discrete mathematics 1_6

Discrete mathematics 1From Wikipedia, the free encyclopedia

Contents

1 Antimatroid 11.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Paths and basic words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Convex geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Join-distributive lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 Supersolvable antimatroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 Join operation and convex dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.8 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Closure (mathematics) 82.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 P closures of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Closure operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Composite number 123.1 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Cutting sequence 174.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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5 DIMACS 185.1 The DIMACS Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Discrete mathematics 206.1 Grand challenges, past and present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2 Topics in discrete mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6.2.1 Theoretical computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2.2 Information theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2.3 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2.4 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2.5 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2.6 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2.7 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2.8 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2.9 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2.10 Calculus of nite dierences, discrete calculus or discrete analysis . . . . . . . . . . . . . . 276.2.11 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2.12 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2.13 Operations research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2.14 Game theory, decision theory, utility theory, social choice theory . . . . . . . . . . . . . . 286.2.15 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2.16 Discrete analogues of continuous mathematics . . . . . . . . . . . . . . . . . . . . . . . . 296.2.17 Hybrid discrete and continuous mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7 Discrete Mathematics (journal) 327.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.2 Abstracting and indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.3 Notable publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8 Divisibility rule 348.1 Divisibility rules for numbers 120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.2 Step-by-step examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

8.2.1 Divisibility by 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.2.2 Divisibility by 3 or 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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8.2.3 Divisibility by 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.2.4 Divisibility by 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.2.5 Divisibility by 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.2.6 Divisibility by 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.2.7 Divisibility by 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

8.3 Beyond 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.3.1 Composite divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.3.2 Prime divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.3.3 Notable examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

8.4 Generalized divisibility rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

8.5.1 Proof using basic algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.5.2 Proof using modular arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

8.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9 Divisor 459.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.2 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.4 Further notions and facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.5 In abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

10 Eulers four-square identity 5010.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

11 Formal language 5211.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.2 Words over an alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.3 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

11.4.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.5 Language-specication formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.6 Operations on languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

11.7.1 Programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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11.7.2 Formal theories, systems and proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

11.9.1 Citation footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.9.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

11.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

12 Greedoid 5912.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.2 Classes of greedoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.4 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

13 Half-integer 6213.1 Notation and algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213.2 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

13.2.1 Sphere packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213.2.2 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6313.2.3 Sphere volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

13.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

14 Integer 6414.1 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6414.2 Order-theoretic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6514.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6514.4 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6714.5 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6714.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6814.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6814.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6814.9 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6914.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

15 Inversion (discrete mathematics) 7015.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7015.2 Weak order of permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7015.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7115.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

15.4.1 Source bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7315.4.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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15.4.3 Presortedness measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

16 Outline of discrete mathematics 7416.1 Subjects in discrete mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7416.2 Discrete mathematical disciplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7516.3 Concepts in discrete mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

16.3.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7516.3.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.3.3 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.3.4 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7616.3.5 Elementary algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.3.6 Mathematical relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7716.3.7 Mathematical phraseology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7816.3.8 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7816.3.9 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7916.3.10 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

16.4 Discrete mathematicians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7916.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7916.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7916.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

17 Portal:Discrete mathematics 8117.1 Discrete mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8117.2 Selected article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8217.3 WikiProjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8317.4 Selected picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8417.5 Did you know? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8417.6 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8417.7 Topics in Discrete mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8517.8 Related portals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8517.9 Wikimedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

18 Reduced residue system 8818.1 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8818.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8918.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8918.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

19 Singly and doubly even 9019.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9019.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

19.2.1 Safer outs in darts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

vi CONTENTS

19.2.2 Irrationality of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9119.2.3 Geometric topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9119.2.4 Other appearances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

19.3 Related classications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9219.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9219.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9219.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

20 Table of divisors 9420.1 Key to the tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9420.2 English Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9420.3 1 to 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.4 101 to 200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.5 201 to 300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.6 301 to 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.7 401 to 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.8 501 to 600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.9 601 to 700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.10701 to 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.11801 to 900 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.12901 to 1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9620.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

21 Table of prime factors 9821.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9821.2 1 to 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.3 101 to 200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.4 201 to 300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.5 301 to 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.6 401 to 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.7 501 to 600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.8 601 to 700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.9 701 to 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.10801 to 900 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10021.11901 to 1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

22 Trinomial triangle 10122.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10122.2 Recursion formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10222.3 The middle entries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10222.4 Chess mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

CONTENTS vii

22.5 Importance in combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10322.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10422.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10422.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 105

22.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10522.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10822.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Chapter 1

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 of matroids, but whereas matroids 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.

1

2 CHAPTER 1. 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.

1.1 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]

1.2 Examples A chain antimatroid has as its formal language the prexes of a single word, and as its feasible sets the sets

of 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]

1.3. PATHS AND BASIC WORDS 3

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

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

1.3 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 we may 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.

1.4 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]

1.5 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

1.6. SUPERSOLVABLE ANTIMATROIDS 5

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, we may 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.

