fact that the ordered pair (x,y) satises may be expressed with theshorthand notation (x) = y .Another approach is taken by the von NeumannBernaysGdel axioms (NBG); classes are the basic objects in thistheory, and a set is then dened to be a class that is an element of some other class. However, the class existenceaxioms of NBG are restricted so that they only quantify over sets, rather than over all classes. This causes NBG tobe a conservative extension of ZF.MorseKelley set theory admits proper classes as basic objects, like NBG, but also allows quantication over allproper classes in its class existence axioms. This causes MK to be strictly stronger than both NBG and ZF.In other set theories, such as New Foundations or the theory of semisets, the concept of proper class still makessense (not all classes are sets) but the criterion of sethood is not closed under subsets. For example, any set theorywith a universal set has proper classes which are subclasses of sets.

4.4 References[1] http://us.metamath.org/mpegif/abeq2.html

Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (third millennium ed.), Berlin, NewYork: Springer-Verlag, ISBN 978-3-540-44085-7

Levy, A. (1979), Basic Set Theory, Berlin, New York: Springer-Verlag

4.5 External links Weisstein, Eric W., Set Class, MathWorld.

Chapter 5

Combinatorial design

Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, constructionand properties of systems of nite sets whose arrangements satisfy generalized concepts of balance and/or symmetry.These concepts are not made precise so that a wide range of objects can be thought of as being under the sameumbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other timesit could involve the spatial arrangement of entries in an array as in Sudoku grids.Combinatorial design theory can be applied to the area of design of experiments. Some of the basic theory ofcombinatorial designs originated in the statistician Ronald Fisher's work on the design of biological experiments.Modern applications are also found in a wide gamut of areas including; Finite geometry, tournament scheduling,lotteries, mathematical biology, algorithm design and analysis, networking, group testing and cryptography.[1]

5.1 ExampleGiven a certain number n of people, is it possible to assign them to sets so that each person is in at least one set, eachpair of people is in exactly one set together, every two sets have exactly one person in common, and no set containseveryone, all but one person, or exactly one person? The answer depends on n.This has a solution only if n has the form q2 + q + 1. It is less simple to prove that a solution exists if q is a primepower. It is conjectured that these are the only solutions. It has been further shown that if a solution exists for qcongruent to 1 or 2 mod 4, then q is a sum of two square numbers. This last result, the BruckRyser theorem, isproved by a combination of constructive methods based on nite elds and an application of quadratic forms.When such a structure does exist, it is called a nite projective plane; thus showing how nite geometry and combi-natorics intersect. When q = 2, the projective plane is called the Fano plane.

5.2 Fundamental combinatorial designsThe classical core of the subject of combinatorial designs is built around balanced incomplete block designs (BIBDs),Hadamard matrices and Hadamard designs, symmetric BIBDs, Latin squares, resolvable BIBDs, dierence sets, andpairwise balanced designs (PBDs).[2] Other combinatorial designs are related to or have been developed from thestudy of these fundamental ones.

A balanced incomplete block design or BIBD (usually called for short a block design) is a collection B ofb subsets (called blocks) of a nite set X of v elements, such that any element of X is contained in the samenumber r of blocks, every block has the same number k of elements, and each pair of distinct elements appeartogether in the same number of blocks. BIBDs are also known as 2-designs and are often denoted as 2-(v,k,)designs. As an example, when = 1 and b = v, we have a projective plane: X is the point set of the plane andthe blocks are the lines.

A symmetric balanced incomplete block design or SBIBD is a BIBD in which v = b (the number of pointsequals the number of blocks). They are the single most important and well studied subclass of BIBDs. Pro-

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5.2. FUNDAMENTAL COMBINATORIAL DESIGNS 15

The Fano plane

jective planes, biplanes and Hadamard 2-designs are all SBIBDs. They are of particular interest since they arethe extremal examples of Fishers inequality (b v).

