Intro. ANN & Fuzzy SystemsLecture 30 Fuzzy Set Theory (II)(C) 2001 by Yu Hen Hu2Intro. ANN & Fuzzy SystemsOutline• Membership Function, Fuzzy Set Supports• Fuzzy Set Operations, Alternative Operations• Fuzzy Propositions• Linguistic Variable Modifiers (Hedges)• Fuzzy Relations(C) 2001 by Yu Hen Hu3Intro. ANN & Fuzzy SystemsFuzzy Proposition• Fuzzy Proposition – Propositions that include fuzzy predicates. For example,– It will be sunny today.– Dow Jones is closed higher yesterday.• Canonical Form – (Unconditional fuzzy proposition)x is AA (a fuzzy set) is a fuzzy predicate called the fuzzy variable or the linguistic variable. The values of a linguistic variable are words or sentences in a natural or synthetic language.(C) 2001 by Yu Hen Hu4Intro. ANN & Fuzzy SystemsFuzzy Proposition• Note that a linguistic variable is a fuzzy (sub)set defined on a Universe of discourse. For example, Janet is young implies the AGE of Janet is Young. Here, Young is a fuzzy set defined on the axis "Age". • Other fuzzy sets may be defined on the same universe include "Old", "Mid-age", etc.• Age is a property of "Janet", and Young is a specific subset of "Age".(C) 2001 by Yu Hen Hu5Intro. ANN & Fuzzy SystemsLinguistic Variable Modifiers• Modifiers (hedges) are words like "extremely", "very" which changes the predicate. For example,"It is cold today" becomes "It is very cold today".Some possible implementations of modifiers are: Very, somewhat, Not, positively, etc.•CONcentration and DILution –transform original membership function µ(x) → µn(x), n > 1 (concentration) and n < 1 (dilution).(C) 2001 by Yu Hen Hu6Intro. ANN & Fuzzy SystemsLinguistic Variable Modifiers • Examples: VERY (µ2(x)), EXTREMELY (µ3(x)), SOMEWHAT, MORE_OR_LESS (µ0.5(x))coldsomewhat coldvery coldnot coldtemperatureµ (temperature)cold1(C) 2001 by Yu Hen Hu7Intro. ANN & Fuzzy SystemsLinguistic Variable Modifiers• INTensify –µint(x) =Aα= {x| µ(x) ≤α} is the α-cut of µ(x). • For example, let n = 2, α = 0.5. The fuzzy sets Tall and POSITIVELY Tall are illustrated below: nµn( x );x ∈Aα1− nµn( x );x ∉Aα 10.5positively talltall(C) 2001 by Yu Hen Hu8Intro. ANN & Fuzzy SystemsLinguistic Variable Modifiers• AROUND, ABOUT, APPROXIMATE – Broaden µ(x).• BELOW, ABOVE – (see illustration below)10.5tallbelow tallabove talltall10.5talltallabout tall(C) 2001 by Yu Hen Hu9Intro. ANN & Fuzzy SystemsFuzzy Relations• Fuzzy relation R from set X to set Y is a fuzzy set in the direct product X × Y = {(x,y)| x ∈ X, y ∈ Y}, and is characterized by a membership functionµR: X × Y → [ 0, 1]When X = Y, R is a fuzzy relation on X.• Relations defined in crisp logic: xyx>yx<yx=y(C) 2001 by Yu Hen Hu10Intro. ANN & Fuzzy SystemsFuzzy Relation Example•Example. The relation y >> x can be defined as:<−+≥=.)(10011;0),(2yxxyyxyxRµµ (y-x)y - xR0(C) 2001 by Yu Hen Hu11Intro. ANN & Fuzzy SystemsOther Fuzzy Relations• "x is A" AND "x is B" = "x is (A AND B)”"x is A" OR "x is B" = "x is (A OR B)”"x is A" AND "y is B" = "(x,y) is A × B”"x is A" OR "y is B" = "(x,y) is A × Y ∪ X × B"xxxyBAyxAB(C) 2001 by Yu Hen Hu12Intro. ANN & Fuzzy SystemsFuzzy Matrices• If both X and Y consists of finite, countable elements, then µR(x,y) can be represented by a matrix called a Fuzzy matrix. •Example. X = {a, b, c}. Then a fuzzy relation R on X may be:R = 0.2/(a,a) + 1/(a,b) + 0.4/(a,c) + 0.6/(b,b) + 0.3/(b,c) + 1/(c,b) + 0.8/(c,c) orR = 8.0103.06.004.012.0(C) 2001 by Yu Hen Hu13Intro. ANN & Fuzzy SystemsFuzzy Graph•A Fuzzy Graph consists of nodes {xi} ≈ {yj}, and arcs µR(xi,yj) from xito yj.Example: Let {xi} = {yj} = {a, b, c}R =
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