Abstract
The extension of fuzzy sets in which membership degrees are intervals in [0, 1], not only real numbers, is called interval-valued fuzzy sets (IVFSs). They include traditional fuzzy sets as particular cases, if looking at real membership degrees as at degenerated intervals. It is useful when real membership degrees are hard or even impossible to be determined, e.g. when experts differ in their opinions on membership degrees: the answer is correct to a degree of 0.5 up to 0.7. The paper presents applications of IVFSs and their cylindric extensions to representation of linguistic data. We introduce cylindric extensions of IVFSs and we formalize unions and intersections of IVFSs in different universes of discourse via cylindric extensions, to represent properties of objects described by attributes having their values in different domains, e.g. tall in cm and rich in €. We apply the described methods to data linguistic summarization in the sense of Yager (1982) but based on IVFSs. In particular, we propose to re-express some algorithms using cylindric extensions to make computations simplified and more human-oriented.
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Notes
Fuzzy sets in which membership degrees are fuzzy sets themselves, usually, fuzzy numbers. In case of IVFSs, those fuzzy numbers have rectangular and normal membership functions.
Actually, in computations, we do not differ objects and tuples being their models, because a system does not take into account any other attributes or properties but only those stored in the database.
The interval-valued fuzzy set \({ce(S_{j})\cap {\mathcal{D}}}\) is represented by the membership functions \({\underline \mu_{{\textnormal{ce}}(S_j)}\restriction{\mathcal{D}}\colon {\mathcal{D}}\to [0,1], (\underline \mu_{{\rm{ce}}(S_{j})}\restriction {\mathcal{D}})(d_{i}) = \underline \mu_{{\rm{ce}}(S_{j})}(d_{i}), }\) and \({\overline\mu_{{\rm{ce}}(S_{j})}\restriction {\mathcal{D}}}\)—analogously.
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Niewiadomski, A. Cylindric extensions of interval-valued fuzzy sets in data linguistic summaries. J Ambient Intell Human Comput 4, 369–376 (2013). https://doi.org/10.1007/s12652-011-0098-3
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DOI: https://doi.org/10.1007/s12652-011-0098-3