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Knowledge-based representation of fuzzy sets

A fuzzy set is considered to represent deterministic uncertainty called fuzziness. In deterministic uncertainty of fuzzy sets, one may subjectively determine the membership function of a given element by his knowledge. Different persons with different knowledge may provide different membership functions for elements in a universe with respect to a given fuzzy set. Here, knowledge plays important roles in determining or defining a fuzzy set. By adding the component of knowledge, we generalize the definition of fuzzy set based on probability theory. Some basic operations are re-defined. In addition, by using a fuzzy conditional probability relation, granularity of knowledge is given in two frameworks, crisp granularity and fuzzy granularity. Also, two asymmetric similarity classes or subsets are considered. When fuzzy sets represent problems or situations, a granule of knowledge might describe a class (group) of knowledge (persons) who has similar point of view in dealing the problems. Objectivity and individuality measures are proposed in order to calculate degree of objectivity and individuality, respectively of a given element of knowledge.