UNION AND MEET OF AN INFINITE NUMBER OF TYPE-2 FUZZY SETS Текст научной статьи по специальности «Математика»

2021 ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА Т. 17. Вып. 2

ПРИКЛАДНАЯ МАТЕМАТИКА. ИНФОРМАТИКА. ПРОЦЕССЫ УПРАВЛЕНИЯ

ПРИКЛАДНАЯ МАТЕМАТИКА

UDC 004.8 MSC 03Е72

Union and meet of an infinite number of type-2 fuzzy sets* О. V. Baskov

St. Petersburg State University, 7-9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation

For citation: Baskov О. V. Union and meet of an infinite number of type-2 fuzzy sets. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2021, vol. 17, iss. 2, pp. 108-119. https://doi.org/10.21638/11701/spbul0.2021.201

The article examines the infimum and supremum of an infinite number of fuzzy numbers. It is shown that familiar properties of these operations, which are valid for real numbers, may apply to fuzzy numbers only under certain conditions. A formula for computing the infimum and supremum of any set of fuzzy numbers is provided. Since the union and meet of type-2 fuzzy sets are defined via the infimum and supremum of fuzzy numbers, all the results obtained are applicable to these operations as well.

Keywords: type-2 fuzzy sets, union, meet, fuzzy numbers, infimum, supremum.

1. Introduction. Fuzzy sets introduced by L. A. Zadeh fl] have numerous applications in various fields of research due to their ability to deal with uncertain information. Usual crisp sets may be characterized by an indicator function that takes two values: 1 for elements belonging to a set, and 0 for all others. In fuzzy sets this function is called membership function, and it is permitted to take any value between 0 and I. Higher values of the membership function correspond to higher degrees of certainty in whether an element should be part of a set. When the elements are the reals, fuzzy sets are called fuzzy numbers.

The next step in generalization of membership is allowing the values of the membership function themselves to be fuzzy [2]. So in type-2 fuzzy sets membership functions map elements to the fuzzy numbers. Some studies [3, 4] have found that increased fuzziness makes type-2 fuzzy sets better suited for certain tasks than type-1 fuzzy sets. Thus, it is undoubted that type-2 fuzzy sets will have many applications, so studying their properties is an important task.

M. Mizumoto and K. Tanaka [5] examined the set-theoretic operations on a finite number of type-2 fuzzy sets. In some areas, for instance, in an axiomatic approach to

* The work is supported by Russian Foundation for Basic Research (project N 20-07-00298-a). © St. Petersburg State University, 2021

Pareto set reduction [6], the need arises to operate on infinite number of type-2 fuzzy sets. In this case the results from [5, 7] are not directly applicable. The aim of this paper is to determine the conditions under which the set-theoretic operations preserve their usual properties when applied to an infinite number of type-2 fuzzy sets. And since these operations are closely related to operations on the values of membership functions, i. e. on fuzzy numbers, we simultaneously study the properties of infimum and supremum of an infinite set of fuzzy numbers.

Operations on type-2 fuzzy sets are derived using Zadeh's extension principle. Such definitions are inconvenient for direct computations, so various algorithms and formulae exist to simplify performing set-theoretic operations on certain classes of type-2 fuzzy sets. One such formula was given by N. N. Karnik and J. M. Mendel [7] for computing join and meet of a finite number of type-2 fuzzy sets. It is applicable to a broad class of type-2 fuzzy sets, namely, the sets with strongly normal and convex values of membership functions. Our main result generalizes this formula to accept infinite, possibly uncountable number of type-2 fuzzy sets.

2. Preliminaries. Let X be the universal set of objects of any kind. A (type-1)

fuzzy set A over X is a set of pairs (x,yA(x)), where x G X, yA(x) G [0; 1]. The number yA(x) is called a degree of membership of an element x in the fuzzy set A. The statement "x certainly belongs to A" corresponds to yA(x) = 1, the assert ion x G A is written as yA(x) = 0. Values of yA(x) between 0 and 1 represent uncertainty about whether x should be part of the set A or not: the higher the value цA(x), the more confident we are that x G A. The function цА : X ^ [0; 1] is called the membership function of the fuzzy set A. Since a fuzzy set is fully determined by its membership function, we will often use them interchangeably.

