Fuzzy mathematics


Fuzzy mathematics forms a branch of mathematics including fuzzy set theory and fuzzy logic. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work Fuzzy sets.

Definition

A fuzzy subset A of a set X is a function A:X→L, where L is the interval . 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 =. More generally, one can use a complete lattice L in a definition of a fuzzy subset A

Fuzzification

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 member functions. Let A and B be two fuzzy subsets of X.
Intersection A ∩ B and union A ∪ B are defined as follows: = min,B), = max,B) for all x ∈ X. Instead of min and max one can use t-norm and t-conorm, respectively
, for example, min 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 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 ≥ min,A). Let 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 ≥ min,A).
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 ≥ min,R).

Fuzzy analogues

Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld
Analogues of other mathematical subjects have been translated to fuzzy mathematics, such as fuzzy field theory and fuzzy Galois theory, fuzzy topology, fuzzy geometry, fuzzy orderings, and fuzzy graphs.