Discrete mathematics 1From Wikipedia, the free encyclopedia

Contents

1 Antimatroid 11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Discrete mathematics 123.1 Grand challenges, past and present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Topics in discrete mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Theoretical computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Information theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.4 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.5 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.6 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.7 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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ii CONTENTS

3.2.8 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.9 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.10 Calculus of finite differences, discrete calculus or discrete analysis . . . . . . . . . . . . . . 193.2.11 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.12 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.13 Operations research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.14 Game theory, decision theory, utility theory, social choice theory . . . . . . . . . . . . . . 203.2.15 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.16 Discrete analogues of continuous mathematics . . . . . . . . . . . . . . . . . . . . . . . . 213.2.17 Hybrid discrete and continuous mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Formal language 244.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Words over an alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.4.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.5 Language-specification formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.6 Operations on languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.7.1 Programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.7.2 Formal theories, systems and proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.9.1 Citation footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.9.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Greedoid 315.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Classes of greedoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Inversion (discrete mathematics) 34

CONTENTS iii

6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 Weak order of permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.4.1 Source bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.4.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.4.3 Presortedness measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 386.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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 different sequences in which elements may be included.Dilworth (1940) was the first 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 defining antimatroids as set systems are very similar to those of matroids, but whereas matroids are definedby an exchange axiom (e.g., the basis exchange, or independent set exchange axioms), antimatroids are defined 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

https://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Formal_systemhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Cryptomorphismhttps://en.wikipedia.org/wiki/Set_systemhttps://en.wikipedia.org/wiki/Formal_languagehttps://en.wikipedia.org/wiki/Robert_P._Dilworthhttps://en.wikipedia.org/wiki/Lattice_(order)https://en.wikipedia.org/wiki/Matroidhttps://en.wikipedia.org/wiki/Matroid#Independent_sets,_bases,_and_circuitshttps://en.wikipedia.org/wiki/Greedoidhttps://en.wikipedia.org/wiki/Semimodular_latticehttps://en.wikipedia.org/wiki/Partial_orderhttps://en.wikipedia.org/wiki/Distributive_latticehttps://en.wikipedia.org/wiki/Complement_(set_theory)https://en.wikipedia.org/wiki/Antimatroid#Convex_geometrieshttps://en.wikipedia.org/wiki/Convex_sethttps://en.wikipedia.org/wiki/Geometry

2 CHAPTER 1. ANTIMATROID

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

1.1 Definitions

An antimatroid can be defined as a finite 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 definition as a formal language, that is, as a set of strings defined from a finitealphabet of symbols. A language L defining 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 prefix 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 defined 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 prefix of s is feasible, then L defines an antimatroid. Thus, thesetwo definitions lead to mathematically equivalent classes of objects.[2]

1.2 Examples

A chain antimatroid has as its formal language the prefixes of a single word, and as its feasible sets the setsof symbols in these prefixes. For instance the chain antimatroid defined 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 finite partially ordered set. By Birkhoffs 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 finite 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 prefixes of perfect eliminationorderings of a chordal graph form an antimatroid.[3]

https://en.wikipedia.org/wiki/Job_shop_schedulinghttps://en.wikipedia.org/wiki/Artificial_intelligencehttps://en.wikipedia.org/wiki/Union_(set_theory)https://en.wikipedia.org/wiki/Closure_(mathematics)https://en.wikipedia.org/wiki/Accessible_set_systemhttps://en.wikipedia.org/wiki/Formal_languagehttps://en.wikipedia.org/wiki/String_(computer_science)https://en.wikipedia.org/wiki/Symbolhttps://en.wikipedia.org/wiki/Prefix_(computer_science)https://en.wikipedia.org/wiki/Lower_sethttps://en.wikipedia.org/wiki/Partially_ordered_sethttps://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theoremhttps://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theoremhttps://en.wikipedia.org/wiki/Total_orderhttps://en.wikipedia.org/wiki/Euclidean_planehttps://en.wikipedia.org/wiki/Euclidean_spacehttps://en.wikipedia.org/wiki/Line_(geometry)https://en.wikipedia.org/wiki/Hyperplanehttps://en.wikipedia.org/wiki/Convex_hullhttps://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Complement_(set_theory)https://en.wikipedia.org/wiki/Perfect_elimination_orderinghttps://en.wikipedia.org/wiki/Chordal_graphhttps://en.wikipedia.org/wiki/Clique_(graph_theory)

1.3. PATHS AND BASIC WORDS 3

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

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 find 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 defined from B as the set of prefixes of words in B. It is often convenientto define antimatroids from basic words in this way, but it is not straightforward to write an axiomatic definition of

https://en.wikipedia.org/wiki/Convex_hullhttps://en.wikipedia.org/wiki/Proper_subsethttps://en.wikipedia.org/wiki/Linear_extension

4 CHAPTER 1. ANTIMATROID

antimatroids in terms of their basic words.

