glossary of set theory 2

Glossary of set theory 2From Wikipedia, the free encyclopedia

Contents

1 Additively indecomposable ordinal 11.1 Multiplicatively indecomposable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Cardinal number 22.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Cardinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4.1 Successor cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.2 Cardinal addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.3 Cardinal multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.4 Cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 The continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Cartesian product 103.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 A deck of cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.2 A two-dimensional coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Most common implementation (set theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.1 Non-commutativity and non-associativity . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.2 Intersections, unions, and subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.3 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 n-ary product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.1 Cartesian power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.2 Finite n-ary product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.3 Innite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Other forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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3.4.1 Abbreviated form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.2 Cartesian product of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5 Denitions outside of Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5.1 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5.2 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Diagonal intersection 184.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Diamond principle 195.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Properties and use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 0 216.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2 Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.3 Surreal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7 Equivalence relation 257.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.5 Well-denedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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7.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.10 Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.11 Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.12 Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.13 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.14 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.16 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8 FefermanSchtte ordinal 348.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

9 Innitary combinatorics 359.1 Ramsey theory for innite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.2 Large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10 Lvy hierarchy 3710.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

10.2.1 0=0=0 formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.2.2 1-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.2.3 1-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.2.4 1-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.2.5 2-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.2.6 2-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.2.7 2-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.2.8 3-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.2.9 3-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.2.10 3-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.2.11 4-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

10.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

11 Measure (mathematics) 4011.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

11.3.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.3.2 Measures of innite unions of measurable sets . . . . . . . . . . . . . . . . . . . . . . . . 4211.3.3 Measures of innite intersections of measurable sets . . . . . . . . . . . . . . . . . . . . . 42

11.4 Sigma-nite measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.6 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.7 Non-measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4411.11Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

12 Order type 4812.1 Order type of well-orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4812.2 Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

13 Ordered pair 5013.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.2 Dening the ordered pair using set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

13.2.1 Wieners denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.2.2 Hausdors denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.2.3 Kuratowski denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.2.4 Quine-Rosser denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5313.2.5 Morse denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

13.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5413.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

14 Ordinal number 5514.1 Ordinals extend the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

14.2.1 Well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5814.2.2 Denition of an ordinal as an equivalence class . . . . . . . . . . . . . . . . . . . . . . . 5814.2.3 Von Neumann denition of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5814.2.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

14.3 Transnite sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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14.4 Transnite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5914.4.1 What is transnite induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5914.4.2 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.4.3 Successor and limit ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.4.4 Indexing classes of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6014.4.5 Closed unbounded sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

14.5 Arithmetic of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6114.6 Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

14.6.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6214.6.2 Conality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

14.7 Some large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6214.8 Topology and ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6314.9 Downward closed sets of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6314.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6314.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6314.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6314.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

15 Rational number 6515.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6515.2 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

15.2.1 Embedding of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6615.2.2 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6615.2.3 Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6615.2.4 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6615.2.5 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6615.2.6 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6615.2.7 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6615.2.8 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6715.2.9 Exponentiation to integer power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

15.3 Continued fraction representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6715.4 Formal construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6715.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6915.6 Real numbers and topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7015.7 p-adic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7015.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7015.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7015.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

16 Real number 7216.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7316.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

vi CONTENTS

16.2.1 Axiomatic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7416.2.2 Construction from the rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

16.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7516.3.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7516.3.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7516.3.3 The complete ordered eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.3.4 Advanced properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

16.4 Applications and connections to other areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.4.1 Real numbers and logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.4.2 In physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7816.4.3 In computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7816.4.4 Reals in set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

16.5 Vocabulary and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7816.6 Generalizations and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7916.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7916.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7916.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8016.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

17 Square principle 8117.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.2 Variant relative to a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

18 Stoneech compactication 8218.1 Universal property and functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8218.2 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

18.2.1 Construction using products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8318.2.2 Construction using the unit interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8318.2.3 Construction using ultralters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8318.2.4 Construction using C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

18.3 The Stoneech compactication of the natural numbers . . . . . . . . . . . . . . . . . . . . . . . 8418.3.1 An application: the dual space of the space of bounded sequences of reals . . . . . . . . . 8418.3.2 Addition on the Stoneech compactication of the naturals . . . . . . . . . . . . . . . . . 85

18.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8518.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8618.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8618.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

19 Sunower (mathematics) 8719.1 lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8819.2 lemma for !2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

CONTENTS vii

19.3 Sunower lemma and conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8819.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

20 Veblen function 8920.1 The Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

20.1.1 Fundamental sequences for the Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . 8920.1.2 The function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

20.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9020.2.1 Finitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9020.2.2 Transnitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

20.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

21 Von Neumann universe 9221.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

21.1.1 Finite and low cardinality stages of the hierarchy . . . . . . . . . . . . . . . . . . . . . . . 9321.2 Applications and interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

21.2.1 Applications of V as models for set theories . . . . . . . . . . . . . . . . . . . . . . . . . 9321.2.2 Interpretation of V as the set of all sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 9321.2.3 V and the axiom of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9321.2.4 The existential status of V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

21.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9421.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9421.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9421.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

22 set 9722.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9722.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9722.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9722.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

23 -compact space 9923.1 Properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9923.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9923.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10023.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10023.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 101

23.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10123.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10423.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Chapter 1

Additively indecomposable ordinal

In set theory, a branch of mathematics, an additively indecomposable ordinal is any ordinal number that is not0 such that for any ; < , we have + < : The class of additively indecomposable ordinals (aka gammanumbers) is denoted H:From the continuity of addition in its right argument, we get that if < and is additively indecomposable, then + = :

Obviously 1 2 H , since 0 + 0 < 1: No nite ordinal other than 1 is in H: Also, ! 2 H , since the sum of two niteordinals is still nite. More generally, every innite cardinal is in H:H is closed and unbounded, so the enumerating function of H is normal. In fact, fH() = !:The derivative f 0H() (which enumerates xed points of fH) is written : Ordinals of this form (that is, xed pointsof fH ) are called epsilon numbers. The number 0 = !!

!

is therefore the rst xed point of the sequence!; !!; !!

!

; : : :

1.1 Multiplicatively indecomposableA similar notion can be dened for multiplication. The multiplicatively indecomposable ordinals (aka delta numbers)are those of the form !! for any ordinal . Every epsilon number is multiplicatively indecomposable; and everymultiplicatively indecomposable ordinal is additively indecomposable. The delta numbers are the same as the primeordinals that are limits.