Let A and В be two fuzzy sets over the universal set X. Their union AuB, intersection А П B, and complement A are defined as follows: /мив(х) = maх{/лА(х); /лв(х)}, ЦАс\в{х) = min {цА(х)\ дв(ж)}, /~1д(х) = 1—/лА(х). It is said that А С В if цА(х) < Цв{х) for every x G X.

An a-cut of a fuzzy set цА is the crisp set Aa = {x G X: yA(x) > a}.

A height of a fuzzy set is hA = sup yA(x). If hA = 1, the fuzzy set цА is called

xex

normal. If furthermore yA(x) = 1 for some x G X, the fuzzy set цА is called strongly normal.

A fuzzy set ^a over a convex set X is itself called convex if for any Л g (0; 1) and x,y G X ft is toe that цА (Лx + (1 — Л)у) > min {уA(x); цА(у)}.

A fuzzy number is a fuzzy set over the reals R. For fuzzy numbers the convexity condition may be rewritten in an easier form: a fuzzy number ц is convex if ц(у) > min{^(x); ц(г)} for any x <y < z.

A fuzzy number ц is upper semi continuous if its a-cuts are closed for any a G [0; 1]. Observe that upper semicontinuous normal fuzzy numbers are necessarily strongly normal.

Operations on fuzzy numbers are derived using Zadeh's extension principle [2]: given some binary operation о от reals, one may define hp0q(x) = sup min (u); цq(v)} for

uov=x

fuzzy numbers P and Q. Thus, if ц and v are two fuzzy numbers, their minimum ц Л v and maximum ц V v are

(у, Л v)(x) = sup min {p(u); v(v)} , (^ V v)(x) = sup min {p(u); v(v)} .

min{u,v}=x max{u,v} = x

We will also use the complement operator: (—y)(x) = ц(1 — x).

A type-2 fuzzy set B over the universal set X is given by its membership function jB : X x [0; 1] ^ [0; 1], which now takes two arguments. The value jB (x,u) may be thought of as a degree of certainty that the membership degree of x in B should be equal to u. If the element x is fixed, then jB (x, u) can be viewed as a fuzzy number over [0; 1]. Thus, it can be said that in type-2 fuzzy sets the degree of membership of every element is given by a fuzzy number, while in type-1 fuzzy sets membership degrees were ordinary-real numbers. Hence, it is natural to define union and intersection of type-2 fuzzy sets via maximum and minimum of fuzzy numbers.

B

every x G X its membership function jB (x,u) is a normal (convex, etc.) fuzzy number over [0; 1].

Following [5], we define a meet of type-2 fuzzy sets A and B as the type-2 fuzzy-set A n B given by jAnB (x, u) = jA(x, u) A jB (x, u). Their jo in A U B is jAuB (x, u) = jA(x,u)V jB (x, u). The complement —A of a type-2 fuzzy set A is j-A(x,u) = jA(x, 1-u).

The inclusion relation is defined as follows: A Z B, if A n B = A Mid A U B = B. This

j

greater than a fuzzy number v, j Z v, if j A v = j and j V v = v. For normal and convex fuzzy numbers the conditions j A v = j and j V v = v are equivalent [5].

3. Definitions. We will employ Zadeh's extension principle to define infimum and supremum of a set of fuzzy numbers. Let ji, i G J, be fuzzy numbers. The index set I maybe finite or infinite, countable or uncountable. Their infimum /\ ji and supremum \J ji

iei iei

are fuzzy numbers with the following membership functions:

(A MM (x) = SUP inf V-

\ г- г J inf Xi = X ieI \ w- г

(x) = sup mi fii(xi), I у mm (x) = sup inf ¡ii(xi).

inf xi = xieI / sup Xi = Xi^I

iei i \i€T / id i

xi

supremum.

