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Fuzzy set theory has long been introduced to handle inexact and imprecise data by Zadeh (1973), since in the real world there is vague information about different applications, such as in sensor databases, we can formalize the measurements from different sensors to a vague set. In fuzzy set theory, each object u ∈ U is assigned a single real value, called the grade of membership, between zero and one. (Here U is a classical set of objects, called the universe of discourse). Gau and Buehrer (1994) pointed out that the drawback of using the single membership value in fuzzy set theory is that the evidence for u ∈ U and the evidence against u ∈ U are in fact mixed together. In order to tackle this problem, Gau and Buehrer (1994) proposed the notion of vague sets (VSs), which allows using interval-based membership instead of using point-based membership as in fuzzy sets (FSs). The interval-based membership generalization in VSs is more expressive in capturing the vagueness of data. However, VSs are shown to be equivalent to that of intuitionistic fuzzy sets (IFSs) (Bustince & Burillo, 1996). For this reason, the interesting features for handling vague data that are unique to VSs are largely ignored. We find that there are many interesting features of VSs from a data modelling point of view. Essentially, due to the fact that a VS corresponds to a more intuitive graphical view of data sets, it is much easier to define and visualize the relationship of vague data objects. The classical nulls representing incompleteness can be viewed as a special case of a vague set and then be generalized to vague data. In addition, we show that the notions of crisp and imprecision in vague data can be captured by interval relationships.
Since Zadeh introduced fuzzy set theory, several new concepts of higher-order FSs have been proposed. Among them, IFSs, proposed by Atanassov (1989), provide a flexible mathematical framework to cope, besides the presence of vagueness, with the hesitancy originating from imperfect or imprecise information. IFSs use two characteristic functions to express the degree of membership (belongingness) and the degree of non-membership (non-belongingness) of elements of the universe to the IFSs. Therefore, the idea of using positive and (independently) negative information becomes the core of IFSs. This idea is natural in real life human discourse and action, and as an obvious consequence, is well-known and widely studied in psychology and other social sciences. In fact, IFSs, interval-valued fuzzy sets (IVFSs) and vague sets can be viewed as three equivalent generalizations of fuzzy sets (Bustince & Burillo, 1996). However, they are different as IFSs force a user to explicitly consider positive and negative information independently. On the other hand, while employing IVFSs, the user’s attention is forced on positive information (in an interval) only. So the two concepts, IFSs and IVFSs, are different in applications.
In the real world there are vaguely specified data values in many applications, such as sensor information. Fuzzy set theory has been proposed to handle such vagueness by generalizing the notion of membership in a set. Essentially, in a FS each element is associated with a point value selected from the unit interval [0,1], which is termed the grade of membership in the set. A VS, as well as an IFS, is a further generalization of an FS. Instead of using point-based membership as in FSs, interval-based membership is used in a VS. The interval-based membership in VSs is more expressive in capturing vagueness of data. In the literature, the notions of IFSs and VSs are regarded as equivalent, in the sense that an IFS is isomorphic to a VS. Furthermore, due to such equivalence and IFSs being earlier known as a tradition, the interesting features for handling vague data that are unique to VSs are largely ignored. In this paper, we attempt to make a comparison between VSs and IFSs from various perspectives of algebraic properties, graphical representations and practical applications.