1.2 See also Ordinal arithmetic

1.3 References Sierpiski, Wacaw (1958), Cardinal and ordinal numbers., Polska Akademia NaukMonograeMatematyczne34, Warsaw: Pastwowe Wydawnictwo Naukowe, MR 0095787

This article incorporates material from Additively indecomposable on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

1

Chapter 2

Cardinal number

This article is about the mathematical concept. For number words indicating quantity (three apples, four birds,etc.), see Cardinal number (linguistics).In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to

measure the cardinality (size) of sets. The cardinality of a nite set is a natural number: the number of elements inthe set. The transnite cardinal numbers describe the sizes of innite sets.Cardinality is dened in terms of bijective functions. Two sets have the same cardinality if, and only if, there is aone-to-one correspondence (bijection) between the elements of the two sets. In the case of nite sets, this agrees withthe intuitive notion of size. In the case of innite sets, the behavior is more complex. A fundamental theorem due toGeorg Cantor shows that it is possible for innite sets to have dierent cardinalities, and in particular the cardinalityof the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a propersubset of an innite set to have the same cardinality as the original set, something that cannot happen with propersubsets of nite sets.There is a transnite sequence of cardinal numbers:

0; 1; 2; 3; : : : ; n; : : : ;@0;@1;@2; : : : ;@; : : : :This sequence starts with the natural numbers including zero (nite cardinals), which are followed by the alephnumbers (innite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under theassumption of the axiom of choice, this transnite sequence includes every cardinal number. If one rejects thataxiom, the situation is more complicated, with additional innite cardinals that are not alephs.Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics includingcombinatorics, abstract algebra, and mathematical analysis. In category theory, the cardinal numbers form a skeletonof the category of sets.

2.1 HistoryThe notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 18741884. Cardinality can be used to compare an aspect of nite sets; e.g. the sets {1,2,3} and {4,5,6} are not equal,but have the same cardinality, namely three (this is established by the existence of a bijection, i.e. a one-to-onecorrespondence, between the two sets; e.g. {1->4, 2->5, 3->6}).Cantor applied his concept of bijection to innite sets;[1] e.g. the set of natural numbers N = {0, 1, 2, 3, ...}. Thus,all sets having a bijection with N he called denumerable (countably innite) sets and they all have the same cardinalnumber. This cardinal number is called@0 , aleph-null. He called the cardinal numbers of these innite sets, transnitecardinal numbers.Cantor proved that any unbounded subset of N has the same cardinality as N, even though this might appear to runcontrary to intuition. He also proved that the set of all ordered pairs of natural numbers is denumerable (whichimplies that the set of all rational numbers is denumerable), and later proved that the set of all algebraic numbers isalso denumerable. Each algebraic number zmay be encoded as a nite sequence of integers which are the coecients

2

2.1. HISTORY 3

X 1

2

3

4

YD

B

C

A

A bijective function, f: X Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to thecardinal number 4.

in the polynomial equation of which it is the solution, i.e. the ordered n-tuple (a0, a1, ..., an), ai Z together with apair of rationals (b0, b1) such that z is the unique root of the polynomial with coecients (a0, a1, ..., an) that lies inthe interval (b0, b1).In his 1874 paper, Cantor proved that there exist higher-order cardinal numbers by showing that the set of real numbershas cardinality greater than that of N. His original presentation used a complex argument with nested intervals, but inan 1891 paper he proved the same result using his ingenious but simple diagonal argument. The new cardinal numberof the set of real numbers is called the cardinality of the continuum and Cantor used the symbol c for it.Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallesttransnite cardinal number ( @0 , aleph-null) and that for every cardinal number, there is a next-larger cardinal

(@1;@2;@3; ):

His continuum hypothesis is the proposition that c is the same as @1 . This hypothesis has been found to be inde-pendent of the standard axioms of mathematical set theory; it can neither be proved nor disproved from the standardassumptions.

4 CHAPTER 2. CARDINAL NUMBER

Aleph null, the smallest innite cardinal

2.2 Motivation

In informal use, a cardinal number is what is normally referred to as a counting number, provided that 0 is included:0, 1, 2, .... They may be identied with the natural numbers beginning with 0. The counting numbers are exactlywhat can be dened formally as the nite cardinal numbers. Innite cardinals only occur in higher-level mathematicsand logic.More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe theposition of an element in a sequence. For nite sets and sequences it is easy to see that these two notions coincide,since for every number describing a position in a sequence we can construct a set which has exactly the right size,e.g. 3 describes the position of 'c' in the sequence , and we can construct the set {a,b,c} which has3 elements. However when dealing with innite sets it is essential to distinguish between the two the two notionsare in fact dierent for innite sets. Considering the position aspect leads to ordinal numbers, while the size aspect isgeneralized by the cardinal numbers described here.The intuition behind the formal denition of cardinal is the construction of a notion of the relative size or bigness ofa set without reference to the kind of members which it has. For nite sets this is easy; one simply counts the numberof elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more subtle notions.

2.3. FORMAL DEFINITION 5

A set Y is at least as big as a set X if there is an injective mapping from the elements of X to the elements of Y.An injective mapping identies each element of the set X with a unique element of the set Y. This is most easilyunderstood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size wewould observe that there is a mapping:

1 a2 b3 c

which is injective, and hence conclude that Y has cardinality greater than or equal to X. Note the element d has noelement mapping to it, but this is permitted as we only require an injective mapping, and not necessarily an injectiveand onto mapping. The advantage of this notion is that it can be extended to innite sets.We can then extend this to an equality-style relation. Two sets X and Y are said to have the same cardinality if thereexists a bijection between X and Y. By the SchroederBernstein theorem, this is equivalent to there being both aninjective mapping from X to Y and an injective mapping from Y to X. We then write |X| = |Y |. The cardinal numberofX itself is often dened as the least ordinal awith |a| = |X|. This is called the von Neumann cardinal assignment; forthis denition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement isthe well-ordering principle. It is however possible to discuss the relative cardinality of sets without explicitly assigningnames to objects.The classic example used is that of the innite hotel paradox, also called Hilberts paradox of the Grand Hotel.Suppose you are an innkeeper at a hotel with an innite number of rooms. The hotel is full, and then a new guestarrives. It is possible to t the extra guest in by asking the guest who was in room 1 to move to room 2, the guest inroom 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:

1 22 33 4...n n + 1...

In this way we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...} since a bijection betweenthe rst and the second has been shown. This motivates the denition of an innite set being any set which has aproper subset of the same cardinality; in this case {2,3,4,...} is a proper subset of {1,2,3,...}.When considering these large objects, we might also want to see if the notion of counting order coincides with thatof cardinal dened above for these innite sets. It happens that it doesn't; by considering the above example we cansee that if some object one greater than innity exists, then it must have the same cardinality as the innite setwe started out with. It is possible to use a dierent formal notion for number, called ordinals, based on the ideasof counting and considering each number in turn, and we discover that the notions of cardinality and ordinality aredivergent once we move out of the nite numbers.It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described.This can be visualized using Cantors diagonal argument; classic questions of cardinality (for instance the continuumhypothesis) are concerned with discovering whether there is some cardinal between some pair of other innite car-dinals. In more recent times mathematicians have been describing the properties of larger and larger cardinals.Since cardinality is such a common concept in mathematics, a variety of names are in use. Sameness of cardinality issometimes referred to as equipotence, equipollence, or equinumerosity. It is thus said that two sets with the samecardinality are, respectively, equipotent, equipollent, or equinumerous.