1.4 Convex geometries

See also: Convex set, Convex geometry and Closure operator

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

G = {U \ S | S F}

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 define antimatroids, the set system defining 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 defined 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 define 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 lattices

Any 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 finite distributive lattices, and can be characterized in several different 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

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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 defined 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 finite 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 finite join-distributive lattice, and any finite join-distributive lattice comes from anantimatroid in this way.[6] Another equivalent characterization of finite 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 finite 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 finite join-distributive lattice as an accessible family of sets closed under unions (that is, asan antimatroid) may be viewed as an analogue of Birkhoffs representation theorem under which any finite distributivelattice has a representation as a family of sets closed under unions and intersections.

1.6 Supersolvable antimatroids

Motivated by a problem of defining partial orders on the elements of a Coxeter group, Armstrong (2007) studied an-timatroids which are also supersolvable lattices. A supersolvable antimatroid is defined 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 difference 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 dimension

If 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 = {S T | S A T B}.

Note that this is a different 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.

https://en.wikipedia.org/wiki/Atom_(order_theory)https://en.wikipedia.org/wiki/Boolean_algebra_(structure)https://en.wikipedia.org/wiki/Power_sethttps://en.wikipedia.org/wiki/Semimodular_latticehttps://en.wikipedia.org/wiki/Semimodular_latticehttps://en.wikipedia.org/wiki/Semimodular_latticehttps://en.wikipedia.org/wiki/Graded_posethttps://en.wikipedia.org/wiki/Birkhoff%2527s_representation_theoremhttps://en.wikipedia.org/wiki/Coxeter_grouphttps://en.wikipedia.org/wiki/Antimatroid#CITEREFArmstrong2007https://en.wikipedia.org/wiki/Total_orderhttps://en.wikipedia.org/wiki/Family_of_setshttps://en.wikipedia.org/wiki/Set-theoretic_differencehttps://en.wikipedia.org/wiki/Semilattice

6 CHAPTER 1. ANTIMATROID

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 prefixes 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 prefix 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 verydifferent: there exist antimatroids with order dimension three but with arbitrarily large convex dimension.

1.8 Enumeration

The 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 Applications

Both 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 artificial 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 defining 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 first to use the term antimatroid.Monjardet (1985) surveys the history of rediscovery of antimatroids.

[2] Korte et al., Theorem 1.4.

https://en.wikipedia.org/wiki/Chain_(order_theory)https://en.wikipedia.org/wiki/Dilworth%2527s_theoremhttps://en.wikipedia.org/wiki/Order_dimensionhttps://oeis.org/A119770https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://en.wikipedia.org/wiki/Notation_for_theoretic_scheduling_problemshttps://en.wikipedia.org/wiki/Antimatroid#CITEREFBoydFaigle1990https://en.wikipedia.org/wiki/Greedy_algorithmhttps://en.wikipedia.org/wiki/Eugene_Lawlerhttps://en.wikipedia.org/wiki/Eugene_Lawlerhttps://en.wikipedia.org/wiki/Antimatroid#CITEREFGlassermanYao1994https://en.wikipedia.org/wiki/Discrete_event_simulationhttps://en.wikipedia.org/wiki/Antimatroid#CITEREFParmar2003https://en.wikipedia.org/wiki/Artificial_intelligencehttps://en.wikipedia.org/wiki/Automated_planning_and_schedulinghttps://en.wikipedia.org/wiki/Mathematical_psychologyhttps://en.wikipedia.org/wiki/Knowledge_spacehttps://en.wikipedia.org/wiki/Antimatroid#CITEREFEdelman1980https://en.wikipedia.org/wiki/Antimatroid#CITEREFJamison1980https://en.wikipedia.org/wiki/Antimatroid#CITEREFMonjardet1985

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 efficiently 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.

Birkhoff, 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 Efficient Planning, AAAI Spring Sympo-sium on Logical Formalization of Commonsense Reasoning (PDF).