A resolvable BIBD is a BIBD whose blocks can be partitioned into sets (called parallel classes), each of whichforms a partition of the point set of the BIBD. The set of parallel classes is called a resolution of the design. Asolution of the famous 15 schoolgirl problem is a resolution of a BIBD with v = 15, k = 3 and = 1.[3]

A Latin rectangle is an r n matrix that has the numbers 1, 2, 3, ..., n as its entries (or any other set of ndistinct symbols) with no number occurring more than once in any row or column where r n. An n n Latinrectangle is called a Latin square. If r < n, then it is possible to append n r rows to an r n Latin rectangleto form a Latin square, using Halls marriage theorem.[4]

Two Latin squares of order n are said to be orthogonal if the set of all ordered pairs consisting of thecorresponding entries in the two squares has n2 distinct members (all possible ordered pairs occur). Aset of Latin squares of the same order forms a set of mutually orthogonal Latin squares (MOLS)if every pair of Latin squares in the set are orthogonal. There can be at most n 1 squares in a set ofMOLS of order n. A set of n 1 MOLS of order n can be used to construct a projective plane of ordern (and conversely).

16 CHAPTER 5. COMBINATORIAL DESIGN

A (v, k, ) dierence set is a subset D of a group G such that the order of G is v, the size of D is k, and everynonidentity element of G can be expressed as a product d1d21 of elements of D in exactly ways (when G iswritten with a multiplicative operation).[5]

If D is a dierence set, and g in G, then g D = {gd: d in D} is also a dierence set, and is called atranslate of D. The set of all translates of a dierence set D forms a symmetric block design. In sucha design there are v elements and v blocks. Each block of the design consists of k points, each point iscontained in k blocks. Any two blocks have exactly elements in common and any two points appeartogether in blocks. This SBIBD is called the development of D.[6]

In particular, if = 1, then the dierence set gives rise to a projective plane. An example of a (7,3,1)dierence set in the group Z/7Z (an abelian group written additively) is the subset {1,2,4}. The devel-opment of this dierence set gives the Fano plane.Since every dierence set gives an SBIBD, the parameter set must satisfy the BruckRyserChowlatheorem, but not every SBIBD gives a dierence set.

AnHadamardmatrix of orderm is anm mmatrixHwhose entries are 1 such thatHH =mI, whereHis the transpose of H and I is the m m identity matrix. An Hadamard matrix can be put into standardizedform (that is, converted to an equivalent Hadamard matrix) where the rst row and rst column entries are all+1. If the order m > 2 then m must be a multiple of 4.

Given an Hadamard matrix of order 4a in standardized form, remove the rst row and rst column andconvert every 1 to a 0. The resulting 01 matrixM is the incidence matrix of a symmetric 2 (4a 1,2a 1, a 1) design called anHadamard 2-design.[7] This construction is reversible, and the incidencematrix of a symmetric 2-design with these parameters can be used to form an Hadamard matrix of order4a. When a = 2 we obtain the, by now familiar, Fano plane as an Hadamard 2-design.

A pairwise balanced design (or PBD) is a set X together with a family of subsets of X (which need not havethe same size and may contain repeats) such that every pair of distinct elements of X is contained in exactly (a positive integer) subsets. The set X is allowed to be one of the subsets, and if all the subsets are copies of X,the PBD is called trivial. The size of X is v and the number of subsets in the family (counted with multiplicity)is b.

Fishers inequality holds for PBDs:[8] For any non-trivial PBD, v b.

This result also generalizes the famous ErdsDe Bruijn theorem: For a PBDwith = 1 having no blocksof size 1 or size v, v b, with equality if and only if the PBD is a projective plane or a near-pencil.[9]

5.3 A wide assortment of other combinatorial designsThe Handbook of Combinatorial Designs (Colbourn & Dinitz 2007) has, amongst others, 65 chapters, each devotedto a combinatorial design other than those given above. A partial listing is given below:

Association schemes A balanced ternary design BTD(V, B; 1, 2, R; K, ) is an arrangement of V elements into B multisets(blocks), each of cardinality K (K V), satisfying:

1. Each element appears R = 1 + 22 times altogether, with multiplicity one in exactly 1 blocks and multiplicitytwo in exactly 2 blocks.