2.3 Formal denitionFormally, assuming the axiom of choice, the cardinality of a set X is the least ordinal such that there is a bijectionbetween X and . This denition is known as the von Neumann cardinal assignment. If the axiom of choice is not


assumed we need to do something dierent. The oldest denition of the cardinality of a set X (implicit in Cantor andexplicit in Frege and Principia Mathematica) is as the class [X] of all sets that are equinumerous with X. This doesnot work in ZFC or other related systems of axiomatic set theory because if X is non-empty, this collection is toolarge to be a set. In fact, for X there is an injection from the universe into [X] by mapping a set m to {m} Xand so by the axiom of limitation of size, [X] is a proper class. The denition does work however in type theory andin New Foundations and related systems. However, if we restrict from this class to those equinumerous with X thathave the least rank, then it will work (this is a trick due to Dana Scott:[2] it works because the collection of objectswith any given rank is a set).Formally, the order among cardinal numbers is dened as follows: |X| |Y | means that there exists an injectivefunction from X to Y. The CantorBernsteinSchroeder theorem states that if |X| |Y | and |Y | |X| then |X| = |Y |.The axiom of choice is equivalent to the statement that given two sets X and Y, either |X| |Y | or |Y | |X|.[3][4]

A set X is Dedekind-innite if there exists a proper subset Y of X with |X| = |Y |, and Dedekind-nite if such a subsetdoesn't exist. The nite cardinals are just the natural numbers, i.e., a set X is nite if and only if |X| = |n| = n forsome natural number n. Any other set is innite. Assuming the axiom of choice, it can be proved that the Dedekindnotions correspond to the standard ones. It can also be proved that the cardinal @0 (aleph null or aleph-0, where alephis the rst letter in the Hebrew alphabet, represented @ ) of the set of natural numbers is the smallest innite cardinal,i.e. that any innite set has a subset of cardinality @0: The next larger cardinal is denoted by @1 and so on. For everyordinal there is a cardinal number @; and this list exhausts all innite cardinal numbers.

2.4 Cardinal arithmeticWe can dene arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers.It can be shown that for nite cardinals these operations coincide with the usual operations for natural numbers.Furthermore, these operations share many properties with ordinary arithmetic.

2.4.1 Successor cardinalFor more details on this topic, see Successor cardinal.

If the axiom of choice holds, every cardinal has a successor + > , and there are no cardinals between and itssuccessor. (Without the axiom of choice, using Hartogs theorem, it can be shown that, for any cardinal number ,there is a minimal cardinal +, so that + : ) For nite cardinals, the successor is simply + 1. For innitecardinals, the successor cardinal diers from the successor ordinal.

2.4.2 Cardinal additionIf X and Y are disjoint, addition is given by the union of X and Y. If the two sets are not already disjoint, then theycan be replaced by disjoint sets of the same cardinality, e.g., replace X by X{0} and Y by Y{1}.

jXj+ jY j = jX [ Y j:Zero is an additive identity + 0 = 0 + = .Addition is associative ( + ) + = + ( + ).Addition is commutative + = + .Addition is non-decreasing in both arguments:

( )! ((+ + ) and ( + + )):Assuming the axiom of choice, addition of innite cardinal numbers is easy. If either or is innite, then

+ = maxf; g :

2.4. CARDINAL ARITHMETIC 7

Subtraction

Assuming the axiom of choice and, given an innite cardinal and a cardinal , there exists a cardinal such that + = if and only if . It will be unique (and equal to ) if and only if < .

2.4.3 Cardinal multiplicationThe product of cardinals comes from the cartesian product.

jXj jY j = jX Y j0 = 0 = 0. = 0 ( = 0 or = 0).One is a multiplicative identity 1 = 1 = .Multiplication is associative () = ().Multiplication is commutative = .Multiplication is non-decreasing in both arguments: ( and ).Multiplication distributes over addition: ( + ) = + and ( + ) = + .Assuming the axiom of choice, multiplication of innite cardinal numbers is also easy. If either or is innite andboth are non-zero, then

= maxf; g:

Division

Assuming the axiom of choice and, given an innite cardinal and a non-zero cardinal , there exists a cardinal such that = if and only if . It will be unique (and equal to ) if and only if < .

2.4.4 Cardinal exponentiationExponentiation is given by

jXjjY j = XY where XY is the set of all functions from Y to X.

0 = 1 (in particular 00 = 1), see empty function.If 1 , then 0 = 0.1 = 1.1 = . + = . = ().() = .

Exponentiation is non-decreasing in both arguments:

(1 and ) ( ) and( ) ( ).


Note that 2|X| is the cardinality of the power set of the set X and Cantors diagonal argument shows that 2|X| > |X| forany set X. This proves that no largest cardinal exists (because for any cardinal , we can always nd a larger cardinal2). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.)All the remaining propositions in this section assume the axiom of choice:

If and are both nite and greater than 1, and is innite, then = .If is innite and is nite and non-zero, then = .

If 2 and 1 and at least one of them is innite, then:

Max (, 2) Max (2, 2).

Using Knigs theorem, one can prove < cf() and < cf(2) for any innite cardinal , where cf() is the conalityof .

Roots

Assuming the axiom of choice and, given an innite cardinal and a nite cardinal greater than 0, the cardinal satisfying = will be .

Logarithms

Assuming the axiom of choice and, given an innite cardinal and a nite cardinal greater than 1, there may ormay not be a cardinal satisfying = . However, if such a cardinal exists, it is innite and less than , and anynite cardinality greater than 1 will also satisfy = .The logarithm of an innite cardinal number is dened as the least cardinal number such that 2. Logarithmsof innite cardinals are useful in some elds of mathematics, for example in the study of cardinal invariants oftopological spaces, though they lack some of the properties that logarithms of positive real numbers possess.[5][6][7]

2.5 The continuum hypothesisThe continuum hypothesis (CH) states that there are no cardinals strictly between @0 and 2@0 : The latter cardinalnumber is also often denoted by c ; it is the cardinality of the continuum (the set of real numbers). In this case2@0 = @1: The generalized continuum hypothesis (GCH) states that for every innite set X, there are no cardinalsstrictly between | X | and 2| X |. The continuum hypothesis is independent of the usual axioms of set theory, theZermelo-Fraenkel axioms together with the axiom of choice (ZFC).

