neutrosophic set

neutrosophic set

(logic)
A generalisation of the intuitionistic set, classical set, fuzzy set, paraconsistent set, dialetheist set, paradoxist set, tautological set based on Neutrosophy. An element x(T, I, F) belongs to the set in the following way: it is t true in the set, i indeterminate in the set, and f false, where t, i, and f are real numbers taken from the sets T, I, and F with no restriction on T, I, F, nor on their sum n=t+i+f.

The neutrosophic set generalises:

- the intuitionistic set, which supports incomplete set theories (for 0<n<100 and i=0, 0<=t,i,f<=100);

- the fuzzy set (for n=100 and i=0, and 0<=t,i,f<=100);

- the classical set (for n=100 and i=0, with t,f either 0 or 100);

- the paraconsistent set (for n>100 and i=0, with both t,f<100);

- the dialetheist set, which says that the intersection of some disjoint sets is not empty (for t=f=100 and i=0; some paradoxist sets can be denoted this way).

http://gallup.unm.edu/~smarandache/NeutSet.txt.

["Neutrosophy / Neutrosophic Probability, Set, and Logic", Florentin Smarandache, American Research Press, 1998].
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As a result, Smarandache [19, 20] has introduced the concept of Neutrosophic Set (NS) which is a generalisation of classical sets, fuzzy set, intuitionistic fuzzy set etc.
To practice NSs in real life situations efficiently,The subclass of the neutrosophic sets called single-valued neutrosophic set (in short SVNS) was defined by Smarandache in [4].
9] introduced the concept of the single-valued neutrosophic set (SVNS), a subclass of the neutrosophic sets.
On the basis of Zadeh's work, several high-order fuzzy sets have been proposed as an extension of fuzzy sets, including interval-valued fuzzy set, type-2 fuzzy set, type-n fuzzy set, soft set, rough set, intuitionistic fuzzy set, interval-valued intuitionistic fuzzy set, hesitant fuzzy set, and neutrosophic set (NS) [2-6].
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Afterwards, one introduces the neutrosophic set operations (complement, intersection, union, difference, Cartesian product, inclusion, and n-ary relationship), some generalizations and comments on them, and finally the distinctions between the neutrosophic set and the intuitionistic fuzzy set.
Neutrosophic set is a powerful general formal framework which generalizes the concept of the classic set, fuzzy set [12], Vague set [11] etc.
Emphasizing advancements and applications to neutrosophics, this text introduces the interval neutrosophic set, which is an instance of the neutrosophic set, describes the interval neutrosophic logic based on neutrosophic sets, a situation which allows for modeling of fuzzy, incomplete, and inconsistent information, and gives a neutrosophic relational data model with a relational data base, and another model in the form of a soft semantic web services agent.
The neutrosophic set developed by Smarandache [7, 8, 9] is a formal framework which generalizes the concept of the classic set, fuzzy set [14], interval valued fuzzy set, intuitionistic fuzzy set [1], interval valued intuitionistic fuzzy set and paraconsistent set etc.
The neutrosophic set (NS) was proposed by Smarandache [8] as a general ization of the fuzzy sets [14], intuitionistic fuzzy sets [12], interval valued fuzzy set [11] and interval-valued intuitionistic fuzzy sets [13] theories, and it is a powerful mathematical tool for dealing with incomplete, indeterminate and inconsistent information in the real world.
5] introduced the idea of single valued neutrosophic set in many practical problems.
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