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Fuzzy sets are an extension of classical set theory and are used in fuzzy logic. In classical set theory the membership of elements in relation to a set is assessed in binary terms according to a crisp 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 relation to a set; this is described with the aid of a membership function valued in the real unit interval . Fuzzy sets are an extension of classical set theory since, for a certain universe, a membership function may act as an indicator function, mapping all elements to either 1 or 0, as in the classical notion. Fuzzy sets have been introduced by Lotfi A. Zadeh (1965).

Definition

Specifically, a fuzzy set on a classical set ${\displaystyle \mathrm {X} }$ is defined as follows:

${\displaystyle {\tilde {\mathit {A}}}=\{(x,\mu _{A}(x))\mid x\in \mathrm {X} \}}$

The membership function ${\displaystyle \mu _{A}(x)}$ quantifies the grade of membership of the elements ${\displaystyle x}$ to the fundamental set ${\displaystyle \mathrm {X} }$. An element mapping to the value 0 means that the member is not included in the given set, 1 describes a fully included member. Values strictly between 0 and 1 characterize the fuzzy members.

Sometimes, a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure ${\displaystyle L}$; usually it is required that ${\displaystyle L}$ be at least a poset or lattice. The usual membership functions with values in are then called -valued membership functions.

Applications

The fuzzy set B, where B = {(3,0.3), (4,0.7), (5,1), (6,0.4)} would be enumerated as B = {0.3/3, 0.7/4, 1/5, 0.4/6} using standard fuzzy notation. Note that any value with a membership grade of zero does not appear in the expression of the set. The standard notation for finding the membership grade of the fuzzy set B at 6 is μB(6) = 0.4.

Fuzzy logic

As an extension of the case of multi-valued logic, valuations (${\displaystyle \mu :{\mathit {V}}_{o}\to {\mathit {W}}}$) of propositional variables (${\displaystyle {\mathit {V}}_{o}}$) into a set of membership degrees (${\displaystyle {\mathit {W}}}$) can be thought of as membership functions mapping predicates into fuzzy sets (or more formally, into an ordered set of fuzzy pairs, called a fuzzy relation). With these valuations, many-valued logic can be extended to allow for fuzzy premises from which graded conclusions may be drawn.

This extension is sometimes called "fuzzy logic in the narrow sense" as opposed to "fuzzy logic in the wider sense," which originated in the engineering fields of automated control and knowledge engineering, and which encompasses many topics involving fuzzy sets and "approximated reasoning."

Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at fuzzy logic.

Fuzzy number

A fuzzy number is a convex, normalized fuzzy set ${\displaystyle {\tilde {\mathit {A}}}\subseteq \mathbb {R} }$ whose membership function is at least segmentally continuous and has the functional value ${\displaystyle \mu _{A}(x)=1}$ at precisely one element. This can be likened to the funfair game "guess your weight," where someone guesses the contestants weight, with closer guesses being more correct, and where the guesser "wins" if they guess near enough to the contestant's weight, with the actual weight being completely correct (mapping to 1 by the membership function).

Fuzzy interval

A fuzzy interval is an uncertain set ${\displaystyle {\tilde {\mathit {A}}}\subseteq \mathbb {R} }$ with a mean interval whose elements possess the membership function value ${\displaystyle \mu _{A}(x)=1}$. As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous.

See also

• Fuzzy measure theory
• Alternative set theory
• Defuzzification
• Fuzzy logic
• Fuzzy set operations
• Neuro-fuzzy
• Rough set
• Uncertainty
• Rough fuzzy hybridization
• Fuzzy subalgebra

External links

• Uncertainty model Fuzziness
• The Algorithm of Fuzzy Analysis
• Fuzzy Image Processing

References

• Goguen, Joseph A., 1967, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174
• Gottwald, Siegfried, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies Press Ltd.
• Zadeh, Lotfi A., 1965, "Fuzzy sets," Information and Control 8: 338–353.
• --------, 1975, "The concept of a linguistic variable and its application to approximate reasoning," Information Sciences 8: 199–249, 301–57; 9: 43–80.
• --------, 1978, "Fuzzy sets as a basis for a theory of possibility," Fuzzy Sets and Systems 1: 3–28.

• Fuzzy logic
• Systems of set theory