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General Concept Lattice
The General Concept Lattice (GCL) proposes a novel general construction of concept hierarchy from formal context, where the conventional Formal Concept Lattice based on Formal concept analysis, Formal Concept Analysis (FCA) only serves as a substructure. The formal context is a data table of Binary relation, heterogeneous relations illustrating how objects carrying attributes. By analogy with Truth table, truth-value table, every formal context can develop its fully extended version including all the columns corresponding to attributes constructed, by means of Boolean operations, out of the given attribute set. The GCL is based on the ''extended'' formal context which comprehends the full information content of formal context in the sense that it incorporates whatever the formal context should consistently imply. Noteworthily, different formal contexts may give rise to the same extended formal context. Background The GCL claims to take into account the extended formal context ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poset
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is Reflexive relation, reflexive, antisymmetric relation, antisymmetric, and Transitive relation, transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Material Implication (rule Of Inference)
A material is a substance or mixture of substances that constitutes an object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical and chemical properties, or on their geological origin or biological function. Materials science is the study of materials, their properties and their applications. Raw materials can be processed in different ways to influence their properties, by purification, shaping or the introduction of other materials. New materials can be produced from raw materials by synthesis. In industry, materials are inputs to manufacturing processes to produce products or more complex materials, and the nature and quantity of materials used may form part of the calculation for the cost of a product or delivery under contract, such as where contract costs are calculated on a " time and materials" basis. Historical elements Materials chart the history of humanity. The system of the three prehist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Armstrong Axioms
Armstrong's axioms are a set of axioms (or, more precisely, inference rules) used to infer all the functional dependencies on a relational database. They were developed by William W. Armstrong in his 1974 paper. The axioms are sound in generating only functional dependencies in the closure of a set of functional dependencies (denoted as F^) when applied to that set (denoted as F). They are also complete in that repeated application of these rules will generate all functional dependencies in the closure F^+. More formally, let \langle R(U), F \rangle denote a relational scheme over the set of attributes U with a set of functional dependencies F. We say that a functional dependency f is logically implied by F, and denote it with F \models f if and only if for every instance r of R that satisfies the functional dependencies in F, r also satisfies f. We denote by F^ the set of all functional dependencies that are logically implied by F. Furthermore, with respect to a set of inferen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implications From Single Attribute To Single Attribute
Implication may refer to: Logic * Logical consequence (also entailment or logical implication), the relationship between statements that holds true when one logically "follows from" one or more others * Material conditional (also material implication), a logical connective and binary truth function typically interpreted as "If ''p'', then ''q''" ** Material implication (rule of inference), a logical rule of replacement ** Implicational propositional calculus, a version of classical propositional calculus that uses only the material conditional connective * Strict conditional or strict implication, a connective of modal logic that expresses necessity * ''modus ponens'', or implication elimination, a simple argument form and rule of inference summarized as "''p'' implies ''q''; ''p'' is asserted to be true, so therefore ''q'' must be true" Linguistics * Implicature, what is suggested in an utterance, even though neither expressed nor strictly implied * Implicational universal o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjunctive Normal Form
In Boolean algebra, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs. In automated theorem proving, the notion "''clausal normal form''" is often used in a narrower sense, meaning a particular representation of a CNF formula as a set of sets of literals. Definition A logical formula is considered to be in CNF if it is a conjunction of one or more disjunctions of one or more literals. As in disjunctive normal form (DNF), the only propositional operators in CNF are or (\vee), and (\and), and not (\neg). The ''not'' operator can only be used as part of a literal, which means that it can only precede a propositional variable. The following is a context-free grammar for CNF: : ''CNF'' \, \to \, ''Disjunct'' \, \mid \, ''Disjunct'' \, \land \, ''CNF'' : ''Disjunct'' \, \to \, ''Literal'' \, \mid\, ''Literal'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as well as the reason of the notation denoting the power set are demonstrated in the below. : An indicator function or a characteristic function of a subset of a set with the cardinality is a function from to the two-element set , denoted as , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse Diagram
In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set (S,\le) one represents each element of S as a vertex in the plane and draws a line segment or curve that goes ''upward'' from one vertex x to another vertex y whenever y covers x (that is, whenever x\ne y, x\le y and there is no z distinct from x and y with x\le z\le y). These curves may cross each other but must not touch any vertices other than their endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order. Hasse diagrams are named after Helmut Hasse (1898–1979); according to Garrett Birkhoff, they are so called because of the effective use Hasse made of them. However, Hasse was not the first to use these diagrams. One example that predates Hasse can be found in an 1895 work by Henri Gustave Vogt. Although Hasse diagrams were orig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum ( join) and an infimum ( meet). A conditionally complete lattice satisfies at least one of these properties for bounded subsets. For comparison, in a general lattice, only ''pairs'' of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete. Complete lattices appear in many applications in mathematics and computer science. Both order theory and universal algebra study them as a special class of lattices. Complete lattices must not be confused with complete partial orders (CPOs), a more general class of partially ordered sets. More specific complete lattices are complete Boolean algebras and complete Heyting algebras (locales). Formal definition A ''complete lattice'' is a partially ordered set (''L'', ≤) such that every subset ''A'' of ''L'' has both a greatest lower bound (the infimum, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
