On the fuzzy cardinality of a fuzzy set

We look at problem of determining the number of elements of a fuzzy set and its cardinality. It is noted that the simple adding of membership grades often does not lead to a satisfactory result. We investigate the use of a fuzzy subset of the non-negative integers to represent the cardinality of a fuzzy set. We prove a fundamental relationship between the degree of fuzziness of a set A and the specificity of the fuzzy subset representing its cardinality. The problem of determining whether the cardinality of a fuzzy set satisfies a given linguistic expressed quantity is also investigated.