![Probabilistic diameter and its properties
www.ijmsi.org 44 | Page
)(,)( uFSupInfuFSupInfTSup qr
BpCr
qr
CrBpyv
=T(FAC(x), FBC(y)).
References:
[1] R. Egbert, Cartesian products of statistical metric spaces, Amer. Math. Soc. Notices 10 (1963), 266-267.
[2] V. Istratescu and I. Vaduva, Products of statistical metric spaces (Romanian), Acad. R. P. Romine Stud. Cerc. Mat. 12 (1961),
567-574.
[3] K. Menger, Statistical metrics, Proc. Nat. Acad. Of Science. U.S.A. 28 (1942), 535-537.
[4] K.Menger, Probabilistic geometry, Pro. Nat. Acad. Of Science. U.S.A. 37 (1951), 226-229.
[5] K.Menger, Geometrie generale (Chap. VII), Memorial des Sciences Mathematiques, No. 124, Paris 1954.
[6] B. Schweizer, Equivalence relations in probabilistic metric spaces, Bulletin of the Polytechnic Inst. Of Jassy 10 (1964), 67-70.
*******](https://image.slidesharecdn.com/g037040044-160112085233/85/Probabilistic-diameter-and-its-properties-5-320.jpg)
Probabilistic diameter and its properties.
- 1. International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 3 Issue 7 || November. 2015 || PP-40-44 www.ijmsi.org 40 | Page Probabilistic diameter and its properties. Dr. Ayaz Ahmad Reader,Deptt. of Mathematics Millat College, Darbhanga, BIHAR, PIN;846004 Abstract: In this paper ,we discuss on probabilistic diameter and some of its basic properties. Key Words: Probabilistic diameter, Probabilistic distance, Distribution function. I. Introduction Probabilistic metric spaces were first introduced by K. Menger in 1942 and reconsidered by him in the early 1950’s B. Schweizer and A. Sklar have been studying these spaces, and have developed their theory in depth. In probabilistic metric spaces the notion of distance between two points x and y is replaced by a distribution function Fxy. Thus, the distance between points as being probabilistic with Fxy(t) representing the probability that the distance between x and y is less than t. Definition: 1.1. Let (S, F,T) denote a Menger space with a continuous t-norm and A be a nonempty subset of S. The function DA, defined by DA (x) = )( , tFInfSup pq Aqpxt , is called the probabilistic diameter of A. Properties of the probabilistic diameter. Defenition.1.2.A nonempty subset A of S is bounded if Supx DA(x) = 1, semi-bounded if 0 < Sup DA(x) < 1, and unbounded if DA = 0. P1 ) The function DA is a distribution function P2 ) If A is a nonempty subset of S, then DA = H if and only if A consists of a single point. P3 ) If A and B are nonempty subsets of S and A B, then DA DB. Theorem.1.3. If A and B are two nonempty subsets of S such that A B = , then DAUB(x + y)T(DA(x), DB(y))…………. (1.1) Proof. Let x and y be given. To establish (1.1) we first show that ))(),(()( ,,, yFInfxFInfTyxFInf pq Bqp pq Aqp pq BAqp ………… (1.2) There are two distinct cases to consider: Case (1). )()( , yxFInfyxFInf pq Bq AP pq BAqp ………………(1.3) Now for any triple of points p,q and r in S, we have Fpq(x + y) T(Fpr(x), Frq(y)). Taking the Infimum of both sides of this inequality as p ranges over A, q ranges over B and r ranges over A B BAr Bq Ap rqprpq BAqp yFxFTInfyxFInf )).(),(()( , However, since T is continuous and non decreasing ,we obtain, )(),()( ,,, yFrqInfxFInfTysFInf Bqr pr Arp pq BAqp . Case (2). )()( , yxFInfyxFInf pq Bq Ap pq BAqp . In this case of the equalities, )()( ,, yxFInfyxFInf pq Aqp pq BAqp Or
- 2. Probabilistic diameter and its properties www.ijmsi.org 41 | Page )()( ,, yxFInfyxFInf pq Bqp pq BAqp must hold. If the first equality holds, we have )(),()( ,, yHxFInfTyxFInf pq Aqp pq BAqp )(),( ,, yFInfxFInfT pq Bqp pq Aqp The same argument works for the second equality. This establishes (1.2.). Finally using the fact that the rectangle {(s, t): 0 ytx 0, } is contained in the triangle {(s, t): s, t 0, s + t < x + y}, the inequality (1.2) and the continuity of T. we have DAB(x+y) = )( , tsFInfSup pq BAqpyxtS )( , txFInfSup pq BAqp yt xS )(,)( ,, tFInfSupsFInfSupT pq Bqpvt pq AqpXS = T(DA(x), DB(y)). Theorem.1.4. If A is a nonempty subset of S, then DA = D A , where A denotes the closure of A in the ε – λ topology on S. Proof. Since A A , if follows from property P1 that DA A D . Let > 0 be given. In view of the uniform continuity of with respect to the Levy metric L there exists an ε > 0 and a λ > 0 such that for any four points p1, p2, p3 and p4 in S, L(Fp1p2, Fp3p4) < When ever Fp1p3 (ε) > 1 – λ and Fp2p4(ε) > 1 – λ. Next, with each point P and A associate a point P( p ) in A such that Fp pp > 1 – λ. Then, in view of the above for any pair of points p and q A, L(F ppF qqpp ),()( ) < . In particular, for all t we have, F )()( )()( tFt qpqqpp . Let A = {p( ):) App . Then since A A, ))(()()( ,, tqqppInftFInf Aqp qP AqP = )()( ,, tFInftFInf pq Aqp pq Aqp . Now, taking the suprimum for t < x of the above inequality yields D )()()( ,, tFInfSuptFInfSupx pq Aqpxt qp Aqpxt A = )( , tFInfSup pq Aqpxt - = DA (x - -) - Since the above inequality is valid for all and, since DA is left continuous. It follows that A D (x) DA(x).Hence A D (x) = DA(x),