Using these operations, it is possible to define the join and meet of an arbitrary-number of type-2 fuzzy sets. Let A^, i G I, be type-2 fuzzy sets. Their meet Ai and

iei

Ai

iei

M П Ai (x,u) = A MAi (x,u), m у Ai (x,u)=\J MAi (x,u). (1)

iei iei iei

4. De Morgan's laws. Hereafter, without loss of generality, we will suppose that all fuzzy numbers under consideration are defined over [0; 1].

ji i G I

— A ji = V —Ji, — V ji = A —ji■

iei iei iei iei

Proof. By definition, I — /\ jJ (x) = I /\ jJ (1 — x) = sup inf ji(xi) =

\ iei J \iei J inf xi=i-xieI

sup inf ji(xi) = sup inf — ji(1 — xi) = I \f —ji I (x). The second equation

1 —inf xi=xiei sup(1 — xi)=xieI Kiel J

can be proved similarly. □

Theorem 1. For any type-2 fuzzy sets Ai, i G I,

- П Ai = U-Ai, - U Ai = П-Ai.

iei iei iei iei

Proof. From (1), у^ ^ Ai (x, u) = — Д fiAi (x, u), у у -A. (x, u) = \J (x, u), so iei iei iei iei

the first equation directly follows from lemma 1. So does the second equation. □

Thus, De Morgan's laws hold for any number of type-2 fuzzy sets. In the following discussion, we will concentrate on properties of infimum of fuzzy numbers, and, therefore, meet of type-2 fuzzy sets. Using De Morgan's laws, it will be easy to transfer the obtained results to supremum of fuzzy numbers and join of type-2 fuzzy sets.

5. Properties of infimum of fuzzy numbers. In this section we will study whether the infimum operation preserves the properties of normality, convexity and upper semicontinuity.

Lemma 2. If у,, i G I, are normal fuzzy numbers, then their infimum Д yi and,

iei

supremum \J yi are also normal fuzzy numbers.

ie i

Proof. Denote у = Д у,. Take arbitrary e > 0. As yi are normal, there exist

iei

such xi that уi(xi) > 1 — e. Then inf уi(xi) > 1 — e. If we denote inf xi = x, then

ie i ie i

у(x) = sup inf у^у-С) > inf уi(xi) > 1 — e. As this is true for any e > 0, we may

inf Vi = x ieI ieI

iei y

conclude that sup у(x) = 1. The proof for supremum is similar. □

x

Lemma 3. If у,, i G I, are strongly normal fuzzy numbers, then so are their infimum

Д у, and supremum \J у,. iei iei

Proof. Consider the infimum у = Д у,, the reasoning for supremum is similar.

ie i

As у, are strongly normal, there exist such x* that уi(x*) = 1. Let x* = inf x*. Then

i i ie i i

у(x*) = sup inf уi(xi) ^ inf уi(x*) = 1. □

inf xi = x* ieI ieI

iei

Lemma 4. If у,, i G I, are convex fuzzy numbers, then their infimum Д у, and

iei

supremum \J у, are also convex fuzzy numbers.

ie i

Proof. Consider the infimum у = Д у,, the proof for supremum is similar.

ie i

A fuzzy number у, over [0; 1] is convex if and only if its membership function is nondecreasing on [0; x*) and nonincreasing on [x*;1] or nondecreasing on [0; x*] and

nonincreasing on (x*; 1] for some x*. Let x* = inf x*.

i i iei i

Take x < у < x* and suppose that y(x) > y(y). Let e = \ (y(x) — y{y)) > 0. As у^) = sup inf у, (xi), there exist such xi that inf xi = x and inf у^,) > у^) — e.

inf xi=x iei iei iei

iei

Let уi = max{xi; y} > x,. Since x* < x*, all functions у, are nondecreasing on [0; x*).

Hence, у,(у,) > yi(xi), and inf у,(у,) > inf у^,). ks inf у^,) = x < y, there exists

iei iei iei

such index j that xj < y, and then yj = y. At the same time, y, > y for all i G I. Thus,

inf yi = y. Therefore, y(y) > inf yi(yi) > inf yi(xi) > y(x) - e = h (y(x) + y{y)) > y(y).

iei iei iei 2

This contradiction proves that у is nondecreasing on [0; x*).

