Fuzzification of set inclusion: Theory and applications
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Fuzzification of set inclusion for fuzzy sets is developed in terms of an indicator for set inclusion, the indicator giving the degree to which a fuzzy set is a subset of another fuzzy set. To date, such indicators have been called 'inclusion grades'; however, in contrast to most existing indicators, it is proposed in the present paper that the indicator must be two-valued for crisp sets. The approach, herein, is to begin by postulating desired properties of indicators for fuzzified set inclusion, to then assume a specific mathematical form for such indicators, and then derive the necessary and sufficient conditions under which the specified formula gives rise to indicators possessing the desired properties. The investigation results in a very general class of indicators based on the bold union operation, and, most importantly, in a complete measure-theoretic characterization of this class. The characterization takes the form of a constrained representation providing explicit formulation for all indicators in the class of interest. The paper closes with applications of fuzzified set inclusion to shape recognition via image processing, in particular, mathematical morphology, and to the measurement of fuzziness in fuzzy sets by means of entropy. 1993.