https://en.wikipedia.org/wiki/Antimatroid#CITEREFAdarichevaGorbunovTumanov2003https://en.wikipedia.org/wiki/Antimatroid#CITEREFArmstrong2007https://en.wikipedia.org/wiki/Antimatroid#CITEREFEdelman1980https://en.wikipedia.org/wiki/Antimatroid#CITEREFArmstrong2007https://en.wikipedia.org/wiki/Antimatroid#CITEREFMonjardet1985https://en.wikipedia.org/wiki/Antimatroid#CITEREFEdelmanSaks1988https://en.wikipedia.org/wiki/Antimatroid#CITEREFDoignonFalmagne1999https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1016%252FS0001-8708%252802%252900011-7https://en.wikipedia.org/wiki/ArXivhttps://arxiv.org/abs/0712.1047https://en.wikipedia.org/wiki/Garrett_Birkhoffhttps://en.wikipedia.org/wiki/Order_(journal)https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252FBF00333128https://en.wikipedia.org/wiki/Anders_Bj%C3%B6rnerhttps://en.wikipedia.org/wiki/G%C3%BCnter_M._Zieglerhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1017%252FCBO9780511662041.009https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-521-38165-7https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=1165537https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1016%252F0166-218X%252890%252990002-Thttp://old-www.cis.uoguelph.ca/~sawada/papers/chordal.pdfhttp://old-www.cis.uoguelph.ca/~sawada/papers/chordal.pdfhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1016%252FS0304-3975%252803%252900221-4https://dx.doi.org/10.1016%252FS0304-3975%252803%252900221-4https://en.wikipedia.org/wiki/Robert_P._Dilworthhttps://en.wikipedia.org/wiki/Annals_of_Mathematicshttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.2307%252F1968857https://en.wikipedia.org/wiki/JSTORhttps://www.jstor.org/stable/1968857https://en.wikipedia.org/wiki/Jean-Paul_Doignonhttps://en.wikipedia.org/wiki/Jean-Claude_Falmagnehttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/3-540-64501-2https://en.wikipedia.org/wiki/Special:BookSources/3-540-64501-2https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252FBF02482912https://en.wikipedia.org/wiki/Order_(journal)https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252FBF00143895https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-471-58041-6http://www.combinatorics.org/Volume_4/Abstracts/v4i1r13.htmlhttps://en.wikipedia.org/wiki/Electronic_Journal_of_Combinatoricshttps://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=1445628https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=608454https://en.wikipedia.org/wiki/Bernhard_Kortehttps://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1szhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/3-540-18190-3https://en.wikipedia.org/wiki/Order_(journal)https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252FBF00582748http://www-formal.stanford.edu/aarati/papers/SS603AParmar.pdfhttp://www-formal.stanford.edu/aarati/papers/SS603AParmar.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 properties

A 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-theoreticdefinitions usually define operations as maps between sets, so adding closure to a structure as an axiom is superfluous;however in practice operations are often defined 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 find 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 defined. In the preceding example, itis important that the reals are closed under subtraction; in the domain of the natural numbers subtraction is not alwaysdefined.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 setsatisfies a closure property.

2.2 Closed sets

A 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 define 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 definition of an operation makes this axiom superfluous; an n-ary operation on S is just a subset of Sn+1. By

8

https://en.wikipedia.org/wiki/Closure_(disambiguation)https://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Operation_(mathematics)https://en.wikipedia.org/wiki/Integershttps://en.wikipedia.org/wiki/Subtractionhttps://en.wikipedia.org/wiki/Integershttps://en.wikipedia.org/wiki/Topological_closurehttps://en.wikipedia.org/wiki/Galois_connectionhttps://en.wikipedia.org/wiki/Monad_(category_theory)https://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/N-aryhttps://en.wikipedia.org/wiki/Operation_(mathematics)

2.3. P CLOSURES OF BINARY RELATIONS 9

its very definition, 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 different sort is that of finding the limit points of a subset of a topological space (if the space isfirst-countable, it suffices 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 qualification, 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 defined similarly.

2.3 P closures of binary relations

The 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 define 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 reflexive transitive closure R*the smallest preorder containing R, or the reflexivetransitive 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 defined 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 reflexivity,transitivity and symmetry are closed under arbitrary intersections. In such cases, the P closure can be directly definedas 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 reflexive 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 fixedarity.

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 reflexivity. 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 operator

Main article: closure operator

https://en.wikipedia.org/wiki/Subgrouphttps://en.wikipedia.org/wiki/Unary_operationhttps://en.wikipedia.org/wiki/Unary_operationhttps://en.wikipedia.org/wiki/Inverse_elementhttps://en.wikipedia.org/wiki/Limit_pointhttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/First-countable_spacehttps://en.wikipedia.org/wiki/Limit_of_a_sequencehttps://en.wikipedia.org/wiki/Net_(mathematics)https://en.wikipedia.org/wiki/Closed_sethttps://en.wikipedia.org/wiki/Topologyhttps://en.wikipedia.org/wiki/Closed_intervalhttps://en.wikipedia.org/wiki/Upper_sethttps://en.wikipedia.org/wiki/Ordinal_numberhttps://en.wikipedia.org/wiki/Binary_relationhttps://en.wikipedia.org/wiki/Predicate_(mathematical_logic)https://en.wikipedia.org/wiki/Symmetric_closurehttps://en.wikipedia.org/wiki/Symmetric_relationhttps://en.wikipedia.org/wiki/Rewritinghttps://en.wikipedia.org/wiki/Preorderhttps://en.wikipedia.org/wiki/Equivalence_relation#Comparing_equivalence_relationshttps://en.wikipedia.org/wiki/Term_algebrahttps://en.wikipedia.org/wiki/Congruence_relation#Universal_algebrahttps://en.wikipedia.org/wiki/Closure_operator

10 CHAPTER 2. CLOSURE (MATHEMATICS)

Given an operation on a set X, one can define 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 defines 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 define 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 thevector 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 field.

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 thatcan 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 normalsubgroup containing the set.

