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Implication (information Science)
In formal concept analysis (FCA) ''implications'' relate sets of properties (or, synonymously, of attributes). An implication ''A''→''B'' ''holds'' in a given domain when every object having all attributes in ''A'' also has all attributes in ''B''. Such implications characterize the concept hierarchy in an intuitive manner. Moreover, they are "well-behaved" with respect to algorithms. The knowledge acquisition method called ''attribute exploration'' uses implications.Ganter, Bernhard and Obiedkov, Sergei (2016) ''Conceptual Exploration''. Springer, Definitions An implication ''A''→''B'' is simply a pair of sets ''A''⊆''M'', ''B''⊆''M'', where ''M'' is the set of attributes under consideration. ''A'' is the ''premise'' and ''B'' is the ''conclusion'' of the implication ''A''→''B'' . A set C respects the implication ''A''→''B'' when ¬(''C''⊆''A'') or ''C''⊆''B''. A ''formal context'' is a triple ''(G,M,I)'', where ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure Operator
In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are determined by their closed sets, i.e., by the sets of the form cl(''X''), since the closure cl(''X'') of a set ''X'' is the smallest closed set containing ''X''. Such families of "closed sets" are sometimes called closure systems or "Moore families", in honor of E. H. Moore who studied closure operators in his 1910 ''Introduction to a form of general analysis'', whereas the concept of the closure of a subset originated in the work of Frigyes Riesz in connection with topological spaces. Though not formalized at the time, the idea of closure originated in the late 19th century with notable contributions by Ernst Schröder, Richard Dedekind and Georg Cantor. Closure operators are also called "hull operators", which prevents confusion with the "c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Armstrong Axioms
Armstrong's axioms are a set of references (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 Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ... 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 wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |