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Fuzzy Mathematics
Fuzzy mathematics is the branch of mathematics including fuzzy set theory and fuzzy logic that deals with partial inclusion of elements in a set on a spectrum, as opposed to simple binary "yes" or "no" (0 or 1) inclusion. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work ''Fuzzy sets''. Linguistics is an example of a field that utilizes fuzzy set theory. Definition A ''fuzzy subset'' ''A'' of a set ''X'' is a function ''A'': ''X'' → ''L'', where ''L'' is the interval , 1 This function is also called a membership function. A membership function is a generalization of an indicator function (also called a characteristic function) of a subset defined for ''L'' = . More generally, one can use any complete lattice ''L'' in a definition of a fuzzy subset ''A''. Fuzzification The evolution of the fuzzification of mathematical concepts can be broken down into three stages: :# straightforward fuzzification during the sixties and seventies, :# ... [...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 Operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary operation ''on a set'' is a binary operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, and multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. An operation of arity two that involves several sets is sometimes also called a ''binary operation''. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. Such binary operations may be called simply binary functions. Binary operations are the keystone of most algebraic structures that are studie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scholarpedia
''Scholarpedia'' is an English-language wiki-based online encyclopedia with features commonly associated with open-access online academic journals, which aims to have quality content in science and medicine. ''Scholarpedia'' articles are written by invited or approved expert authors and are subject to peer review. ''Scholarpedia'' lists the real names and affiliations of all authors, curators and editors involved in an article: however, the peer review process (which can suggest changes or additions, and has to be satisfied before an article can appear) is anonymous. ''Scholarpedia'' articles are stored in an online repository, and can be cited as conventional journal articles (''Scholarpedia'' has the ISSN number ). ''Scholarpedia''s citation system includes support for revision numbers. The project was created in February 2006 by Eugene M. Izhikevich, while he was a researcher at the Neurosciences Institute, San Diego, California. Izhikevich is also the encyclopedia's editor-i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T-norm
In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic. The name ''triangular norm'' refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize the triangle inequality of ordinary metric spaces. Definition A t-norm is a function T: , 1× , 1→ , 1that satisfies the following properties: * Commutativity: T(''a'', ''b'') = T(''b'', ''a'') * Monotonicity: T(''a'', ''b'') ≤ T(''c'', ''d'') if ''a'' ≤ ''c'' and ''b'' ≤ ''d'' * Associativity: T(''a'', T(''b'', ''c'')) = T(T(''a'', ''b''), ''c'') * The number 1 acts as identity element: T(''a'', 1) = ''a'' Since a t-norm is a binary algebraic operation on the interval , 1 infix algebraic notation is also common, with the t-nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Possibility Theory
Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessary to necessary, respectively. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. Didier Dubois and Henri Prade further contributed to its development. Earlier in the 1950s, economist G. L. S. Shackle proposed the min/max algebra to describe degrees of potential surprise. Formalization of possibility For simplicity, assume that the universe of discourse Ω is a finite set. A possibility measure is a function \operatorname from 2^\Omega to , 1such that: :Axiom 1: \operatorname(\varnothing) = 0 :Axiom 2: \operatorname(\Omega) = 1 :Axiom 3: \operatorname(U \cup V) = \max \left( \operatorname(U), \operatorname(V) \right) for any disjoint subsets U and V. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal T-norm Logic
In mathematical logic, monoidal t-norm based logic (or shortly MTL), the logic of left-continuous t-norms, is one of the t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices;Ono (2003). it extends the logic of commutative bounded integral residuated lattices (known as Höhle's monoidal logic, Ono's FLew, or intuitionistic logic without contraction) by the axiom of prelinearity. Motivation In fuzzy logic, rather than regarding statements as being either true or false, we associate each statement with a numerical ''confidence'' in that statement. By convention the confidences range over the unit interval ,1/math>, where the maximal confidence 1 corresponds to the classical concept of true and the minimal confidence 0 corresponds to the classical concept of false. T-norms are binary functions on the real unit interval , 1 which in fuzzy logic are often used to represent a conjunction connective; if a,b \in ,1/math> a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Subalgebra
