By R. Lowen
The objective of this booklet is to supply the reader who's attracted to functions of fuzzy set conception, within the first position with a textual content to which she or he can refer for the fundamental theoretical rules, recommendations and strategies during this box and within the moment position with an enormous and recent account of the literature. even supposing there are actually many books approximately fuzzy set thought, and in general approximately its purposes, e. g. on top of things idea, there isn't fairly a publication on hand which introduces the hassle-free thought of fuzzy units, in what i want to name "a strong measure of generality". to jot down a booklet which might deal with the full variety of effects in regards to the easy theoretical ideas in nice aspect and which might additionally take care of all attainable versions and choices of the speculation, similar to e. g. tough units and L-fuzzy units for arbitrary lattices L, with the possibility-probability theories and interpretations, with the root of fuzzy set idea through multi-valued good judgment or through specific equipment etc, could were an altogether varied undertaking. This publication is way extra modest in its mathematical content material and in its scope.
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Extra info for Fuzzy Set Theory: Basic Concepts, Techniques and Bibliography
With range the unit interval. However there does exist a generalization of the original eoneept defined by Zadeh, where the range is replaeed by a more general lattice. We shall not be eoneerned with this type of generalization in this work. 3 Examples I. A "probabilistic" example. This example uses some eoneepts from prob ability theory. g. P. Billingsley "Convergenee of probability measures" [Wiley and Sons, 1968]. If X is an arbitrary measurable spaee, then we let M (X) stands for the set of all probability measures on X.
The idea of using a-levels of fuzzy sets goes back to the very beginning of fuzzy set theory. In all possible applications and calculations it has proved to be an interesting way of looking at things and a useful tool for calculations. We refer to an interesting paper of De Baets and Kerre [FSS 62, 1994, 295-309] where the use of a-Ievels is clearly demonstrated. We also refer to the section dealing with Lagrange interpolation for an example of the effective use of a-levels in the solution of a problem.
Let Bx and By stand for the measurable sets in X and Y respectively. Then f* : M(X) --+ M(Y) : P --+ f*(P) is defined by letting VP E M(X), VB E By : f*(P)(B) ~ P(J-l(B)). If Bx contains the singletons then we can embed the original set X into M(X) by X --+ M(X) : x --+ PX ' Thus we see that we can interpret M(X), as an extension of X. The points of X, or the Dirac measures, are deterministic points, and the points of M (X) \ X are points with uncertainty. For instance, the height of man is not areal number.