Consider the set J(x) = {i £ I: x* < x}. All functions jj with j £ J(x) are

nonincreasing on [x;1]^f x > x* = inf x*, then J(x) = 0. Take x* < x < y and

iei

suppose that /x(x) < /x(y). Let e = \ {p-{y) — m(x)) > 0- There exist such yi that inf yi =y

iei

and inf ji(yi) > j(y) — e. Let xi = x for i £ J(i^d xi = yi for i £ J(x). As J(x) = 0,

iei

inf xi = x. Then j(x) ^ inf ji(xi). For i £ J(x), as yi ^ inf yi = y > x = xit due

iei iei iei

to ji being nonincreasing on [x; 1], Ji(xi) > ¡¿(y^. For i £ J(x), we have xi = yi,

so Ji(xi) = ji(yi). Thus, inf Ji(xi) > inf Ji(yi). Collecting all inequalities, we obtain

iei ie i

j(x) > j(y) — e = j(x) + e, a contradiction. Hence, the function j must be nonincreasing on (x*; 1].

Let K = {i £ I: x* = x*}. If i £ K, then x* > x*, and the function ji is nondecreasing on [0; x*].

Consider the case where for Vi £ K the functions ji are nondecreasing on [0; x*], or K = 0. Then all functions i £ I, Me nondecreasing on [0; x*]. Take x < x* and suppose

that /x(x) > n(x*). Let e = b (/x(x*) — /x(x)) > 0. Then there exist Xj such that inf Xj = x

ie i

and inf Ji(xi) > j(x) — be in dices i £ I for which xi < x*. For these

iei

i let yi = x^en ji(yi) > Ji(xi). For all other indices, in other words, when xi > x*,

let yi = x^^en inf yi = x*, and j(x*) ^ inf Ji(yi) ^ inf Ji(xi) > j(x) — e = j(x*) + e,

ie i ie i iei

a contradiction. Thus, in this case the function j is nondecreasing on [0; x*].

Now consider the case where for some k £ K the function jk is nondecreasing on [0;x*) and nonincreasing on [x*; 1]. Suppose that for some y > x* we have j(x*) < j(y).

Let e = \ {p-iy) — m(x*)) > 0- There exist such y^ that inf y^ = y and inf /¿¿(j/j) > n{y) — e.

ie i iei

Let xi = yi for i = k, and xk = x* < y < yk, so that jk(xk) ^ Jk(yk^^en inf xi = x*,

ie i

and j(x*) > inf ji(xi) > inf ji(yi) > j(y) — e > j(x) + e, a contradiction. Thus, in this

iei ie i

case the function j is nonincreasing on [x*; 1].

Summing up, the function j is either nondecreasing on [0; x*] and nonincreasing on (x*; 1], or nondecreasing on [0; x*) and nonincreasing on [x*; 1]. Thus, j is convex. □

Theorem 2. The join and meet of any number of type-2 fuzzy sets with normal (strongly normal, convex) fuzzy grades are also type-2 fuzzy sets with normal (strongly normal, convex) fuzzy grades.

□

Unfortunately, the set of upper semicontinuous fuzzy numbers is not closed with respect to the infimum and supremum operators, as the following example demonstrates. Let

m(x) = <(o, ±-±<x<f,

1, | < X < 1,

for i e N. It is easy to verify that

1, 0 < x < i,

Д/i; W= 0, i < x <

2

2 a, \ 3 ,

Vi=1 / I 1,

Thus, despite every /ц being upper semicontinuous, their infimum is not upper semicontinuous.

Lemma 5. If yu i € I, are normal convex upper semicontinuous fuzzy numbers,

then their infimum /\ yi and supremum /\ yi are strongly normal, convex, and upper

iei iei

semicontinuous.

Proof. Consider the infimum y = /\ yi, as the proof for supremum is similar.

iei

As normal upper semicontinuous fuzzy numbers are strongly normal, there exist x*

such that yi(x*) = 1. Let x* = inf x^hen y(x*) = 1.

iei

y

to show that y is upper semicontinuous. Consider an a-cut A = {x: y(x) > a}. As y is strongly normal, it is nonempty. Let x' = inf ^^d x" = sup x. If x' = x", then the

set A = {x'} consists of a single point and is thus closed. Consider the case x' < x''. By convexity of y, (x'; x'') C A. As y(x*) = 1 ^ a, x' ^ x* ^ x''.