In mathematical analysis and in probability theory, the closure of a collection of subsets of X under countablymany 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))

https://en.wikipedia.org/wiki/Generating_set_of_a_grouphttps://en.wikipedia.org/wiki/Idempotenthttps://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/Topologyhttps://en.wikipedia.org/wiki/Topological_closurehttps://en.wikipedia.org/wiki/Kuratowski_closure_axiomshttps://en.wikipedia.org/wiki/Linear_algebrahttps://en.wikipedia.org/wiki/Linear_spanhttps://en.wikipedia.org/wiki/Vector_spacehttps://en.wikipedia.org/wiki/Linear_combinationhttps://en.wikipedia.org/wiki/Linear_subspacehttps://en.wikipedia.org/wiki/Matroidhttps://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Transitive_set#Transitive_closurehttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Transitive_closurehttps://en.wikipedia.org/wiki/Binary_relationhttps://en.wikipedia.org/wiki/Abstract_algebrahttps://en.wikipedia.org/wiki/Algebraic_closurehttps://en.wikipedia.org/wiki/Field_(algebra)https://en.wikipedia.org/wiki/Commutative_algebrahttps://en.wikipedia.org/wiki/Integral_closurehttps://en.wikipedia.org/wiki/Tight_closurehttps://en.wikipedia.org/wiki/Geometryhttps://en.wikipedia.org/wiki/Convex_hullhttps://en.wikipedia.org/wiki/Convex_sethttps://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Formal_languagehttps://en.wikipedia.org/wiki/Kleene_closurehttps://en.wikipedia.org/wiki/Group_theoryhttps://en.wikipedia.org/wiki/Conjugate_closurehttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Mathematical_analysishttps://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Countablehttps://en.wikipedia.org/wiki/Countablehttps://en.wikipedia.org/wiki/Algebra_of_setshttps://en.wikipedia.org/wiki/Sigma-algebrahttps://en.wikipedia.org/wiki/Open_sethttps://en.wikipedia.org/wiki/Clopen_set

2.8. REFERENCES 11

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

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

https://en.wikipedia.org/wiki/Franz_Baaderhttps://en.wikipedia.org/wiki/Tobias_Nipkowhttp://books.google.com/books/about/Term_Rewriting_and_All_That.html?id=N7BvXVUCQk8Chttps://en.wikipedia.org/wiki/Garrett_Birkhoff

Chapter 3

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 definition 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 finite or infinite. The term finite mathematics is sometimesapplied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant tobusiness.

12

https://en.wikipedia.org/wiki/Discrete_Mathematics_(journal)https://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Mathematical_structurehttps://en.wikipedia.org/wiki/Discrete_spacehttps://en.wiktionary.org/wiki/continuoushttps://en.wikipedia.org/wiki/Graph_(mathematics)https://en.wikipedia.org/wiki/Graph_propertyhttps://en.wikipedia.org/wiki/Algorithmhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Graph_(mathematics)https://en.wikipedia.org/wiki/Statement_(logic)https://en.wikipedia.org/wiki/Mathematical_logichttps://en.wikipedia.org/wiki/Calculushttps://en.wikipedia.org/wiki/Mathematical_analysishttps://en.wikipedia.org/wiki/Enumerationhttps://en.wikipedia.org/wiki/Countable_set

3.1. GRAND CHALLENGES, PAST AND PRESENT 13

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 significant 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 toefforts 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.

3.1 Grand challenges, past and present

The history of discrete mathematics has involved a number of challenging problems which have focused attentionwithin areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem,first 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 coefficients 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 first 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 verification 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 fields 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 offered a $1million USD prize for the first correct proof, along with prizes for six other mathematical problems.[12]