2.6 See also

2.7 Notes

2.8 ReferencesNotes

[1] Dauben 1990, pg. 54

[2] Deiser, Oliver (May 2010). On the Development of the Notion of a Cardinal Number. History and Philosophy of Logic31 (2): 123143. doi:10.1080/01445340903545904.

[3] Enderton, Herbert. Elements of Set Theory, Academic Press Inc., 1977. ISBN 0-12-238440-7

2.9. EXTERNAL LINKS 9

[4] Friedrich M. Hartogs (1915), Felix Klein, Walther von Dyck, David Hilbert, Otto Blumenthal, ed., "ber das Problem derWohlordnung, Math. Ann (Leipzig: B. G. Teubner), Bd. 76 (4): 438443, ISSN 0025-5831

[5] Robert A. McCoy and Ibula Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathe-matics 1315, Springer-Verlag.

[6] Eduard ech, Topological Spaces, revised by Zdenek Frolk and Miroslav Katetov, John Wiley & Sons, 1966.

[7] D.A. Vladimirov, Boolean Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.

Bibliography

Dauben, Joseph Warren (1990), Georg Cantor: His Mathematics and Philosophy of the Innite, Princeton:Princeton University Press, ISBN 0691-02447-2

Hahn, Hans, Innity, Part IX, Chapter 2, Volume 3 of The World of Mathematics. New York: Simon andSchuster, 1956.

Halmos, Paul, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).

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

Weisstein, Eric W., Cardinal Number, MathWorld. Cardinality at ProvenMath proofs of the basic theorems on cardinality.

Chapter 3

Cartesian product

Cartesian square redirects here. For Cartesian squares in category theory, see Cartesian square (category theory).In mathematics, a Cartesian product is a mathematical operation which returns a set (or product set or simply

(z,1) (z,2) (z,3)

(y,1) (y,2) (y,3)

(x,1) (x,2) (x,3)

1 2 3

z

y

x

BA

AB

Cartesian product AB of the sets A=fx;y;zg and B=f1;2;3g

product) from multiple sets. That is, for sets A and B, the Cartesian product A B is the set of all ordered pairs (a,b) where a A and b B. Products can be specied using set-builder notation, e.g.

AB = f (a; b) j a 2 A and b 2 B g: [1]

A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian productrows columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).More generally, a Cartesian product of n sets, also known as a n-fold Cartesian product, can be represented by anarray of n dimensions, where each element is an n-tuple. An ordered pair is a 2-tuple or couple.

10

3.1. EXAMPLES 11

The Cartesian product is named after Ren Descartes,[2] whose formulation of analytic geometry gave rise to theconcept, which is further generalized in terms of direct product.

3.1 Examples

3.1.1 A deck of cards

Standard 52-card deck

An illustrative example is the standard 52-card deck. The standard playing card ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5,4, 3, 2} form a 13-element set. The card suits {, , , } form a 4-element set. The Cartesian product of thesesets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.Ranks Suits returns a set of the form {(A, ), (A, ), (A, ), (A, ), (K, ), ..., (3, ), (2, ), (2, ), (2, ), (2,)}.Suits Ranks returns a set of the form {(, A), (, K), (, Q), (, J), (, 10), ..., (, 6), (, 5), (, 4), (, 3), (,2)}.Both sets are distinct, even disjoint.

3.1.2 A two-dimensional coordinate system

The main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in anumerical way and extract numerical information from shapes numerical representations, Ren Descartes assigned toeach point in the plane a pair of real numbers, called its coordinates. Usually, such a pairs rst and second componentis called its x and y coordinate, respectively, cf. picture. The set of all such pairs, i.e. the Cartesian product with denoting the real numbers, is thus assigned to the set of all points in the plane.

3.2 Most common implementation (set theory)Main article: Implementation of mathematics in set theory

A formal denition of the Cartesian product from set-theoretical principles follows from a denition of ordered pair.The most common denition of ordered pairs, the Kuratowski denition, is (x; y) = ffxg; fx; ygg . Note that,under this denition, X Y P(P(X [ Y )) , where P represents the power set. Therefore, the existence of theCartesian product of any two sets in ZFC follows from the axioms of pairing, union, power set, and specication.

12 CHAPTER 3. CARTESIAN PRODUCT

3 2 1 1 2 3

3

2

1

3

2

1

(3,1)

(1.5,2.5)

(2,3)

(0,0) x

y

Cartesian coordinates of example points

Since functions are usually dened as a special case of relations, and relations are usually dened as subsets of theCartesian product, the denition of the two-set Cartesian product is necessarily prior to most other denitions.

3.2.1 Non-commutativity and non-associativityLet A, B, C, and D be sets.The Cartesian product A B is not commutative,

AB 6= B A;because the ordered pairs are reversed except if at least one of the following conditions is satised:[3]

A is equal to B, or A or B is the empty set.

For example:

3.2. MOST COMMON IMPLEMENTATION (SET THEORY) 13

A = {1,2}; B = {3,4}

A B = {1,2} {3,4} = {(1,3), (1,4), (2,3), (2,4)}B A = {3,4} {1,2} = {(3,1), (3,2), (4,1), (4,2)}

A = B = {1,2}

A B = B A = {1,2} {1,2} = {(1,1), (1,2), (2,1), (2,2)}

A = {1,2}; B =

A B = {1,2} = B A = {1,2} =

Strictly speaking, the Cartesian product is not associative (unless one of the involved sets is empty).

(AB) C 6= A (B C)

If for example A = {1}, then (A A) A = { ((1,1),1) } { (1,(1,1)) } = A (A A).

3.2.2 Intersections, unions, and subsets

The Cartesian product behaves nicely with respect to intersections, cf. left picture.

(A \B) (C \D) = (A C) \ (B D) [4]

In most cases the above statement is not true if we replace intersection with union, cf. middle picture.

(A [B) (C [D) 6= (A C) [ (B D)

In fact, we have that:

(A C) [ (B D) = [(A nB) C] [ [(A \B) (C [D)] [ [(B nA)D]

For the set dierence we also have the following identity:

(A C) n (B D) = [A (C nD)] [ [(A nB) C]

Here are some rules demonstrating distributivity with other operators (cf. right picture):[3]

A (B \ C) = (AB) \ (A C);A (B [ C) = (AB) [ (A C);A (B n C) = (AB) n (A C);(AB)c = (Ac Bc) [ (Ac B) [ (ABc): [4]

Other properties related with subsets are:

ifA B then A C B C;both ifA;B 6= ; then AB C D () A C ^B D: [5]


3.2.3 Cardinality

See also: Cardinal arithmetic

The cardinality of a set is the number of elements of the set. For example, dening two sets: A = {a, b} and B = {5,6}. Both set A and set B consist of two elements each. Their Cartesian product, written as A B, results in a new setwhich has the following elements:

A B = {(a,5), (a,6), (b,5), (b,6)}.

