Abstract
Fuzzy sets, formalized by Zadeh in 1965, generalizes the classical idea of sets. The idea itself was generalized in 1975 when Zadeh introduced the interval-valued fuzzy sets. In this paper, we generalize further the above concepts by introducing interval-valued fuzzy on ideal sets, where an ideal is a nonempty collection of sets with a property describing the notion of smallness. We develop its basic concepts and properties and consider how one can create mappings of interval-valued fuzzy on ideal sets from mappings of ordinary sets. We then consider topology and continuity with respect to these sets.
Highlights
In classical set theory, an element either belongs or does not belong to a given set
The membership of elements to a given set is assessed in binary terms
A fuzzy set is a mapping from U into the unit interval [0, 1]
Summary
An element either belongs or does not belong to a given set. There are informations that cannot be precisely assessed as belonging to or not to a given set, like the set of young people in a group To address this problem, in 1965, Zadeh [9] and Klaua [4] introduced fuzzy sets, where elements have degrees of membership, not just 0 or 1. We could not just give it a value equal to the minimum of the two separate possibilities Expressing information like this motivated the introduction of fuzzy on ideal sets in [6] by Mernilo and Caga-anan. Encapsulates the preceding idea that the possibility that tomorrow there will be rain and at the same time it will be sunny should not just be equal to the minimum of the separate possibilies as it could be far less. We introduce and develop the interval-valued fuzzy on ideal sets.
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