Product Possibility Space with Finitely Many Independent Fuzzy Vectors (Information and mathematics of non-additivity and non-extensivity : non-additivity and convex analysis)

Product

Possibility

Space with

Finitely

Many

Independent Fuzzy

Vectors

aKakuzoIwamura,

bMasami

Yasuda, cMasayuki Kageyama,

dMasami

Kurano

aDepartmentofMathematics, Josai University,Japan

bFacultyof Science, Chiba University,Japan

cGraduate SchoolofMathematics andInformatics,ChibaUniversity, Japan

dFacultyofEducation, ChibaUniversity, Japan

kiwamura@josai.ac.jp, yasuda@math.s.chiba-u.ac.jp, kage@graduate.chiba-u.ac.jp, kurano@faculty.chiba-u.jp

Abstract

This paper discussesabout productpossibilityspaoeand independentfuzzyvariables.

Keywords: uncertainty theory, possibility measure,productpossibility space, fuzzy variable, fuzzy

vector,independence

1

Introduction

Fuzzyset

was

first introducedby Zadeh [24] in 1965. Thisnotion has beenveryuseful in human decision

making under uncertainty. We

can

see

lotsof papers which

use

this fuzzyset theory in K.Iwamura and

B.Liu [3] [4], B.Liu and K.Iwamura [18] [19] [20], X.Ji and K.Iwamura [9], X.Gao and K.Iwamura [2],

G.WangandK.Iwamura [22], M.Wen and K.Iwamura [23]and others. We also have

some

books

on

fuzzy

decision making under fuzzy environments such as D.Dubois and H.Prade [1], H-J.Zimmermann [26],

M.Sakawa [27],J.Kacprzyk [10], B.Liu andA.O.Esogbue [17].

RecentlyB.Liu has founded a frequentionist fuzzy theorywith hugeamount of applications in fuzzy

mathematical programming. Wesee it in books such

as

B.Liu $[$12] [16] [13]. In his book $[$13] published

in 2004, B.Liu [13] has succeeded in establishing

an

axiomaticfoundation for uncertainty theory, where

they proposed a notionof independent fuzzy variables.

In this paper, wediscuss onpossibility axioms, $\infty nstmct$product possibility space andshowthatwe

not only have finitely many independent fuzzyvariables$[$6$]$ but also

we

have finitely manyindependent

The rest of the paper is organized

as

follows. The nextsection providesabrief review ontheresults

ofpossibility

measure

axioms with definitions offuzzy variables, fuzzyvectors, independenceof thetwo.

Section 3 presents how

we

canget finitely manymdependent fuzzy vectors under possibility

measures.

2

Possibility

Measure

and

Product

Possibility Space

We start with the axiomatic definitionof possibility

measure

given byB. Liu [13] in 2004. Let $\Theta$ be

an

arbitrary nonempty set,

and

let $\varphi(e)$ be the powerset of$\Theta$.

The three axioms for possibility

measure are

listed

as

follows:

Axiom 1. Pos$\{\Theta\}=1$

.

Axiom 2. Pos$t\emptyset\}=0$.

Axiom 3. Pos$\{\bigcup_{t}A_{i}\}=\sup_{i}$Pos$\{A_{\iota}\}$

for

any collection $\{A_{i}\}$ in$\varphi(\Theta)$

.

We call Pos a possibility

measure

over $\Theta$ if it satisfies these three axioms. We call the triplet

$(\Theta,$ $\varphi(\Theta)$,Pos$)$ a possibilityspace.

Theorem 2.1 $(B. Liu, 2004)$

.

Let $e_{i}$ be nonempty sets on which $Pos_{i}\{\cdot\}SatiS\mathfrak{h}r$ the three axioms,

$i=1,2,$$\cdots,$$n$, respectively, and let $\Theta=\Theta_{1}\cross\Theta_{2}\cross\cdots x\Theta_{n}$

.

DefinePos$\{\cdot\}$ by

$Pos\{A\}=$

(1)

$\sup_{(\theta_{1},\theta_{2},\cdots,\theta_{n})\in A}Pos_{1}\{\theta_{1}\}\wedge Pos_{2}\{\theta_{2}\}\wedge\cdots\wedge Pos_{n}\{\theta_{n}\}$

for each $A\in\varphi(e)$

.

Then Pos$\{\cdot\}$ satisfies the three axioms for possibility

measure.

Therefore, anewly definedtriplet $(\Theta,$$\varphi(\Theta)$,Pos$)$ isapossibilityspace. Pos$\{\cdot\}$isapossibility

measure

on $\varphi(e)$. We callit product possibility

measure

derived from $(\Theta_{i}, Pos_{i}\{\cdot\}, \varphi(\Theta_{i})),i=1,$ $\cdots$,$n$

.

We call

$(e,$$\varphi(e)$, Pos$)$ theproduct possibilityspace derived from $(\Theta_{i}, Pos_{i}\{\cdot\}, \varphi(\Theta_{i})),$$i=1,$$\cdots$,$n$.

Lemma 2.1 Let$0\leq a_{i}\leq 1$ and let$\epsilon>0$,

for

_{$1\leq i\leq n$}

.

Thenwe get

$(\epsilon+a_{1})\wedge(\epsilon+a_{2})\wedge\cdots\wedge(\epsilon+a_{n})\leq\epsilon+a_{1}\wedge a_{2}\wedge\cdots\wedge a_{n}$ (2)

Lemma 2.2 Let$A_{i}\in\varphi(e_{i})$

for

$1\leq i\leq n$. Thenwe get$A_{1}\cross\cdots\cross A_{n}\in\varphi(\Theta)$ and

Pos$\{A_{1}\cross\cdots\cross A_{n}\}=$

Posl

$\{A_{1}\}\wedge Pos_{2}\{A_{2}\}\wedge\cdots\wedge Pos_{n}\{A_{n}\}$

.