Each element of A is paired with each element of B. Each pair makes up one element of the output set. The numberof values in each element of the resulting set is equal to the number of sets whose cartesian product is being taken; 2in this case. The cardinality of the output set is equal to the product of the cardinalities of all the input sets. That is,

|A B| = |A| |B|.

Similarly

|A B C| = |A| |B| |C|

and so on.The set A B is innite if either A or B is innite and the other set is not the empty set.[6]

3.3 n-ary product

3.3.1 Cartesian power

The Cartesian square (or binary Cartesian product) of a set X is the Cartesian product X2 = X X. An exampleis the 2-dimensional plane R2 = R R where R is the set of real numbers: R2 is the set of all points (x,y) where xand y are real numbers (see the Cartesian coordinate system).The cartesian power of a set X can be dened as:

Xn = X X X| {z }n

= f(x1; : : : ; xn) j xi 2 X all for i = 1; : : : ; ng:

An example of this is R3 = R R R, with R again the set of real numbers, and more generally Rn.The n-ary cartesian power of a set X is isomorphic to the space of functions from an n-element set to X. As a specialcase, the 0-ary cartesian power of X may be taken to be a singleton set, corresponding to the empty function withcodomain X.

3.3.2 Finite n-ary product

The Cartesian product can be generalized to the n-ary Cartesian product over n sets X1, ..., Xn:

X1 Xn = f(x1; : : : ; xn) : xi 2 Xig:

It is a set of n-tuples. If tuples are dened as nested ordered pairs, it can be identied to (X1 ... Xn1) Xn.

3.4. OTHER FORMS 15

3.3.3 Innite productsIt is possible to dene the Cartesian product of an arbitrary (possibly innite) indexed family of sets. If I is any indexset, and fXi j i 2 Ig is a collection of sets indexed by I, then the Cartesian product of the sets in X is dened to be

Yi2I

Xi =

(f : I !

[i2I

Xi

(8i)(f(i) 2 Xi)) ;that is, the set of all functions dened on the index set such that the value of the function at a particular index i isan element of Xi. Even if each of the Xi is nonempty, the Cartesian product may be empty if the axiom of choice(which is equivalent to the statement that every such product is nonempty) is not assumed.For each j in I, the function

j :Yi2I

Xi ! Xj ;

dened by j(f) = f(j) is called the jth projection map.An important case is when the index set is N , the natural numbers: this Cartesian product is the set of all innitesequences with the ith term in its corresponding set Xi. For example, each element of

1Yn=1

R = R R

can be visualized as a vector with countably innite real number components. This set is frequently denoted R! , orRN .The special case Cartesian exponentiation occurs when all the factors Xi involved in the product are the same setX. In this case,

Yi2I

Xi =Yi2I

X

is the set of all functions from I to X, and is frequently denoted XI. This case is important in the study of cardinalexponentiation.The denition of nite Cartesian products can be seen as a special case of the denition for innite products. In thisinterpretation, an n-tuple can be viewed as a function on {1, 2, ..., n} that takes its value at i to be the ith element ofthe tuple (in some settings, this is taken as the very denition of an n-tuple).

3.4 Other forms

3.4.1 Abbreviated formIf several sets are being multiplied together, e.g. X1, X2, X3, , then some authors[7] choose to abbreviate theCartesian product as simply Xi.

3.4.2 Cartesian product of functionsIf f is a function from A to B and g is a function from X to Y, their Cartesian product f g is a function from A X to B Y with

(f g)(a; b) = (f(a); g(b)):This can be extended to tuples and innite collections of functions. Note that this is dierent from the standardcartesian product of functions considered as sets.


3.5 Denitions outside of Set theory

3.5.1 Category theoryAlthough the Cartesian product is traditionally applied to sets, category theory provides a more general interpretationof the product of mathematical structures. This is distinct from, although related to, the notion of a Cartesian squarein category theory, which is a generalization of the ber product.Exponentiation is the right adjoint of the Cartesian product; thus any category with a Cartesian product (and a nalobject) is a Cartesian closed category.

3.5.2 Graph theoryIn graph theory the Cartesian product of two graphs G and H is the graph denoted by G H whose vertex set is the(ordinary) Cartesian product V(G) V(H) and such that two vertices (u,v) and (u,v) are adjacent in G H if andonly if u = u and v is adjacent with v in H, or v = v and u is adjacent with u in G. The Cartesian product of graphsis not a product in the sense of category theory. Instead, the categorical product is known as the tensor product ofgraphs.

3.6 See also Exponential object Binary relation Coproduct Empty product Product (category theory) Concatenation of sets is deceptively similar but dierent concept Product topology Finitary relation Ultraproduct Product type Euclidean space Orders on Rn

Join (SQL), Cross join

3.7 References[1] Warner, S: Modern Algebra, page 6. Dover Press, 1990.

[2] cartesian. (2009). InMerriam-Webster OnlineDictionary. RetrievedDecember 1, 2009, from http://www.merriam-webster.com/dictionary/cartesian

[3] Singh, S. (2009, August 27). Cartesian product. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/

[4] CartesianProduct at PlanetMath.org.

[5] Cartesian Product of Subsets. (2011, February 15). ProofWiki. Retrieved 05:06, August 1, 2011 from https://proofwiki.org/w/index.php?title=Cartesian_Product_of_Subsets&oldid=45868

3.8. EXTERNAL LINKS 17

[6] Peter S. (1998). A Crash Course in the Mathematics Of Innite Sets. St. Johns Review, 44(2), 3559. Retrieved August1, 2011, from http://www.mathpath.org/concepts/infinity.htm

[7] Osborne, M., and Rubinstein, A., 1994. A Course in Game Theory. MIT Press.

3.8 External links Cartesian Product at ProvenMath Hazewinkel, Michiel, ed. (2001), Direct product, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

How to nd the Cartesian Product, Education Portal Academy

Chapter 4

Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory.If is an ordinal number and hX j < i is a sequence of subsets of , then the diagonal intersection, denoted by

Chapter 5

Diamond principle

In mathematics, and particularly in axiomatic set theory, the diamond principle is a combinatorial principle in-troduced by Ronald Jensen in Jensen (1972) that holds in the constructible universe and that implies the continuumhypothesis. Jensen extracted the diamond principle from his proof that the Axiom of constructibility (V=L) impliesthe existence of a Suslin tree.

5.1 DenitionsThe diamond principle says that there exists a -sequence, in other words sets A for

20 CHAPTER 5. DIAMOND PRINCIPLE

Shelah showed that the diamond principle solves the Whitehead problem by implying that every Whitehead group isfree.

