Fuzzy cardinality

An important property of a fuzzy set is its cardinality. While for crisp sets the cardinality is simply the number of elements in a set, the cardinality of a fuzzy set A, CardA, gives the sum of the values of the membership function of A, as in Eq. (9). 

Besides, when jealous local doctors forced them, on n November, to move away to the nearby town of Trent, Seraphina s recent act of piety proved handy. A letter of introduction from their landlord in Roveredo brought a warm welcome from the ruler of the town, Prince Pierre-Vigil Thun, bishop of Trent. A current of attraction crackled between Cagliostro and this bishop, recapitulating in a lower key his first encounter with Cardinal Rohan. It helped that Thun shared Rohan s fuzzy love of the occult. 

Fuzzy Measures of Molecular Shape and Size 159 a fuzzy set A, also called the cardinality of /I, is defined as the integral... 

We shall require an auxiliary measure, the cardinality m(A) of a fuzzy set A. This measure m(A) that can be regarded as a generalization of the concept of mass for fuzzy sets. The measure m(A) of a fuzzy set A can be defined as the integral... 

The number of elements in a set is termed cardinality in classical theory. The latter is obtained for fuzzy sets from summation or integration over the membership function of that set ... 

Figure 8.22 Intersection (a) and union (b) of two fuzzy sets and the cardinality of a fuzzy set (c).

The relative cardinality of the fuzzy set in Figure 8.22c is to be calculated. The membership function for this set is... 

In the first step of fuzzy control model, the incoming and outcome variable are approximated within the input values. The input values do not have to be completed and they can be different than cardinal. Based on input values, a membership function is built for every variable. Thus the results of the first step in the fuzzy control model are membership functions of the variables. The elaboration of the membership function is the main disadvantage of the fuzzy control model in... 

For classical sets and multisets, cardinality is just the total number of elements in the set, while for fuzzy sets no such simple interpretation is possible. Nonetheless, it is clear that in all three cases, cardinality provides a measure of the size of the sets. 

Fono, L. A., Gwet, H. Bouchon-Meunier, B. (2007). Fuzzy implication operators for difference operations for fuzzy sets and cardinality-based measures of comparison, European Journal of Operational Research (183) 314-326. 

The cardinality of a set can be useful in giving a measure as a single number. The cardinality with respect to a fuzzy set... 

A modified approach was used to characterize the performance of a fuzzy similarity measure in comparison with the estimation of the correlation coefficient. Here the variance weighted relative cardinality was superior to the classical method for identification of UV spectra. ... 

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