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Reflexive Closure
In mathematics, the reflexive closure of a binary relation ''R'' on a set ''X'' is the smallest reflexive relation on ''X'' that contains ''R''. For example, if ''X'' is a set of distinct numbers and ''x R y'' means "''x'' is less than ''y''", then the reflexive closure of ''R'' is the relation "''x'' is less than or equal to ''y''". Definition The reflexive closure ''S'' of a relation ''R'' on a set ''X'' is given by :S = R \cup \left\ In English, the reflexive closure of ''R'' is the union of ''R'' with the identity relation on ''X''. Example As an example, if :X = \left\ :R = \left\ then the relation R is already reflexive by itself, so it does not differ from its reflexive closure. However, if any of the pairs in R was absent, it would be inserted for the reflexive closure. For example, if on the same set X :R = \left\ then the reflexive closure is :S = R \cup \left\ = \left\ . See also * Transitive closure * Symmetric closure References * Franz Baader and Tobi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elements in and in . It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element is ''related'' to an element , if and only if the pair belongs to the set of ordered pairs that defines the ''binary relation''. A binary relation is the most studied special case of an Finitary relation, -ary relation over sets , which is a subset of the Cartesian product X_1 \times \cdots \times X_n. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime is related to each integer that is a Divisibility, multiple of , but not to an integer that is not a multiple of . In this relation, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Relation
In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations. Definitions Let R be a binary relation on a set X, which by definition is just a subset of X \times X. For any x, y \in X, the notation x R y means that (x, y) \in R while "not x R y" means that (x, y) \not\in R. The relation R is called if x R x for every x \in X or equivalently, if \operatorname_X \subseteq R where \operatorname_X := \ denotes the identity relation on X. The of R is the union R \cup \operatorname_X, which can equivalently be defined as the smallest (with respect to \subseteq) reflexive relation on X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Relation
In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation over ''X''". An example of a homogeneous relation is the relation of kinship, where the relation is over people. Common types of endorelations include orders, graphs, and equivalences. Specialized studies order theory and graph theory have developed understanding of endorelations. Terminology particular for graph theory is used for description, with an ordinary graph presumed to correspond to a symmetric relation, and a general endorelation corresponding to a directed graph. An endorelation ''R'' corresponds to a logical matrix of 0s and 1s, where the expression ''xRy'' corresponds to an edge between ''x'' and ''y'' in the graph, and to a 1 in the square matrix of ''R''. It is called an adjacency matrix in graph terminology. Particular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Closure
In mathematics, the transitive closure of a binary relation on a set is the smallest relation on that contains and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets it is the unique minimal transitive superset of . For example, if is a set of airports and means "there is a direct flight from airport to airport " (for and in ), then the transitive closure of on is the relation such that means "it is possible to fly from to in one or more flights". Informally, the ''transitive closure'' gives you the set of all places you can get to from any starting place. More formally, the transitive closure of a binary relation on a set is the transitive relation on set such that contains and is minimal; see . If the binary relation itself is transitive, then the transitive closure is that same binary relation; otherwise, the transitive closure is a different relation. Conversely, transitive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Closure
In mathematics, the symmetric closure of a binary relation R on a set X is the smallest symmetric relation on X that contains R. For example, if X is a set of airports and xRy means "there is a direct flight from airport x to airport y", then the symmetric closure of R is the relation "there is a direct flight either from x to y or from y to x". Or, if X is the set of humans and R is the relation 'parent of', then the symmetric closure of R is the relation "x is a parent or a child of y". Definition The symmetric closure S of a relation R on a set X is given by S = R \cup \. In other words, the symmetric closure of R is the union of R with its converse relation, R^. See also * * {{annotated link, Reflexive closure References * Franz Baader and Tobias Nipkow Tobias Nipkow (born 1958) is a German computer scientist. Career Nipkow received his Diplom (MSc) in computer science from the Department of Computer Science of the Technische Hochschule Darmstadt in 1982, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Franz Baader
Franz Baader (15 June 1959, Spalt) is a German computer scientist at Dresden University of Technology. He received his PhD in Computer Science in 1989 from the University of Erlangen-Nuremberg, Germany, where he was a teaching and research assistant for 4 years. In 1989, he went to the German Research Centre for Artificial Intelligence (DFKI) as a senior researcher and project leader. In 1993 he became associate professor for computer science at RWTH Aachen, and in 2002 full professor for computer science at TU Dresden. He received the Herbrand Award The Herbrand Award for Distinguished Contributions to Automated Reasoning is an award given by the Conference on Automated Deduction (CADE), Inc., (although it predates the formal incorporation of CADE) to honour persons or groups for important cont ... for the year 2020 "in recognition of his significant contributions to unification theory, combinations of theories and reasoning in description logics". Works * * * * References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tobias Nipkow
Tobias Nipkow (born 1958) is a German computer scientist. Career Nipkow received his Diplom (MSc) in computer science from the Department of Computer Science of the Technische Hochschule Darmstadt in 1982, and his Ph.D. from the University of Manchester in 1987. He worked at MIT from 1987, changed to Cambridge University in 1989, and to Technical University Munich in 1992, where he was appointed professor for programming theory. He is chair of the Logic and Verification group since 2011. He is known for his work in interactive and automatic theorem proving, in particular for the Isabelle proof assistant; he was the editor of the '' Journal of Automated Reasoning'' up to January 1, 2021. Moreover, he focuses on programming language semantics, type systems and functional programming In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Relations
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in . It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element is ''related'' to an element , if and only if the pair belongs to the set of ordered pairs that defines the ''binary relation''. A binary relation is the most studied special case of an -ary relation over sets , which is a subset of the Cartesian product X_1 \times \cdots \times X_n. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime is related to each integer that is a multiple of , but not to an integer that is not a multiple of . In this relation, for instance, the prime number 2 is related to number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure Operators
Closure may refer to: Conceptual Psychology * Closure (psychology), the state of experiencing an emotional conclusion to a difficult life event Computer science * Closure (computer programming), an abstraction binding a function to its scope * Relational database model: Set-theoretic formulation and Armstrong's axioms for its use in database theory Mathematics * Closure (mathematics), the result of applying a closure operator * Closure (topology), for a set, the smallest closed set containing that set Philosophy * Epistemic closure, a principle in epistemology * Deductive closure, a principle in logic * Cognitive closure, a principle in philosophy of mind * ''Closure: A Short History of Everything'', a philosophical book by Hilary Lawson Sociology * Closure (sociology) * Closure, a concept in the social construction of technology Physical objects * Closure (container) used to seal a bottle, jug, jar, can, or other container ** Closure (wine bottle), a stopper * Hook-and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