5.3 See also Statements true in L

5.4 References Akemann, Charles; Weaver, Nik (2004), Consistency of a counterexample toNaimarks problem, Proceedingsof the National Academy of Sciences of the United States of America 101 (20): 75227525, arXiv:math.OA/0312135, doi:10.1073/pnas.0401489101, MR 2057719

Jensen, R. Bjrn (1972), The ne structure of the constructible hierarchy, Annals of Mathematical lLogic 4:229308, doi:10.1016/0003-4843(72)90001-0, MR 0309729

Rinot, Assaf (2011), Jensens diamond principle and its relatives, Set theory and its applications, Contemp.Math. 533, Providence, RI: Amer. Math. Soc., pp. 125156, arXiv:0911.2151, ISBN 978-0-8218-4812-8,MR 2777747

Shelah, S. (1974), Innite Abelian groups, Whitehead problem and some constructions, Israel Journal ofMathematics 18 (3): 243256, doi:10.1007/BF02757281, MR 0357114

Chapter 6

0

This article is about an ordinal in mathematics. For the physical constant 0, see vacuum permittivity.

In mathematics, the epsilon numbers are a collection of transnite numbers whose dening property is that they arexed points of an exponential map. Consequently, they are not reachable from 0 via a nite series of applicationsof the chosen exponential map and of weaker operations like addition and multiplication. The original epsilonnumbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers thatsatisfy the equation

" = !";

in which is the smallest innite ordinal. Any solution to this equation has Cantor normal form " = !" .The least such ordinal is 0 (pronounced epsilon nought or epsilon zero), which can be viewed as the limit obtainedby transnite recursion from a sequence of smaller limit ordinals:

"0 = !!!

= supf!; !!; !!! ; !!!!

; : : : gLarger ordinal xed points of the exponential map are indexed by ordinal subscripts, resulting in "1; "2; : : : ; "!; "!+1; : : : ; ""0 ; : : : ; ""1 ; : : : ; """ ; : : :. The ordinal 0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals,and uncountable epsilon numbers whose index is an uncountable ordinal).The smallest epsilon number 0 is very important in many induction proofs, because for many purposes, transniteinduction is only required up to 0 (as in Gentzens consistency proof and the proof of Goodsteins theorem). Itsuse by Gentzen to prove the consistency of Peano arithmetic, along with Gdels second incompleteness theorem,show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with thisproperty, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peanoarithmetic).Many larger epsilon numbers can be dened using the Veblen function.A more general class of epsilon numbers has been identied by John Horton Conway and Donald Knuth in the surrealnumber system, consisting of all surreals that are xed points of the base exponential map x x.Hessenberg (1906) dened gamma numbers (see additively indecomposable ordinal) to be numbers >0 such that+= whenever 1 such that =whenever 0

22 CHAPTER 6. 0

0 = 1 ; +1 = ; = lim sup 1, the mapping 7! is a normal function, so it hasarbitrarily large xed points by the xed-point lemma for normal functions. When = ! , these xed points areprecisely the ordinal epsilon numbers. The smallest of these, , is the supremum of the sequence

0; !0 = 1; !1 = !; !!; !!!

; : : : ; ! "" k; : : :

in which every element is the image of its predecessor under the mapping 7! ! . (The general term is given usingKnuths up-arrow notation; the "" operator is equivalent to tetration.) Just as is dened as the supremum of { k} for natural numbers k, the smallest ordinal epsilon number may also be denoted ! "" ! ; this notation is muchless common than .The next epsilon number after "0 is

"1 = supf"0 + 1; !"0+1; !!"0+1 ; !!!"0+1

; : : : g;

in which the sequence is again constructed by repeated base exponentiation but starts at "0 + 1 instead of at 0.Notice

!"0+1 = !"0 !1 = "0 ! ;

!!"0+1

= !("0!) = (!"0)! = "!0 ;

!!!"0+1

= !"0!

= !"01+!

= !("0"0!) = (!"0)

"0!

= "0"0!

:

A dierent sequence with the same supremum, "1 , is obtained by starting from 0 and exponentiating with base instead:

"1 = supf0; 1; "0; "0"0 ; "0"0"0 ; : : :g;

The epsilon number "+1 indexed by any successor ordinal +1 is constructed similarly, by base exponentiationstarting from " + 1 (or by base " exponentiation starting from 0).

"+1 = supf" + 1; !"+1; !!"+1 ; : : : g = supf0; 1; "; "" ; """ ; : : : g

An epsilon number indexed by a limit ordinal is constructed dierently. The number " is the supremum of the setof epsilon numbers f" ; < g . The rst such number is "! . Whether or not the index is a limit ordinal, " isa xed point not only of base exponentiation but also of base exponentiation for all ordinals 1 < < " .Since the epsilon numbers are an unbounded subclass of the ordinal numbers, they are enumerated using the ordinalnumbers themselves. For any ordinal number , " is the least epsilon number (xed point of the exponential map)not already in the set f" ; < g . It might appear that this is the non-constructive equivalent of the constructivedenition using iterated exponentiation; but the two denitions are equally non-constructive at steps indexed by limitordinals, which represent transnite recursion of a higher order than taking the supremum of an exponential series.The following facts about epsilon numbers are very straightforward to prove:

Although it is quite a large number, "0 is still countable, being a countable union of countable ordinals; in fact," is countable if and only if is countable.

The union (or supremum) of any nonempty set of epsilon numbers is an epsilon number; so for instance

6.2. VEBLEN HIERARCHY 23

"! = supf"0; "1; "2; : : :g

is an epsilon number. Thus, the mapping n 7! "n is a normal function.

Every uncountable cardinal number is an epsilon number.

1 ! "! = ! :

6.2 Veblen hierarchyMain article: Veblen function

The xed points of the epsilon mapping x 7! "x form a normal function, whose xed points form a normal function,whose ; this is known as the Veblen hierarchy (the Veblen functions with base 0() = ). In the notation of theVeblen hierarchy, the epsilon mapping is 1, and its xed points are enumerated by 2.Continuing in this vein, one can dene maps for progressively larger ordinals (including, by this rareed formof transnite recursion, limit ordinals), with progressively larger least xed points (0). The least ordinal notreachable from 0 by this procedurei. e., the least ordinal for which (0)=, or equivalently the rst xed pointof the map ! (0)is the FefermanSchtte ordinal 0. In a set theory where such an ordinal can be provento exist, one has a map that enumerates the xed points 0, 1, 2, ... of ! (0) ; these are all still epsilonnumbers, as they lie in the image of for every 0, including of the map 1 that enumerates epsilon numbers.

