By Xiaodong Liu, Witold Pedrycz

In the age of computing device Intelligence and automatic choice making, we need to take care of subjective imprecision inherently linked to human belief and defined in ordinary language and uncertainty captured within the kind of randomness. This treatise develops the basics and method of Axiomatic Fuzzy units (AFS), within which fuzzy units and likelihood are handled in a unified and coherent model. It deals an effective framework that bridges genuine international issues of summary constructs of arithmetic and human interpretation services forged within the surroundings of fuzzy sets.

In the self-contained quantity, the reader is uncovered to the AFS being taken care of not just as a rigorous mathematical idea but in addition as a versatile improvement technique for the advance of clever systems.

The approach within which the speculation is uncovered is helping show and rigidity linkages among the basics and well-delineated and sound layout practices of sensible relevance. The algorithms being provided in an in depth demeanour are conscientiously illustrated via numeric examples to be had within the realm of layout and research of data systems.

The fabric are available both effective to the readers occupied with the idea and perform of fuzzy units in addition to these drawn to arithmetic, tough units, granular computing, formal idea research, and using probabilistic tools.

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C. Inductive Condition: For any property ε , if (1) Every minimal element (if it exists) has property ε , (2) For ∀a, x ∈ S, x < a, x has property ε ⇒ a has property ε . Then every element in S has also property ε . The duality hold; we have A . Maximal Condition: Every non-empty subset of S must have maximal elements. B . Descending Chain Condition: For every sequence of elements {ai | i = 1, 2, . }, if a1 ≤ a2 ≤ . . ≤ an ≤ . . , then there exists a positive integer m such that am = am+n , n = 1, 2, .

Let E be the real plane, and define for ε > 0, Sε (x, y) = {(u, v) | (u, v) ∈ E, |x − u| < ε } and U(x,y) = {U | U ⊇ Sε (x, y) for some ε > 0}. 13 we can verify that U(x,y) is a neighborhood system for each (x, y) ∈ E. 12. Let {(xn , yn )} be a sequence in E, with the topology, T , and let (x0 , y0 ) be a limit of the sequence {(xn , yn )}, then (x0 , z) for any z is also a limit of the sequence {(xn , yn )}, and observe thus that limits of sequence need not be unique. We have noticed that, in general, a net in a topological space may converging to several different points.

E. C) is satisfied. Let M = {a | a ∈ S and a has no property ε }, 20 1 Fundamentals then M ⊆ S. If M = ∅, there exists a minimal element a ∈ M by A, but a is not a minimal element in S from the premises of inductive condition. However, if x < a and x ∈ P, then x ∈ / M and has property ε , consequently, a has also property ε from the premise (2) of inductive condition. This contradicts that a ∈ M and M = ∅. e. C) holds. C⇒B. e. C). Definite: a(a ∈ S) has property ε if and only if for every descending chain a = a1 ≥ a2 ≥.