"Fuzzy math" redirects here. For the controversies about mathematics education curricula that are sometimes disparaged as "fuzzy math," see Math wars.

Fuzzy mathematics form a branch of mathematics related to fuzzy logic. It started in 1965 after publication by Lotfi Asker Zadeh of his seminal work Fuzzy sets ^[1] . A fuzzy subset A of a set X is a function A:X?L, where L is the interval [0,1]. This function 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 ^[2].

The evolution of the fuzzification of mathematical concepts can be broken into three stages ^[3] :

1. straightforward fuzzification during the sixties and seventies,

1. the explosion of the possible choices in the generalization process during the eighties,

1. 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 U B are defined as follows: (A ? B)(x) = min(A(x),B(x)), (A U B)(x) = max(A(x),B(x)) for all x \in X. Instead of min and max one can use t-norm and t-conorm, respectively ^[4], 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 \in 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 ^[5]. Hundreds of papers on related topics have been published. Recent results and references can be found in ^[6], ^[7].

Main results in fuzzy fields and fuzzy Galois theory are published in ^[8].

Fuzzy topology was introduced by C.L. Chang ^[9] in 1968 and further was studied in many papers ^[10].

Main concepts of fuzzy geometry were introduced by A. Rosenfeld in 1974 and by J.J. Buckley and E. Eslami in 1997 ^[11].

Basic types of fuzzy relations were introduced in ^[12].

The properties of fuzzy graphs have been studied by A. Kaufman ^[13], A. Rosenfeld ^[14] and by R.T. Yeh and S.Y. Bang ^[15]. Recent results can be found in ^[16].

Possibility theory, nonadditive measures, fuzzy measure theory and fuzzy integrals are studied in ^[17], ^[18], ^[19], ^[20], ^[21].

Main results and references on formal fuzzy logic can be found in ^[22], ^[23].

See also

• Fuzzy subalgebra
• t-norm
• Monoidal t-norm logic
• Possibility theory
• Fuzzy measure theory

References

External links

• Zadeh, L.A. Fuzzy Logic - article at Scholarpedia
• Hajek, P. Fuzzy Logic - article at Stanford Encyclopedia of Philosophy
• Navara, M. Triangular Norms and Conorms - article at Scholarpedia
• Dubois, D., Prade H. Possibility Theory - article at Scholarpedia

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