# Fuzzy Set

A Fuzzy Set is a dataset whose elements are linguist variables values and have degrees of membership.

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**Example(s):**- ("cold", "comfortable", "warm") is a fuzzy set whose elements are values of the linguist variable representing temperature.
- ("small", "medium", "large") is a fuzzy set whose elements are values of the linguist variable representing size.

**See:**Fuzzy System, Unit Interval, Bioinformatics, Mathematics, Set (Mathematics), Element (Mathematics), Linguistics, Decision Making, Cluster Analysis, Set Theory, Principle of Bivalence.

## References

### 2017a

- (Sammut & Webb, 2017) ⇒ (2017) "Fuzzy Sets". In: Sammut, C., Webb, G.I. (eds) Encyclopedia of Machine Learning and Data Mining. Springer, Boston, MA.
- QUOTE: Fuzzy sets were introduced by Lofti Zadeh as a generalization of the concept of a regular set. A fuzzy set is characterized by a membership function that assigns a degree (or grade) of membership to all the elements in the universe of discourse. The membership value is a real number in the range [0, 1], where 0 denotes no definite membership, 1 denotes definite membership, and intermediate values denote partial membership to the set. In this way, the transition from nonmembership to membership in a fuzzy set is gradual and not abrupt like in a regular set, allowing the representation of imprecise concepts like “small,” “cold,” “large,” or “very” for example.
A variable with its values defined by fuzzy sets is called a linguistic variable. For example, a linguistic variable used to represent a temperature can be defined as taking the values “cold,” “comfortable,” and “warm,” each one of them defined as a fuzzy set. These linguistic labels, which are imprecise by their own nature, are, however, defined very precisely by using fuzzy set concepts.

Based on the concepts of fuzzy sets and linguistic variables, it is possible to define a complete fuzzy logic, which is an extension of the classical logic but appropriate to deal with approximate knowledge, uncertainty, and imprecision.

- QUOTE: Fuzzy sets were introduced by Lofti Zadeh as a generalization of the concept of a regular set. A fuzzy set is characterized by a membership function that assigns a degree (or grade) of membership to all the elements in the universe of discourse. The membership value is a real number in the range [0, 1], where 0 denotes no definite membership, 1 denotes definite membership, and intermediate values denote partial membership to the set. In this way, the transition from nonmembership to membership in a fuzzy set is gradual and not abrupt like in a regular set, allowing the representation of imprecise concepts like “small,” “cold,” “large,” or “very” for example.

### 2017b

- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Fuzzy_set Retrieved:2017-7-16.
- In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh
^{[1]}and Dieter Klaua^{[2]}in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an*L*-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are used now in different 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.^{[3]}In fuzzy set theory, classical bivalent sets are usually called*crisp*sets. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as bioinformatics.^{[4]}

- In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh

- ↑ L. A. Zadeh (1965) "Fuzzy sets".
*Information and Control*8 (3) 338–353. - ↑ Klaua, D. (1965) Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsb. Deutsch. Akad. Wiss. Berlin 7, 859–876. A recent in-depth analysis of this paper has been provided by
- ↑ D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.
- ↑ Lily R. Liang, Shiyong Lu, Xuena Wang, Yi Lu, Vinay Mandal, Dorrelyn Patacsil, and Deepak Kumar, "FM-test: A Fuzzy-Set-Theory-Based Approach to Differential Gene Expression Data Analysis", BMC Bioinformatics, 7 (Suppl 4): S7. 2006.