For every i € I we define Bi = {x: yi(x) > a} Since all yi are upper semicontinuous, these sets are closed, and since all yi are strongly normal and convex, Bi = [a^; bi], and ai < x* < bi.

If 3j € I: yj (x') > a, then, taking xj = x^d xi = x* > x* > x' for i = j, we

obtain inf xi = x^d inf yi(xi) = yj(x') > a, so y(x') > a. Suppose that yj(x') < a for

iei iei

all j € I. As x' < x* < x\ < b^ x' < a^. Consider a = inf a^. We have yi(ai) > a, so

i ie i

inf yi(ai) ^ a. Since x' < ai for yi € I, x' ^ ^f a = x', then immediately y(x') ^ a.

iei

1 f^v ^ ^ — J_i •_i

Consider the case x' < a. Take e > 0, yn = x' + ^ for n > no = max jo a;,, x„_x, j, so that x' < yn < min{a; x''}. As y2n € (x'; x''), y(y2n) ^ a. Then there exist xn such

that inf xn = y2n and inf yi(xrn) > a — e. Since y2n < yn, there exists such jn € I that

iei i iei i

y2n ^ xnn < y^d yjn (xnn) > a — e. Let now xi = inf xnn for i € |J {jn}, and xi = x*

jn=i n>n0

for the remaining indices i € I\ |J {jn}. If i € |J {jn}, then yi(xnn) > a — e whenever

n>no n>no

jn = i, so using upper semicontinuity we get yi(xi) > a — e. For the remaining indices

yi(xi) = ^^us, inf yi(xi) > a — e, while inf xi = inf xn = x'. Therefore, y(x') > a. iei iei n>no jn

Consider the set J = {i € I: x* < x''}. If J = 0, x'' < x* for all i € I, hence,

x'' < x*. But x* < x'', so x* = x'', and y(x'') = y(x*) = 1 > a. Consider the case J = 0.

Let b = inf bj. If b > x'', then bj > x'' > x* for every j € J, so yj(x'') > yj(bj) > a as yj jeJ j

are nonincreasing on (x*; 1]. ^^^ taking xi = x'' for i € J and xi = x* > x'' for i € J■,

we obtain inf yi(xi) > a and inf xi = x'', so y(x'') > a the case b < x''. Take

iei ie i

y € (m'Ax{xJ; b}; x''). As b < y, 3j € J: bj < ^^en yj(y) < a. As yj is nonincreasing

on (x*; 1], yj (xj) < yj (y) for Vxj > y. Then for any xi ^^^ that inf xi = y we will have j ie i

xj > y, so inf yi(xi) < yj (xj) < yj (y). Then y(y) < yj (y) < a. But y € (x'; x''), so

ie i

y(y) > a a contradiction. Thus, b < x'' is impossible.

Summing up, we have shown that y(x') ^ ^rnd y(x'') ^ ^oA = [ x'; x''] is ct closed set. Thus, y is upper semicontinuous. □

Theorem 3. The join and meet of type-2 fuzzy sets with normal convex upper semicontinuous fuzzy grades are type-2 fuzzy sets with strongly normal convex upper semicontinuous fuzzy grades.

□

6. Idempotency of inflmum. Consider the following example. Let

ц(х) = 1, i < x < §,

It is a strongly normal and convex fuzzy number, but it is not upper semicontinuous. It is easy to verify that / Л / = But if we take an infinite number of identical fuzzy numbers

w w

/i = i e N, and compute their infimum v = Д then we will find that Д /i = /

i=i i=i

v{x) = 1, 5 < X < §,

Indeed, due to infinite number of arguments, we may take Xi = A + i, so that inf Xi = A,

3 г ieN 3

mi^jj,i{xi) = 1, and thus yield v = 1. The infimum v is still not upper semicontinuous,

however. This example demonstrates that using not upper semicontinuous fuzzy numbers may violate one of intuitive properties of infimum, namely, idempotency: for real numbers,

if хг = x for yi e N, then inf хг = x, but this may not hold for fuzzy numbers unless they

ieN

are upper semicontinuous.

