Complementary set (in the case of a fuzzy setX : The whole setLawsofexcludedmiddleandcontradictiondonotholdACXA : Asubset of XAUAX, ANA+ΦA : The complementary set of AWhy?AcXNegation of Aμ(x)μ (x)The membership function of AHotHotμ(x)=1-μ(x)℃02628300≤μ(x)≤1Membershipfunction (multivalued logic11
Complementary set (in the case of a fuzzy set) 11 A ⊂ X A ⊂ X X ：The whole set A ： A subset of X A ：The complementary set of A The membership function of A Negation of A (x) 1 (x) µ A = − µ A 0 ≤ (x) ≤1 A µ A A ≠ X , A A ≠ φ Laws of excluded middle and contradiction do not hold. 0 1 28 ℃ 26 30 Hot Hot (x) µ A (x) A µ Membership function (multivalued logic) Why?
Theory that accepts contradictionμAuz(x) = max(μua(x), μ-(x))Lawsofexcludedmiddleandcontradictiondonotholdμx(x)=1.01AUA+X, ANAΦμAuz(x)Why?x0μ-(x)μ (x)1μAn(x)= min(ua(x),μ(x)HotHot1℃0μAnz(x)2628300xMembershipfunction(multivaluedlogic)μ(x)=0.012
Theory that accepts contradiction 12 A A ≠ X , A A ≠ φ Laws of excluded middle and contradiction do not hold. 0 1 28 ℃ 26 30 Hot Hot (x) µ A (x) A µ Membership function (multivalued logic) Why? (x) max{ (x), (x)} A A A A µ = µ µ (x) min{ (x), (x)} A A A A µ = µ µ 0 1 (x) AA µ x 0 1 (x) AA µ x µ X (x) =1.0 µφ (x) = 0.0
Draw membership functions of“lukewarm"and"not lukewarm."ThetemperatureofthebathHowisthewatertemperature?A=lukewarmis lowerthan theproperIsitslightlylukewarm?temperature.B=not lukewarmA=complementarysetofANo,itisnot lukewarmThe meaning of words1canchangeaccordingtothecontexts.The negationof a worddoes not necessarilycorrespondtothemathematical negation.0℃35404513
13 A=lukewarm B=not lukewarm 0 1 40 ℃ 35 45 • The meaning of words can change according to the contexts. • The negation of a word does not necessarily correspond to the mathematical negation. Draw membership functions of “lukewarm” and “not lukewarm.” A =complementary set of A How is the water temperature? Is it slightly lukewarm? No, it is not lukewarm. The temperature of the bath is lower than the proper temperature
Union and intersection of “lukewarm"and“not lukewarm"A=lukewarmμAus(x)=max(ua(x),μg(x)AUBB=not lukewarmAB0℃354045HAnB(x)=min(ua(x), μg(x)0354045ANB0℃35454014
14 0 1 40 ℃ 35 45 A B 0 1 40 ℃ 35 45 0 1 40 ℃ 35 45 A B A B Union and intersection of “lukewarm” and “not lukewarm” A=lukewarm B=not lukewarm µ AB (x) = min{µ A (x),µ B (x)} µ AB (x) = max{µ A (x),µ B (x)}
Degree-of-fit (Closeness)A:Comfortabletemperature(Afuzzyset)B:About25degree(Afuzzyset)μ(x)μg(x)xx15C20°℃25°℃C15°C20°℃25°℃The degree-of-fit that"About25degree"is"Comfortable temperature"max (min(μa(x), μg(x) ))= max μanB(x)xXis the set of temperatures.15℃20°℃25°C15
15 Degree-of-fit (Closeness) 0 1 15 C 20 C 25 C A: Comfortable temperature (A fuzzy set) 0 1 15 C 20 C 25 C B: About 25 degree (A fuzzy set) The degree-of-fit that “About 25 degree” is “Comfortable temperature” 0 1 15 C 20 C 25 C max{min{ (x), (x) }} max (x) A B x X A B x X µ µ µ ∈ ∈ = X is the set of temperatures. (x) µ A (x) µ B x x x