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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] |
Crisp Set
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] |
Operation (mathematics)
In mathematics, an operation is a function which takes zero or more input values (also called "''operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations (i.e., operations of arity 2), such as addition and multiplication, and unary operations (i.e., operations of arity 1), such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is a constant. The mixed product is an example of an operation of arity 3, also called ternary operation. Generally, the arity is taken to be finite. However, infinitary operations are sometimes considered, in which case the "usual" operations of finite arity are called finitary operations. A partial operation is defined similarly to an operation, but with a partial function in place of a function. Types of operation There are two common types of operations: unary and binar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Sets
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] |
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 t-nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x \mapsto -x), reciprocation (x \mapsto 1/x), and complex conjugation (z \mapsto \bar z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition of two involutions ''f'' and ''g'' is an involution if and only if they commute: . Involutions on finite sets The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: :a_0 = a_1 = 1 and a_n = a_ + ... [...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] |
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] |
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] |
De Morgan Algebra
__NOTOC__ In mathematics, a De Morgan algebra (named after Augustus De Morgan, a British mathematician and logician) is a structure ''A'' = (A, ∨, ∧, 0, 1, ¬) such that: * (''A'', ∨, ∧, 0, 1) is a bounded distributive lattice, and * ¬ is a De Morgan involution: ¬(''x'' ∧ ''y'') = ¬''x'' ∨ ¬''y'' and ¬¬''x'' = ''x''. (i.e. an involution that additionally satisfies De Morgan's laws) In a De Morgan algebra, the laws * ¬''x'' ∨ ''x'' = 1 (law of the excluded middle), and * ¬''x'' ∧ ''x'' = 0 (law of noncontradiction) do not always hold. In the presence of the De Morgan laws, either law implies the other, and an algebra which satisfies them becomes a Boolean algebra. Remark: It follows that ¬(x ∨ y) = ¬x ∧ ¬y, ¬1 = 0 and ¬0 = 1 (e.g. ¬1 = ¬1 ∨ 0 = ¬1 ∨ ¬¬0 = ¬(1 ∧ ¬0) = ¬¬0 = 0). Thus ¬ is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
