Distinctions between Various Types of Fuzzy-Extension HyperSoft Sets

: We define the universes of discourses for all fuzzy and fuzzy-extension sets. Then present many types of Plithogenic Universes of discourse and their connections to HyperSoft Sets. Afterward, we make distinctions between various hybrid forms of HyperSoft Sets


Introduction
We provide concrete examples for each type of fuzzy and fuzzy-extension HyperSoft Set and their many hybrid forms, including those with plithogenic sets.Gradually, we list the types of corresponding fuzzy and fuzzy-extensions universes of discourses in connection to the HyperSoft Sets.

Fuzzy and Fuzzy-extension Universes of Discourses
Let U be a classical (discrete or continuous, non-empty) Universe of discourse.
(i) The fuzzy Universe (FU) of discourse is defined as: { ( ), } FU x t x U  , where t (that is the degree of truth-membership) of a generic element x from FU, is either a single number, an interval, or in general a subset of [0, 1].
(ii) The Intuitionistic Fuzzy Universe (IFU) of discourse is: x t f x U  , where t (that is the degree of truth-membership), and f (that is the degree of falsehood-nonmembership), of a generic element x from IFU, are either single numbers, intervals, or in general subsets from [0, 1], with sup( ) sup( ) x t i f x U  , where t (that is the degree of truth-membership), i (that is the degree of indeterminacy), f (that is the degree of falsehood-nonmembership), of a generic element x from NU, are either single numbers, or intervals, or in general subsets from [0, 1], with sup( ) sup( ) sup( ) 3 (iv) In general, a fuzzy-extension Universe (FEU) of discourse, is: where d is the fuzzy-extension degree of appurtenance of the generic element x to the universe FEU, and d should be included in [0, 1].
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Florentin Smarandache, Distinctions between Various Types of Fuzzy-Extension HyperSoft Sets
{Exception to this restriction is for the fuzzy and fuzzy-extension over-under-off-sets [6], where the degrees are allowed to be outside of the interval [0, 1]}.

Informal Definition of Plithogenic Set [1]
A plithogenic set P is a set such that each element x is characterized by one or more attributes (parameters), and each attribute (parameter) may have many attribute values.With respect to each attribute-value v, a generic element x has a corresponding degree of appurtenance d(x, v) of the element x to the set P. These attributes (parameters) and their values may be independent, dependent, or partially independent and dependent -according to the applications to solve.
The degree of appurtenance d(x, v) may be fuzzy, intuitionistic fuzzy, neutrosophic, or any fuzzyextension type.
(i) Plithogenic Universe (PU) of discourse is: While d1, d2, …, dn are the fuzzy or fuzzy-extension degrees of appurtenance of the generic element x, with respect to the attribute-values a1, a2, …, an respectively, to the set PU.
(ii) The Plithogenic Fuzzy Universe (PFU) of discourse is

a t a t x U 
It's a particular case of the Plithogenic Universe, where the degrees of appurtenances (truth-memberships) are fuzzy, with t ∊ [0, 1].

PNU x a t i f a t i f a t i f x U 
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(v) In general, the Plithogenic fuzzy-extension Universe (PFEU) of discourse is: x U  Also, PFEU is a particular case of the Plithogenic Universe, for the case when the degrees of appurtenance d are fuzzy extensions.

. ( ) n F A A A P FU    
The Cartesian product 12 ... n A A A    ensures the HyperSoft-ness of the set, while the FU (Fuzzy Universe) ensures the fuzzy-ness degree of appurtenance of the elements x to the set FU.

. ( ) n F A A A P IFU    
Similarly, the Cartesian product 12 ... n A A A    ensures the HyperSoft-ness of the set, while the IFU (Intuitionistic Fuzzy Universe) ensures the intuitionistic_fuzzy-ness degree of appurtenance of the elements x to the set IFU.

(iv)
Neutrosophic HyperSoft Set 12 : ... ( ) The same, the Cartesian product 12 ... n A A A    ensures the HyperSoft-ness of the set, while the NU (Neutrosophic Universe) ensures the neutrosophic-ness degree of appurtenance of the elements x to the set NU.
(vi) Plithogenic HyperSoft Set 12 : ... ( ) The cartesian product 12 ... n A A A    ensures, in the same way, the HyperSoft-ness, while the set PU ensures the plithogeny of the elements, i.e. each element x is characterized as in our real life by An International Journal of Computational Intelligence Methods, and Applications Florentin Smarandache, Distinctions between Various Types of Fuzzy-Extension HyperSoft Sets many attribute-values a1, a2, …, and the element x belongs to the set PU in a certain degree dij with respect to each individual attribute-value.
The degrees of appurtenance dij of an element x to the set PU may be fuzzy, intuitionistic fuzzy, neutrosophic, or any other fuzzy-extension degrees.Which means that, the element x1 belongs to the set PU in a neutrosophic degree of (0.7, 0.1.0.6) with respect to its attribute-value big, in a neutrosophic degree (0.4, 0.3, 0.1) with respect to its attribute-value white, and in a neutrosophic degree (0.9, 0.0, 0.1) with respect to its attributevalue central.
Similarly for the element x2.

Acknowledgement and Conclusion
The author would like to thank Dr. Nivetha Martin who determined the author to write this paper after her question about How to differentiate between Neutrosophic Hypersoft Set and Plithogenic Neutrosophic Hypersoft Set?
The answer follows below, using a simple concrete example, according to the previous explanations.
(i) Neutrosophic HyperSoft Set: F(big, white, central) = {x1(0.7,0.3, 0.2), x2(0.8,0.4, 0.1)}, which means that house x1 is in a neutrosophic degree of 70% true 30% indeterminate and 20% false with respect to all three attribute-values together (big and white and central), while the house x2 is in a neutrosophic degree of 80% true and 40% indeterminate and 10% false also with respect to all three attribute-values together (big and white and central).