1.6 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.

1.7 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.

1.8 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).

1.9 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]

1.10 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.

1.11. REFERENCES 7

[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).

1.11 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 2

Closure (mathematics)

For other uses, see Closure (disambiguation).

A set has closure under an operation if performance of that operation on members of the set always produces amember of the same set; in this case we also say that the set is closed under the operation. For example, the integersare closed under subtraction, but the positive integers are not: 1 and 2 are both positive integers, but the result ofsubtracting 2 from 1 is not a positive integer. Another example is the set containing only the number zero, which isclosed under addition, subtraction and multiplication.Similarly, a set is said to be closed under a collection of operations if it is closed under each of the operationsindividually.

2.1 Basic propertiesA set that is closed under an operation or collection of operations is said to satisfy a closure property. Often aclosure property is introduced as an axiom, which is then usually called the axiom of closure. Modern set-theoreticdenitions usually dene operations as maps between sets, so adding closure to a structure as an axiom is superuous;however in practice operations are often dened initially on a superset of the set in question and a closure proof isrequired to establish that the operation applied to pairs from that set only produces members of that set. For example,the set of even integers is closed under addition, but the set of odd integers is not.When a set S is not closed under some operations, one can usually nd the smallest set containing S that is closed.This smallest closed set is called the closure of S (with respect to these operations). For example, the closure undersubtraction of the set of natural numbers, viewed as a subset of the real numbers, is the set of integers. An importantexample is that of topological closure. The notion of closure is generalized by Galois connection, and further bymonads.The set S must be a subset of a closed set in order for the closure operator to be dened. In the preceding example, itis important that the reals are closed under subtraction; in the domain of the natural numbers subtraction is not alwaysdened.The two uses of the word closure should not be confused. The former usage refers to the property of being closed,and the latter refers to the smallest closed set containing one that may not be closed. In short, the closure of a setsatises a closure property.

2.2 Closed setsA set is closed under an operation if that operation returns a member of the set when evaluated on members of theset. Sometimes the requirement that the operation be valued in a set is explicitly stated, in which case it is known asthe axiom of closure. For example, one may dene a group as a set with a binary product operator obeying severalaxioms, including an axiom that the product of any two elements of the group is again an element. However themodern denition of an operation makes this axiom superuous; an n-ary operation on S is just a subset of Sn+1. By

8

2.3. P CLOSURES OF BINARY RELATIONS 9

its very denition, an operator on a set cannot have values outside the set.Nevertheless, the closure property of an operator on a set still has some utility. Closure on a set does not necessarilyimply closure on all subsets. Thus a subgroup of a group is a subset on which the binary product and the unaryoperation of inversion satisfy the closure axiom.An operation of a dierent sort is that of nding the limit points of a subset of a topological space (if the space isrst-countable, it suces to restrict consideration to the limits of sequences but in general one must consider at leastlimits of nets). A set that is closed under this operation is usually just referred to as a closed set in the context oftopology. Without any further qualication, the phrase usually means closed in this sense. Closed intervals like [1,2]= {x : 1 x 2} are closed in this sense.A partially ordered set is downward closed (and also called a lower set) if for every element of the set all smallerelements are also in it; this applies for example for the real intervals (, p) and (, p], and for an ordinal numberp represented as interval [ 0, p); every downward closed set of ordinal numbers is itself an ordinal number.Upward closed and upper set are dened similarly.

2.3 P closures of binary relationsThe notion of a closure can be applied for an arbitrary binary relation R SS, and an arbitrary property P in thefollowing way: the P closure of R is the least relation Q SS that contains R (i.e. R Q) and for which propertyP holds (i.e. P(Q) is true). For instance, one can dene the symmetric closure as the least symmetric relationcontaining R. This generalization is often encountered in the theory of rewriting systems, where one often uses morewordy notions such as the reexive transitive closure R*the smallest preorder containing R, or the reexivetransitive symmetric closure Rthe smallest equivalence relation containing R, and therefore also known as theequivalence closure. When considering a particular term algebra, an equivalence relation that is compatible withall operations of the algebra [note 1] is called a congruence relation. The congruence closure of R is dened as thesmallest congruence relation containing R.For arbitrary P and R, the P closure of R need not exist. In the above examples, these exist because reexivity,transitivity and symmetry are closed under arbitrary intersections. In such cases, the P closure can be directly denedas the intersection of all sets with property P containing R.[1]

Some important particular closures can be constructively obtained as follows:

cl(R) = R { x,x : x S } is the reexive closure of R, cl(R) = R { y,x : x,y R } is its symmetry closure, cl(R) = R { x1,xn : n >1 x1,x2, ..., xn,xn R } is its transitive closure, cl,(R) = R { f(x1,,xi,xi,xi,,xn), f(x1,,xi,y,xi,,xn) : xi,y R f n-ary 1 i n x1,...,xn S } is its embedding closure with respect to a given set of operations on S, each with a xedarity.

