page_title: Fuzzy set
section_title: Neutrosophic fuzzy sets
hierarchical_section_title: Fuzzy set / Definition / Neutrosophic fuzzy sets
caption_attribution_description: English: Some Key Developments in the Introduction of Fuzzy Set Concepts.
context_page_description: In mathematics, fuzzy sets are somewhat like sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set.
  At the same time, Salii defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and has applications in areas such as linguistics, decision-making, and clustering, are special cases of L-relations when L is the unit interval [0, 1].
  In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1.
context_section_description: The concept of IFS has been extended into two major models. The two extensions of IFS are neutrosophic fuzzy sets and Pythagorean fuzzy sets.
  Neutrosophic fuzzy sets were introduced by Smarandache in 1998. Like IFS, neutrosophic fuzzy sets have the previous two functions: one for membership  and another for non-membership . The major difference is that neutrosophic fuzzy sets have one more function: for indeterminate . This value indicates that the degree of undecidedness that the entity x belongs to the set. This concept of having indeterminate  value can be particularly useful when one cannot be very confident on the membership or non-membership values for item x. In summary, neutrosophic fuzzy sets are associated with the following functions:
  1.  - degree of membership of x
  2.  - degree of non-membership of x
  3.  - degree of indeterminate value of x
caption: Some Key Developments in the Introduction of Fuzzy Set Concepts.[11]
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