- 3. Probabilistic diameter and its properties www.ijmsi.org 42 | Page Hence, the proof is complete. Definition.1.5. Let A and B be nonempty subsets for S. The probabilistic distance between A and B is the function FAB defined by FAB(x) = )(,)( tFSupInftFSupInfTSup pq BpBq pq BqApxt ……… (1.4) The following are the properties of FAB. P4. FAB is a distribution function. P5. If A and B are nonempty subsets of S, then FAB = FBA. P6. If A is a nonempty subset of S, then FAA = H. Theorem.1.6. If A and B are nonempty subsets of S, then FAB = F BA . Proof. It is sufficient to show that FAB = F BA since this result together with property P5 yields. Hence, FAB = FA BAABABB FFF . Now, we first show that BBSinceFF ABBA . for all t, )()( tFSupInftFSupInf qp ApBq pq ApBq ………….. (1.5) Let > 0 be given. The argument given in the proof of Theorem( 1.4) establishes that for each point Bq , there exists a point q( q ) in B such that for all t, )()( tFtF qpqqp . Let B = {q( q ) : Bq }. Since B B we have, )()()( tFSuptSuptFSup pq BqBq qp Bq )(tFSup pq Bq . Consequently, )()(sup tFSupInftFInf pq BpAp qp BqAp . Now ,taking the supremum on t < x of the above inequality,yields for any , f(x) )()( tFSupInfSuptFSupInfSup qp BqApxx pq BqApxt df = )()( xgdftFSupInfSup qp BqAqxt . since both f and g are left-continuous and is arbitrary, it follows that f(x) g(x). This together with (1.2.5), and the continuity of T yields. FAB(x) = )(,)( yFSupInfSuptFSupInfSupT pq ApBqxt pq BqApxt )(,)( tFSupInfSuptFSupInfSupT qp ApBqxt qp BqApxt = )()(,)( xFtFSupInftFSupInfTSup BAqp ApBq qp BqApxt . A similar argument shows that F BA FAB. Theorem.1.7. If A and B are nonempty subsets of S,
- 4. Probabilistic diameter and its properties www.ijmsi.org 43 | Page then FAB = H, if and only if BA . Proof. Suppose FAB = H and let ε > 0 be given. Then 1 = FAB (ε) = T )(,)( tFSupInfSuptFSupInfSup pq ApBqt pq BqApt = )()( pq ApBq pq ApBqt FSupInftFSupInfSup . So that for any q B and every λ > 0 there exists a point p in A for which Fpq (ε) > 1 – λ. Consequently, q is an accumulation point of A and we have B A . A similar argument shows that A B . Conversely, suppose BA . Then in view of P6 and Theorem 1.7 We have, FAB = AABA FF = H. Theorem.1.8. If A, B and C are nonempty subsets of S, then for any x and y FAB (x - y) T(FAC(x), FBC(y)). Proof. Let u and v be given. Then for any triple of points p, q and r in S we have Fpq(u+v) T(Fpr(u), Fqr(v)). Making use of the continuity and monotonicity of T we have the following inequality : )(),()( vFSupInfuFSupTvuFSup qr BqCr pr Cr pq Bq . Consequently, )(,)()( vFSupInfuFSupInfTvuFSupInf rq BqCr pr CrAp pq BqAp . Similarly, )(,)()( vFSupInfuFSupInfTvuFSupInf rq CqBr pr ArCp pq AqBp Therefore, since T is associative, we have )_(,)( vuFSupInfvuFSupInfT pq AqBr pq BrAp )(,)( uFSupInfuFSupInfTT pr ApCr pr CrAp , T )(,)( vFSupInfvFSupInf qr BqCr qr CrBq . So, we have FAB(x+ y) = ,)( vuFSupInfTSup pq BqApyxvu )( vuFSupInf pq ApBq )(,)( vuFSupInfvuFSupInfTSup pq BqBq pq BqAp yv xu =T ,)(,)( uFSupInfuFSupInfTSup pr ApCr pr CrApxu
- 5. Probabilistic diameter and its properties www.ijmsi.org 44 | Page )(,)( uFSupInfuFSupInfTSup qr BpCr qr CrBpyv =T(FAC(x), FBC(y)). References: [1] R. Egbert, Cartesian products of statistical metric spaces, Amer. Math. Soc. Notices 10 (1963), 266-267. [2] V. Istratescu and I. Vaduva, Products of statistical metric spaces (Romanian), Acad. R. P. Romine Stud. Cerc. Mat. 12 (1961), 567-574. [3] K. Menger, Statistical metrics, Proc. Nat. Acad. Of Science. U.S.A. 28 (1942), 535-537. [4] K.Menger, Probabilistic geometry, Pro. Nat. Acad. Of Science. U.S.A. 37 (1951), 226-229. [5] K.Menger, Geometrie generale (Chap. VII), Memorial des Sciences Mathematiques, No. 124, Paris 1954. [6] B. Schweizer, Equivalence relations in probabilistic metric spaces, Bulletin of the Polytechnic Inst. Of Jassy 10 (1964), 67-70. *******