In this paper our investigation on aimed information, started in 2011, will be completed on fuzzy setting. Here will be given a form of information for fuzzy setting, when it is conditioned and aimed. This information is called "general", because it is defined without using probability or fuzzy measure

Maria, Divari

Vivona, Doretta

English

VIVONA, Doretta

Maria Divari

Archivio della ricerca- Università di Roma La Sapienza

General conditioned and aimed information on fuzzy settingDORETTA VIVONA”Sapienza” - University of RomeDepartment of Basic andApplied Sciences for Engeneeringvia A.Scarpa n.16, 00161 ROMAITALYdoretta.vivona@sbai.uniroma1.itMARIA DIVARI”Sapienza” - University of RomeDepartment of Basic andApplied Sciences for Engeneeringvia A.Scarpa n.16, 00161 ROMAITALYmaria.divari@alice.itAbstract: In this paper our investigation on aimed information, started in 2011, will be completed on fuzzy setting.Here will be given a form of information for fuzzy sets, when it is conditioned and aimed. This information iscalled general,because it is defined without using probability or fuzzy measure.Key–Words: Fuzzy sets, information, conditioning information, aimed information1 IntroductionBy using the concept of general information (i.e. in-formation without probability or fuzzy measure [7, 8,5]), the definition of conditional information [10, 11]and aimed information [12] have been introduced forcrisp sets.It is possible to move to fuzzy setting. In fact thegoal of this paper is to introduce a form of generalinformation J conditioned and aimed by two differ-ent sets, independent of each other with respect to J(J−independence).This measure can be useful when we want to mea-sure information of a set of people with different lev-els of the same illness, treated with different dose of amedicament.The paper is organized in the following way.Sect.2 contains some preliminaries. In Sect.3 in fuzzysetting will be introduced the definition of generalconditional information with a given aim, by meansof axioms. The properties of this information aretraslated in a system of functional equations [1, 2]. InSect.4 the problem is solved, finding a class of solu-tions and a particular solution in J−independent case.Sect.5 is devoted to the conclusion.2 PreliminairesLet X be an abstact space and F the σ−algebra of allfuzzy sets ofX, such that (X,F) is measurable. Basicnotions, notations and operation on fuzzy sets can befound in [14, 9]. Now, the definition of measure ofgeneral information for fuzzy sets is recalled [6].Definition 1 Measure of general information J(·) isa mapping J(·) : F → [0,+∞] such that ∀F, F ′ ∈F :(i)F ⊃ F ′ → J(F ) ≤ J(F ′),(ii)J(∅) = +∞, J(X) = 0.Given a measure of general information J andK,K ′ ∈ F with K 6= K ′,K ∩K ′ 6= ∅, K and K ′ aresaid J− independent (i.e. independent of each otherwith respect to J) if(iii) J(K ∩K ′) = J(K) + J(K ′).3 Statement of the problemIn this paragraph will be introduced measure of gen-eral information when it is conditioned by a givenevent H and it is aimed by a different event S . Fromnow on, the following assumption is considered:let H,S ∈ F ,H 6= S, (1)J(H) 6= +∞, J(S) 6= +∞,H and S are calling conditioning and aiming events,respectively. Now, given a conditioning and aimingsets as in (1), it is introduced the definition of gen-eral information of the set F ∈ F conditioned by Hwith the aim S: this information will be denoted