Axiomatic Fuzzy Set Theory and Its Applications
Springer Science & Business Media, Apr 7, 2009 - Mathematics - 514 pages
It is well known that “fuzziness”—informationgranulesand fuzzy sets as one of its formal manifestations— is one of important characteristics of human cognitionandcomprehensionofreality. Fuzzy phenomena existinnature and are encountered quite vividly within human society. The notion of a fuzzy set has been introduced by L. A. , Zadeh in 1965 in order to formalize human concepts, in connection with the representation of human natural language and computing with words. Fuzzy sets and fuzzy logic are used for mod- ing imprecise modes of reasoning that play a pivotal role in the remarkable human abilities to make rational decisions in an environment a?ected by - certainty and imprecision. A growing number of applications of fuzzy sets originated from the “empirical-semantic” approach. From this perspective, we were focused on some practical interpretations of fuzzy sets rather than being oriented towards investigations of the underlying mathematical str- tures of fuzzy sets themselves. For instance, in the context of control theory where fuzzy sets have played an interesting and practically relevant function, the practical facet of fuzzy sets has been stressed quite signi?cantly. However, fuzzy sets can be sought as an abstract concept with all formal underpinnings stemming from this more formal perspective. In the context of applications, it is worth underlying that membership functions do not convey the same meaning at the operational level when being cast in various contexts.
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AFS algebra AFS formal concept AFS fuzzy logic AFS structure AFS theory analysis applied attributes Axiomatic Fuzzy Set B.H & VH Based on AFS belonging binary relation Boolean matrix classiﬁer coherence membership functions complete lattice completely distributive lattice concept lattice countable data sets decision tree deﬁned defined as follows Deﬁnition denoted E#I algebra element Exercise exists ﬁnite following assertions hold formal concept analysis fuzzy clustering fuzzy concepts fuzzy description fuzzy rough sets hence IEEE implies Iris Data logic operations mathematical membership degrees membership functions minimal family node non-empty sets partially ordered set probability space Proof prove real numbers semantics set of simple Sets and Systems simple concepts subset Theorem threshold topological molecular lattice topological space training samples Wang weight function Wine Data