The relation R is said to have closure under some cl, if R = cl(R); for example R is called symmetric if R =cl(R).Any of these four closures preserves symmetry, i.e., if R is symmetric, so is any cl(R). [note 2] Similarly, all fourpreserve reexivity. Moreover, cl preserves closure under cl, for arbitrary . As a consequence, the equivalenceclosure of an arbitrary binary relationR can be obtained as cl(cl(cl(R))), and the congruence closure with respectto some can be obtained as cl(cl,(cl(cl(R)))). In the latter case, the nesting order does matter; e.g. if S isthe set of terms over = { a, b, c, f } and R = { a,b, f(b),c }, then the pair f(a),c is contained in the congruenceclosure cl(cl,(cl(cl(R)))) of R, but not in the relation cl,(cl(cl(cl(R)))).

2.4 Closure operatorMain article: closure operator

10 CHAPTER 2. CLOSURE (MATHEMATICS)

Given an operation on a set X, one can dene the closure C(S) of a subset S of X to be the smallest subset closedunder that operation that contains S as a subset, if any such subsets exist. Consequently, C(S) is the intersection ofall closed sets containing S. For example, the closure of a subset of a group is the subgroup generated by that set.The closure of sets with respect to some operation denes a closure operator on the subsets of X. The closed sets canbe determined from the closure operator; a set is closed if it is equal to its own closure. Typical structural propertiesof all closure operations are: [2]

The closure is increasing or extensive: the closure of an object contains the object. The closure is idempotent: the closure of the closure equals the closure. The closure is monotone, that is, if X is contained in Y, then also C(X) is contained in C(Y).

An object that is its own closure is called closed. By idempotency, an object is closed if and only if it is the closureof some object.These three properties dene an abstract closure operator. Typically, an abstract closure acts on the class of allsubsets of a set.If X is contained in a set closed under the operation then every subset of X has a closure.

2.5 Examples In topology and related branches, the relevant operation is taking limits. The topological closure of a set is the

corresponding closure operator. The Kuratowski closure axioms characterize this operator. In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of the

vector space that includes X and is closed under the operation of linear combination. This subset is a subspace. In matroid theory, the closure of X is the largest superset of X that has the same rank as X. In set theory, the transitive closure of a set. In set theory, the transitive closure of a binary relation. In algebra, the algebraic closure of a eld. In commutative algebra, closure operations for ideals, as integral closure and tight closure. In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset. In the theory of formal languages, the Kleene closure of a language can be described as the set of strings that

can be made by concatenating zero or more strings from that language. In group theory, the conjugate closure or normal closure of a set of group elements is the smallest normal

subgroup containing the set. In mathematical analysis and in probability theory, the closure of a collection of subsets of X under countably

many set operations is called the -algebra generated by the collection.

2.6 See also Open set Clopen set

2.7 Notes[1] that is, such that e.g. xRy implies f(x,x2) R f(y,x2) and f(x1,x) R f(x1,y) for any binary operation f and arbitrary x1,x2 S

[2] formally: if R = cl(R), then cl(R) = cl(cl(R))

2.8. REFERENCES 11

2.8 References[1] Baader, Franz; Nipkow, Tobias (1998). Term Rewriting and All That. Cambridge University Press. pp. 89.

[2] Birkho, Garrett (1967). Lattice Theory. Colloquium Publications 25. Am. Math. Soc. p. 111.

Chapter 3

Composite number

A composite number is a positive integer that has at least one positive divisor other than one or the number itself.In other words, a composite number is any integer greater than one that is not a prime number.[1][2]

So, if n > 0 is an integer and there are integers 1 < a, b < n such that n = a b, then n is composite. By denition,every integer greater than one is either a prime number or a composite number. The number one is a unit;[3][4] it isneither prime nor composite. For example, the integer 14 is a composite number because it can be factored as 2 7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one anditself.The rst 114 composite numbers (all the composite numbers less than or equal to 150) (sequence A002808 in OEIS)are

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42,44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77,78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110,111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133,134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150.

Every composite number can be written as the product of two or more (not necessarily distinct) primes,[5] for example,the composite number 299 can be written as 13 23, and that the composite number 360 can be written as 23 32 5; furthermore, this representation is unique up to the order of the factors. This fact is called the fundamentaltheorem of arithmetic.[6][7][8][9]

There are several known primality tests that can determine whether a number is prime or composite, without neces-sarily revealing the factorization of a composite input.

3.1 TypesOne way to classify composite numbers is by counting the number of prime factors. A composite number withtwo prime factors is a semiprime or 2-almost prime (the factors need not be distinct, hence squares of primes areincluded). A composite number with three distinct prime factors is a sphenic number. In some applications, it isnecessary to dierentiate between composite numbers with an odd number of distinct prime factors and those withan even number of distinct prime factors. For the latter

(n) = (1)2x = 1(where is the Mbius function and x is half the total of prime factors), while for the former

(n) = (1)2x+1 = 1:However for prime numbers, the function also returns 1 and (1) = 1 . For a number n with one or more repeatedprime factors,

12

3.2. FACTORIZATION 13

(n) = 0 .[10]

If all the prime factors of a number are repeated it is called a powerful number. If none of its prime factors arerepeated, it is called squarefree. (All prime numbers and 1 are squarefree.)Another way to classify composite numbers is by counting the number of divisors. All composite numbers have atleast three divisors. In the case of squares of primes, those divisors are f1; p; p2g . A number n that has more divisorsthan any x < n is a highly composite number (though the rst two such numbers are 1 and 2).

