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.

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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.

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

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