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Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets have been introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion of set.[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.[2]

Definition Edit

A fuzzy set is a pair (A, m) where A is a set and m : A \rightarrow [0,1].

For each x\in A, m(x) is the grade of membership of x. x\in (A, m)\iff x\in A\wedge m(x)\neq 0. If A=\{x_1,...,x_n\} the fuzzy set (A, m) can be denoted \{m(x_1)/x_1,...,m(x_n)/x_n\}.

An element mapping to the value 0 means that the member is not included in the fuzzy set, 1 describes a fully included member. Values strictly between 0 and 1 characterize the fuzzy members.[3]

Sometimes, a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure L; usually it is required that L be at least a poset or lattice. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. This generalization was first considered in 1967 by Joseph Goguen, who was a student of Zadeh.[4]

Fuzzy logic Edit

Main article: Fuzzy logic

As an extension of the case of multi-valued logic, valuations (\mu : \mathit{V}_o \to \mathit{W}) of propositional variables (\mathit{V}_o) into a set of membership degrees (\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.[5]

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."[6]

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

Fuzzy number Edit

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

Fuzzy interval Edit

A fuzzy interval is an uncertain set \tilde{\mathit{A}}\subseteq\mathbb{R} with a mean interval whose elements possess the membership function value \mu_{A}(x)=1. As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous.[7]


Application of fuzzy set theory in clinical settingsEdit

Application in roboticsEdit

JournalsEdit

See alsoEdit

ReferencesEdit

  1. # L.A. Zadeh (1965) Fuzzy sets. Information and Control 8 (3) 338--353<
  2. D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.
  3. AAAI http://www.aaai.org/aitopics/pmwiki/pmwiki.php/AITopics/FuzzyLogic
  4. Goguen, Joseph A., 1967, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174
  5. Gottwald, Siegfried, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies Press Ltd., ISBN 978-0863802621
  6. "The concept of a linguistic variable and its application to approximate reasoning," Information Sciences 8: 199–249, 301–357; 9: 43–80.
  7. "Fuzzy sets as a basis for a theory of possibility," Fuzzy Sets and Systems 1: 3–28

Further readingEdit

BooksEdit

  • Kay, P., & McDaniel, C. K. (1997). The linguistic significance of the meanings of basic color terms. Cambridge, MA: The MIT Press.
  • Magdalena, L. (1997). A first approach to a taxonomy of fuzzy-neural systems. Mahwah, NJ: Lawrence Erlbaum Associates Publishers.
  • Reyna, V. F., & Brainerd, C. J. (1993). Fuzzy memory and mathematics in the classroom. Amsterdam, Netherlands: North-Holland/Elsevier Science Publishers.

PapersEdit

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  • Alami, J., & El Imrani, A. (2008). Dielectric composite multimodal optimization using a multipopulation cultural algorithm: Intelligent Data Analysis Vol 12(4) 2008, 359-378.
  • Alliger, G. M., Feinzig, S. L., & Janak, E. A. (1993). Fuzzy sets and personnel selection: Discussion and an application: Journal of Occupational and Organizational Psychology Vol 66(2) Jun 1993, 163-169.
  • Alonso, J. M., Magdalena, L., Guillaume, S., Sotelo, M. A., Bergasa, L. M., Ocana, M., et al. (2007). Knowledge-based intelligent diagnosis of ground robot collision with non detectable obstacles: Journal of Intelligent & Robotic Systems Vol 48(4) Apr 2007, 539-566.
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  • Barai, R. K., & Nonami, K. (2008). Locomotion control of a hydraulically actuated hexapod robot by robust adaptive fuzzy control with self-tuned adaptation gain and dead zone fuzzy pre-compensation: Journal of Intelligent & Robotic Systems Vol 53(1) Sep 2008, 35-56.
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DissertationsEdit

  • Dorsey, D. W. (1997). Managerial merit pay allocation: An analysis of alternative judgment models using regression and fuzzy expert system techniques. Dissertation Abstracts International: Section B: The Sciences and Engineering.
  • Fulginiti, J. V. (1995). Ambiguity, precision, and choice: A fuzzy trace theory analysis of framing effects in decision-making under uncertainty. Dissertation Abstracts International: Section B: The Sciences and Engineering.
  • Gogus, O. M. (1998). Fuzzy multi-criteria decision making. Dissertation Abstracts International: Section B: The Sciences and Engineering.
  • Joo, Y. (1996). A fuzzy relational approach to cognitive structures of an urban knowledge-based system. Dissertation Abstracts International: Section B: The Sciences and Engineering.
  • Juliano, B. J. A. (1993). A fuzzy logic approach for cognitive diagnosis: Dissertation Abstracts International.
  • Kim, E. (1995). Analysis and meta-analysis of fuzzy relational structures by means of generalized morphism and their computer-aided tool support. Dissertation Abstracts International: Section B: The Sciences and Engineering.
  • Laudeman, I. V. (1995). An application of unsupervised neural networks and fuzzy clustering in the identification of structure in personality data. Dissertation Abstracts International: Section B: The Sciences and Engineering.
  • Laviolette, M. J. (1990). On the efficacy of fuzzy representations of uncertainty: Dissertation Abstracts International.
  • Lotan, T. (1993). Modeling route choice behavior in the presence of information using concepts from fuzzy set theory and approximate reasoning: Dissertation Abstracts International.
  • Macmillan, M. R. (1993). Measuring the effectiveness of different methods of user-based document clustering for information retrieval: A simulation: Dissertation Abstracts International.
  • McDowell, B. D. (1997). Word recognition and the independent evaluation of letters. Dissertation Abstracts International: Section B: The Sciences and Engineering.
  • Meleiro, L. A. C., Campello, R. J. G. B., Filho, R. M., & Amaral, W. C. (2006).
  • Takayanagi, S. (1992). An examination of how researchers make statistical judgments: Statistical and fuzzy set approaches: Dissertation Abstracts International.
  • Tang, J. (1997). Prototypes in Lesser Seal Scripts. Dissertation Abstracts International Section A: Humanities and Social Sciences.
  • Wells, T. W. (1997). Fuzzy performance evaluation and risk assessment. Dissertation Abstracts International: Section B: The Sciences and Engineering.
  • Willson, I. A. (1993). Fuzzy sets and uncertain choice processes: An empirical test of models: Dissertation Abstracts International.
  • Yeung, F. Y. (2008). Developing a Partnering Performance Index (PPI) for construction projects---a Fuzzy Set Theory approach. Dissertation Abstracts International Section A: Humanities and Social Sciences.

External linksEdit


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