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Encyclopedia > Fuzzy set

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 mu to [0,1]. 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. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Fuzzy logic is derived from fuzzy set theory dealing with reasoning that is approximate rather than precisely deduced from classical predicate logic. ... The membership function of a fuzzy set corresponds to the indicator function of classical sets. ... In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ... In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ...

Contents


Definition

Specifically, a fuzzy set on a classical set Χ is defined as follows:

tilde{mathit{A}}={(x,mu_{A}(x))mid x in Chi}

The membership function μA(x) quantifies the grade of membership of the elements x to the fundamental set Χ. 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.

Fuzzy set and crisp set

The following holds for the functional values of the membership function μA(x) Image File history File links Summary comparision fuzzy set and crisp set Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

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 (mu : mathit{V}_o to mathit{W}) of propositional variables (Vo) into a set of membership degrees (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. Multi-valued logics are logical calculi in which there are more than two possible truth values. ... The membership function of a fuzzy set corresponds to the indicator function of classical sets. ... It has been suggested that Predicate calculus be merged into this article or section. ... The word premise came from Latin praemisus meaning placed in front. See Premise (film) for an article discussing the use of the word in the film industry A premise (sometimes spelled premiss in philosophy) is a statement, usually put forth as part of a logical argument, that will be presumed...


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." Engineering is the application of scientific and technical knowledge to solve human problems. ... Automation (ancient Greek: = self dictated) or industrial automation or numerical control is the use of control systems (e. ... Knowledge engineering is a relatively new branch of software engineering. ...


Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at fuzzy logic. Fuzzy logic is derived from fuzzy set theory dealing with reasoning that is approximate rather than precisely deduced from classical predicate logic. ...


Fuzzy Number

A fuzzy number is a convex, normalized fuzzy set tilde{mathit{A}}subseteqmathbb{R} whose membership function is at least segmentally continuous and has the functional value μA(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 contestants weight, with the actual weight being completely correct (mapping to 1 by the membership function). In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... A travelling funfair has many attractions, run by different showmen, who all converge for the duration of the fair, then go their separate ways to set up at fairs in other towns. ...


Fuzzy Interval

A fuzzy interval is an uncertain set tilde{mathit{A}}subseteqmathbb{R} with a mean interval whose elements possess the membership function value μA(x) = 1. As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous. In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... Broadly, normalization (also spelled normalisation) is any process that makes something more normal, which typically means conforming to some regularity or rule, or returning from some state of abnormality. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...


See also

Fuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. ... An alternative set theory is an alternative mathematical approach to the concept of set. ... Defuzzification is the process of producing a quantifiable result in fuzzy logic. ... Fuzzy logic is derived from fuzzy set theory dealing with reasoning that is approximate rather than precisely deduced from classical predicate logic. ... This article needs to be cleaned up to conform to a higher standard of quality. ... In the field of artificial intelligence, neuro-fuzzy refers to hybrids of artificial neural networks and fuzzy logic. ... In mathematical logic, a rough set is an imprecise representation of a crisp set (conventional set) in terms of two subsets, a lower approximation and upper approximation. ... // Relation between uncertainty, probability and risk In his seminal work Risk, Uncertainty, and Profit, Frank Knight (1921) established the important distinction between risk and uncertainty: … Uncertainty must be taken in a sense radically distinct from the familiar notion of Risk, from which it has never been properly separated. ... Rough fuzzy hybridization is a method of hybrid intelligent system or soft computing, where Fuzzy set theory is used for linguistic representation of patterns, leading to a fuzzy granulation of the feature space. ...

External links

References

  • Gottwald, Siegfried, A Treatise on Many-Valued Logics. Research Studies Press LTD. (2001) Baldock, Hertfordshire, England.
  • Zadeh, L. A., Fuzzy sets. Information and Control, Vol. 8, pp. 338-353. (1965).
  • Zadeh, L. A., The concept of a linguistic variable and its application to approximate reasoning. Information Sciences, Vol. 8, pp. 199–249, 301–357; Vol. 9, pp. 43–80. (1975).
  • Zadeh, L. A., Fuzzy Sets as a Basis for a Theory of Possibility, Fuzzy Sets and Systems, Vol. 1, No. 1, pp. 3–28 (1978).

 
 

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