Abstract
Multiple definitions have been put forward in the literature to measure the differences between two interval-valued fuzzy sets. However, in most cases, the outcome is just a real value, although an interval could be more appropriate in this environment. This is the starting point of this contribution. Thus, we revisit the axioms that a measure of the difference between two interval-valued fuzzy sets should satisfy, paying special attention to the condition of monotonicity in the sense that the closer the intervals are, the smaller the measure of difference between them is. Its formalisation leads to very different concepts: distances, divergences and dissimilarities. We have proven that distances and divergences lead to contradictory properties for this kind of sets. Therefore, we conclude that dissimilarities are the only appropriate measures to measure the difference between two interval-valued fuzzy sets when the outcome is an interval.
Highlights
Once we have introduced the basic concepts about different operations between Interval-valued fuzzy sets (IVFSs), we can start to think about the necessary requirements of a measure to be an appropriate way to quantify the difference between two IVFSs
We have provided examples that show that this is the case for the fifth axioms associated with distances and divergences, but we have not found any example that leads to a contradiction with the notion of dissimilarity given in Definition 7
We have recalled the basic conditions that a function should satisfy in order to formalise the differences between IVFSs
Summary
It us usually understood that knowledge of comparisons of objects, opinions, etc. are incomplete. It is possible to consider a dual approach, based on measuring the difference (see, e.g., [21]) Another previous study related to this topic can be obtained from the related concept of intuitionistic fuzzy sets. Measures to compare intuitionistic fuzzy sets have already been introduced (see, for example, [25,26]) These proposals could provide us with an initial idea on the way to compare two interval-valued fuzzy sets. We will consider the different approaches, compare them and conclude which ones are the best axioms in order to characterise a measure of the difference between two interval-valued fuzzy sets. We put forward some questions that remain open
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