By Prof. Dr. Maciej Wygralak (auth.)

ISBN-10: 3540363823

ISBN-13: 9783540363828

ISBN-10: 3642535143

ISBN-13: 9783642535147

Counting is without doubt one of the easy undemanding mathematical actions. It comes with complementary points: to figure out the variety of components of a collection - and to create an ordering among the gadgets of counting simply by counting them over. For finite units of gadgets those elements are discovered by way of an analogous form of num­ bers: the common numbers. That those complementary features of the counting seasoned­ cess might have other kinds of numbers turns into obvious if one extends the method of counting to countless units. As common instruments to figure out numbers of components the cardinals were created in set conception, and set theorists have in parallel created the ordinals to count number over any set of gadgets. For either kinds of numbers it isn't in simple terms counting they're used for, it's also the strongly comparable means of calculation - particularly addition and, derived from it, multiplication or even exponentiation - that is dependent upon those numbers. For fuzzy units the assumption of counting, in either elements, looses its naive origin: since it is to a wide volume based upon of the concept that there's a transparent distinc­ tion among these gadgets that have to remember - and people ones that have to be overlooked for the actual counting process.

Show description

Read or Download Cardinalities of Fuzzy Sets PDF

Similar operations research books

Download e-book for kindle: Regression Analysis Under A Priori Parameter Restrictions by Pavel S. Knopov, Arnold S. Korkhin

This monograph makes a speciality of the development of regression versions with linear and non-linear constrain inequalities from the theoretical viewpoint. not like earlier guides, this quantity analyses the houses of regression with inequality constrains, investigating the flexibleness of inequality constrains and their skill to conform within the presence of extra a priori details The implementation of inequality constrains improves the accuracy of types, and reduces the chance of error.

Read e-book online Multi-objective Management in Freight Logistics: Increasing PDF

The complexity of contemporary offer chains calls for selection makers in logistics to paintings with a suite of effective (Pareto optimum) recommendations, customarily to trap diverse financial points for which one optimum answer with regards to a unmarried goal functionality isn't really capable of catch solely. encouraged by way of this, and through fresh alterations in worldwide markets and the provision of latest transportation companies, Multi-objective administration in Freight Logistics offers an in depth learn of freight transportation structures, with a particular concentrate on multi-objective modeling.

Continuous-time Markov chains and applications : a - download pdf or read online

Prologue and Preliminaries: creation and review- Mathematical preliminaries. - Markovian types. - Two-Time-Scale Markov Chains: Asymptotic Expansions of options for ahead Equations. - profession Measures: Asymptotic houses and Ramification. - Asymptotic Expansions of suggestions for Backward Equations.

Get Flexible and Generalized Uncertainty Optimization: Theory PDF

This e-book provides the idea and strategies of versatile and generalized uncertainty optimization. really, it describes the idea of generalized uncertainty within the context of optimization modeling. The publication starts off with an summary of versatile and generalized uncertainty optimization. It covers uncertainties which are either linked to lack of knowledge and that extra normal than stochastic idea, the place well-defined distributions are assumed.

Extra info for Cardinalities of Fuzzy Sets

Sample text

41) generated by at-norm t and a negation v. 3. 43) Theory of generalized FGCounts, FLCounts and FECounts is presented in detail in Chapter 4, inc1uding questions of equipotency, ordering relations, and arithmetical operations. Chapter 3. 5. An axiomatization of scalar cardinalities is the starting point of our study. We like to investigate their properties, including the valuation property, the cartesian productrule and the complementarity rule. The question of the simultaneous fulfilment of these properties will also be discussed.

One then defines I A I := an ordinary cardinal number. Possible candidates for I A I are in this case I core(A) I, I supp(A) I and, more generally , with a suitable threshold point t (see GOTIWALD (1980), KAUFMANN (1977), WYGRALAK (1997a, 1997b». If AeFFS, another way of defining the cardinality of A in the scalar manner can be used besides the above nonnegative integers, namely we put I A I := a nonnegative real number. The first concepts of this kind was proposed and discussed in DE LUCAlTERMINI (1972, 1977, 1979).

E. a nonempty set of arbitrary cardinality, finite or not. Single bold italicized capitals A, B, C, ... will symbolize sets in M. e. ID: M~ {O,I} with ID(x):= 1 for xeD and ID(x) :=0 for xf1D. Sets as some objects in M can be graphically represented by means of Venn diagrams. Those diagrams, trivially speaking, are black-and-white pictures presenting sets as sharp, black "stains" on a white background. Let us imagine more complex, nebular or nebulous objects in M whose (analogues of) Venn diagrams become M.

Download PDF sample

Cardinalities of Fuzzy Sets by Prof. Dr. Maciej Wygralak (auth.)


by John
4.2

Rated 4.44 of 5 – based on 18 votes