2. Every pair of distinct elements appears times (counted with multiplicity); that is, if m is the multiplicityof the element v in block b, then for every pair of distinct elements v and w,PBb=1mvbmwb = .For example, one of the only two nonisomorphic BTD(4,8;2,3,8;4,6)s (blocks are columns) is:[10]

5.3. A WIDE ASSORTMENT OF OTHER COMBINATORIAL DESIGNS 17

The incidence matrix of a BTD (where the entries are the multiplicities of the elements in the blocks)can be used to form a ternary error-correcting code analogous to the way binary codes are formed fromthe incidence matrices of BIBDs.[11]

A balanced tournament design of order n (a BTD(n)) is an arrangement of all the distinct unordered pairsof a 2n-set V into an n (2n 1) array such that

1. every element of V appears precisely once in each column, and

2. every element of V appears at most twice in each row.

An example of a BTD(3) is given by

The columns of a BTD(n) provide a 1-factorization of the complete graph on 2n vertices, K.[12]BTD(n)s can be used to schedule round robin tournaments: the rows represent the locations, the columnsthe rounds of play and the entries are the competing players or teams.

Bent functions Costas arrays Factorial designs A frequency square (F-square) is a higher order generalization of a Latin square. Let S = {s1,s2, ..., s} be aset of distinct symbols and (1, 2, ...,) a frequency vector of positive integers. A frequency square of ordern is an n n array in which each symbol s occurs times, i = 1,2,...,m, in each row and column. The ordern = 1 + 2 + ... + . An F-square is in standard form if in the rst row and column, all occurrences of sprecede those of s whenever i < j.

A frequency square F1 of order n based on the set {s1,s2, ..., s} with frequency vector (1, 2, ...,)and a frequency square F2, also of order n, based on the set {t1,t2, ..., t} with frequency vector (1,2, ...,) are orthogonal if every ordered pair (s, t) appears precisely times when F1 and F2 aresuperimposed.

Hall triple systems (HTSs) are Steiner triple systems (STSs) (but the blocks are called lines) with the propertythat the substructure generated by two intersecting lines is isomorphic to the nite ane plane AG(2,3).

Any ane space AG(n,3) gives an example of an HTS. Such an HTS is an ane HTS. Nonane HTSsalso exist.The number of points of an HTS is 3m for some integer m 2. Nonane HTSs exist for any m 4 anddo not exist for m = 2 or 3.[13]Every Steiner triple system is equivalent to a Steiner quasigroup (idempotent, commutative and satisfying(xy)y = x for all x and y). A Hall triple system is equivalent to a Steiner quasigroup which is distributive,that is, satises a(xy) = (ax)(ay) for all a,x,y in the quasigroup.[14]

Let S be a set of 2n elements. A Howell design, H(s,2n) (on symbol set S) is an s s array such that:

1. Each cell of the array is either empty or contains an unordered pair from S,

2. Each symbol occurs exactly once in each row and column of the array, and

3. Every unordered pair of symbols occurs in at most one cell of the array.


An example of an H(4,6) is

An H(2n 1, 2n) is a Room square of side 2n 1, and thus the Howell designs generalize the conceptof Room squares.

The pairs of symbols in the cells of a Howell design can be thought of as the edges of an s regular graphon 2n vertices, called the underlying graph of the Howell design.

Cyclic Howell designs are used as Howell movements in duplicate bridge tournaments. The rows of thedesign represent the rounds, the columns represent the boards, and the diagonals represent the tables.[15]

Linear spaces An (n,k,p,t)-lotto design is an n-set V of elements together with a set of k-element subsets of V (blocks),so that for any p-subset P of V, there is a block B in for which |P B | t. L(n,k,p,t) denotes the smallestnumber of blocks in any (n,k,p,t)-lotto design. The following is a (7,5,4,3)-lotto design with the smallestpossible number of blocks:[16]

{1,2,3,4,7} {1,2,5,6,7} {3,4,5,6,7}.Lotto designs model any lottery that is run in the following way: Individuals purchase tickets consistingof k numbers chosen from a set of n numbers. At a certain point the sale of tickets is stopped and aset of p numbers is randomly selected from the n numbers. These are the winning numbers. If any soldticket contains t or more of the winning numbers, a prize is given to the ticket holder. Larger prizes goto tickets with more matches. The value of L(n,k,p,t) is of interest to both gamblers and researchers, asthis is the smallest number of tickets that are needed to be purchased in order to guarantee a prize.