6.3 Surreal numbersIn On Numbers and Games, the classic exposition on surreal numbers, John Horton Conway provided a number ofexamples of concepts that had natural extensions from the ordinals to the surreals. One such function is the ! -mapn 7! !n ; this mapping generalises naturally to include all surreal numbers in its domain, which in turn provides anatural generalisation of the Cantor normal form for surreal numbers.It is natural to consider any xed point of this expanded map to be an epsilon number, whether or not it happens tobe strictly an ordinal number. Some examples of non-ordinal epsilon numbers are

"1 = f0; 1; !; !!; : : : j "0 1; !"01; : : :g

and

" 12= f"0 + 1; !"0+1; : : : j "1 1; !"11; : : :g:

There is a natural way to dene "n for every surreal number n, and the map remains order-preserving. Conwaygoes on to dene a broader class of irreducible surreal numbers that includes the epsilon numbers as a particularly-interesting subclass.

6.4 See also Ordinal arithmetic Large countable ordinal

24 CHAPTER 6. 0

6.5 References J.H. Conway, On Numbers and Games (1976) Academic Press ISBN 0-12-186350-6 Section XIV.20 of Sierpiski, Wacaw (1965), Cardinal and ordinal numbers (Second revised ed.), PWN Polish Scientic Publishers

6.6 External links Fusible numbers

Chapter 7

Equivalence relation

This article is about the mathematical concept. For the patent doctrine, see Doctrine of equivalents.In mathematics, an equivalence relation is the relation that holds between two elements if and only if they are

members of the same cell within a set that has been partitioned into cells such that every element of the set is amember of one and only one cell of the partition. The intersection of any two dierent cells is empty; the union ofall the cells equals the original set. These cells are formally called equivalence classes.

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

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

a ~ a. (Reexivity)

if a ~ b then b ~ a. (Symmetry)

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

X together with the relation ~ is called a setoid. The equivalence class of a under ~, denoted [a], is dened as[a] = fb 2 X j a bg .

7.3 Examples

7.3.1 Simple example

Let the set fa; b; cg have the equivalence relation f(a; a); (b; b); (c; c); (b; c); (c; b)g . The following sets are equivalenceclasses of this relation:[a] = fag; [b] = [c] = fb; cg .The set of all equivalence classes for this relation is ffag; fb; cgg .

25

26 CHAPTER 7. EQUIVALENCE RELATION

7.3.2 Equivalence relations

The following are all equivalence relations:

Has the same birthday as on the set of all people.

Is similar to on the set of all triangles.

Is congruent to on the set of all triangles.

Is congruent to, modulo n" on the integers.

Has the same image under a function" on the elements of the domain of the function.

Has the same absolute value on the set of real numbers

Has the same cosine on the set of all angles.

7.3.3 Relations that are not equivalences

The relation "" between real numbers is reexive and transitive, but not symmetric. For example, 7 5 doesnot imply that 5 7. It is, however, a partial order.

The relation has a common factor greater than 1 with between natural numbers greater than 1, is reexiveand symmetric, but not transitive. (Example: The natural numbers 2 and 6 have a common factor greater than1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1).

The empty relation R on a non-empty set X (i.e. aRb is never true) is vacuously symmetric and transitive, butnot reexive. (If X is also empty then R is reexive.)

The relation is approximately equal to between real numbers, even if more precisely dened, is not an equiv-alence relation, because although reexive and symmetric, it is not transitive, since multiple small changes canaccumulate to become a big change. However, if the approximation is dened asymptotically, for example bysaying that two functions f and g are approximately equal near some point if the limit of f g is 0 at that point,then this denes an equivalence relation.

7.4 Connections to other relations A partial order is a relation that is reexive, antisymmetric, and transitive.

A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraicstructure, and which respects the additional structure. In general, congruence relations play the role of kernelsof homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many importantcases congruence relations have an alternative representation as substructures of the structure on which theyare dened. E.g. the congruence relations on groups correspond to the normal subgroups.

Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set that isreexive, symmetric and antisymmetric.

A strict partial order is irreexive, transitive, and asymmetric.

A partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reexive if and onlyif for all a X, there exists a b X such that a ~ b.

A reexive and symmetric relation is a dependency relation, if nite, and a tolerance relation if innite.

A preorder is reexive and transitive.

7.5. WELL-DEFINEDNESS UNDER AN EQUIVALENCE RELATION 27

7.5 Well-denedness under an equivalence relationIf ~ is an equivalence relation on X, and P(x) is a property of elements of X, such that whenever x ~ y, P(x) is true ifP(y) is true, then the property P is said to be well-dened or a class invariant under the relation ~.A frequent particular case occurs when f is a function from X to another set Y ; if x1 ~ x2 implies f(x1) = f(x2) thenf is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of nite groups. The latter case with the function f can be expressed by a commutative triangle. Seealso invariant. Some authors use compatible with ~" or just respects ~" instead of invariant under ~".More generally, a function may map equivalent arguments (under an equivalence relation ~A) to equivalent values(under an equivalence relation ~B). Such a function is known as a morphism from ~A to ~B.

7.6 Equivalence class, quotient set, partitionLet a; b 2 X . Some denitions:

7.6.1 Equivalence classMain article: Equivalence class

A subset Y of X such that a ~ b holds for all a and b in Y, and never for a in Y and b outside Y, is called an equivalenceclass of X by ~. Let [a] := fx 2 X j a xg denote the equivalence class to which a belongs. All elements of Xequivalent to each other are also elements of the same equivalence class.

7.6.2 Quotient setMain article: Quotient set

The set of all possible equivalence classes of X by ~, denoted X/ := f[x] j x 2 Xg , is the quotient set of X by~. If X is a topological space, there is a natural way of transforming X/~ into a topological space; see quotient spacefor the details.

7.6.3 ProjectionMain article: Projection (relational algebra)

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

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

7.6.4 Equivalence kernelThe equivalence kernel of a function f is the equivalence relation ~ dened by x y () f(x) = f(y) . Theequivalence kernel of an injection is the identity relation.

7.6.5 PartitionMain article: Partition of a set


A partition of X is a set P of nonempty subsets of X, such that every element of X is an element of a single elementof P. Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their unionis X.

Counting possible partitions

Let X be a nite set with n elements. Since every equivalence relation over X corresponds to a partition of X, andvice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which isthe nth Bell number Bn:

Bn =1

e

1Xk=0

kn

k!;

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

7.7 Fundamental theorem of equivalence relationsA key result links equivalence relations and partitions:[2][3]

An equivalence relation ~ on a set X partitions X.

Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.

In both cases, the cells of the partition of X are the equivalence classes of X by ~. Since each element of X belongsto a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by~, each element of X belongs to a unique equivalence class of X by ~. Thus there is a natural bijection from the setof all possible equivalence relations on X and the set of all partitions of X.