3.2 Topics in discrete mathematics

3.2.1 Theoretical computer science

Main 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,

https://en.wikipedia.org/wiki/Digital_computershttps://en.wikipedia.org/wiki/Programming_languagehttps://en.wikipedia.org/wiki/Cryptographyhttps://en.wikipedia.org/wiki/Automated_theorem_provinghttps://en.wikipedia.org/wiki/Software_developmenthttps://en.wikipedia.org/wiki/Software_developmenthttps://en.wikipedia.org/wiki/Operations_researchhttps://en.wikipedia.org/wiki/Computer_sciencehttps://en.wikipedia.org/wiki/Mathematical_maturityhttps://en.wikipedia.org/wiki/Precalculushttps://en.wikipedia.org/wiki/Fulkerson_Prizehttps://en.wikipedia.org/wiki/Four_color_theoremhttps://en.wikipedia.org/wiki/Mathematical_logichttps://en.wikipedia.org/wiki/Hilbert%2527s_second_problemhttps://en.wikipedia.org/wiki/David_Hilberthttps://en.wikipedia.org/wiki/Hilbert%2527s_problemshttps://en.wikipedia.org/wiki/Axiomshttps://en.wikipedia.org/wiki/Arithmetichttps://en.wikipedia.org/wiki/Consistenthttps://en.wikipedia.org/wiki/G%C3%B6del%2527s_second_incompleteness_theoremhttps://en.wikipedia.org/wiki/Hilbert%2527s_tenth_problemhttps://en.wikipedia.org/wiki/Diophantine_equationhttps://en.wikipedia.org/wiki/Yuri_Matiyasevichhttps://en.wikipedia.org/wiki/Matiyasevich%2527s_theoremhttps://en.wikipedia.org/wiki/Cryptanalysishttps://en.wikipedia.org/wiki/World_War_IIhttps://en.wikipedia.org/wiki/Cryptographyhttps://en.wikipedia.org/wiki/Theoretical_computer_sciencehttps://en.wikipedia.org/wiki/Colossus_computerhttps://en.wikipedia.org/wiki/Bletchley_Parkhttps://en.wikipedia.org/wiki/Alan_Turinghttps://en.wikipedia.org/wiki/Operations_researchhttps://en.wikipedia.org/wiki/Cold_Warhttps://en.wikipedia.org/wiki/Public-key_cryptographyhttps://en.wikipedia.org/wiki/Critical_path_methodhttps://en.wikipedia.org/wiki/Telecommunicationhttps://en.wikipedia.org/wiki/Information_theoryhttps://en.wikipedia.org/wiki/Formal_verificationhttps://en.wikipedia.org/wiki/Software_developmenthttps://en.wikipedia.org/wiki/Software_developmenthttps://en.wikipedia.org/wiki/Safety-critical_systemhttps://en.wikipedia.org/wiki/Automated_theorem_provinghttps://en.wikipedia.org/wiki/Computational_geometryhttps://en.wikipedia.org/wiki/Computer_graphics_(computer_science)https://en.wikipedia.org/wiki/Video_gamehttps://en.wikipedia.org/wiki/Computer-aided_designhttps://en.wikipedia.org/wiki/Combinatoricshttps://en.wikipedia.org/wiki/Bioinformaticshttps://en.wikipedia.org/wiki/Phylogenetic_treehttps://en.wikipedia.org/wiki/P_=_NP_problemhttps://en.wikipedia.org/wiki/Complexity_classhttps://en.wikipedia.org/wiki/P_(complexity)https://en.wikipedia.org/wiki/NP_(complexity)https://en.wikipedia.org/wiki/Clay_Mathematics_Institutehttps://en.wikipedia.org/wiki/USDhttps://en.wikipedia.org/wiki/Millennium_Prize_Problemshttps://en.wikipedia.org/wiki/Theoretical_computer_sciencehttps://en.wikipedia.org/wiki/Graph_theoryhttps://en.wikipedia.org/wiki/Mathematical_logichttps://en.wikipedia.org/wiki/Computability

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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.

https://en.wikipedia.org/wiki/Graph_theoryhttps://en.wikipedia.org/wiki/Graph_coloringhttps://en.wikipedia.org/wiki/Four_color_theoremhttps://en.wikipedia.org/wiki/Kenneth_Appelhttps://en.wikipedia.org/wiki/Wolfgang_Hakenhttps://en.wikipedia.org/wiki/Automata_theoryhttps://en.wikipedia.org/wiki/Formal_languagehttps://en.wikipedia.org/wiki/Petri_nethttps://en.wikipedia.org/wiki/Process_algebrahttps://en.wikipedia.org/wiki/VLSIhttps://en.wikipedia.org/wiki/Computational_geometryhttps://en.wikipedia.org/wiki/Computer_image_analysis

3.2. TOPICS IN DISCRETE MATHEMATICS 15

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

3.2.2 Information theory

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

3.2.3 Logic

Main 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 verified with a truth table. The study of mathematical proof is par-ticularly important in logic, and has applications to automated theorem proving and formal verification of software.Logical formulas are discrete structures, as are proofs, which form finite 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 finite set, generally restricted to two values: true and false, butlogic can also be continuous-valued, e.g., fuzzy logic. Concepts such as infinite proof trees or infinite derivation treeshave also been studied,[16] e.g. infinitary logic.