3.2 FactorizationHere are all the composite numbers less than or equal to 150 and their factorization:4 = 22

6 = 2 38 = 23

9 = 32

10 = 2 512 = 22 314 = 2 715 = 3 516 = 24

18 = 2 32

20 = 22 521 = 3 722 = 2 1124 = 23 325 = 52

26 = 2 1327 = 33

28 = 22 730 = 2 3 532 = 25

33 = 3 1134 = 2 1735 = 5 736 = 22 33

38 = 2 1939 = 3 1340 = 23 542 = 2 3 744 = 22 1145 = 32 546 = 2 2348 = 24 3

14 CHAPTER 3. COMPOSITE NUMBER

49 = 72

50 = 2 52

51 = 3 1752 = 22 1354 = 2 33

55 = 5 1156 = 23 757 = 3 1958 = 2 2960 = 22 3 562 = 2 3163 = 32 764 = 26

65 = 5 1366 = 2 3 1168 = 22 1769 = 3 2370 = 2 5 772 = 23 32

74 = 2 3775 = 3 52

76 = 22 1977 = 7 1178 = 2 3 1380 = 24 581 = 34

82 = 2 4184 = 22 3 785 = 5 1786 = 2 4387 = 3 2988 = 23 1190 = 2 32 591 = 7 1392 = 22 2393 = 3 3194 = 2 4795 = 5 1996 = 25 398 = 2 72

99 = 32 11

3.2. FACTORIZATION 15

100 = 22 52

102 = 2 3 17104 = 23 13105 = 3 5 7106 = 2 53108 = 22 33

110 = 2 5 11111 = 3 37112 = 24 7114 = 2 3 19115 = 5 23116 = 22 29117 = 32 13118 = 2 59119 = 7 17120 = 23 3 5121 = 112

122 = 2 61123 = 3 41124 = 4 31125 = 53

126 = 2 32 7128 = 27

129 = 3 43130 = 2 5 13132 = 22 3 11133 = 7 19134 = 2 67135 = 33 5136 = 23 17138 = 2 3 23140 = 22 5 7141 = 3 47142 = 2 71143 = 11 13144 = 24 32

145 = 5 29146 = 2 73147 = 3 72

148 = 22 37150 = 2 3 52

16 CHAPTER 3. COMPOSITE NUMBER

3.3 See also Table of prime factors Integer factorization Canonical representation of a positive integer

3.4 Notes[1] Pettofrezzo (1970, pp. 2324)

[2] Long (1972, p. 16)

[3] Fraleigh (1976, pp. 198,266)

[4] Herstein (1964, p. 106)

[5] Long (1972, p. 16)

[6] Fraleigh (1976, p. 270)

[7] Long (1972, p. 44)

[8] McCoy (1968, p. 85)

[9] Pettofrezzo (1970, p. 53)

[10] Long (1972, p. 159)

3.5 References Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-

201-01984-1 Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016 Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and

Company, LCCN 77-171950

McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN68-15225

Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Clis: PrenticeHall, LCCN 77-81766

3.6 External links An integer factorizer, can factor all integers less than 1060

Java applet: Factorization using the Elliptic Curve Method to nd very large composites Lists of composites with prime factorization (rst 100, 1,000, 10,000, 100,000, and 1,000,000) Divisor Plot (patterns found in large composite numbers)

Chapter 4

Cutting sequence

The Fibonacci word is an example of a Sturmian word. The start of the cutting sequence shown here illustrates the start of the word0100101001.

In digital geometry, a cutting sequence is a sequence of symbols whose elements correspond to the individual gridlines crossed (cut) as a curve crosses a square grid.[1]

Sturmian words are a special case of cutting sequences where the curves are straight lines of irrational slope.[2]

4.1 References[1] Monteil, T. (2011). The complexity of tangent words. Electronic Proceedings in Theoretical Computer Science 63: 152.

doi:10.4204/EPTCS.63.21.[2] Pytheas Fogg (2002) p.152

Pytheas Fogg, N. (2002). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathemat-ics 1794. Editors Berth, Valrie; Ferenczi, Sbastien; Mauduit, Christian; Siegel, A. Berlin: Springer-Verlag.ISBN 3-540-44141-7. Zbl 1014.11015.

