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Fuzzy Rule
Fuzzy rules are used within fuzzy logic systems to infer an output based on input variables. Modus ponens and modus tollens are the most important rules of inference. A modus ponens rule is in the form :Premise: ''x is A'' :Implication: ''IF x is A THEN y is B'' :Consequent: ''y is B'' In crisp logic, the premise ''x is A'' can only be true or false. However, in a fuzzy rule, the premise ''x is A'' and the consequent ''y is B'' can be true to a degree, instead of entirely true or entirely false. This is achieved by representing the linguistic variables ''A'' and ''B'' using fuzzy sets. In a fuzzy rule, modus ponens is extended to ''generalised modus ponens:.'' :Premise: ''x is A''* :Implication: ''IF x is A THEN y is B'' :Consequent: ''y is B''* The key difference is that the premise ''x is A'' can be only partially true. As a result, the consequent ''y is B'' is also partially true. Truth is represented as a real number between 0 and 1, where 0 is false and 1 is true. Compari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1. The term ''fuzzy logic'' was introduced with the 1965 proposal of fuzzy set theory by Iranian Azerbaijani mathematician Lotfi Zadeh. Fuzzy logic had, however, been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz and Tarski. Fuzzy logic is based on the observation that people make decisions based on imprecise and non-numerical information. Fuzzy models or sets are mathematical means of representing vagueness and imprecise information (hence the term fuzzy). These models have the capability of recognising, representing, manipulating, interpreting, and using data and information that are vague and lack ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modus Ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "''P implies Q.'' ''P'' is true. Therefore ''Q'' must also be true." ''Modus ponens'' is closely related to another valid form of argument, ''modus tollens''. Both have apparently similar but invalid forms such as affirming the consequent, denying the antecedent, and evidence of absence. Constructive dilemma is the disjunctive version of ''modus ponens''. Hypothetical syllogism is closely related to ''modus ponens'' and sometimes thought of as "double ''modus ponens''." The history of ''modus ponens'' goes back to antiquity. The first to explicitly describe the argument form ''modus ponens'' was Theophrastus. It, along with ''modus tollens'', is one of the standard patterns of inference that can be applied to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modus Tollens
In propositional logic, ''modus tollens'' () (MT), also known as ''modus tollendo tollens'' (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference. ''Modus tollens'' takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from ''P implies Q'' to ''the negation of Q implies the negation of P'' is a valid argument. The history of the inference rule ''modus tollens'' goes back to antiquity. The first to explicitly describe the argument form ''modus tollens'' was Theophrastus. ''Modus tollens'' is closely related to '' modus ponens''. There are two similar, but invalid, forms of argument: affirming the consequent and denying the antecedent. See also contraposition and proof by contrapositive. Explanation The form of a ''modus tollens'' argument resembles a syllog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Set
In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an ''L''-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of ''L''-relations when ''L'' is the unit interval , 1 In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition—an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval , 1 Fuzzy sets generali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Set Operations
Fuzzy set operations are a generalization of crisp set operations for fuzzy sets. There is in fact more than one possible generalization. The most widely used operations are called ''standard fuzzy set operations''; they comprise: fuzzy complements, fuzzy intersections, and fuzzy unions. Standard fuzzy set operations Let A and B be fuzzy sets that A,B ⊆ U, u is any element (e.g. value) in the U universe: u ∈ U. ;Standard complement :\mu_(u) = 1 - \mu_A(u) The complement is sometimes denoted by ∁A or A∁ instead of ¬A. ;Standard intersection :\mu_(u) = \min\ ;Standard union :\mu_(u) = \max\ In general, the triple (i,u,n) is called De Morgan Triplet iff * i is a t-norm, * u is a t-conorm (aka s-norm), * n is a strong negator, so that for all ''x'',''y'' ∈ , 1the following holds true: :''u''(''x'',''y'') = ''n''( ''i''( ''n''(''x''), ''n''(''y'') ) ) (generalized De Morgan relation).Ismat Beg, Samina AshrafSimilarity measures for fuzzy sets at: Applied and Compu ... [...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 ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : A^\complement = U \setminus A. Or formally: A^\complement = \. The absolute complement of is u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