3.2.4 Set theory

Main article: Set theory

https://en.wikipedia.org/wiki/Computational_complexity_theoryhttps://en.wikipedia.org/wiki/Algorithmhttps://en.wikipedia.org/wiki/Quicksorthttps://en.wikipedia.org/wiki/Information_theoryhttps://en.wikipedia.org/wiki/Informationhttps://en.wikipedia.org/wiki/Coding_theoryhttps://en.wikipedia.org/wiki/Analog_signalhttps://en.wikipedia.org/wiki/Analog_codinghttps://en.wikipedia.org/wiki/Analog_encryptionhttps://en.wikipedia.org/wiki/Mathematical_logichttps://en.wikipedia.org/wiki/Inferencehttps://en.wikipedia.org/wiki/Consistencyhttps://en.wikipedia.org/wiki/Soundnesshttps://en.wikipedia.org/wiki/Completeness_(logic)https://en.wikipedia.org/wiki/Intuitionistic_logichttps://en.wikipedia.org/wiki/Peirce%2527s_lawhttps://en.wikipedia.org/wiki/Truth_tablehttps://en.wikipedia.org/wiki/Mathematical_proofhttps://en.wikipedia.org/wiki/Automated_theorem_provinghttps://en.wikipedia.org/wiki/Formal_verificationhttps://en.wikipedia.org/wiki/Well-formed_formulahttps://en.wikipedia.org/wiki/Proof_theoryhttps://en.wikipedia.org/wiki/Tree_structurehttps://en.wikipedia.org/wiki/Directed_acyclic_graphhttps://en.wikipedia.org/wiki/Directed_acyclic_graphhttps://en.wikipedia.org/wiki/Rule_of_inferencehttps://en.wikipedia.org/wiki/Premisehttps://en.wikipedia.org/wiki/Truth_valuehttps://en.wikipedia.org/wiki/Fuzzy_logichttps://en.wikipedia.org/wiki/Infinitary_logichttps://en.wikipedia.org/wiki/Set_theory

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101 0111110 1001110 1011110 1001111 0000110 0101110 0100110 1001110 0001

WikipediaThe 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 (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications inseveral areas.In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as abranch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infiniteset, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outsidethe scope of discrete mathematics. Indeed, contemporary work in descriptive set theory makes extensive use oftraditional continuous mathematics.

https://en.wikipedia.org/wiki/ASCIIhttps://en.wikipedia.org/wiki/Binary_numeral_systemhttps://en.wikipedia.org/wiki/Information_theoryhttps://en.wikipedia.org/wiki/Algorithmhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Prime_numberhttps://en.wikipedia.org/wiki/Partially_ordered_sethttps://en.wikipedia.org/wiki/Relation_(mathematics)https://en.wikipedia.org/wiki/Countable_sethttps://en.wikipedia.org/wiki/Finite_sethttps://en.wikipedia.org/wiki/Georg_Cantorhttps://en.wikipedia.org/wiki/Infinite_sethttps://en.wikipedia.org/wiki/Infinite_sethttps://en.wikipedia.org/wiki/Descriptive_set_theory

3.2. TOPICS IN DISCRETE MATHEMATICS 17

3.2.5 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 unifiedframework 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 field. Order theory is the study of partially ordered sets, both finiteand infinite.

3.2.6 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 flow 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

https://en.wikipedia.org/wiki/Combinatoricshttps://en.wikipedia.org/wiki/Enumerative_combinatoricshttps://en.wikipedia.org/wiki/Twelvefold_wayhttps://en.wikipedia.org/wiki/Permutationshttps://en.wikipedia.org/wiki/Combinationshttps://en.wikipedia.org/wiki/Partition_of_a_sethttps://en.wikipedia.org/wiki/Analytic_combinatoricshttps://en.wikipedia.org/wiki/Complex_analysishttps://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Generating_functionshttps://en.wikipedia.org/wiki/Generating_functionshttps://en.wikipedia.org/wiki/Asymptotic_analysishttps://en.wikipedia.org/wiki/Combinatorial_designhttps://en.wikipedia.org/wiki/Set_intersectionhttps://en.wikipedia.org/wiki/Partition_theoryhttps://en.wikipedia.org/wiki/Integer_partitionhttps://en.wikipedia.org/wiki/Q-serieshttps://en.wikipedia.org/wiki/Special_functionshttps://en.wikipedia.org/wiki/Special_functionshttps://en.wikipedia.org/wiki/Orthogonal_polynomialshttps://en.wikipedia.org/wiki/Number_theoryhttps://en.wikipedia.org/wiki/Analysishttps://en.wikipedia.org/wiki/Order_theoryhttps://en.wikipedia.org/wiki/Partially_ordered_setshttps://en.wikipedia.org/wiki/Graph_theoryhttps://en.wikipedia.org/wiki/Graph_(mathematics)https://en.wikipedia.org/wiki/Network_theoryhttps://en.wikipedia.org/wiki/Graph_theoryhttps://en.wikipedia.org/wiki/Group_theoryhttps://en.wikipedia.org/wiki/Truncated_tetrahedronhttps://en.wikipedia.org/wiki/Alternating_grouphttps://en.wikipedia.org/wiki/Topologyhttps://en.wikipedia.org/wiki/Knot_theoryhttps://en.wikipedia.org/wiki/Algebraic_graph_theoryhttps://en.wikipedia.org/wiki/Continuous_graph

18 CHAPTER 3. DISCRETE MATHEMATICS

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

3.2.7 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 flocks 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.

3.2.8 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

https://en.wikipedia.org/wiki/Discrete_probability_theoryhttps://en.wikipedia.org/wiki/Sample_spaceshttps://en.wikipedia.org/wiki/Normal_distributionhttps://en.wikipedia.org/wiki/Dicehttps://en.wikipedia.org/wiki/Decks_of_cardshttps://en.wikipedia.org/wiki/Enumerative_combinatoricshttps://en.wikipedia.org/wiki/Ulam_spiralhttps://en.wikipedia.org/wiki/Prime_numberhttps://en.wikipedia.org/wiki/Prime_number#Distributionhttps://en.wikipedia.org/wiki/Number_theory

3.2. TOPICS IN DISCRETE MATHEMATICS 19

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 fields.