Let J = N, and Ij = {1} for every j e J. Observe that IJ Ij = {1} is a finite set. Let

jeJ

/1 = / from the previous example. Then we have Д Д /i = Д /i. Thus, even such

jeJ ieij ie U j

seemingly obvious properties as independence of infimum from reordering or grouping its arguments must be carefully examined.

pi i I I = Ij J

jeJ

pi pi = pi pi =

jeJieij iei jeJ ieij

pi

iei

vj = pi v = vj p = pi ieij jeJ iei

Suppose first that ц(х) > v(x). Take e = \ {p>{x) — v{x)) > 0. Then there exist xi such

that inf xi = x, inf /i(xi) > /(x) — e. For yj = inf xi we will have Vj(yj) ^ inf /i(xi), so

iei iei ieij ieij

inf Vj (yj) ^ inf /i(xi) > /(x) — e. As for any S > 0 there exists i e I for which xi < x + S,

jeJ ieij

there exists j e J such that the index i e Ij, so yj < xi < x + S. Thus, inf yj = x.

jeJ

Therefore, v(x) > inf /j (yj) > /(x) — e = v(x) + e, a contradiction. je J

Suppose now that ц(х) < v(x). Take e = j {v{x) — ц{х)) > 0. There exist such Xj

that inf xj = x and inf Vj (xj) > v(x) — e. Then, for every j e J, there exist such xji je J je J

that inf xji = xj and inf /i(xji) > vj (xj) — e > v (x) — 2e. Let yi = inf xji. Since

ieij ieij j: ieij

xji ^ xj > x, yi > x. For any S > 0 we can find such index j that xj < x + S, and

then such i e Ij that xji < xj + S < x + 2S. Then yi ^ xji < x + 2S. Thus, inf yi = x.

ie i

pi i I j i Ij

yi(xji) ^ inf yi(xji) > v(x) — 2e we may conclude that yi(yi) > v(x) — 2e. If J is finite,

ieij

yi = xji for some j € J, so again yi(yi) = yi(xji) > inf yi(xji) > v(x) — 2e. Then

ieij

y(x) > inf yi(yi) > v(x) — 2e = y(x) + 2e, a contradiction.

iei

y = v □

Theorem 4. Let A^ i € I, be type-2 fuzzy sets, and I = |J Ij. If J is finite, or

jeJ

Ai Ai = Ai

jeJieij iei

Ai = Ai

jeJieij iei

□

Theorem 5. Let A^ i € I, be type-2 fuzzy sets with normal convex fuzzy grades. Then

1) n A E Aj E u Ai for Vj € I;

iei iei

2) n yi E n yi for any J C I;

ie i ie J

) yi E yi J C I

ieJ iei

Proof. By theorem 4, Aj n Ai = LI Ai = L[ Ai1 and due to normality and

iei ie{j}ui iei

Ai E Aj □

iei

Theorem 6. For type-2 fuzzy sets with normal convex upper semi continuous fuzzy grades:

1) if A E Bifor Vi € I, then A E LI Bi;

iei

2) if Bi E A for Vi € I, then \J Bi E A;

ie i

3) if Ai E Bi for Vi € I, then U Ai E LI Bu and |J Ai E U Bi.

iei iei iei iei

Proof. Denote A = Bo. If A E B^ then B0 n Bi = B0. Using theorem 4, we

obtain A n n Bi = LI Bi = [I Bi = n (Bo n Bj) ^ Bo = R U Bi =

iei ieiu{0} ie U {0;j} jei jei jei ie{o}

Bi = Bo = A Ai E

ie U {o} iei

j€I

Aj E Bj E U Bi for any j € I. Then by previous properties we get the remaining

iei

□

7. Formulae for join and meet of type-2 fuzzy sets. Let y^ i € I = {1, 2,...,n}, be strongly normal convex fuzzy numbers. Let x* be such points that yi,(x*) = 1, i € I. Without loss of generality, suppose that x* < x| < ••• < x*n. Then infimum and supremum of these numbers can be computed as follows [7]:

max yi(x), x < x*,

n

f\yi) (x) = <{ i mmfc Уi(x), x*k ^ x<x*k+v

min yi(x), x ^ x*n

= l,...,n

min yi(x), x < x*,

= 1,...,n

V yi 1 x = 4 i=mn..,nyi(x), x*k ^ x<x*+1,

i=

^Qax yi (x), x xn.

