- Fuzzy set
**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. [*# L.A. Zadeh (1965) Fuzzy sets. Information and Control 8 (3) 338--353<*] In classicalset 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 theindicator function s of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. [*D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.*]**Definition**A fuzzy set is a pair $(A,\; m)$ where $A$ is a set and $m\; :\; A\; ightarrow\; [0,1]$.

For each $xin\; A$, $m(x)$ is the

**grade**of membership of $x$. $xin\; (A,\; m)iff\; xin\; Awedge\; m(x)\; eq\; 0$. If $A=\{x\_1,...,x\_n\}$ the fuzzy set $(A,\; m)$ can be denoted $\{m(z\_1)/z\_1,...,m(z\_n)/z\_n\}$.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. [

*AAAI http://www.aaai.org/aitopics/pmwiki/pmwiki.php/AITopics/FuzzyLogic*]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 byJoseph Goguen , who was a student of Zadeh. [*Goguen, Joseph A., 1967, "L"-fuzzy sets". "Journal of Mathematical Analysis and Applications"*]**18**: 145–174**Fuzzy logic**As an extension of the case of

multi-valued logic , valuations ($mu\; :\; mathit\{V\}\_o\; o\; mathit\{W\}$) ofpropositional variable s ($mathit\{V\}\_o$) 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 fuzzypremises from which graded conclusions may be drawn. [*Gottwald, Siegfried, 2001. "A Treatise on Many-Valued Logics". Baldock, Hertfordshire, England: Research Studies Press Ltd., ISBN 978-0863802621*]

* Zadeh, Lotfi A.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 andknowledge engineering , and which encompasses many topics involving fuzzy sets and "approximated reasoning." [*"The concept of a linguistic variable and its application to approximate reasoning," "Information Sciences"*]**8**: 199–249, 301–357;**9**: 43–80.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 $ilde\{mathit\{Asubseteqmathbb\{R\}$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 thefunfair 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**A

**fuzzy interval**is an uncertain set $ilde\{mathit\{Asubseteqmathbb\{R\}$ 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. [*"Fuzzy sets as a basis for a theory of possibility," "Fuzzy Sets and Systems"*]**1**: 3–28**ee also**

*Fuzzy measure theory

*Alternative set theory

*Linear partial information

*Defuzzification

*Fuzzy set operations

*Neuro-fuzzy

*Rough set

*Uncertainty

*Rough fuzzy hybridization

*Fuzzy subalgebra **External links*** [

*http://www.uncertainty-in-engineering.net/uncertainty_models/fuzziness Uncertainty model Fuzziness*]* ScholarPedia [

*http://www.scholarpedia.org/article/Fuzzy_sets*]

* [*http://www.uncertainty-in-engineering.net/uncertainty_methods/fuzzy_analysis/ The Algorithm of Fuzzy Analysis*]

* [*http://pami.uwaterloo.ca/tizhoosh/set.htm Fuzzy Image Processing*]

* Zadeh's 1965 paper on [*http://www-bisc.cs.berkeley.edu/Zadeh-1965.pdf Fuzzy Sets*]**References****External links*** Fuzzy Systems Journal http://www.elsevier.com/wps/find/journaldescription.cws_home/505545/description#description

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**fuzzy set**— Math. a generalization of a classical set with the property that each member of a population of objects has associated with it a number, usually from 0 to 1, that indicates the degree to which the object belongs to the set. [1960 65] * * * fuzzy… … Useful english dictionary**Fuzzy Set**— unscharfe Menge, Fuzzy Menge; Menge, deren Elemente bestimmten Mengen zu verschiedenem Grad angehören bzw. in der die Aussage „ein Element x gehört zur Menge X“ zu verschiedenem Grad wahr sein kann. Ein F.S. definiert sich aus (x, μ(x)}, also aus … Lexikon der Economics**fuzzy set**— noun A set whose elements have degrees of membership, instead of only full and empty membership. See Also: fuzzy logic … Wiktionary**fuzzy set**— noun Date: 1964 a mathematical set with the property that an object can be a member of the set, not a member of the set, or any of a continuum of states of being a partial member of the set … New Collegiate Dictionary**fuzzy set**— Math. a generalization of a classical set with the property that each member of a population of objects has associated with it a number, usually from 0 to 1, that indicates the degree to which the object belongs to the set. [1960 65] * * * … Universalium**Fuzzy set operations**— A fuzzy set operation is an operation on fuzzy sets. These operations are generalization of crisp set operations. There is more than one possible generalization. The most widely used operations are called standard fuzzy set operations . There are … Wikipedia**Fuzzy-Menge**— ⇡ Fuzzy Set … Lexikon der Economics**Fuzzy logic**— is a form of multi valued logic derived from fuzzy set theory to deal with reasoning that is approximate rather than precise. Just as in fuzzy set theory the set membership values can range (inclusively) between 0 and 1, in fuzzy logic the degree … Wikipedia**Fuzzy control system**— Fuzzy control and Fuzzy Control redirect here. For the rock band, see Fuzzy Control (band). A fuzzy control system is a control system based on fuzzy logic a mathematical system that analyzes analog input values in terms of logical variables that … Wikipedia**fuzzy logic**— ☆ fuzzy logic n. 〚< fuzzy (set), coined (1965) by L. A. Zadeh, U.S. computer scientist〛 a type of logic used in computers and other electronic devices for processing imprecise or variable data: in place of the traditional binary values, fuzzy… … Universalium