Remarks on the generalized probability of the bifuzzy event

Tadeusz Gerstenkorn , Jo a n n a Gerstenkorn *

REMARKS ON THE GENERALIZED PROBABILITY

OF THE BIFUZZY EVENT

A b str a c t. The presentation is a continuation o f a paper at M S A ’04 (T. Gerstenkorn,

J.

K ey w o rd s: b ifu zzy (intuitionistic) event, generalized-probability, fu zzy set.

I. INTRODU CTIO N

In 1965 L. Zadeh introduced the notion o f a fuzzy set as a generalization of

the Cantor’s set that was dominated to this moment in science. Conception o f the

fuzzy set allowed the mathematical modelling o f not sharp formulated notions,

usable very often in the so-called soft-sciences, as e.g. economy, humanistics,

law, medicine. The characteristics o f the fuzzy set comes after the introducing

the so-called membership function of an element x o f a considered space X to

a fuzzy set A, defined as follows:

A = {(x, /лл (х))1 x ' . x & X ) ,

(1)

where X -> [0,1], i.e. f.tA(x ) is a characteristic function o f the set A, with this

difference to the characteristic function (flA (x) o f the Cantor s set A that it can

take all the values o f the interval [0, 1] and not only the values 0 or 1.

P rofessor em eritus o f the Ł ód ź U n iv., Fac. o f M athem atics, Prof. o f U niv. ot Trade in Łódź.

Already in 1968 L. Zadeh introduced also and discussed the idea o f prob­

ability o f the fuzzy event in relation to the fuzzy set. He has brought out the dif­

ference between the fuzziness and accidentalness.

Assuming a probability space (.X, .с/, P), the probability o f the fuzzy event

A

has been defined by

P ( A ) = \ n A( x ) P ( d x ),

(2)

x

where the meaning o f juA( x ) and o f the space X is as above and . с / is a ď -

Algebra.

One can easy see that the membership function

has replaced in (2)

the characteristic function <pA(x ) o f the normal (crisp) set A.

The considerations relating to the probability of the fuzzy event were con­

tinued in following years by many authors. A review o f these problems one can

find in papers o f T. Gerstenkorn and J. Mańko (1994, 1996).

The idea o f the fuzzy set has been developed, extended and generalized for

the so-called bifuzzy set, called by K. Atanassov (1983, 1985, 1986) the in-

tuitionistic set.

The generalization consisted in introducing to considerations, besides the

membership function, also the so-called non-membership function o f the ele­

ment x to the set A, i.e. one has proposed the following definition o f that set:

A = {(M

( * ) . ( * ) ) / x : x e X } ,

(3)

where /лл ( х ) , v A( x ) : X —> [0,1] with the condition

0 < /uA( x ) + vA(x) < 1 f o r x e X .

(4)

The above statement assumes the existence o f the function

* л ( х ) = 1 - М л ( х ) - ул ( х )

called the intuitionistic index and the number я A{x) e [0,1] is treated as

a measure o f the hesitancy (indecision) connected with a valuation o f the degree

o f membership or non-membership o f the element x to the set A. Examples of

such interpretation and procedure one can find, e.g. in T. Gerstenkorn and

J. Mańko (2006a, 2006b).

Alike as for the fuzzy set, the probability o f the intuitionistic fuzzy set has

been introduced. The review o f some standpoints in this question was presented,

e.g. in T. Gerstenkorn and J. Gerstenkorn (2007) and in papers o f T. Gerstenkorn

and J. Mańko (1995-2006).

II. THE GENERALIZED PROBABILITY OF TIIE FUZZY EVENT

In 1978 Philippe Smets proposed the so-called g-probability o f the fuzzy

event as a generalization o f the probability of that event given by L. Zadeh

(1968, (5), p. 423). The definition is the following:

P { A ) = \g { /.iA( x ) ) d F ( x ) ,

(6)

x

where g is a monotonic and non-decreasing function with conditions:

g (0 ) = 0, g ( l) = 1 and F ( x ) is a distribution function in the probability space

(*, s f, P).

But two years later Stanisław Heilpem considered also the g-probability and

analysed inquiringly its properties.

In 1982 Ph. Smets came back to his considerations with the g-probability but

in this case he presented the axiomatic grounds o f its correctness and concor­

dance with the axiomatics o f Kolmogoroff.

III. THE GENERALIZED PROBABILITY OF THE BIFUZZY

EVENT

Definition. Let X be any set and S f a cr -Algebra o f its bifuzzy sets in X.

