The Concept of Fuzzy Set and Membership Function and Basic Properties of Fuzzy Set Operation

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1 FORMALIZED MATHEMATICS Volume 9, Number 2, 2001 University of Białystok The Concept of Fuzzy Set and Membership Function and Basic Properties of Fuzzy Set Operation Takashi Mitsuishi Shinshu University Nagano Noboru Endou Shinshu University Nagano Yasunari Shidama Shinshu University Nagano Summary. This article introduces the fuzzy theory. At first, the definition of fuzzy set characterized by membership function is described. Next, definitions ofemptyfuzzysetanduniversalfuzzysetandbasicoperationsofthesefuzzy sets are shown. At last, exclusive sum and absolute difference which are special operation are introduced. MML Identifier: FUZZY 1. The terminology and notation used in this paper have been introduced in the followingarticles:[8],[1],[5],[9],[3],[4],[6],[7],and[2]. 1. Definition of Membership Function and Fuzzy Set Inthispaper Cisanonemptysetand cisanelementof C. We now state the proposition (1) rng(χ C,C ) [0, 1]. Letusconsider C.Apartialfunctionfrom Cto Rissaidtobeamembership functionof Cif: (Def.1) domit = Candrngit [0, 1]. The following proposition is true (2) χ C,C isamembershipfunctionof C. 351 c 2001 University of Białystok ISSN

2 352 takashi mitsuishi et al. Inthesequel f, h, g, h 1 denotemembershipfunctionsof C. Let Cbeanonemptysetandlet hbeamembershipfunctionof C.Aset iscalledafuzzysetof C, hif: (Def.2) It = [:C, h C :]. afuzzysetof C, h,andlet BbeaFuzzySetof C, g.thepredicate A = Bis defined as follows: (Def.3) Foreveryelement cof Cholds h(c) = g(c). afuzzysetof C, h,andlet BbeaFuzzySetof C, g.thepredicate A Bis defined by: (Def.4) Foreveryelement cof Cholds h(c) g(c). Inthesequel AdenotesaFuzzySetof C, f, BdenotesaFuzzySetof C, g, DdenotesaFuzzySetof C, h,and D 1 denotesafuzzysetof C, h 1. One can prove the following propositions: (3) A = Biff A Band B A. (4) A A. (5) If A Band B D,then A D. 2. Intersection, Union and Complement Let Cbeanonemptysetandlet h, gbemembershipfunctionsof C.The functor min(h, g) yielding a membership function of C is defined by: (Def.5) Foreveryelement cof Cholds (min(h,g))(c) =min(h(c),g(c)). Let Cbeanonemptysetandlet h, gbemembershipfunctionsof C.The functor max(h, g) yields a membership function of C and is defined by: (Def.6) Foreveryelement cof Cholds (max(h,g))(c) =max(h(c),g(c)). Let Cbeanonemptysetandlet hbeamembershipfunctionof C.The functor 1-minus h yielding a membership function of C is defined by: (Def.7) Foreveryelement cof Cholds (1-minush)(c) = 1 h(c). a FuzzySetof C, h,andlet BbeaFuzzySetof C, g.thefunctor A Byielding afuzzysetof C,min(h,g)isdefinedasfollows: (Def.8) A B = [:C, (min(h,g)) C :]. afuzzysetof C, h,andlet BbeaFuzzySetof C, g.thefunctor A Byields afuzzysetof C,max(h,g)andisdefinedby: (Def.9) A B = [:C, (max(h,g)) C :].

3 theconceptoffuzzysetandmembership Let Cbeanonemptyset,let hbeamembershipfunctionof C,andlet A beafuzzysetof C, h.thefunctor A c yieldingafuzzysetof C,1-minushis defined by: (Def.10) A c = [:C, (1-minush) C :]. We now state a number of propositions: (6) min(h(c), g(c)) = (min(h, g))(c) and max(h(c), g(c)) = (max(h, g))(c). (7) max(h,h) = handmin(h,h) = handmax(h,h) = min(h,h)and min(f, g) =min(g,f)andmax(f, g) =max(g, f). (8) f = giff A = B. (9) A A = Aand A A = A. (10) A B = B Aand A B = B A. (11) max(max(f, g), h) = max(f, max(g, h)) and min(min(f, g), h) = min(f, min(g, h)). (12) (A B) D = A (B D). (13) (A B) D = A (B D). (14) max(f,min(f, g)) = fandmin(f,max(f, g)) = f. (15) A A B = Aand A (A B) = A. (16) min(f,max(g, h)) =max(min(f, g),min(f,h))andmax(f,min(g,h)) = min(max(f, g), max(f, h)). (17) A B D = (A B) (A D)and A (B D) = A B A D. (18) 1-minus1-minush = h. (19) (A c ) c = A. (20) 1-minus max(f, g) = min(1-minus f, 1-minus g) and 1-minus min(f, g) = max(1-minus f, 1-minus g). (21) (A B) c = A c B c and (A B) c = A c B c. 3.EmptyFuzzySetandUniversalFuzzySet Let Cbeanonemptyset.AsetiscalledanEmptyFuzzySetof Cif: (Def.11) It = [:C, (χ,c ) C :]. Let Cbeanonemptyset.AsetiscalledaUniversalFuzzySetof Cif: (Def.12) It = [:C, (χ C,C ) C :]. Inthesequel XisaUniversalFuzzySetof Cand EisanEmptyFuzzySet of C. One can prove the following two propositions: (22) rng(χ,c ) [0, 1]. (23) χ,c isamembershipfunctionof C.