The Hungarian Lottery is a (90,5,5,t)-lotto design and it is known that L(90,5,5,2) = 100. Lotterieswith parameters (49,6,6,t) are also popular worldwide and it is known that L(49,6,6,2) = 19. In generalthough, these numbers are hard to calculate and remain unknown.[17]

A geometric construction of one such design is given in Transylvanian lottery.

Magic squares

A (v,k,)-Mendelsohn design, or MD(v,k,),is a v-set V and a collection of ordered k-tuples of distinctelements of V (called blocks), such that each ordered pair (x,y) with x y of elements of V is cyclicallyadjacent in blocks. The ordered pair (x,y) of distinct elements is cyclically adjacent in a block if the elementsappear in the block as (...,x,y,...) or (y,...,x). An MD(v,3,) is a Mendelsohn triple system, MTS(v,). Anexample of an MTS(4,1) on V = {0,1,2,3} is:

(0,1,2) (1,0,3) (2,1,3) (0,2,3)Any triple system can be made into a Mendelson triple system by replacing the unordered triple {a,b,c}with the pair of ordered triples (a,b,c) and (a,c,b), but as the example shows, the converse of this state-ment is not true.If (Q,) is an idempotent semisymmetric quasigroup, that is, x x = x (idempotent) and x (y x) = y(semisymmetric) for all x, y in Q, let = {(x,y,x y): x, y in Q}. Then (Q, ) is a Mendelsohn triplesystem MTS(|Q|,1). This construction is reversible.[18]

Orthogonal arrays

A quasi-3 design is a symmetric design (SBIBD) in which each triple of blocks intersect in either x or y points,for xed x and y called the triple intersection numbers (x < y). Any symmetric design with 2 is a quasi-3design with x = 0 and y = 1. The point-hyperplane design of PG(n,q) is a quasi-3 design with x = (qn2 1)/(q 1) and y = = (qn1 1)/(q 1). If y = for a quasi-3 design, the design is isomorphic to PG(n,q) or aprojective plane.[19]

5.3. A WIDE ASSORTMENT OF OTHER COMBINATORIAL DESIGNS 19

A t-(v,k,) design D is quasi-symmetric with intersection numbers x and y (x < y) if every two distinct blocksintersect in either x or y points. These designs naturally arise in the investigation of the duals of designs with = 1. A non-symmetric (b > v) 2-(v,k,1) design is quasisymmetric with x = 0 and y = 1. A multiple (repeatall blocks a certain number of times) of a symmetric 2-(v,k,) design is quasisymmetric with x = and y = k.Hadamard 3-designs (extensions of Hadamard 2-designs) are quasisymmetric.[20]

Every quasisymmetric block design gives rise to a strongly regular graph (as its block graph), but not allSRGs arise in this way.[21]

The incidence matrix of a quasisymmetric 2-(v,k,) design with k x y (mod 2) generates a binaryself-orthogonal code (when bordered if k is odd).[22]

Room squares

A spherical design is a nite set X of points in a (d 1)-dimensional sphere such that, for some integer t, theaverage value on X of every polynomial

f(x1; : : : ; xd)

of total degree at most t is equal to the average value of f on the whole sphere, i.e., the integral of fdivided by the area of the sphere.

Turn systems

An r n tuscan-k rectangle on n symbols has r rows and n columns such that:

1. each row is a permutation of the n symbols and

2. for any two distinct symbols a and b and for each m from 1 to k, there is at most one row in which b is m stepsto the right of a.