byJH(F → S).Definition 2 Given H and S as in (1), measure ofgeneral information conditioned by H with the aimS is a mappingJH(· → S) : F → [0,+∞]such that ∀F, F ′ ∈ F :(l) F ⊃ F ′ → JH(F → S) ≤ JH(F ′ → S),(ll) JH(∅ → S) = +∞, JH(X → S) = 0.Given a measure JH(· → S) as in Def.3,K,K ′ ∈F with K 6= K ′,K ∩K ′ 6= ∅, K and K ′ are said J−conditional independent with the aim S (i.e. indepen-dent of each other with respect to J conditioned by Hwith the aim S) if(lll) JH((K ∩K ′)→ S) =JH(K → S) + JH(K ′ → S).3.1 The function ΦWith the assumption (1), our study considers thatmeasure JH(· → S) of F ∈ F depends onJ(F ), J(H), J(S), J(F ∩H), J(F ∩S). So, one willfind a function Φ such that:JH(F → S) = (2)ΦJ(F ), J(H), J(S), J(F ∩H), J(F ∩ S),with Φ : T → [0,+∞] and T will be specified later.Putting: x = J(F ), y = J(H), z = J(S), u = J(F∩H), v = J(F ∩ S), with x, u, v ∈ [0,+∞], y, z ∈[0,+∞), x ≤ u, y ≤ u, x ≤ v, z ≤ v, from (2) it isJH(F → S) = Φx, y, z, u, v (3)and T = {(x, y, z, u, v)/x, u, v ∈ [0,+∞], y, z ∈[0,+∞), x ≤ u, y ≤ u, x ≤ v, z ≤ v}.Moreover, setting x′ = J(F ′), u′ = J(F ′ ∩H), v′ = J(F ′ ∩ S), with x′, u′, v′ ∈ [0,+∞], x′ ≤u′, x′ ≤ v′, the properties [(l)−(ll)] of JH(· → S) aretraslated in the following system of functional equa-tions:(e1) Φ(x, y, z, u, v) ≤ Φ(x′, y, z, u′, v′)if x ≤ x′, u ≤ u′, v ≤ v′ ,(e2) Φ(+∞, y, z,+∞,+∞) = +∞ ,(e3) Φ(0, y, z, y, z) = 0 .4 Solution of the problem4.1 General caseFor the system [(e1)− (e3)] it isProposition 3 A class of solution of the system[(e1)− (e3)] isΦh(x, y, z, u, v) = (4)h−1h(x)− h(y)− h(z) + h(u) + h(v)where h is any continuous, strictly increasing functionh : [0,+∞] → [0,+∞] with h(0) = 0, h(+∞) =+∞.Proof: The prof follows easily from the properties ofthe function h.From (3) and (4), given H and S as in (1), mea-sure of general information of any fuzzy set F condi-tioned by H with the aim S isJH(F → S) = h−1h(J(F ))− h(J(H))− (5)h(J(S)) + h(J(F ∩H) + h(J(F ∩ S)where h is any continuous, strictly increasing functionh : [0,+∞] → [0,+∞] with h(0) = 0, h(+∞) =+∞.4.2 J−independenceIn the case of J−independence the system [(e1) −(e3)] must be completed with an extra equation de-duced by the property (lll) :(e4) Φt+ t′, y, z, t+ t′ + y, t+ t′ + z =Φt, y, z, t+ y, t+ z + Φt′, y, z, t′+ y, t′+ z,where t = J(K), t′ = J(K ′), t, t′ ∈ [0,+∞].Among all h of the Proposition 3, only differen-tiable functions are considered. Here it is used thesame procedure of [13].The equation [(e4)] ish−1(h(t+t′)−h(y)−h(z)+h(t+t′+y)+h(t+t′+z))= h−1(h(t)− h(y)− h(z) + h(t+ y) + h(t+ z))+h−1(h(t′)− h(y)− h(z) + h(t′ + y) + h(t′ + z)).Now, the function h will be characterized.Putting y = z,h(h(t+t′)−h(y)−h(y)+h(t+t′+y)+h(t+t′+y))= h−1(h(t)− h(y)− h(y) + h(t+ y) + h(t+ y))+h−1(h(t′)− h(y)− h(y) + h(t′ + y) + h(t′ + y)),i.e. it ish−12 h(t+ t′ + y) + h(t+ t)− 2 h(y) = (6)h−12 h(t+ y) + h(t)− 2 h(y)+h−12 h(t′ + y) + h(t′)− 2 h(y).Settingϕ(t, y) = h−12 h(t+ y) + h(t)− 2 h(y) (7)the equation (6) becomesϕ(t+ t′, y) = ϕ(t, y) + ϕ(t′, y). (8)Fixed y = y∗, the (8) is the classical Cauchy equation[1], whose solution is the continuous function ϕ :ϕ(t, y∗) = λ(y∗)t. (9)So, from (7),λ(y∗)t = h−12 h(t+ y∗) + h(t)− 2 h(y∗) i .e.hλ(y∗)t = 2 h(t+ y∗) + h(t)− 2 h(y∗). (10)If y∗ = 0, as h(0) = 0, from (10), one hashλ(0)t = 2 h(t) + h(t), i .e.hλ(0)t = 3 h(t). (11)Taking inspiration by [1, 2, 3, 4] one will prove thathλ(0)t = 3 h(t) =⇒ λ(0) = 3. (12)Set λ(0) = c, from (11), one