7.8 Comparing equivalence relationsIf ~ and are two equivalence relations on the same set S, and a~b implies ab for all a,b S, then is said to be acoarser relation than ~, and ~ is a ner relation than . Equivalently,

~ is ner than if every equivalence class of ~ is a subset of an equivalence class of , and thus every equivalenceclass of is a union of equivalence classes of ~.

~ is ner than if the partition created by ~ is a renement of the partition created by .

The equality equivalence relation is the nest equivalence relation on any set, while the trivial relation that makes allpairs of elements related is the coarsest.The relation "~ is ner than " on the collection of all equivalence relations on a xed set is itself a partial orderrelation.

7.9 Generating equivalence relations Given any set X, there is an equivalence relation over the set [XX] of all possible functions XX. Two suchfunctions are deemed equivalent when their respective sets of xpoints have the same cardinality, correspondingto cycles of length one in a permutation. Functions equivalent in this manner form an equivalence class on[XX], and these equivalence classes partition [XX].

7.10. ALGEBRAIC STRUCTURE 29

An equivalence relation ~ on X is the equivalence kernel of its surjective projection : X X/~.[4] Conversely,any surjection between sets determines a partition on its domain, the set of preimages of singletons in thecodomain. Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are threeequivalent ways of specifying the same thing.

The intersection of any collection of equivalence relations over X (binary relations viewed as a subset of X X)is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given anybinary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containingR. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x1, x2, ..., xn in Xsuch that a = x1, b = xn, and (xi,xi )R or (xi,xi)R, i = 1, ..., n1.

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

Any total order on X has exactly one equivalence class, X itself, because x ~ y for all x and y; Any subset of the identity relation on X has equivalence classes that are the singletons of X.

Equivalence relations can construct new spaces by gluing things together. Let X be the unit Cartesian square[0,1] [0,1], and let ~ be the equivalence relation on X dened by a, b [0,1] ((a, 0) ~ (a, 1) (0, b) ~ (1, b)).Then the quotient space X/~ can be naturally identied (homeomorphism) with a torus: take a square piece ofpaper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder soas to glue together its two open ends, resulting in a torus.

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

7.10.1 Group theoryJust as order relations are grounded in ordered sets, sets closed under pairwise supremum and inmum, equivalencerelations are grounded in partitioned sets, which are sets closed under bijections and preserve partition structure.Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations. Hencepermutation groups (also known as transformation groups) and the related notion of orbit shed light on the mathe-matical structure of equivalence relations.Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G denotethe set of bijective functions over A that preserve the partition structure of A: x A g G (g(x) [x]). Then thefollowing three connected theorems hold:[5]

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

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

Given a transformation groupG overA, there exists an equivalence relation ~ over A, whose equivalence classesare the orbits of G.[6][7]

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


A. Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a setof bijections, A A.Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a~ b (ab1 H). The equivalence classes of ~also called the orbits of the action of H on Gare the right cosetsof H in G. Interchanging a and b yields the left cosets.Proof.[8] Let function composition interpret group multiplication, and function inverse interpret group inverse. ThenG is a group under composition, meaning that x A g G ([g(x)] = [x]), because G satises the following fourconditions:

G is closed under composition. The composition of any two elements of G exists, because the domain andcodomain of any element of G is A. Moreover, the composition of bijections is bijective;[9]

Existence of identity function. The identity function, I(x)=x, is an obvious element of G; Existence of inverse function. Every bijective function g has an inverse g1, such that gg1 = I; Composition associates. f(gh) = (fg)h. This holds for all functions over all domains.[10]

Let f and g be any two elements of G. By virtue of the denition of G, [g(f(x))] = [f(x)] and [f(x)] = [x], so that[g(f(x))] = [x]. Hence G is also a transformation group (and an automorphism group) because function compositionpreserves the partitioning of A. Related thinking can be found in Rosen (2008: chpt. 10).

7.10.2 Categories and groupoidsLet G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing thisequivalence relation as follows. The objects are the elements of G, and for any two elements x and y of G, there existsa unique morphism from x to y if and only if x~y.The advantages of regarding an equivalence relation as a special case of a groupoid include:

Whereas the notion of free equivalence relation does not exist, that of a free groupoid on a directed graphdoes. Thus it is meaningful to speak of a presentation of an equivalence relation, i.e., a presentation of thecorresponding groupoid;

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

In many contexts quotienting, and hence the appropriate equivalence relations often called congruences, areimportant. This leads to the notion of an internal groupoid in a category.[11]

7.10.3 LatticesThe possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called ConX by convention. The canonical map ker: X^X Con X, relates the monoid X^X of all functions on X and Con X.ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f: XX toits kernel ker f. Likewise, ker(ker) is an equivalence relation on X^X.

7.11 Equivalence relations and mathematical logicEquivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation withexactly two innite equivalence classes is an easy example of a theory which is -categorical, but not categorical forany larger cardinal number.An implication of model theory is that the properties dening a relation can be proved independent of each other(and hence necessary parts of the denition) if and only if, for each property, examples can be found of relationsnot satisfying the given property while satisfying all the other properties. Hence the three dening properties ofequivalence relations can be proved mutually independent by the following three examples:

7.12. EUCLIDEAN RELATIONS 31

Reexive and transitive: The relation on N. Or any preorder; Symmetric and transitive: The relation R on N, dened as aRb ab 0. Or any partial equivalence relation; Reexive and symmetric: The relation R on Z, dened as aRb "a b is divisible by at least one of 2 or 3.Or any dependency relation.

Properties denable in rst-order logic that an equivalence relation may or may not possess include:

The number of equivalence classes is nite or innite; The number of equivalence classes equals the (nite) natural number n; All equivalence classes have innite cardinality; The number of elements in each equivalence class is the natural number n.

7.12 Euclidean relationsEuclid's The Elements includes the following Common Notion 1":

Things which equal the same thing also equal one another.

Nowadays, the property described by Common Notion 1 is called Euclidean (replacing equal by are in relationwith). By relation is meant a binary relation, in which aRb is generally distinct from bRa. An Euclidean relationthus comes in two forms:

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

The following theorem connects Euclidean relations and equivalence relations:

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

Proof for a left-Euclidean relation

(aRc bRc) aRb [a/c] = (aRa bRa) aRb [reexive; erase T] = bRa aRb. Hence R is symmetric.

(aRc bRc) aRb [symmetry] = (aRc cRb) aRb. Hence R is transitive.

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

7.13 See also Partition of a set Equivalence class Up to Conjugacy class Topological conjugacy


7.14 Notes[1] Garrett Birkho and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 35, Th. 19. Chelsea.

[2] Wallace, D. A. R., 1998. Groups, Rings and Fields. p. 31, Th. 8. Springer-Verlag.

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

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

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

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

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

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

[9] Wallace, D. A. R., 1998. Groups,

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