If r = n and k = 1 these are referred to as Tuscan squares, while if r = n and k = n - 1 they are Florentinesquares. A Roman square is a tuscan square which is also a latin square (these are also known as rowcomplete latin squares). A Vatican square is a orentine square which is also a latin square.

The following example is a tuscan-1 square on 7 symbols which is not tuscan-2:[23]

A tuscan square on n symbols is equivalent to a decomposition of the complete graph with n verticesinto n hamiltonian directed paths.[24]

In a sequence of visual impressions, one ash card may have some eect on the impression given by thenext. This bias can be cancelled by using n sequences corresponding to the rows of an n n tuscan-1square.[25]

A t-wise balanced design (or t BD) of type t (v,K,) is a v-set X together with a family of subsets of X(called blocks) whose sizes are in the set K, such that every t-subset of distinct elements of X is contained inexactly blocks. If K is a set of positive integers strictly between t and v, then the t BD is proper. If all thek-subsets of X for some k are blocks, the t BD is a trivial design.[26]

Notice that in the following example of a 3-{12,{4,6},1) design based on the set X = {1,2,...,12}, somepairs appear four times (such as 1,2) while others appear ve times (6,12 for instance).[27]


1 2 3 4 5 6 1 2 7 8 1 2 9 11 1 2 10 12 3 5 7 8 3 5 9 11 3 5 10 12 4 6 7 8 4 6 9 11 4 6 10 127 8 9 10 11 12 2 3 8 9 2 3 10 7 2 3 11 12 4 1 8 9 4 1 10 7 4 1 11 12 5 6 8 9 5 6 10 7 5 6 11123 4 9 10 3 4 11 8 3 4 7 12 5 2 9 10 5 2 11 8 5 2 7 12 1 6 9 10 1 6 11 8 1 6 7 124 5 10 11 4 5 7 9 4 5 8 12 1 3 10 11 1 3 7 9 1 3 8 12 2 6 10 11 2 6 7 9 2 6 8 125 1 11 7 5 1 8 10 5 1 9 12 2 4 11 7 2 4 8 10 2 4 9 12 3 6 11 7 3 6 8 10 3 6 9 12

A Youden square is a k v rectangular array (k < v) of v symbols such that each symbol appears exactly oncein each row and the symbols appearing in any column form a block of a symmetric (v, k, ) design, all theblocks of which occur in this manner. A Youden square is a Latin rectangle. The term square in the namecomes from an older denition which did use a square array.[28] An example of a 4 7 Youden square is givenby:

The seven blocks (columns) form the order 2 biplane (a symmetric (7,4,2)-design).

5.4 See also Algebraic statistics Hypergraph

5.5 Notes[1] Stinson 2003, pg.1

[2] Stinson 2003, pg. IX

[3] Beth, Jungnickel & Lenz 1986, pg. 40 Example 5.8

[4] Ryser 1963, pg. 52, Theorem 3.1

[5] When the group G is an abelian group (or written additively) the dening property looks like d1 d2 from which the termdierence set comes from.

[6] Beth, Jungnickel & Lenz 1986, pg. 262, Theorem 1.6

[7] Stinson 2003, pg. 74, Theorem 4.5



[10] Colbourn & Dinitz 2007, pg. 331, Example 2.2

[11] Colbourn & Dinitz 2007, pg. 331, Remark 2.8


[13] Colbourn & Dinitz 2007, pg. 496, Theorem 28.5



[16] Colbourn & Dinitz 2007, pg. 512, Example 32.4




5.6. REFERENCES 21

[20] Colbourn & Dinitz 2007, pp. 578-579[21] Colbourn & Dinitz 2007, pg. 579, Theorem 48.10[22] Colbourn & Dinitz 2007, pg. 580, Lemma 48.22[23] Colbourn & Dinitz 2007, pg. 652, Examples 62.4[24] Colbourn & Dinitz 2007, pg. 655, Theorem 62.24[25] Colbourn & Dinitz 2007, pg. 657, Remark 62.29[26] Colbourn & Dinitz 2007, pg. 657[27] Colbourn & Dinitz 2007, pg. 658, Example 63.5[28] Colbourn & Dinitz 2007, pg. 669, Remark 65.3

5.6 References Assmus, E.F.; Key, J.D. (1992), Designs and Their Codes, Cambridge: Cambridge University Press, ISBN0-521-41361-3

Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried (1986), Design Theory, Cambridge: Cambridge UniversityPress. 2nd ed. (1999) ISBN 978-0-521-44432-3.

