Fuzzy mathematics
Encyclopedia
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:
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),B(x)). 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)).
. Hundreds of papers on related topics have been published. Recent results and references can be found in
,
.
Main results in fuzzy fields and fuzzy Galois theory are published in a 1998 paper.
Fuzzy topology was introduced by C.L. Chang in 1968 and further was studied in many papers.
Main concepts of fuzzy geometry were introduced by A. Rosenfeld in 1974 and by J.J. Buckley and E. Eslami in 1997.
Basic types of fuzzy relations were introduced by Zadeh in 1971.
The properties of fuzzy graphs have been studied by A. Kaufman, A. Rosenfeld, and by R.T. Yeh and S.Y. Bang. Recent results can be found in a 2000 article.
Possibility theory
, nonadditive measures, fuzzy measure theory
and fuzzy integrals are studied in the cited articles and treatises.
Main results and references on formal fuzzy logic can be found in these citations.
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
. It started in 1965 after the publication of Lotfi Asker Zadeh
Lotfi Asker Zadeh
Lotfali Askar Zadeh , better known as Lotfi A. Zadeh, is a mathematician, electrical engineer, computer scientist, artifical intelligence researcher and professor emeritus of computer science at the University of California, Berkeley...
'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
Characteristic function
In mathematics, characteristic function can refer to any of several distinct concepts:* The most common and universal usage is as a synonym for indicator function, that is the function* In probability theory, the characteristic function of any probability distribution on the real line is given by...
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
T-norm
In mathematics, a t-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...
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),B(x)). 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)).
Some fields of mathematics using fuzzy set theory
Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld. Hundreds of papers on related topics have been published. Recent results and references can be found in
,
.
Main results in fuzzy fields and fuzzy Galois theory are published in a 1998 paper.
Fuzzy topology was introduced by C.L. Chang in 1968 and further was studied in many papers.
Main concepts of fuzzy geometry were introduced by A. Rosenfeld in 1974 and by J.J. Buckley and E. Eslami in 1997.
Basic types of fuzzy relations were introduced by Zadeh in 1971.
The properties of fuzzy graphs have been studied by A. Kaufman, A. Rosenfeld, and by R.T. Yeh and S.Y. Bang. Recent results can be found in a 2000 article.
Possibility theory
Possibility theory
Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. D. Dubois and H. Prade further...
, nonadditive measures, fuzzy measure theory
Fuzzy measure theory
Fuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures, fuzzy set membership function and the classical probability measures...
and fuzzy integrals are studied in the cited articles and treatises.
Main results and references on formal fuzzy logic can be found in these citations.
See also
- Fuzzy measure theoryFuzzy measure theoryFuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures, fuzzy set membership function and the classical probability measures...
- Fuzzy subalgebraFuzzy subalgebraFuzzy 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. Indeed, consider a first order language for algebraic structures with a monadic predicate...
- InrecoLan FuzzyMathInrecoLan FuzzyMath- Overview :InrecoLAN FuzzyMath is a fuzzy logic add-in for OpenOffice.org Calc.InrecoLAN FuzzyMath allows to use uncertain or approximate values in OpenOffice.org Calc...
- Monoidal t-norm logicMonoidal t-norm logicMonoidal t-norm based logic , the logic of left-continuous t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic of commutative bounded integral residuated lattices by the axiom of...
- Possibility theoryPossibility theoryPossibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. D. Dubois and H. Prade further...
- T-normT-normIn mathematics, a t-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...
External links
- Zadeh, L.A. Fuzzy Logic - article at ScholarpediaScholarpediaScholarpedia is an English-language online wiki-based encyclopedia that uses the same MediaWiki software as Wikipedia, but has features more commonly associated with open-access online academic journals....
- Hajek, P. Fuzzy Logic - article at Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
- Navara, M. Triangular Norms and Conorms - article at ScholarpediaScholarpediaScholarpedia is an English-language online wiki-based encyclopedia that uses the same MediaWiki software as Wikipedia, but has features more commonly associated with open-access online academic journals....
- Dubois, D., Prade H. Possibility Theory - article at ScholarpediaScholarpediaScholarpedia is an English-language online wiki-based encyclopedia that uses the same MediaWiki software as Wikipedia, but has features more commonly associated with open-access online academic journals....
- Center for Mathematics of Uncertainty Fuzzy Math Research - Web site hosted at Creighton University