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- Fuzzy_mathematics abstract "Fuzzy mathematics forms a branch of mathematics related to fuzzy set theory and fuzzy logic. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work Fuzzy sets. A fuzzy subset A of a set X is a function A:X→L, where L is the interval [0,1]. This function is also called a membership function. A membership function is a generalization of a characteristic function or an indicator function of a subset defined for L = {0,1}. More generally, one can use a complete lattice L in a definition of a fuzzy subset A.The evolution of the fuzzification of mathematical concepts can be broken down into three stages: straightforward fuzzification during the sixties and seventies, the explosion of the possible choices in the generalization process during the eighties, the standardization, axiomatization and L-fuzzification in the nineties.Usually, a fuzzification of mathematical concepts is based on a generalization of these concepts from characteristic functions to membership functions. Let A and B be two fuzzy subsets of X. Intersection A ∩ B and union A ∪ B are defined as follows: (A ∩ B)(x) = min(A(x),B(x)), (A ∪ B)(x) = max(A(x),B(x)) for all x ∈ X. Instead of min and max one can use t-norm and t-conorm, respectively , for example, min(a,b) can be replaced by multiplication ab. A straightforward fuzzification is usually based on min and max operations because in this case more properties of traditional mathematics can be extended to the fuzzy case. A very important generalization principle used in fuzzification of algebraic operations is a closure property. Let * be a binary operation on X. The closure property for a fuzzy subset A of X is that for all x,y ∈ X, A(x*y) ≥ min(A(x),A(y)). Let (G,*) be a group and A a fuzzy subset of G. Then A is a fuzzy subgroup of G if for all x,y in G, A(x*y−1) ≥ min(A(x),A(y−1)). A similar generalization principle is used, for example, for fuzzification of the transitivity property. Let R be a fuzzy relation in X, i.e. R is a fuzzy subset of X×X. Then R is transitive if for all x,y,z in X, R(x,z) ≥ min(R(x,y),R(y,z)).".
- Fuzzy_mathematics wikiPageExternalLink logic-fuzzy.
- Fuzzy_mathematics wikiPageExternalLink fuzzymath.
- Fuzzy_mathematics wikiPageExternalLink ref.php?cid=C84b541099.
- Fuzzy_mathematics wikiPageExternalLink Fuzzy_Logic.
- Fuzzy_mathematics wikiPageExternalLink Possibility_Theory.
- Fuzzy_mathematics wikiPageExternalLink Triangular_Norms_and_Conorms.
- Fuzzy_mathematics wikiPageExternalLink 978-3-540-71794-2.
- Fuzzy_mathematics wikiPageID "14075975".
- Fuzzy_mathematics wikiPageLength "8395".
- Fuzzy_mathematics wikiPageOutDegree "18".
- Fuzzy_mathematics wikiPageRevisionID "681412511".
- Fuzzy_mathematics wikiPageWikiLink Category:Fuzzy_logic.
- Fuzzy_mathematics wikiPageWikiLink Fuzzy_logic.
- Fuzzy_mathematics wikiPageWikiLink Fuzzy_measure_theory.
- Fuzzy_mathematics wikiPageWikiLink Fuzzy_set.
- Fuzzy_mathematics wikiPageWikiLink Fuzzy_subalgebra.
- Fuzzy_mathematics wikiPageWikiLink Indicator_function.
- Fuzzy_mathematics wikiPageWikiLink Lotfi_A._Zadeh.
- Fuzzy_mathematics wikiPageWikiLink Monoidal_t-norm_logic.
- Fuzzy_mathematics wikiPageWikiLink Possibility_theory.
- Fuzzy_mathematics wikiPageWikiLink Scholarpedia.
- Fuzzy_mathematics wikiPageWikiLink Stanford_Encyclopedia_of_Philosophy.
- Fuzzy_mathematics wikiPageWikiLink T-norm.
- Fuzzy_mathematics wikiPageWikiLinkText "Fuzzy mathematics".
- Fuzzy_mathematics wikiPageWikiLinkText "fuzzy math".
- Fuzzy_mathematics wikiPageWikiLinkText "fuzzy mathematics".
- Fuzzy_mathematics wikiPageWikiLinkText "fuzzy or non-fuzzy mathematics".
- Fuzzy_mathematics wikiPageWikiLinkText "fuzzy topology".
- Fuzzy_mathematics wikiPageWikiLinkText "fuzzy".
- Fuzzy_mathematics wikiPageUsesTemplate Template:Other_uses.
- Fuzzy_mathematics wikiPageUsesTemplate Template:Reflist.
- Fuzzy_mathematics wikiPageUsesTemplate Template:Technical.
- Fuzzy_mathematics subject Category:Fuzzy_logic.
- Fuzzy_mathematics type Redirect.
- Fuzzy_mathematics comment "Fuzzy mathematics forms a branch of mathematics related to fuzzy set theory and fuzzy logic. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work Fuzzy sets. A fuzzy subset A of a set X is a function A:X→L, where L is the interval [0,1]. This function is also called a membership function. A membership function is a generalization of a characteristic function or an indicator function of a subset defined for L = {0,1}.".
- Fuzzy_mathematics label "Fuzzy mathematics".
- Fuzzy_mathematics sameAs Q5511129.
- Fuzzy_mathematics sameAs m.03cst5h.
- Fuzzy_mathematics sameAs Q5511129.
- Fuzzy_mathematics sameAs 模糊数学.
- Fuzzy_mathematics wasDerivedFrom Fuzzy_mathematics?oldid=681412511.
- Fuzzy_mathematics isPrimaryTopicOf Fuzzy_mathematics.