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- "Fuzzy math" redirects here. For the controversies about mathematics education curricula that are sometimes disparaged as "fuzzy math," see Math wars. 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 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 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,B), (A ∪ B)(x) = max(A,B) 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,B). 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) ≥ min(A,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(x,z) ≥ min(R,R). For the controversies about mathematics education curricula that are sometimes disparaged as "fuzzy math," see Math wars.
- 模糊数学，亦称弗晰数学或模糊性数学。1965年以后，在模糊集合、模糊逻辑的基础上发展起来的模糊拓扑、模糊测度论等数学领域的统称。是研究现实世界中许多界限不分明甚至是很模糊的问题的数学工具。在模式识别、人工智能等方面有广泛的应用。
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