R. C. Bose, A Note on Fishers Inequality for Balanced Incomplete Block Designs, Annals of MathematicalStatistics, 1949, pages 619620.

Caliski, Tadeusz and Kageyama, Sanpei (2003). Block designs: A Randomization approach, Volume II: De-sign. Lecture Notes in Statistics 170. New York: Springer-Verlag. ISBN 0-387-95470-8.

Colbourn, Charles J.; Dinitz, Jerey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton:Chapman & Hall/ CRC, ISBN 1-58488-506-8

R. A. Fisher, An examination of the dierent possible solutions of a problem in incomplete blocks, Annalsof Eugenics, volume 10, 1940, pages 5275.

Hall, Jr., Marshall (1986), Combinatorial Theory (2nd ed.), New York: Wiley-Interscience, ISBN 0-471-09138-3

Hughes, D.R.; Piper, E.C. (1985), Design theory, Cambridge: Cambridge University Press, ISBN 0-521-25754-9

Lander, E. S. (1983), Symmetric Designs: An Algebraic Approach, Cambridge: Cambridge University Press Lindner, C.C.; Rodger, C.A. (1997), Design Theory, Boca Raton: CRC Press, ISBN 0-8493-3986-3 Raghavarao, Damaraju (1988). Constructions and Combinatorial Problems in Design of Experiments (correctedreprint of the 1971 Wiley ed.). New York: Dover.

Raghavarao, Damaraju and Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications.World Scientic.

Ryser, Herbert John (1963), Chapter 8: Combinatorial Designs, Combinatorial Mathematics (Carus Mono-graph #14), Mathematical Association of America

S. S. Shrikhande, and Vasanti N. Bhat-Nayak, Non-isomorphic solutions of some balanced incomplete blockdesigns I Journal of Combinatorial Theory, 1970

Stinson, Douglas R. (2003), Combinatorial Designs: Constructions and Analysis, New York: Springer, ISBN0-387-95487-2

Street, Anne Penfold and Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P.[Clarendon]. pp. 400+xiv. ISBN 0-19-853256-3.

van Lint, J.H., and R.M. Wilson (1992), A Course in Combinatorics. Cambridge, Eng.: Cambridge UniversityPress.


5.7 External links Design DB: A comprehensive database of combinatorial, statistical, experimental block designs

Chapter 6

Delta-ring

In mathematics, a nonempty collection of setsR is called a -ring (pronounced delta-ring) if it is closed under union,relative complementation, and countable intersection:

1. A [B 2 R if A;B 2 R2. AB 2 R if A;B 2 R3. T1n=1An 2 R if An 2 R for all n 2 N

If only the rst two properties are satised, then R is a ring but not a -ring. Every -ring is a -ring, but not every-ring is a -ring.-rings can be used instead of -elds in the development of measure theory if one does not wish to allow sets ofinnite measure.

6.1 See also Ring of sets Sigma eld Sigma ring

6.2 References Cortzen, Allan. Delta-Ring. From MathWorldA Wolfram Web Resource, created by Eric W. Weisstein.http://mathworld.wolfram.com/Delta-Ring.html

23

Chapter 7

Disjoint sets

This article is about the mathematical concept. For the data structure, see Disjoint-set data structure.In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are

A BTwo disjoint sets.

sets whose intersection is the empty set.[1] For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and{3, 4, 5} are not.

7.1 Generalizations

This denition of disjoint sets can be extended to any family of sets. A family of sets is pairwise disjoint ormutuallydisjoint if every two dierent sets in the family are disjoint.[1] For example, the collection of sets { {1}, {2}, {3}, ...} is pairwise disjoint.Two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two innite setswhose intersect

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