3.2.9 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 finite versions ofgroups, rings and fields are important in algebraic coding theory; discrete semigroups and monoids appear in thetheory of formal languages.

3.2.10 Calculus of finite differences, discrete calculus or discrete analysis

Main article: finite difference

A function defined on an interval of the integers is usually called a sequence. A sequence could be a finite sequencefrom a data source or an infinite sequence from a discrete dynamical system. Such a discrete function could be definedexplicitly by a list (if its domain is finite), or by a formula for its general term, or it could be given implicitly by arecurrence relation or difference equation. Difference equations are similar to a differential equations, but replacedifferentiation by taking the difference between adjacent terms; they can be used to approximate differential equationsor (more often) studied in their own right. Many questions and methods concerning differential equations havecounterparts for difference 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 finite metric spaces and finite topological spaces.

3.2.11 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.

3.2.12 Topology

Although topology is the field 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, finite topological space, topology (chemistry).

3.2.13 Operations research

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

https://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Cryptographyhttps://en.wikipedia.org/wiki/Cryptanalysishttps://en.wikipedia.org/wiki/Cryptologyhttps://en.wikipedia.org/wiki/Modular_arithmetichttps://en.wikipedia.org/wiki/Diophantine_equationshttps://en.wikipedia.org/wiki/Primality_testhttps://en.wikipedia.org/wiki/Geometry_of_numbershttps://en.wikipedia.org/wiki/Analytic_number_theoryhttps://en.wikipedia.org/wiki/Transcendental_numberhttps://en.wikipedia.org/wiki/Diophantine_approximationhttps://en.wikipedia.org/wiki/P-adic_analysishttps://en.wikipedia.org/wiki/Function_field_of_an_algebraic_varietyhttps://en.wikipedia.org/wiki/Abstract_algebrahttps://en.wikipedia.org/wiki/Algebraic_structurehttps://en.wikipedia.org/wiki/Boolean_algebra_(logic)https://en.wikipedia.org/wiki/Boolean_algebra_(logic)https://en.wikipedia.org/wiki/Logic_gatehttps://en.wikipedia.org/wiki/Relational_algebrahttps://en.wikipedia.org/wiki/Databaseshttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Ring_(mathematics)https://en.wikipedia.org/wiki/Field_(mathematics)https://en.wikipedia.org/wiki/Algebraic_coding_theoryhttps://en.wikipedia.org/wiki/Semigrouphttps://en.wikipedia.org/wiki/Monoidhttps://en.wikipedia.org/wiki/Formal_languageshttps://en.wikipedia.org/wiki/Finite_differencehttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Sequencehttps://en.wikipedia.org/wiki/Discrete_dynamical_systemhttps://en.wikipedia.org/wiki/Recurrence_relationhttps://en.wikipedia.org/wiki/Difference_equationhttps://en.wikipedia.org/wiki/Differential_equationhttps://en.wikipedia.org/wiki/Derivativehttps://en.wikipedia.org/wiki/Integral_transformshttps://en.wikipedia.org/wiki/Harmonic_analysishttps://en.wikipedia.org/wiki/Discrete_transformhttps://en.wikipedia.org/wiki/Discrete_metrichttps://en.wikipedia.org/wiki/Finite_metric_spacehttps://en.wikipedia.org/wiki/Finite_topological_spacehttps://en.wikipedia.org/wiki/Discrete_geometryhttps://en.wikipedia.org/wiki/Computational_geometryhttps://en.wikipedia.org/wiki/Discrete_geometryhttps://en.wikipedia.org/wiki/Tessellationhttps://en.wikipedia.org/wiki/Topologyhttps://en.wikipedia.org/wiki/Topological_invarianthttps://en.wikipedia.org/wiki/Topological_invarianthttps://en.wikipedia.org/wiki/Combinatorial_topologyhttps://en.wikipedia.org/wiki/Topological_graph_theoryhttps://en.wikipedia.org/wiki/Topological_combinatoricshttps://en.wikipedia.org/wiki/Topological_combinatoricshttps://en.wikipedia.org/wiki/Computational_topologyhttps://en.wikipedia.org/wiki/Discrete_topological_spacehttps://en.wikipedia.org/wiki/Finite_topological_spacehttps://en.wikipedia.org/wiki/Topology_(chemistry)https://en.wikipedia.org/wiki/Operations_researchhttps://en.wikipedia.org/wiki/Linear_programminghttps://en.wikipedia.org/wiki/Optimization_(mathematics)https://en.wikipedia.org/wiki/Queuing_theoryhttps://en.wikipedia.org/wiki/Scheduling_algorithmhttps://en.wikipedia.org/wiki/Network_theory

20 CHAPTER 3. DISCRETE MATHEMATICS

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.

