- •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.
42 Complexity studies the time taken by algorithms, such as this sorting routine.
43 Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and logic. Included within theoretical computer science is the study of algorithms for computing mathematical results. Computability studies what can be computed in principle, and has close ties to logic, while complexity studies the time taken by computations. Automata theory and formal language theory are closely related to computability. Petri nets and process algebras are used to model computer systems, and methods from discrete mathematics are used in analyzing VLSI electronic circuits. Computational geometry applies algorithms to geometrical problems, while computer image analysis applies them to representations of images. Theoretical computer science also includes the study of various continuous computational topics.
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.
106 Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields, vector spaces and lattices. The term universal algebra is used for structures with no relation symbols.
170 The closed subsets (or induced substructures) of a structure form a lattice. The meet of two subsets is their intersection. The join of two subsets is the closed subset generated by their union. Universal algebra studies the lattice of substructures of a structure in detail.
subsets
5 Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. 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 – do not vary smoothly in this way, but have distinct, separated values. 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 characterized as the branch of mathematics dealing with countable sets (sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers). However, there is no exact, universally agreed, definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
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.
115 2 Induced substructures and closed subsets
161 Induced substructures and closed subsets
168 For every subset there is a smallest closed subset of that contains B. It is called the closed subset generated by B, or the hull of B, and denoted by or . The operator is a finitary closure operator on the set of subsets of .
170 The closed subsets (or induced substructures) of a structure form a lattice. The meet of two subsets is their intersection. The join of two subsets is the closed subset generated by their union. Universal algebra studies the lattice of substructures of a structure in detail.
binary