17

Chapter 5

DIMACS

TheCenter for DiscreteMathematics and Theoretical Computer Science (DIMACS) is a collaboration betweenRutgers University, Princeton University, and the research rms AT&T, Bell Labs, Applied Communication Sciences,and NEC. It was founded in 1989 with money from the National Science Foundation. Its oces are located on theRutgers campus, and 250 members from the six institutions form its permanent members.DIMACS is devoted to both theoretical development and practical applications of discrete mathematics and theoret-ical computer science. It engages in a wide variety of evangelism including encouraging, inspiring, and facilitatingresearchers in these subject areas, and sponsoring conferences and workshops.Fundamental research in discrete mathematics has applications in diverse elds including Cryptology, Engineering,Networking, and Management Decision Support.The current director of DIMACS is Rebecca Wright. Past directors were Fred S. Roberts, Daniel Gorenstein andAndrs Hajnal.[1]

5.1 The DIMACS ChallengesDIMACS sponsors implementation challenges to determine practical algorithm performance on problems of interest.There have been eleven DIMACS challenges so far.

1990-1991: Network Flows and Matching 1992-1992: NP-Hard Problems: Max Clique, Graph Coloring, and SAT 1993-1994: Parallel Algorithms for Combinatorial Problems 1994-1995: Computational Biology: Fragment Assembly and Genome Rearrangement 1995-1996: Priority Queues, Dictionaries, and Multidimensional Point Sets 1998-1998: Near Neighbor Searches 2000-2000: Semidenite and Related Optimization Problems 2001-2001: The Traveling Salesman Problem 2005-2005: The Shortest Path Problem 2011-2012: Graph Partitioning and Graph Clustering 2013-2014: Steiner Tree Problems

5.2 References[1] A history of mathematics at Rutgers, Charles Weibel.

18

5.3. EXTERNAL LINKS 19

5.3 External links DIMACS Website

Chapter 6

Discrete mathematics

For the mathematics journal, see Discrete Mathematics (journal).Discretemathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.

123

546

Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulnessas models of real-world problems, and their importance in developing computer algorithms.

In contrast to real numbers that have the property of varying smoothly, the objects studied in discrete mathematics such as integers, graphs, and statements in logic[1] do not vary smoothly in this way, but have distinct, separatedvalues.[2] Discrete mathematics therefore excludes topics in continuous mathematics such as calculus and analysis.Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterizedas the branch of mathematics dealing with countable sets[3] (sets that have the same cardinality as subsets of thenatural numbers, including rational numbers but not real numbers). However, there is no exact denition of the termdiscrete mathematics.[4] Indeed, discrete mathematics is described less by what is included than by what is excluded:continuously varying quantities and related notions.The set of objects studied in discrete mathematics can be nite or innite. The term nite mathematics is sometimesapplied to parts of the eld of discrete mathematics that deals with nite sets, particularly those areas relevant tobusiness.

20

6.1. GRAND CHALLENGES, PAST AND PRESENT 21

Research in discrete mathematics increased in the latter half of the twentieth century partly due to the developmentof digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations fromdiscrete mathematics are useful in studying and describing objects and problems in branches of computer science,such as computer algorithms, programming languages, cryptography, automated theorem proving, and software de-velopment. Conversely, computer implementations are signicant in applying ideas from discrete mathematics toreal-world problems, such as in operations research.Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuousmathematics are often employed as well.In the university curricula, Discrete Mathematics appeared in the 1980s, initially as a computer science supportcourse; its contents was somewhat haphazard at the time. The curriculum has thereafter developed in conjunction toeorts by ACM and MAA into a course that is basically intended to develop mathematical maturity in freshmen; assuch it is nowadays a prerequisite for mathematics majors in some universities as well.[5][6] Some high-school-leveldiscrete mathematics textbooks have appeared as well.[7] At this level, discrete mathematics it is sometimes seen apreparatory course, not unlike precalculus in this respect.[8]

The Fulkerson Prize is awarded for outstanding papers in discrete mathematics.

6.1 Grand challenges, past and presentThe history of discrete mathematics has involved a number of challenging problems which have focused attentionwithin areas of the eld. In graph theory, much research was motivated by attempts to prove the four color theorem,rst stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computerassistance).[9]

In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axiomsof arithmetic are consistent. Gdels second incompleteness theorem, proved in 1931, showed that this was notpossible at least not within arithmetic itself. Hilberts tenth problem was to determine whether a given polynomialDiophantine equation with integer coecients has an integer solution. In 1970, Yuri Matiyasevich proved that thiscould not be done.The need to break German codes in World War II led to advances in cryptography and theoretical computer science,with the rst programmable digital electronic computer being developed at Englands Bletchley Park with the guid-ance of Alan Turing and his seminal work, On Computable Numbers.[10] At the same time, military requirementsmotivated advances in operations research. The Cold War meant that cryptography remained important, with fun-damental advances such as public-key cryptography being developed in the following decades. Operations researchremained important as a tool in business and project management, with the critical path method being developedin the 1950s. The telecommunication industry has also motivated advances in discrete mathematics, particularlyin graph theory and information theory. Formal verication of statements in logic has been necessary for softwaredevelopment of safety-critical systems, and advances in automated theorem proving have been driven by this need.Computational geometry has been an important part of the computer graphics incorporated into modern video gamesand computer-aided design tools.Several elds of discrete mathematics, particularly theoretical computer science, graph theory, and combinatorics,are important in addressing the challenging bioinformatics problems associated with understanding the tree of life.[11]

