By Prof. Dr. Maciej Wygralak (auth.)
Counting is likely one of the uncomplicated common mathematical actions. It comes with complementary features: to figure out the variety of parts of a suite - and to create an ordering among the gadgets of counting simply by counting them over. For finite units of items those elements are discovered via an identical kind of num bers: the common numbers. That those complementary facets of the counting professional 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 parts the cardinals were created in set idea, and set theorists have in parallel created the ordinals to count number over any set of items. For either different types of numbers it's not merely counting they're used for, it's also the strongly comparable technique of calculation - in particular addition and, derived from it, multiplication or even exponentiation - that's dependent upon those numbers. For fuzzy units the assumption of counting, in either facets, looses its naive beginning: since it is to a wide volume based upon of the concept that there's a transparent distinc tion among these items that have to count - and people ones that have to be missed for the actual counting process.
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Extra info for Cardinalities of Fuzzy Sets
Example text
Of having the same cardinality, the same number of elements. That notion, expressed in the language of bijections, is crucial in the classical cardinality theory as weIl as in the theory of generalized cardinals of fuzzy sets in Chapter 4. Its position in the presented theory of scalar cardinalities is much weaker. The scalar cardinality of a fuzzy set principally plays the role of an index of cardinality of that fuzzy set. An equipotency relation - such that for each A, BE FFS with a cardinality pattern 1 is generally difficult to defme, especially in terms of some bijections.
However, if 1 is finite and, say, 1 = {I, 2, ... ,k} with k ~ 1, let us introduce the following notation: ~ Ai := Al U fil Ai := Al jE! and jE! S A 2U S ••• U s Ak n t A 2 n t ... n t A k, where sand I are quite arbitrary. 9), we then get Aus ( ( ~ A)V jE! = n B) = n AusBi' jE! rul Ajv iE! with u := SV, jE! Ant ( ( fil U B) jE! A;)V = lj jE! (ViEl: AjcB) => ( fil iE! Aj c fil B j & jE! = U AntBi, iE! Aiv with v:= IV, JEJ ~ jE! Ai C ~ B j), iE! 4. Other Elements 0/ the Language 0/ Fuzzy Sets 'Vjel: (i) Ai e Aj e iEJ where V 31 eJ Ai' iEJ is a strong negation.
On the other hand, there exists another group of constructive approaches in which that cardinality is itself fuzzy. e. by means of some weighted family of usual cardinals with the weights from [0,1]. This alternative "fuzzy" optics leads to a very complete and adequate cardinal description of A at the price of relatively high complexity. Generalized cardinals will be denoted by lowercase letters a, ß, y, ... l from the first half of the Greek alphabet. If a generalized cardinal a does express the cardinality of A, we write 1 A 1 = a, and we say that the cardinality 01 A is equal to a.