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Fuzzy Fuzzy Sets Jan Jantzen A set is a collection of objects A special kind of set

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Summary: A set... 2 This object is not a member of the set This object is a member of the set A classical set has a sharp boundary

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... and a fuzzy set 3 A fuzzy set has a graded boundary This object is not a member of the set This object is a member of the set to a degree, for instance 0.8. The membership is between 0 and 1.

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4 Example: High and low pressures

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5 around Example: "Find books from around 1980" This could include 1978, 1979, 1980, 1981, and 1982

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6 Maybe even 41 or 42 could be all right?

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Example: A fuzzy washing machine If you fill it with only a few clothes, it will use shorter time and thus save electricity and water. 7 Samsung J1045AV capacity 7 kg There is a computer inside that makes decisions depending on how full the machine is and other information from sensors.

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A rule (implication) IF the machine is full THEN wash long time 8 Condition Action. The internal computer is able to execute an if—then rule even when the condition is only partially fulfilled.

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IF the machine is full … Example: Load = 3.5 kg clothes 9 true false Three 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 set Linear fuzzy set Nonlinear fuzzy set

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… THEN wash long time long time could be t = 120 minutes 10 The duration depends on the washing program that the user selects.

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Decision Making (inference) Rule. IF the machine is full THEN wash long time Measurement. Load = 3.5 kg Conclusion. Full(Load) × t = 0.5 × 120 = 60 mins 11 Analogy The dots mean 'therefore' The machine is only half full, so it washes half the time.

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A rule base with four rules 1.IF machine is full AND clothes are dirty THEN wash long time 2.IF machine is full AND clothes are not dirty THEN wash medium time 3.IF machine is not full AND clothes are dirty THEN wash medium time 4.IF machine is not full AND clothes are not dirty THEN wash short time 12 There are two inputs that are combined with a logical 'and'.

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THEORY OF FUZZY SETS 13

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14 Lotfi Zadeh’s Challenge much greater than beautifultall Clearly, 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).

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15 Sets {Live dinosaurs in British Museum} = The set of The set of positive integers belonging to for which The empty set

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16 Fuzzy Fuzzy Sets {nice days} {adults} Membership function Much greater than

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17 Tall Tall Persons Height [cm] Membership fuzzy crisp Universe Degree of membership Membership function

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18 Fuzzy Fuzzy (http://www.m-w.com) adjective Synonyms: faint, bleary, dim, ill-defined, indistinct, obscure, shadowy, unclear, undefined, vague Unfortunately, they all carry a negative connotation.

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Around noon 'Around noon' 19 Trapezoidal Triangular Smooth versions of the same sets.

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20 The 4 Seasons Time of the year Membership SpringSummerAutumnWinter we are here Seasons have overlap; the transition is fuzzy.

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Summary 21 A setA fuzzy set

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OPERATIONS ON FUZZY SETS 22

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23 Set Operations Classical Fuzzy UnionIntersectionNegation

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24 Fuzzy Set Operations AB Union Intersection Negation

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25 Example: Age Primary term The square of Young The square root of Old The negation of 'very young'

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26 Operations 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.

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Cartesian Product 27 The AND composition of all possible combinations of memberships from A and B The curves correspond to a cut by a horizontal plane at different levels

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28 Example: Donald Duck's family Suppose, resembles –nephew Huey resembles nephew Dewey –nephew Huey resembles nephew Louie –nephew Dewey resembles uncle Donald –nephew Louie resembles uncle Donald Question: How much does Huey resemble Donald?

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29 Solution: Fuzzy Composition of Relations Dewey Louie Huey Dewey Louie = Donald Huey Donald Huey Dewey Donald Huey Louie Donald Composition Relation ? ? ?

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If—Then Rules 30 1.If x is Neg then y is Neg 2.If x is Pos then y is Pos Rule 2 Rule 1 approximately equal

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31 Key Concepts Universe Membership function Fuzzy variables Set operations Fuzzy relations All of the above are parallels to classical set theory

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32 Application examples Database and WWW searches Matching of buyers and sellers Rule bases in expert systems

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