07 May 2018

Data Science: Fuzzy Set (Definitions)

"A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint." (Lotfi A Zadeh, "Fuzzy Sets", 1965)

"A fuzzy set can be defined mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set. This grade corresponds to the degree to which that individual is similar or compatible with the concept represented by the fuzzy set. Thus, individuals may belong in the fuzzy act to a greater or lesser degree as indicated by a larger or smaller membership grade. As already mentioned, these membership grades are very often represented by real-number values ranging in the closed interval between 0 and 1." (George J Klir & Bo Yuan, "Fuzzy Sets and Fuzzy Logic: Theory and Applications", 1995)

"A set of items whose degree of membership in the set may range from 0 to l." (Joseph P Bigus, "Data Mining with Neural Networks: Solving Business Problems from Application Development to Decision Support", 1996)

"A set whose members belong to it to some degree. In contrast, a standard or nonfuzzy set contains its members all or none. The set of even numbers has no fuzzy members." (Guido Deboeck & Teuvo Kohonen (Eds), "Visual Explorations in Finance with Self-Organizing Maps" 2nd Ed., 2000)

"A fuzzy set is any set that allows its members to have different grades of membership (membership function) in the interval [0,1]. A numerical value between 0 and 1 that represents the degree to which an element belongs to a particular set, also referred to as membership value." (Harish Garg,  "Predicting Uncertain Behavior and Performance Analysis of the Pulping System in a Paper Industry using PSO and Fuzzy Methodology", 2014)

"A set whose elements have degrees of membership, as opposed to a classical set." (Michael Mutingi et al, Fuzzy System Dynamics of Manpower Systems, 2014)

"Any set that allows its members to have different grades of membership (membership function) in the interval [0,1]. A numerical value between 0 and 1 that represents the degree to which an element belongs to a particular set, also referred to as membership value." (Harish Garg, "A Hybrid GA-GSA Algorithm for Optimizing the Performance of an Industrial System by Utilizing Uncertain Data", 2015)

"It is a set of elements which have no strict boundaries." (Alexander P Ryjov & Igor F Mikhalevich, "Hybrid Intelligence Framework for Improvement of Information Security of Critical Infrastructures", 2021)

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