i= ,...,n n

These formulae at the same time allow to compute join and meet of a finite number of type-2 fuzzy sets, as these operations by definition are reduced to maximum and minimum of fuzzy numbers corresponding to each element of the universal set.

oo

Consider the following example. Let ¡in(x) = 1 — (1 — \/x)n, n e N, and y = /\

n=

Take arbitrary e > 0. If xn = 2-n, then lim yn(xn) = lim (1 — 2-n) = 1. Then for some

n0 € N we will have yn(xn) > 1 — e for Vn > n0. Let xi = 2-i for i > n^d xi = 1

for i < n^^en inf xi = ^^d inf yi(xi) > 1 — e, so y(0) > 1 — e. Due to arbitrariness

ieN ieN

of e, y(0) = 1. Now note that yn(0) = 0 for Vn € N, so sup yn(0) = y(0). Thus, the given

neN

formulae cannot be naively generalized to the case of infinite number of fuzzy numbers. Lemma 7. Let yit i € I, be normal convex upper semi continuous fuzzy numbers, and,

y = yi

ie i

{lim sup yi(y), Vi G I ^y < x: m(y) = 1,

v^x+0 iei (2) inf yi(x), 3i € 13y < x: yi(y) = 1.

iei: By^x : Pi(y) = 1

Proof. As normal upper semicontinuous fuzzy numbers are strongly normal, there

exist xi such that yi(xi) = 1. Let x* = inf x^ By upper semicontinuity, yj,(x*) = 1.

mi(xi)=1

Let also x* = inf x*.

ieii

Consider x < x*. By choice of x* we have Vi € I ^y < x: yi(y) = 1. Since all yi

are convex, they are nondecreasing on [x■.; x*]. Then so is p(y) = supyi(y). Indeed, if for

iei

some x' ,x'' such th at x < x' < x'' < x* we h ad x') = sup yi(x') > sup yi(x'') = p(x''),

iei iei

then for e = i (f(x') — <p{x")) > 0 there would exist such j G / that yj(x') > <p(x') — e = p(x'') + e > p(x'') > yj(x''), so yj would not be nondecreasing. Futhermore, p(y) > 0, so the function p(y) is monotone and bounded on [x■.; x*]. Therefore, ^(x) = lim p(y)

y^x+0

exists. As y(y) is nondecreasing, ^(x) < p(y) for Vy € [x; x*].

Let yn = x + x2~x, n G N, be a monotonically decreasing sequence, yn —> x + 0 as n ^ ro. Sinee yn € (x; x*), p(yn) > ^(x). Take arbitrary e > 0. Then for every n € N we can find in € I such that yin (yn) > p(yn) — e > ^(x) — e. Denote IE = {in: n € N}.

Let xi = inf yn for i € Ie, and xi = x* for i € Ie- Since yn ^ x + 0 inf xi = x.

neN : in =i iei

For i € Ie we have yi(yn) > ^(x) — e for n € N: in = i, so due to yi being upper semicontinuous, yi(xi) > ^(x) — e. For other indices i € Ie> simply yi(xi) = 1. Thus,

y(x) > inf yi(xi) > ^(x) — e. Due to arbitrariness of e, y(x) > ^(x).

ie i

Suppose that y(x) > ip(x). Let £ = \ (y(x) — ip(x)) > 0. Then 3(5 > 0: Vy G (x; x + S) ^ \p(y) — y>(x)\ < e. Hence, p(y) < ^(x) + e = y(x) — e for Vy € (x; x + S). As p is nondecreasing, p(x) < p(y) < y(x) — e. Since y(x) = sup inf yi(xi), there exist

inf xi = x iei iei

Consider now x = x*. Taking xi = x*, we get inf xi = x* and inf /i(xi) = 1, so

i ie i iei

such xi that inf xi = x and inf /i(xi) > /(x) — e. For the found S then 3j: xj < x + S,

iei iei

so xj) < /(x) — e < inf /i(xi) < /j (xj) < sup /i(xj) = y(xj), a contradiction. Thus,

iei iei

/(x) = ф(x). Consii

/(x*) = 1.

