Foundation of the SuperHyperSoft Set and the Fuzzy Extension SuperHyperSoft Set: A New Vision

: We introduce for the first time the SuperHyperSoft Set and the Fuzzy and Fuzzy Extension SuperHyperSoft Set. Through a theorem we prove that the SuperHyperSoft Set is composed from many HyperSoft Sets.


Definition of Soft Set
Let  be a universe of discourse, () the power set of , and a set of attributes A. Then, the pair (F, ), where :  → () is called a Soft Set over  [1].

Definition of SuperHyperSoft Set
The SuperHyperSoft Set is an extension of the HyperSoft Set.As for the SuperHyperAlgebra, SuperHyperGraph, SuperHyperTopology and in general for SuperHyperStructure and Neutrosophic SuperHyperStructure (that includes indeterminacy) in any field of knowledge, "Super" stands for working on the powersets (instead of sets) of the attribute value sets.Let  be a universe of discourse, () the powerset of .Let  1 ,  2 , …,   , for  ≥ 1, be  distinct attributes, whose corresponding attribute values are respectively the sets  1 ,  2 , …,   , with   ∩   = ∅, for  ≠ , and ,  ∈ {1, 2, … , }.
Therefore, the SuperHyperSoft Set offers a larger variety of selections, so  1 and  2 may be: either medium, or tall (but not small), either white, or red, or black (but not yellow), mandatory female (not male), and either American, or Italian (but not French, Spanish, Chinese).
In this example there are: Card{medium, tall} • Card{white, red, black} possibilities, where Card{ } means cardinal of the set { }.This is closer to our everyday life, since for example, when selecting something, we have not been too strict, but accepting some variations (for example: medium or tall, white or red or black, etc.).

Example of Fuzzy Extension SuperHyperSoft Set
In the previous example, taking the degree of a generic element ( 0 ) as neutrosophic, one gets the Neutrosophic SuperHyperSoft Set.
Which means that: x1 with respect to the attribute values ({medium or tall} and {white or red or black} and {female}, and {American or Italian}) has the degree An International Journal on Informatics, Decision Science, Intelligent Systems Applications Florentin Smarandache, Foundation of the SuperHyperSoft Set and the Fuzzy Extension SuperHyperSoft Set of appurtenance to the set 0.7, the indeterminate degree of appurtenance 0.4, and the degree of nonappurtenance 0.1.
While x2 has the degree of appurtenance to the set 0.9, the indeterminate degree of appurtenance 0.2, and the degree of non-appurtenance 0.3.

Conclusion
A new type of soft set has been introduced, called SuperHyperSoft Set and an application has been presented.Further work to do is to define the operations (union, intersection, complement) of the SuperHyperSoft Sets.