(3)

Lemma 2.3 Pos$\{\}$

on

$\varphi(e)$ at (1) is monotone with _$7tspect$ to setinclusion, i. e.,

for

any sets $A,$ $B\in$

$\varphi(\Theta)$ with$A\subset B$, weget

Afuzzyvariableis definedas afunction from$\Theta$ ofapossibilityspace $(\Theta,$ $\varphi(\Theta)$,Pos$)$tothe setofreals

$\mathcal{R}$

.

An _$n$-dimensional fuzzy vectoris defined

as

a function from $e$ ofa possibility space $(\ominus,$$\varphi(\Theta)$,Pos$)$

to an $n$-dimensional Eucledian space $\mathcal{R}^{n}$

.

Let $\xi$ be a fuzzy variable defined on the possibility space

$(\Theta,$$\varphi(e)$,Pos$)$.

Then its membership function is derived through the possibility

measure

Posby

$\mu(x)=Pos\{\theta\in e|\xi(\theta)=x\}$, $x\in\Re$

.

(5)

Theorem 2.2 $(B. Liu, 2004)$. Let$\mu$: $\Rearrow[0,1]$ be a

function

utth$\sup\mu(x)=1$. Then there is

a

fuzzy

variable whose membership

_function

is$\mu$

.

The fuzzyvariables$\xi_{1},$$\xi_{2},$$\cdots$ ,$\xi_{m}$

are

said tobe independentifand only if

Pos$\{\xi_{i}\in B_{1}, i=1,2, \cdots, m\}=\min_{1\leq:\leq m}$Pos$\{\xi_{i}\in B.\}$

for any subsets $B_{1},$ $B_{2},$ $\cdots,$$B_{m}$ of the set ofreals $\mathcal{R}$

.

The fuzzy vectors _{$\xi_{i}(1\leq i\leq m)$}

are

said to be

independentifand only if

Pos$\{\xi_{i}\in B_{i},i=1,2, \cdots,m\}=\min_{1\leq:\leq m}$Pos$\{\xi_{i}\in B_{\dot{*}}\}$

for any subsets$B_{i}\in \mathcal{R}^{n_{*}}(1\leq i\leq m)$. Hereafter we use $a\wedge b$inplaceof$\min\{a, b\}$.

Note 2.1 : We have fuzzy variables$\xi_{1},$$\xi_{2}$ whicharenot independent$(Liu[1SJ)$. Let$\Theta=\{\theta_{1}, \theta_{2}\}$, Pos$\{\theta_{1}\}=$

$1$,Pos$\{\theta_{2}\}=0.8$ and

define

$\xi_{1},$$\xi_{2}$ by

$\xi_{1}(\theta)=\{\begin{array}{l}0, if \theta=\theta_{1}1, if \theta=\theta_{2},\end{array}$ $\xi_{2}(\theta)=\{\begin{array}{l}1, if \theta=\theta_{1}0, if \theta=\theta_{2}\end{array}$

Then

we

have Pos$\{\xi_{1}=1,\xi_{2}=1\}=$ Pos$\{\emptyset\}=0\neq 0.8$ A$1=$ Pos$\{\xi_{1}=1\}$ APos$\{\xi_{2}=1\}$.

3

Finitely

Many Independent

Fuzzy

Vectors

Let $\xi$

.

be a fuzzy vector from a possibility space $(e_{i},$$\varphi(e_{i})$,

Pos:

$)$ to the $l_{i}-$th dimensional Eucledian

space $\mathcal{R}^{l_{i}}$, for $i=1,2,$

$\cdots$,$n$. Define $e$ by

$e=e_{1}x\Theta_{2}\cross\cdots\cross\Theta_{n}$ (6)

and$\tilde{\xi}_{1}$ on $e$by

$\overline{\xi}_{i}(\theta)=\xi_{i}(\theta_{i})$ for any $\theta=(\theta_{1}, \theta_{2}, \cdots, \theta_{n})\in e,$ $1\leq i\leq n$. (7)

Theorem 3.1 Thefuzzy vectors$\tilde{\xi}_{i}(1\leq i\leq n)$ given above are independentfuzzy vectors, i. e.,

Pos$\{\tilde{\xi}_{i}(\theta)\in B_{i}(1\leq i\leq n)\}=$Pos$\{\tilde{\xi}_{1}\in B_{1}\}\wedge$ Pos$\{\tilde{\xi}_{2}\in B_{2}\}\wedge\cdots\wedge$Pos$\{\tilde{\xi}_{n}\in B_{n}\}$ (8)

4

Conclusion

We have shownthatAxiom. 4 inB.Liu([13])

can

be proved through Axiom. 1 ,2and 3. Wehaveproved

the fact that there exists finitely manyindependent fuzzyvectors. Through these proohwehave shown

that for possibility

measures

existence of finitely manyindependent fuzzyvectors depends

on

product

possibility space. Although the notion of independence was discovered by L.A.Zadeh and others([25])

under the term of”noninteractiveness” or‘’unrelatedness”, ournotion ofindependencethroughproduct

possibilityspacehave broughtaboutagrandworld offuzzyprocess,hybridprocess and uncertainprocess

of B.Liu([15]).

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