Fuzzy Sets

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A set is a collection of objects. A special kind of set. Fuzzy Sets. Jan Jantzen www.inference.dk 2013. Summary: A set . This object is not a member of the set. This object is a member of the set. A classical set has a sharp boundary. ... and a fuzzy set. - PowerPoint PPT Presentation

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Fuzzy SetsJan Jantzenwww.inference.dk2013A set is a collection of objectsA special kind of set1Summary: A set ...2This object is not a member of the setThis object is a member of the setA classical set has a sharp boundary2... and a fuzzy set3A fuzzy set has a graded boundaryThis object is not a member of the setThis object is a member of the set to a degree, for instance 0.8. The membership is between 0 and 1.34Example: High and low pressures

45

Example: "Find books from around 1980"This could include 1978, 1979, 1980, 1981, and 198256

Maybe even 41 or 42 could be all right?6Example: A fuzzy washing machineIf you fill it with only a few clothes, it will use shorter time and thus save electricity and water.7

Samsung J1045AV capacity 7 kgThere is a computer inside that makes decisions depending on how full the machine is and other information from sensors.A rule (implication)IF the machine is full THEN wash long time

8ConditionAction.The internal computer is able to execute an ifthen rule even when the condition is only partially fulfilled.

IF the machine is full Example: Load = 3.5 kg clothes9truefalseThree examples of functions that define full. The horizontal axis is the weight of the clothes, and the vertical axis is the degree of truth of the statement the machine is full.Classical setLinear fuzzy setNonlinear fuzzy set THEN wash long timelong time could be t = 120 minutes10The duration depends on the washing program that the user selects.Decision Making (inference)Rule. IF the machine is full THEN wash long timeMeasurement. Load = 3.5 kg Conclusion. Full(Load) t = 0.5 120 = 60 mins11

AnalogyThe dots mean 'therefore'The machine is only half full, so it washes half the time.A rule base with four rulesIF machine is full AND clothes are dirty THEN wash long timeIF machine is full AND clothes are not dirty THEN wash medium timeIF machine is not full AND clothes are dirty THEN wash medium timeIF machine is not full AND clothes are not dirty THEN wash short time

12There are two inputs that are combined with a logical 'and'.Theory of Fuzzy Sets1314Lotfi Zadehs ChallengeClearly, the class of all real numbers which are much greater than 1, or the class of beautiful women, or the class of tall men, do not constitute classes or sets in the usual mathematical sense of these terms (Zadeh 1965).

1415Sets

{Live dinosaurs in British Museum} =

The set ofThe set of positive integersbelonging tofor whichThe empty set1516Fuzzy Sets

{nice days}{adults}Membership functionMuch greater than1617Tall Persons15016017018019020000.20.40.60.81Height [cm]MembershipfuzzycrispUniverseDegree of membershipMembership function1718Fuzzy (http://www.m-w.com)adjectiveSynonyms: faint, bleary, dim, ill-defined, indistinct, obscure, shadowy, unclear, undefined, vague

Unfortunately, they all carry a negative connotation.18

'Around noon'19TrapezoidalTriangularSmooth versions of the same sets.20The 4 Seasons00.51Time of the yearMembershipSpringSummerAutumnWinterwe are hereSeasons have overlap; the transition is fuzzy.20Summary21A setA fuzzy set21Operations on Fuzzy Sets22

23Set Operations

ClassicalFuzzyUnionIntersectionNegation2324Fuzzy Set Operations

AB

UnionIntersectionNegation2425Example: Age

Primary termPrimary termThe square of YoungThe square root of OldThe negation of 'very young'2526Operations

Here is a whole vocabulary of seven words.

Each operates on a membership function and returns a membership function. They can be combined serially, one after the other, and the result will be a membership function.26Cartesian Product27

The AND composition of all possible combinations of memberships from A and BThe curves correspond to a cut by a horizontal plane at different levels28Example: Donald Duck's familySuppose,nephew Huey resembles nephew Deweynephew Huey resembles nephew Louienephew Dewey resembles uncle Donaldnephew Louie resembles uncle DonaldQuestion: How much does Huey resemble Donald?

2829Solution: Fuzzy Composition of RelationsDewey LouieHueyDewey Louie=DonaldHueyDonald0.80.90.50.6Huey Dewey DonaldHuey Louie DonaldCompositionRelationRelationRelation0.80.50.90.6???29IfThen Rules30

If x is Neg then y is NegIf x is Pos then y is Pos

Rule 2Rule 1approximately equal31Key ConceptsUniverseMembership functionFuzzy variablesSet operationsFuzzy relations All of the above are parallels to classical set theory3132Application examplesDatabase and WWW searchesMatching of buyers and sellersRule bases in expert systems32