Fuzzy subalgebras theory is a chapter of fuzzy set theory. It is obtained from an interpretation in a multi-valued logic of axioms usually expressing the notion of subalgebra of a given algebraic structure. Definition Consider a first order language for algebraic structures with a monadic predicate symbol S. Then a ''fuzzy subalgebra'' is a fuzzy model of a theory containing, for any ''n''-ary operation h, the axioms \forall x_1, ..., \forall x_n (S(x_1) \land ..... \land S(x_n) \rightarrow S(h(x_1, ..., x_n)) and, for any constant c, S(c). The first axiom expresses the closure of S with respect to the operation h, and the second expresses the fact that c is an element in S. As an example, assume that the valuation structure is defined in ,1and denote by \odot the operation in ,1used to interpret the conjunction. Then a fuzzy subalgebra of an algebraic structure whose domain is D is defined by a fuzzy subset of D such that, for every d1,...,dn in D, if h is the interp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Measure Theory
In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also ''capacity'', see ), which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals. There exists a number of different classes of fuzzy measures including plausibility/belief measures; possibility/necessity measures; and probability measures, which are a subset of classical measures. Definitions Let \mathbf be a universe of discourse, \mathcal be a class of subsets of \mathbf, and E,F\in\mathcal. A function g:\mathcal\to\mathbb where # \emptyset \in \mathcal \Rightarrow g(\emptyset)=0 # E \subseteq F \Rightarrow g(E)\leq g(F) is called a ''fuzzy measure''. A fuzzy measure is called ''normalized'' or ''regular'' if g(\mathbf)=1. Properties of fuzzy measures A fuzzy measure is: * additive if for an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inform
Inform is a programming language and design system for interactive fiction originally created in 1993 by Graham Nelson. Inform can generate programs designed for the Z-code or Glulx virtual machines. Versions 1 through 5 were released between 1993 and 1996. Around 1996, Nelson rewrote Inform from first principles to create version 6 (or Inform 6). Over the following decade, version 6 became reasonably stable and a popular language for writing interactive fiction. In 2006, Nelson released Inform 7 (briefly known as Natural Inform), a completely new language based on principles of natural language and a new set of tools based around a book-publishing metaphor. Z-Machine and Glulx The Inform compilers translate Inform code to story files for Glulx or Z-code, two virtual machines designed specifically for interactive fiction. Glulx, which can support larger games, is the default. The Z-machine was originally developed by Infocom in 1979 for their interactive fiction titles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liu Yingming
Liu Yingming (; 8 October 1940 – 15 July 2016) was a Chinese mathematician. He was an academician of the Chinese Academy of Sciences (CAS). Biography Liu graduated from Peking University in 1963, majoring in mathematics. He was assigned to Sichuan University after his graduation. His research field was topology and fuzzy mathematics, mainly in the algebra problem of unclear topology, embedded theory and nonclear convex sets. Liu was the deputy president of Sichuan University between 1989 and 2005. He was elected an academician of the Chinese Academy of Sciences in 1995. Liu joined Jiusan Society in 1995. He was the vice chairman of the central committee of Jiusan Society from 1997 to 2007. Liu was diagnosed as leukemia in November 2015. He died on 15 July 2016 at the age of 75 in Chengdu Chengdu (, ; simplified Chinese: 成都; pinyin: ''Chéngdū''; Sichuanese pronunciation: , Standard Chinese pronunciation: ), alternatively romanized as Chengtu, is a sub-prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Relation
In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive. Definition A homogeneous relation on the set is a ''transitive relation'' if, :for all , if and , then . Or in terms of first-order logic: :\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc, where is the infix notation for . Examples As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy, too, is an ancestor of Carrie. On the other hand, "is the birth parent of" is not a transitive relation, because if Alice is the birth parent of Brenda, and Brenda is the birth parent of Claire, then this does not imply that Alice is the birth parent of Claire. What is more, it is antitransitive: Alice can ''never'' be the birth parent of Claire. "Is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
