- •1 Discrete mathematics
- •103 Structure (mathematical logic)
- •17 2.6 Graph theory
- •12 2.1 Theoretical computer science
- •36 Computational geometry has been an important part of the computer graphics incorporated into modern video games and computer-aided design tools.
- •42 Complexity studies the time taken by algorithms, such as this sorting routine.
- •46 The ascii codes for the word "Wikipedia", given here in binary, provide a way of representing the word in information theory, as well as for information-processing algorithms.
- •74 Main article: finite difference
- •105 In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it.
- •105 In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it.
- •25 2.14 Game theory, decision theory, utility theory, social choice theory
- •25 2.14 Game theory, decision theory, utility theory, social choice theory
- •32 Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be colored with only four colors. Kenneth Appel and Wolfgang Haken finally proved this in 1976.
- •8 Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.
- •8 Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.
- •32 Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be colored with only four colors. Kenneth Appel and Wolfgang Haken finally proved this in 1976.
- •25 2.14 Game theory, decision theory, utility theory, social choice theory
- •115 2 Induced substructures and closed subsets
- •25 2.14 Game theory, decision theory, utility theory, social choice theory
- •36 Computational geometry has been an important part of the computer graphics incorporated into modern video games and computer-aided design tools.
- •28 2.17 Hybrid discrete and continuous mathematics
- •61 Graph theory has close links to group theory. This truncated tetrahedron graph is related to the alternating group a4.
- •32 Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be colored with only four colors. Kenneth Appel and Wolfgang Haken finally proved this in 1976.
- •162 Is called an (induced) substructure of if
- •25 2.14 Game theory, decision theory, utility theory, social choice theory
- •77 Computational geometry applies computer algorithms to representations of geometrical objects.
- •90 Payoff matrix for the Prisoner's dilemma, a common example in game theory. One player chooses a row, the other a column; the resulting pair gives their payoffs
- •67 The Ulam spiral of numbers, with black pixels showing prime numbers. This diagram hints at patterns in the distribution of prime numbers.
61 Graph theory has close links to group theory. This truncated tetrahedron graph is related to the alternating group a4.
62 Graph theory, the study of graphs and networks, is often considered part of combinatorics, but has grown large enough and distinct enough, with its own kind of problems, to be regarded as a subject in its own right. Graphs are one of the prime objects of study in Discrete Mathematics. They are among the most ubiquitous models of both natural and human-made structures. They can model many types of relations and process dynamics in physical, biological and social systems. In computer science, they represent networks of communication, data organization, computational devices, 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 for the most part research in graph theory falls within the domain of discrete mathematics.
135 9 External links
n
147 The interpretation function I of assigns functions and relations to the symbols of the signature. Each function symbol f of arity n is assigned an n-ary function on the domain. Each relation symbol R of arity n is assigned an n-ary relation on the domain. A nullary function symbol c is called a constant symbol, because its interpretation I(c) can be identified with a constant element of the domain.
167 A subset of the domain of a structure is called closed if it is closed under the functions of , i.e. if the following condition is satisfied: for every natural number n, every n-ary function symbol f (in the signature of ) and all elements , the result of applying f to the n-tuple is again an element of B: .
n-ary
147 The interpretation function I of assigns functions and relations to the symbols of the signature. Each function symbol f of arity n is assigned an n-ary function on the domain. Each relation symbol R of arity n is assigned an n-ary relation on the domain. A nullary function symbol c is called a constant symbol, because its interpretation I(c) can be identified with a constant element of the domain.
167 A subset of the domain of a structure is called closed if it is closed under the functions of , i.e. if the following condition is satisfied: for every natural number n, every n-ary function symbol f (in the signature of ) and all elements , the result of applying f to the n-tuple is again an element of B: .
notion
81 Although topology is the field of mathematics that formalizes and generalizes the intuitive notion of "continuous deformation" of objects, it gives rise to many discrete topics; this can be attributed in part to the focus on topological invariants, which themselves usually take discrete values. See combinatorial topology, topological graph theory, topological combinatorics, computational topology, discrete topological space, finite topological space, topology (chemistry).
107 Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. In model theory a structure is often called just a model, although it is sometimes disambiguated as a semantic model when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as interpretations.
145 Since the signatures that arise in algebra often contain only function symbols, a signature with no relation symbols is called an algebraic signature. A structure with such a signature is also called an algebra; this should not be confused with the notion of an algebra over a field.
ordered
54 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 in several areas.
58 Combinatorics studies the way in which discrete structures can be combined or arranged. Enumerative combinatorics concentrates on counting the number of certain combinatorial objects - e.g. the twelvefold way provides a unified framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration (i.e., determining the number) of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties. Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, partition theory is now considered a part of combinatorics or an independent field. Order theory is the study of partially ordered sets, both finite and infinite.
159 A signature for ordered fields needs an additional binary relation such as < or ?, and therefore structures for such a signature are not algebras, even though they are of course algebraic structures in the usual, loose sense of the word.
programming
7 Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete 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 development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.
72 Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: boolean algebra used in logic gates and programming; relational algebra used in databases; discrete and finite versions of groups, rings and fields are important in algebraic coding theory; discrete semigroups and monoids appear in the theory of formal languages.
85 Operations research provides techniques for solving practical problems in business and other fields — problems such as allocating resources to maximize profit, or scheduling project activities to minimize risk. Operations research techniques include linear programming and other areas of optimization, queuing theory, scheduling theory, network theory. Operations research also includes continuous topics such as continuous-time Markov process, continuous-time martingales, process optimization, and continuous and hybrid control theory.
proof
38 Currently, one of the most famous open problems in theoretical computer science is the P = NP problem, which involves the relationship between the complexity classes P and NP. The Clay Mathematics Institute has offered a $1 million US prize for the first correct proof, along with prizes for six other mathematical problems.
50 Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, and completeness. For example, in most systems of logic (but not in intuitionistic logic) Peirce's law (((P>Q)>P)>P) is a theorem. For classical logic, it can be easily verified with a truth table. The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.
51 Logical formulas are discrete structures, as are proofs, which form finite trees or, more generally, directed acyclic graph structures (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, but logic can also be continuous-valued, e.g., fuzzy logic. Concepts such as infinite proof trees or infinite derivation trees have also been studied, e.g. infinitary logic.
prove