Deriving fuzzy subsethood measures from violations of the implication between elements
The aim of this paper is to present a collection of new measures of subsethood between fuzzy sets. Starting from the relationship between crisp set containment and logical implication, some fuzzy approaches are reviewed. An excerpt of reasonable fuzzy implication operators is used to define fuzzy measures of inclusion using Kosko's fitviolation strategy. We test these measures on two axiomatics and derive, when possible, measures of fuzzy entropy. Once a subsethood measure between fuzzy sets is defined, other operations as set equality, similarity, disjointness, complement,... can be considered. The need for containment measures is present in wide areas as approximate reasoning and inference, image processing or learning.
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