The membership valuedepends on thecontextHD.E.Rumelhart:IntroductiontoHumanInformationProcessingPossibilityisNOTprobabilityJohnWiley&Sons,1977Thesumisnotnecessarily1THE CHTContext 1Context2Possibilitythatμ(H)=0.3μA(H)= 0.8this symbol is H.Possibilitythatthis symbol is A.μ(H)= 0.9μ(H)= 0.1
The membership value depends on the context 6 D.E.Rumelhart: Introduction to Human Information Processing. John Wiley & Sons, 1977. µ A ( ) = 0.3 µ H ( ) = 0.9 µ A ( ) = 0.8 µ H ( ) = 0.1 Context 1 Context 2 Possibility that this symbol is A. Possibility is NOT probability. The sum is not necessarily 1. Possibility that this symbol is H
Fuzzy sets (Fuzzy numbers).Traditional sets in mathematics are crisp sets, which are treated by binary logic(i.e.,Oor1).Seethefigureontheleftbelow.·However,forexample,a setoftemperatures of"Hot"cannotbehandled bybinarylogic.(Thereareindividualdifferencesorregionaldifferences.)Quantifysubjectivitybymembershipfunctions.11Aor(x)fuo (at)HotHot℃℃00262830262830Characteristicfunction (binary logic)Membershipfunction(multivaluedlogic)
• Traditional sets in mathematics are crisp sets, which are treated by binary logic (i.e., 0 or 1). See the figure on the left below. • However, for example, a set of temperatures of “Hot" cannot be handled by binary logic. (There are individual differences or regional differences.) Quantify subjectivity by membership functions. Fuzzy sets (Fuzzy numbers) 7 0 1 28 ℃ 26 30 Hot 0 1 28 ℃ 26 30 Hot Characteristic function (binary logic) Membership function (multivalued logic) (x) (x) µ Hot fHot
Union and common set (in the case of crisp setsfs(x)X : The whole setf.(x)A, B : Two crisp subsets of XAUB : Union0xfaus(x)=max(f (x), fr(x))Thelargervalue of two functions at each point x.fAUB(x)AB : Intersection (Common set)fan(x)fans(x)= min(f (x), fe(x)0xThe smaller value oftwo functions at each pointx8
8 0 1 f (x) AB f (x) AB 0 1 f (x) A f (x) B A, B： Two crisp subsets of X X ： The whole set A B ： Union A B ： Intersection (Common set) f AB (x) = max{f A (x), fB (x)} f AB (x) = min{f A (x), fB (x)} Union and common set (in the case of crisp sets) x x The larger value of two functions at each point x. The smaller value of two functions at each point x
Union and intersection (in the case of fuzzy setsX : The whole setμg(x)μA(x)1A, B : Twofuzzy subsets of XAUB : Union0xμAuB(x) =max(u (x), μg(x)Thelargervalueof two functions at each point x.HAUB(x)AnB :IntersectionμAnB(x)=min(u (x), μg(x))μAn(x)x0Thesmallervalue oftwofunctions ateachpointx.9
9 0 1 (x) µ A (x) µ B 0 1 (x) µ AB (x) µ AB A, B : Two fuzzy subsets ofX X ： The whole set A B ： Union A B ： Intersection µ AB (x) = max{µ A (x),µ B (x)} µ AB (x) = min{µ A (x),µ B (x)} Union and intersection (in the case of fuzzy sets) x x The larger value of two functions at each point x. The smaller value of two functions at each point x
Complementary set (in the case of a crisp set)X : The whole setLawofLaw ofexcluded middlecontradictionACXA : Asubset of XAUA=X, ANA=ΦA : The complementary set of AAcXNegationofAfa(x)fa(x)1The characteristicfunctionofAHotHotfa(x)=1- f(x)℃0if xeA1.262830lo, if x@ACharacteristicfunction (binarylogic)10
Complementary set (in the case of a crisp set) 10 A ⊂ X A ⊂ X ∉ ∈ = = − x A x A f x f x A A 0, if 1, if ( ) 1 ( ) A A = X , A A = φ 0 1 28 ℃ 26 30 Hot f (x) A f (x) A Hot X ： The whole set A ： A subset of X A ：The complementary set of A The characteristic function of A Law of excluded middle Negation of A Characteristic function (binary logic) Law of contradiction