Currently, one of the most famous open problems in theoretical computer science is the P = NP problem, whichinvolves the relationship between the complexity classes P and NP. The Clay Mathematics Institute has oered a $1million USD prize for the rst correct proof, along with prizes for six other mathematical problems.[12]

6.2 Topics in discrete mathematics

6.2.1 Theoretical computer scienceMain article: Theoretical computer scienceTheoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on

graph theory and mathematical logic. Included within theoretical computer science is the study of algorithms forcomputing mathematical results. Computability studies what can be computed in principle, and has close ties to logic,

22 CHAPTER 6. DISCRETE MATHEMATICS

Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be colored using only four colorsso that no areas of the same color touched. Kenneth Appel and Wolfgang Haken proved this in 1976.[9]

while complexity studies the time taken by computations. Automata theory and formal language theory are closelyrelated to computability. Petri nets and process algebras are used to model computer systems, and methods fromdiscrete mathematics are used in analyzing VLSI electronic circuits. Computational geometry applies algorithmsto geometrical problems, while computer image analysis applies them to representations of images. Theoreticalcomputer science also includes the study of various continuous computational topics.

6.2. TOPICS IN DISCRETE MATHEMATICS 23

Complexity studies the time taken by algorithms, such as this sorting routine.

6.2.2 Information theoryMain article: Information theoryInformation theory involves the quantication of information. Closely related is coding theory which is used to designecient and reliable data transmission and storage methods. Information theory also includes continuous topics suchas: analog signals, analog coding, analog encryption.

6.2.3 LogicMain article: Mathematical logic

Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, andcompleteness. For example, in most systems of logic (but not in intuitionistic logic) Peirces law (((PQ)P)P)is a theorem. For classical logic, it can be easily veried with a truth table. The study of mathematical proof is par-ticularly important in logic, and has applications to automated theorem proving and formal verication of software.Logical formulas are discrete structures, as are proofs, which form nite trees[13] or, more generally, directed acyclicgraph structures[14][15] (with each inference step combining one or more premise branches to give a single conclusion).The truth values of logical formulas usually form a nite set, generally restricted to two values: true and false, butlogic can also be continuous-valued, e.g., fuzzy logic. Concepts such as innite proof trees or innite derivation treeshave also been studied,[16] e.g. innitary logic.

6.2.4 Set theoryMain article: Set theory


101 0111110 1001110 1011110 1001111 0000110 0101110 0100110 1001110 0001Wikipedia

The ASCII codes for the word Wikipedia, given here in binary, provide a way of representing the word in information theory, aswell as for information-processing algorithms.

Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red}or the (innite) set of all prime numbers. Partially ordered sets and sets with other relations have applications inseveral areas.In discrete mathematics, countable sets (including nite sets) are the main focus. The beginning of set theory as abranch of mathematics is usually marked by Georg Cantor's work distinguishing between dierent kinds of inniteset, motivated by the study of trigonometric series, and further development of the theory of innite sets is outsidethe scope of discrete mathematics. Indeed, contemporary work in descriptive set theory makes extensive use oftraditional continuous mathematics.


6.2.5 Combinatorics

Main article: Combinatorics

Combinatorics studies the way in which discrete structures can be combined or arranged. Enumerative combinatoricsconcentrates on counting the number of certain combinatorial objects - e.g. the twelvefold way provides a uniedframework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumer-ation (i.e., determining the number) of combinatorial structures using tools from complex analysis and probabilitytheory. In contrast with enumerative combinatorics which uses explicit combinatorial formulae and generating func-tions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Design theory is a studyof combinatorial designs, which are collections of subsets with certain intersection properties. Partition theory studiesvarious enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, specialfunctions and orthogonal polynomials. Originally a part of number theory and analysis, partition theory is now con-sidered a part of combinatorics or an independent eld. Order theory is the study of partially ordered sets, both niteand innite.

6.2.6 Graph theory

Main article: Graph theoryGraph theory, the study of graphs and networks, is often considered part of combinatorics, but has grown large enough

Graph theory has close links to group theory. This truncated tetrahedron graph is related to the alternating group A4.

and distinct enough, with its own kind of problems, to be regarded as a subject in its own right.[17] Graphs are one ofthe prime objects of study in discrete mathematics. They are among the most ubiquitous models of both natural andhuman-made structures. They can model many types of relations and process dynamics in physical, biological andsocial systems. In computer science, they can represent networks of communication, data organization, computationaldevices, the ow of computation, etc. In mathematics, they are useful in geometry and certain parts of topology, e.g.knot theory. Algebraic graph theory has close links with group theory. There are also continuous graphs, however


for the most part research in graph theory falls within the domain of discrete mathematics.