4 354 takashi mitsuishi et al. Let Cbeanonemptyset.ThefunctorEMFCyieldsamembershipfunction of Candisdefinedasfollows: (Def.13) EMFC = χ,c. Let Cbeanonemptyset.ThefunctorUMFCyieldsamembershipfunction of Candisdefinedasfollows: (Def.14) UMFC = χ C,C. One can prove the following propositions: (24) Foreverymembershipfunction hof Csuchthat h = χ C,C holds [:C, (χ C,C ) C :]isafuzzysetof C, h. (25) Foreverymembershipfunction hof Csuchthat h = χ,c holds [:C, (χ,c ) C :]isafuzzysetof C, h. (26) EisaFuzzySetof C,EMFC. (27) XisaFuzzySetof C,UMFC. Let Cbeanonemptyset.WeseethattheEmptyFuzzySetof Cisa FuzzySetof C,EMFC. Let Cbeanonemptyset.WeseethattheUniversalFuzzySetof Cisa FuzzySetof C,UMFC. Inthesequel XdenotesaUniversalFuzzySetof Cand EdenotesanEmpty FuzzySet of C. One can prove the following propositions: (28) Let a, bbeelementsof Rand f beapartialfunctionfrom Cto R. Supposerng f [a,b]anddomf and a b.let xbeanelementof C.If x domf,then a f(x)and f(x) b. (29) E A. (30) A X. (31) Foreveryelement xof Candforeverymembershipfunction hof C holds (EMFC)(x) h(x)and h(x) (UMFC)(x). (32) max(f, UMF C) = UMF C and min(f, UMF C) = f and max(f,emfc) = fandmin(f,emfc) =EMFC. (33) A X = Xand A X = A. (34) A E = Aand A E = E. (35) A A B. (36) If A Dand B D,then A B D. (37) Forallelements a, b, cof Rsuchthat a bholdsmax(a,c) max(b,c). (38) If A B,then A D B D. (39) If A Band D D 1,then A D B D 1. (40) If A B,then A B = B. (41) A B A.

5 theconceptoffuzzysetandmembership (42) A B A B. (43) If D Aand D B,then D A B. (44) Forallelements a, b, c, dof Rsuchthat a band c dholdsmin(a,c) min(b,d). (45) Forallelements a, b, cof Rsuchthat a bholdsmin(a,c) min(b,c). (46) If A B,then A D B D. (47) If A Band D D 1,then A D B D 1. (48) If A B,then A B = A. (49) If A Band A Dand B D = E,then A = E. (50) If A B A D = A,then A B D. (51) If A Band B D = E,then A D = E. (52) If A E,then A = E. (53) A B = Eiff A = Eand B = E. (54) A = B Diff B Aand D Aandforall h 1, D 1 suchthat B D 1 and D D 1 holds A D 1. (55) A = B Diff A Band A Dandforall h 1, D 1 suchthat D 1 B and D 1 Dholds D 1 A. (56) If A B Dand A D = E,then A B. (57) A Biff B c A c. (58) If A B c,then B A c. (59) If A c B,then B c A. (60) (A B) c A c and (A B) c B c. (61) A c (A B) c and B c (A B) c. (62) 1-minusEMFC =UMFCand1-minusUMFC =EMFC. (63) E c = Xand X c = E. 4. Exclusive Sum, Absolute Difference afuzzysetof C, h,andlet BbeaFuzzySetof C, g.thefunctor A. Byields afuzzysetof C,max(min(h,1-minusg),min(1-minush,g))andisdefinedas follows: (Def.15) A. B = [:C, (max(min(h,1-minusg),min(1-minush,g))) C :]. The following propositions are true: (64) A. B = A B c A c B. (65) A. B = B. A. (66) A. E = Aand E. A = A.

6 356 takashi mitsuishi et al. (67) A. X = A c and X. A = A c. (68) A B B D D A = (A B) (B D) (D A). (69) A B A c B c (A. B) c. (70) (A. B) A B A B. (71) A. A = A A c. Let Cbeanonemptysetandlet h, gbemembershipfunctionsof C.The functor h g yieldsamembershipfunctionof Candisdefinedasfollows: (Def.16) Foreveryelement cof Cholds h g (c) = h(c) g(c). a FuzzySetof C, h,andlet BbeaFuzzySetof C, g.thefunctor A B yielding afuzzysetof C, h g isdefinedby: (Def.17) A B = [:C, h g C :]. References [1] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55 65, [2] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47 53, [3] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4): , [4] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2): , [5] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4): , [6] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3): , [7] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67 71, [8] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17 23, [9] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1): , Received May 18, 2000