will solve the equationh(c t) = 3 h(t); (13)by differentiating ch′(c t) = 3 h′(t) from whichc h′(c t)h(c t)=h′(t)h(t). (14)Settingv(t) =h′(t)h(t), (15)the (14) isv(c t) =v(t)c, ∀ t. (16)The function v(t) = 1t is the unique solution admit-ting a Laurent expansion about 0. By substituing in(15), one obtain the equationh′(t)h(t)=1t(17)whose solution ish(t) = k t, t ∈ [0,+∞], k > 0. (18)By substituing (18) in (13), it is c = λ(0) = 3. So, thefunction h satisfies the following condition:h(3 t) = 3 h(t). (19)From (10),ϕ(x, t) = 3 t = h−12 h(t+ y) + h(t)− 2 h(y)i .e. h(3 t) = 2 h(t+ y) + h(t)− 2 h(y),taking into account (19), it is3 h(t) = 2 h(t+ y)− 2 h(y) + h(t)i .e. h(t) + h(y) = h(t+ y),which is the classical Cauchy equation [1], whose so-lution ish(x) = c x, c > 0. (20)Now, it is possible to give the followingProposition 4 The solution of the system [(e1)−(e4)]isΦ(x, y, z, u, v) = x− y − z + u+ v. (21)Proof: It is easy to check that (21) holds, by applying(20) in the (4).In the independent case, given H and S as in (1),from (21), information of any set A ∈ A conditionedby H with the aim S isJH(A→ S) = J(A)− J(H)− J(S)+ (22)J(A ∩H) + J(A ∩ S).5 ConclusionFirst, by axiomatic way, it has been defined generalconditional information with an aim, on fuzy setting.By using its properties, it has been possible to find aclass of this measure (5).Then, taking into account the J−independenceproperty, it has been obtained a particular measure(22).Acknowledgements: This research was supported byresearch centre CRITEVAT of ”Sapienza” Unicersityof Roma (Italy) and GNFMof MIUR (Italy).References:[1] J. Acze´l, Lectures on Functional Equations andtheir Applications, Academic Press, New Jork1969.[2] J. Acze´l, On Applicatios and Theory of Func-tional Equations, Birka˝user Verlag, Basel 1969.[3] J. Acze´l and Z. Daro´czy, On Measures of Infor-mation and Their Characterization, AcademicPress, New Jork 1975.[4] J. Acze´l and J.G. Dhombres, Functional Equa-tions in several variables, Cambridge UniversityPress 1989.[5] P. Benvenuti, L’opera scientifica, Ed.Univ.LaSapienza, Roma – Italia 2004.[6] P. Benvenuti, D. Vivona and M. Divari, AGeneral Information for Fuzzy Sets, Uncer-tainty in Knowledge Bases, Lectures Notes inComputer Sciences, 521, 1990, pp.307-316,http://dx.doi.org/10.1007/BFb0028117.[7] B.Forte, Measure of information.The general ax-iomatic theory, R.I.R.O., 1969, pp.63-90[8] J. Kampe´ De Fe´riet and B. Forte, Information etProbabilite´, C.R.Acad.Sc.Paris, 265, 1967, pp.110-114, 142-146, 350-353.[9] G.J. Klir and T.A. Folger, Fuzzy Sets, Uncer-tainty and Information, Prentice-Hall Interna-tional Editios, Englewood Cliffs, 1988.[10] D. Vivona and M. Divari, On a conditional infor-mation for fuzzy sets, Proc. AGOP05, 2005, pp.147-149.[11] D. Vivona and M. Divari, Aggregation operatorsfor conditional information without probability,Proc.IPMU08, 2008, pp.258-260.[12] D. Vivona and M. Divari, Aggregation operatorsof general aimed information, Proc.EUSFLAT-LFA 2011, 2011, pp.75-78.[13] D. Vivona and M. Divari, General conditional in-formation with an aim, Proc.8th ASM’14, 2014,pp.96-100.[14] L.A. Zadeh, Fuzzy Sets, Information and Con-trol, 1965, 8, pp.338-353.

General conditioned and aimed information on fuzzy setting

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General conditioned and aimed information on fuzzy setting

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