6.2.7 Probability

Main article: Discrete probability theory

Discrete probability theory deals with events that occur in countable sample spaces. For example, count observationssuch as the numbers of birds in ocks comprise only natural number values {0, 1, 2, ...}. On the other hand, continuousobservations such as the weights of birds comprise real number values and would typically be modeled by a continuousprobability distribution such as the normal. Discrete probability distributions can be used to approximate continuousones and vice versa. For highly constrained situations such as throwing dice or experiments with decks of cards,calculating the probability of events is basically enumerative combinatorics.

6.2.8 Number theory

The Ulam spiral of numbers, with black pixels showing prime numbers. This diagram hints at patterns in the distribution of primenumbers.

Main article: Number theory


Number theory is concerned with the properties of numbers in general, particularly integers. It has applications tocryptography, cryptanalysis, and cryptology, particularly with regard to modular arithmetic, diophantine equations,linear and quadratic congruences, prime numbers and primality testing. Other discrete aspects of number theoryinclude geometry of numbers. In analytic number theory, techniques from continuous mathematics are also used.Topics that go beyond discrete objects include transcendental numbers, diophantine approximation, p-adic analysisand function elds.

6.2.9 Algebra

Main article: Abstract algebra

Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: booleanalgebra used in logic gates and programming; relational algebra used in databases; discrete and nite versions ofgroups, rings and elds are important in algebraic coding theory; discrete semigroups and monoids appear in thetheory of formal languages.

6.2.10 Calculus of nite dierences, discrete calculus or discrete analysis

Main article: nite dierence

A function dened on an interval of the integers is usually called a sequence. A sequence could be a nite sequencefrom a data source or an innite sequence from a discrete dynamical system. Such a discrete function could be denedexplicitly by a list (if its domain is nite), or by a formula for its general term, or it could be given implicitly by arecurrence relation or dierence equation. Dierence equations are similar to a dierential equations, but replacedierentiation by taking the dierence between adjacent terms; they can be used to approximate dierential equationsor (more often) studied in their own right. Many questions and methods concerning dierential equations havecounterparts for dierence equations. For instance where there are integral transforms in harmonic analysis forstudying continuous functions or analog signals, there are discrete transforms for discrete functions or digital signals.As well as the discrete metric there are more general discrete or nite metric spaces and nite topological spaces.

6.2.11 Geometry

Main articles: discrete geometry and computational geometry

Discrete geometry and combinatorial geometry are about combinatorial properties of discrete collections of geomet-rical objects. A long-standing topic in discrete geometry is tiling of the plane. Computational geometry appliesalgorithms to geometrical problems.

6.2.12 Topology

Although topology is the eld of mathematics that formalizes and generalizes the intuitive notion of continuous defor-mation of objects, it gives rise to many discrete topics; this can be attributed in part to the focus on topological invari-ants, which themselves usually take discrete values. See combinatorial topology, topological graph theory, topologicalcombinatorics, computational topology, discrete topological space, nite topological space, topology (chemistry).

6.2.13 Operations research

Main article: Operations researchOperations research provides techniques for solving practical problems in business and other elds problems suchas allocating resources to maximize prot, or scheduling project activities to minimize risk. Operations researchtechniques include linear programming and other areas of optimization, queuing theory, scheduling theory, network


Computational geometry applies computer algorithms to representations of geometrical objects.

theory. Operations research also includes continuous topics such as continuous-time Markov process, continuous-time martingales, process optimization, and continuous and hybrid control theory.

6.2.14 Game theory, decision theory, utility theory, social choice theory

Decision theory is concerned with identifying the values, uncertainties and other issues relevant in a given decision,its rationality, and the resulting optimal decision.Utility theory is about measures of the relative economic satisfaction from, or desirability of, consumption of variousgoods and services.Social choice theory is about voting. A more puzzle-based approach to voting is ballot theory.Game theory deals with situations where success depends on the choices of others, which makes choosing the bestcourse of action more complex. There are even continuous games, see dierential game. Topics include auctiontheory and fair division.


PERT charts like this provide a business management technique based on graph theory.

6.2.15 DiscretizationMain article: Discretization

Discretization concerns the process of transferring continuous models and equations into discrete counterparts, oftenfor the purposes of making calculations easier by using approximations. Numerical analysis provides an importantexample.

6.2.16 Discrete analogues of continuous mathematicsThere are many concepts in continuous mathematics which have discrete versions, such as discrete calculus, discreteprobability distributions, discrete Fourier transforms, discrete geometry, discrete logarithms, discrete dierentialgeometry, discrete exterior calculus, discrete Morse theory, dierence equations, discrete dynamical systems, anddiscrete vector measures.In applied mathematics, discrete modelling is the discrete analogue of continuous modelling. In discrete modelling,discrete formulae are t to data. A common method in this form of modelling is to use recurrence relation.In algebraic geometry, the concept of a curve can be extended to discrete geometries by taking the spectra ofpolynomial rings over nite elds to be models of the ane spaces over that eld, and letting subvarieties or spectra ofother rings provide the curves that lie in that space. Although the space in which the curves appear has a nite numberof points, the curves are not so much sets of points as analogues of curves in continuous settings. For example, everypoint of the form V (x c) SpecK[x] = A1 for K a eld can be studied either as SpecK[x]/(x c) = SpecK, a point, or as the spectrum SpecK[x](xc) of the local ring at (x-c), a point together with a neighborhood aroundit. Algebraic varieties also have a well-dened notion of tangent space called the Zariski tangent space, making manyfeatures of calculus applicable even in nite settings.