Then by g-probability o f the bifuzzy event A e S t we call a non-negative func­

tion Р determined on A with the values on [0,1], as follows:

P(A) = Jg C M * )-

M )P ( d x ),

(7)

where g is a monotonic, non-decreasing function with conditions. g(0,0) - 0,

£(1,1) = 1 and P i s a probabilistic measure on X.

Similarly as it was done by Ph. Smets, we show that the function (7) fulfils

the KolmogorofPs axioms.

The fulfilment o f this condition is evident in view o f the postulated condi­

tions for f.i(x) and v(x).

Axiom 2: P ( X ) = \.

It suffices to notice that

(

) = 1 and vx ( x ) = 0 and to take (7).

Axiom 3:

Let

A

X

and

B

X

and

A n B = {<0},

then

P ( A <J B ) = P ( A ) + P ( B )

(the events A and В are bifuzzy ones and being excluded.).

We assume that:

(*) = H

( * ) л M

W ' thc °Peration minimum,

(8)

И

u /i(x ) = /*a(x ) v / /jv(x ) ' the °P eration maximum.

(9)

Proof. The bifuzzy events A and В arc excluded by assumption therefore the

sets A and В are disjoint. It means that Vx e X the membership function

/■1Аглв(х ) ~ 0, i.e. /.

(

)

jUB(x ) = 0, whereas the non-membership function

v ^ g i * ) has the value 1 for the product of the bifuzzy events A and B. If we

assume that the element x <£ A then we form the set X A = { x : /.iA(x) = 0 and

vA (x) = 1}. But if we take x £ B then we set up the set X B = { x : f.iB (x) = 0

and vfi(x) = l}. Let us notice that in account of V x G X A Г \ Х B it will be

/JA^ B(x) = 0

and

at

the

same

time

уж л Д * ) = 1*

Then

als0

g ( / W * ) > vA ,* (* )) = 0 - Next

there is / / ^ ( x ) = f i B( x) and

\ / x e X B there is M

^

(x ) = M

(x ) and similarly for vAuB( x ) (it is evident

regarding (9)).

Therefore

P ( A ) = J g(M A( x ) , v A( x ) ) P { d x ) =

J

g (/J A( x ) , v A( x ) )P (d x ) =

X X H

=

*ЛиВ( х ) ) Р № ) ,

(/.iA( x ) > 0 only if /uB( x ) = 0 , i.e. if x e X B).

Similarly

Therefore

P ( A

B) = \ j + J \ ё ( / лАив(х )‘>

= P ( A ) + P (B ).

IV. T H E G E N ER A LIZED CO ND ITIO N A L PR O B A B ILITY O F TH E

BIFUZZY EVENT

If there are given two bifuzzy events A and В defined on X then we present

the g-probability o f the event A under the given event В traditionally , i.e.

Р ( А \ В ) - П *

п ^ g )- .

0 0 )

assuming the foregoing conditions on g and P ( B ) > 0 .

Proof. It is sufficient to notice that the probability P ( B) fulfils the axioms

° f Kolmogoroff, as it was shown above and in the domain o f bifuzzy sets, we

have

A n B = {(//л (л) л M

W . v/< W v

W ) ' x : x e

■

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Tadeusz Gerstenkorn, Joanna Gerstenkorn

UW AGI O U O G Ó LN IO N Y M PR A W D O PO D O B IEŃ STW IE ZD A R ZEN IA D W O IST O R O Z M Y T E G O

Niniejsza prezentacja jest kontynuacją pracy pt. Probability o f fu zzy event. Review

o f problems (Prawdopodobieństwo zdarzenia rozmytego. Przegląd zagadnień), przed­

stawionej na WAS'05 Acta Univ. Lodz., Folia Oeconomica 2007.

W 1978 r. Philippe Smets zaproponował tzw. ^-prawdopodobieństwo zdarzenia rozmytego jako pewne uogólnienie prawdopodobieństwa tegoż zdarzenia podanego Przez Lotfi Zadeha w 1968 r.

W 1980 r. Stanisław Heilpem także rozważał g-prawdopodobieństwo i analizował jego własności.

W 1982 r. Ph. Smets ponownie i szeroko rozpatrywał ^-prawdopodobieństwo i do­ wodził jego aksjomatycznych własności.

W przedstawianym opracowaniu pragniemy rozpatrzyć g-prawdopodobieństwo zda­ rzenia dwoistorozmytego (intuicjonistycznego) i jego własności jako zgodne z aksjoma- tyką Kołmogorowa.