Fuzzy cardinality concerns the assessment of the number of elements in a fuzzy set. There are different approaches to fuzzy cardinality, dating back to the early works of Zadeh, De Luca and Termini, Dubois and Prade, and Wygralak, among others. But the most important ones can be classified into scalar and fuzzy approaches. The first one yields a scalar number (either integer or real) as the result, whereas the second one represents the cardinality of a fuzzy set as a fuzzy subset of the nonnegative integers.
The cardinality of a fuzzy set is on the basis of two other important issues in fuzzy sets: fuzzy bags and fuzzy quantification. Bags (also called multisets) are algebraic set-like structures that allow for repeated elements. Usually, this information is summarized as a set of pairs (item, number of times it appears). Fuzzy bags where introduced by Yager as fuzzy extensions of the notion of bag in which each element can appear more than once, with different degrees. To obtain the summary, a measure of the fuzzy cardinality of the set of instances of each element is needed.
Fuzzy quantifiers represent imprecise quantities or percentages, such as "approximately between 4 and 6," "around 20%," or "most." They generalize the usual "exists" and "forall" quantifiers of first-order logic, allowing us to represent the meaning of expressions like "Most of the students are tall." The evaluation of such quantified sentences requires determining the compatibility degree between the cardinality of the fuzzy set of young students and the linguistic quantifier "most." Early approaches to this problem are due to Zadeh, Dubois and Prade, Yager, and Sugeno, among others.
Fuzzy cardinality and the closely related issues of fuzzy bags and fuzzy quantification are on the basis of many basic operations associated to most of the more important and promising areas of application of fuzzy sets. A few of them are