3.2.14 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 differential game. Topics include auctiontheory and fair division.

https://en.wikipedia.org/wiki/Network_theoryhttps://en.wikipedia.org/wiki/Network_theoryhttps://en.wikipedia.org/wiki/Algorithmhttps://en.wikipedia.org/wiki/Geometryhttps://en.wikipedia.org/wiki/Continuous-time_Markov_processhttps://en.wikipedia.org/wiki/Martingale_(probability_theory)https://en.wikipedia.org/wiki/Process_optimizationhttps://en.wikipedia.org/wiki/Control_theoryhttps://en.wikipedia.org/wiki/Decision_theoryhttps://en.wikipedia.org/wiki/Utility_theoryhttps://en.wikipedia.org/wiki/Economichttps://en.wikipedia.org/wiki/Social_choice_theoryhttps://en.wikipedia.org/wiki/Votinghttps://en.wikipedia.org/wiki/Ballot_theoryhttps://en.wikipedia.org/wiki/Game_theoryhttps://en.wikipedia.org/wiki/Differential_gamehttps://en.wikipedia.org/wiki/Auction_theoryhttps://en.wikipedia.org/wiki/Auction_theoryhttps://en.wikipedia.org/wiki/Fair_division

3.2. TOPICS IN DISCRETE MATHEMATICS 21

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

3.2.15 Discretization

Main 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.

3.2.16 Discrete analogues of continuous mathematics

There are many concepts in continuous mathematics which have discrete versions, such as discrete calculus, discreteprobability distributions, discrete Fourier transforms, discrete geometry, discrete logarithms, discrete differentialgeometry, discrete exterior calculus, discrete Morse theory, difference 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 fit 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 finite fields to be models of the affine spaces over that field, 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 finite 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 field 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-defined notion of tangent space called the Zariski tangent space, making manyfeatures of calculus applicable even in finite settings.

3.2.17 Hybrid discrete and continuous mathematics

The time scale calculus is a unification of the theory of difference equations with that of differential equations, whichhas applications to fields requiring simultaneous modelling of discrete and continuous data. Another way of modelingsuch a situation is the notion of hybrid dynamical system.

https://en.wikipedia.org/wiki/Program_Evaluation_and_Review_Techniquehttps://en.wikipedia.org/wiki/Graph_theoryhttps://en.wikipedia.org/wiki/Discretizationhttps://en.wikipedia.org/wiki/Numerical_analysishttps://en.wikipedia.org/wiki/Discrete_calculushttps://en.wikipedia.org/wiki/Discrete_probability_distributionhttps://en.wikipedia.org/wiki/Discrete_probability_distributionhttps://en.wikipedia.org/wiki/Discrete_Fourier_transformhttps://en.wikipedia.org/wiki/Discrete_geometryhttps://en.wikipedia.org/wiki/Discrete_logarithmhttps://en.wikipedia.org/wiki/Discrete_differential_geometryhttps://en.wikipedia.org/wiki/Discrete_differential_geometryhttps://en.wikipedia.org/wiki/Discrete_exterior_calculushttps://en.wikipedia.org/wiki/Discrete_Morse_theoryhttps://en.wikipedia.org/wiki/Difference_equationhttps://en.wikipedia.org/wiki/Discrete_dynamical_systemhttps://en.wikipedia.org/wiki/Shapley%E2%80%93Folkman_lemma#Probability_and_measure_theoryhttps://en.wikipedia.org/wiki/Applied_mathematicshttps://en.wikipedia.org/wiki/Discrete_modellinghttps://en.wikipedia.org/wiki/Continuous_modellinghttps://en.wikipedia.org/wiki/Datahttps://en.wikipedia.org/wiki/Recurrence_relationhttps://en.wikipedia.org/wiki/Algebraic_geometryhttps://en.wikipedia.org/wiki/Spectrum_of_a_ringhttps://en.wikipedia.org/wiki/Polynomial_ringhttps://en.wikipedia.org/wiki/Finite_fieldhttps://en.wikipedia.org/wiki/Affine_spacehttps://en.wikipedia.org/wiki/Algebraic_varietyhttps://en.wikipedia.org/wiki/Localization_of_a_ringhttps://en.wikipedia.org/wiki/Tangent_spacehttps://en.wikipedia.org/wiki/Zariski_tangent_spacehttps://en.wikipedia.org/wiki/Time_scale_calculushttps://en.wikipedia.org/wiki/Difference_equationshttps://en.wikipedia.org/wiki/Differential_equationshttps://en.wikipedia.org/wiki/Hybrid_system

22 CHAPTER 3. DISCRETE MATHEMATICS

3.3 See also Outline of discrete mathematics

CyberChase, a show that teaches Discrete Mathematics to children

3.4 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 finite or countably infinite 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 Geoffrey 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 Suffice. 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 Artificial Intelligence: Joint Ger-man/Austrian Conferen