6.2.17 Hybrid discrete and continuous mathematicsThe time scale calculus is a unication of the theory of dierence equations with that of dierential equations, whichhas applications to elds requiring simultaneous modelling of discrete and continuous data. Another way of modelingsuch a situation is the notion of hybrid dynamical system.


6.3 See also Outline of discrete mathematics CyberChase, a show that teaches Discrete Mathematics to children

6.4 References[1] Richard Johnsonbaugh, Discrete Mathematics, Prentice Hall, 2008.

[2] Weisstein, Eric W., Discrete mathematics, MathWorld.

[3] Biggs, Norman L. (2002), Discrete mathematics, Oxford Science Publications (2nd ed.), New York: The Clarendon PressOxford University Press, p. 89, ISBN 9780198507178, MR 1078626, Discrete Mathematics is the branch of Mathematicsin which we deal with questions involving nite or countably innite sets.

[4] Brian Hopkins, Resources for Teaching Discrete Mathematics, Mathematical Association of America, 2008.

[5] Ken Levasseur; Al Doerr. Applied Discrete Structures. p. 8.

[6] Albert Georey Howson, ed. (1988). Mathematics as a Service Subject. Cambridge University Press. pp. 7778. ISBN978-0-521-35395-3.

[7] Joseph G. Rosenstein. Discrete Mathematics in the Schools. American Mathematical Soc. p. 323. ISBN 978-0-8218-8578-9.

[8] http://ucsmp.uchicago.edu/secondary/curriculum/precalculus-discrete/

[9] Wilson, Robin (2002). Four Colors Suce. London: Penguin Books. ISBN 978-0-691-11533-7.

[10] Hodges, Andrew. Alan Turing: the enigma. Random House, 1992.

[11] Trevor R. Hodkinson; John A. N. Parnell (2007). Reconstruction the Tree of Life: Taxonomy And Systematics of Large AndSpecies Rich Taxa. CRC PressINC. p. 97. ISBN 978-0-8493-9579-6.

[12] Millennium Prize Problems. 2000-05-24. Retrieved 2008-01-12.

[13] A. S. Troelstra; H. Schwichtenberg (2000-07-27). Basic Proof Theory. Cambridge University Press. p. 186. ISBN978-0-521-77911-1.

[14] Samuel R. Buss (1998). Handbook of Proof Theory. Elsevier. p. 13. ISBN 978-0-444-89840-1.

[15] Franz Baader; Gerhard Brewka; Thomas Eiter (2001-10-16). KI 2001: Advances in Articial Intelligence: Joint Ger-man/Austrian Conference on AI, Vienna, Austria, September 19-21, 2001. Proceedings. Springer. p. 325. ISBN 978-3-540-42612-7.

[16] Brotherston, J.; Bornat, R.; Calcagno, C. (January 2008). Cyclic proofs of program termination in separation logic. ACMSIGPLAN Notices 43 (1). CiteSeerX: 10 .1 .1 .111 .1105.

[17] Graphs on Surfaces, Bojan Mohar and Carsten Thomassen, Johns Hopkins University press, 2001

6.5 Further reading Norman L. Biggs (2002-12-19). Discrete Mathematics. Oxford University Press. ISBN 978-0-19-850717-8. John Dwyer (2010). An Introduction to Discrete Mathematics for Business & Computing. ISBN 978-1-907934-

00-1. Susanna S. Epp (2010-08-04). Discrete Mathematics With Applications. Thomson Brooks/Cole. ISBN 978-0-

495-39132-6. Ronald Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics. Ralph P. Grimaldi (2004). Discrete and CombinatorialMathematics: An Applied Introduction. Addison Wesley.

ISBN 978-0-201-72634-3.

6.6. EXTERNAL LINKS 31

Donald E. Knuth (2011-03-03). The Art of Computer Programming, Volumes 1-4a Boxed Set. Addison-WesleyProfessional. ISBN 978-0-321-75104-1.

Ji Matouek; Jaroslav Neetil (1998). Discrete Mathematics. Oxford University Press. ISBN 978-0-19-850208-1.

Obrenic, Bojana (2003-01-29). Practice Problems in Discrete Mathematics. Prentice Hall. ISBN 978-0-13-045803-2.

Kenneth H. Rosen; John G. Michaels (2000). Hand Book of Discrete and Combinatorial Mathematics. CRCPressI Llc. ISBN 978-0-8493-0149-0.

Kenneth H. Rosen (2007). Discrete Mathematics: And Its Applications. McGraw-Hill College. ISBN 978-0-07-288008-3.

Andrew Simpson (2002). Discrete Mathematics by Example. McGraw-Hill Incorporated. ISBN 978-0-07-7098

discrete mathematics 1_6

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