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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 C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crisp Set
Crisp may refer to: Foods * Potato chips, known in British English as a potato crisp, a thin slice of a potato deep fried or baked until crispy * Chip (snack), or crisp in British English * Crisp (dessert), a type of American dessert, usually including fruit * Crisp (chocolate bar), a Nestlé brand of wafer candy sold in the United States * Simit, in Ottoman cuisine, a circular bread Places United States * Crisp, North Carolina, an unincorporated community * Crisp, Texas, a ghost town * Crisp, West Virginia, an unincorporated community * Crisp County, Georgia Antarctica * Crisp Glacier, Victoria Land People * Crisp (surname), a list of people * Crisp Gascoyne (1700–1761), English businessman and Lord Mayor of London * Crisp Molineux (1730-1792), English Member of Parliament * Michael Crisp (1950-), Australian botanist whose standard author abbreviation is Crisp Other uses * Crisp (horse), famous for the 1973 Grand National * Crisp baronets, a title in the Baronetage of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operation (mathematics)
In mathematics, an operation is a function from a set to itself. For example, an operation on real numbers will take in real numbers and return a real number. An operation can take 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 operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Sets
Fuzzy or Fuzzies may refer to: Music * Fuzzy (band), a 1990s Boston indie pop band * Fuzzy (composer), Danish composer Jens Vilhelm Pedersen (born 1939) * Fuzzy (album), ''Fuzzy'' (album), 1993 debut album of American rock band Grant Lee Buffalo * "Fuzzy", a song from the 2009 ''Collective Soul (2009 album), Collective Soul'' album by Collective Soul * "Fuzzy", a song from ''Poppy.Computer'', the debut 2017 album by Poppy * Fuzzy, an Australian events company that organises Listen Out, a multi-city Australian music festival Nickname * Faustina Agolley (born 1984), Australian television presenter, host of the Australian television show ''Video Hits'' * Fuzzy Haskins (1941–2023), American singer and guitarist with the doo-wop group Parliament-Funkadelic * Fuzzy Hufft (1901−1973), American baseball player * Fuzzy Knight (1901−1976), American actor * Andrew Levane (1920−2012), American National Basketball Association player and coach * Robert Alfred Theobald (1884−1957), Uni ... [...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 C ... [...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 (set theory), intersection in a lattice (order), lattice and logical conjunction, 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 (mathematics), function T: [0, 1] × [0, 1] → [0, 1] that 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 operatio ... [...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 (), reciprocation (), and complex conjugation () 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 and 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_ + (n - 1)a_ for n > 1. The first few terms of ... [...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 (set theory), intersection in a lattice (order), lattice and logical conjunction, 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 (mathematics), function T: [0, 1] × [0, 1] → [0, 1] that 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 operatio ... [...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 algebra, 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 mathematician Lotfi A. Zadeh, Lotfi Zadeh. Fuzzy logic had, however, been studied since the 1920s, as Łukasiewicz logic, infinite-valued logic—notably by Jan Łukasiewicz, Łukasiewicz and Alfred Tarski, Tarski. Fuzzy logic is based on the observation that people make decisions based on imprecise and non-numerical information. Fuzzy models or fuzzy sets are mathematical means of representing vagueness and imprecise information (hence the term fuzzy). These models have the capability of recognising, representing, manipulating, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Set
Fuzzy or Fuzzies may refer to: Music * Fuzzy (band), a 1990s Boston indie pop band * Fuzzy (composer), Danish composer Jens Vilhelm Pedersen (born 1939) * Fuzzy (album), ''Fuzzy'' (album), 1993 debut album of American rock band Grant Lee Buffalo * "Fuzzy", a song from the 2009 ''Collective Soul (2009 album), Collective Soul'' album by Collective Soul * "Fuzzy", a song from ''Poppy.Computer'', the debut 2017 album by Poppy * Fuzzy, an Australian events company that organises Listen Out, a multi-city Australian music festival Nickname * Faustina Agolley (born 1984), Australian television presenter, host of the Australian television show ''Video Hits'' * Fuzzy Haskins (1941–2023), American singer and guitarist with the doo-wop group Parliament-Funkadelic * Fuzzy Hufft (1901−1973), American baseball player * Fuzzy Knight (1901−1976), American actor * Andrew Levane (1920−2012), American National Basketball Association player and coach * Robert Alfred Theobald (1884−1957), Uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type-2 Fuzzy Sets And Systems
Type-2 fuzzy sets and systems generalize standard fuzzy sets, type-1 fuzzy sets and fuzzy sets and systems, systems so that more uncertainty can be handled. From the beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-1 fuzzy set has no uncertainty associated with it, something that seems to contradict the word ''fuzzy'', since that word has the connotation of much uncertainty. So, what does one do when there is uncertainty about the value of the membership function? The answer to this question was provided in 1975 by the inventor of fuzzy sets, Lotfi A. Zadeh,L. A. Zadeh, "The Concept of a Linguistic Variable and Its Application to Approximate Reasoning–1," ''Information Sciences'', vol. 8, pp. 199–249, 1975. when he proposed more sophisticated kinds of fuzzy sets, the first of which he called a "type-2 fuzzy set". A type-2 fuzzy set lets us incorporate uncertainty about the membership function into fuzzy set theory, and is a way ... [...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] |