If 3i e I: x* = x*, then it is the case 3i e I, 3y < x: /i(y) = 1. For i e I, if 3y < x =

x*: //i(y) = 1, then x* ^ x* ^ ^oy = x*, so /j,(x*) = 1. Then inf /i(x) =

iei: 3y^.x : ¡li (y) = 1

1 = /(x).

If x* > x* for yi e I, then, obviously, yi e I Зу ^ x: /i(y) = 1. Sinee x* = inf x*, we

iei

may choose a sequence ж* —> x* +0. Then sup/¿¿(x*) > цп(х*п) = 1, so lim supHi{y) =

iei y^x*+° iei

1 = /(x*).

Finally, consider x > x*. Let J(x) = {i e I: x* ^ x}. As x* = inf x* 3i € I: x* < x,

i iei i i

so J(x) = 0. Besides, the condition x\ < x is equivalent to 3y < x: /i(y) = 1, so the

J(x)

Let xi = x for i e J(x), and xi = x* for i e J(x). Then inf xi = x and inf /i(xi) =

i iei iei

inf /i(x)^o /(x) ^ inf /i(x).

ieJ(x) ieJ(x)

Suppose that ц(х) > xix) = inf Let e = A {p-{x) — xix)) > 0- Then there

ieJ(x)

exist such xi that inf xi = x and inf /i(xi) > /(x) — e. Hence, /i(xi) > /(x) — e for all

iei iei

i e I. On the other hand, as x(x) = inf /i(x) 3j e J(x): /j(x) < x(x) + e = /(x) — e.

ieJ(x)

As j e J(x), x* < x, so % convexity the function /j is nonincreasing on [x1]. As xj > x, /j (xj) ^ /j (x) < /(x) — e, a contradiction. Thus, /(x) = x(x). □

It must be noted that we cannot completely get rid of the upper limit sign in (2), as the following example demonstrates. Let

_ Г1, Ж = i,

Mx)-\o, жД

be a fuzzy number that represents the crisp number Obviously,

(Д-) {0: x ~л

is the fuzzy counterpart of the crisp zero. But

f ^ J1' X

sup /n(x) = ^ 1

neN [0, ± £ N,

so the ordinary limit lim sup /n(x) does not exist. Fortunately, this can happen only

x^+o neN

1

exists, as follows from the proof.

The presented formula (2) can be used to compute fuzzy grades for join and meet of any number of type-2 fuzzy sets, as in the following result.

Theorem 7. Let Ai} i e I, be type-2 fuzzy sets with normal convex upper semiconti-nuous fuzzy grades. Then for any x e X, u e [0; 1],

lim supyA (x,u), u<u*(x),

v^u+O ieI i

У П Ai (x,u) = < 1,

inf HAi (x,u),

ieJ(x,u)

inf HAi (x,u),

ieJ' (x,u)

У LI Ai (x,u) = \ 1,

u = u*(x), u > u*(x),

u < u0(x), u = u0(x),

lim sup yAi (x,u), u > u0(x),

v^ru—0

iei

u, u (x) = sup u0

i- iei

(x)

u0(x) = sup u,

HAi (x,u)=l

'where u*(x) = inf u*(x), u*(x) = inf

iEl I^Ai (x,u) =

J(x,u) = {i e I: u ^ u*(x)}, J'(x,u) = {i e I: u ^ u0(x)~}.

Proof. The first part directly follows from the proof of lemma 7 and (1). The second formula can be derived from the first with the use of De Morgan's laws. □

8. Conclusions. Zadeh's extension principle gives a natural way to extend known operations on real numbers to fuzzy numbers. However, they might not retain all usual properties. We have studied an extension of infimum and supremum to fuzzy numbers. We have shown that these operations preserve normality and convexity, but do not preserve upper semicontinuity. At the same time, if fuzzy numbers are not upper semicontinuous, their infimum and supremum might behave counter-intuitively, for example, they might be not idempotent. The conditions under which familiar properties of infimum and supremum hold have been given. We have also derived folmulae for computing infimum and supremum of arbitrary number of fuzzy numbers. All obtained results are applicable to type-2 fuzzy sets, as set-theoretic operations on them are defined using infimum and supremum of fuzzy numbers.

References

1. Zadeh L. A. Fuzzy sets. Information and Control, 1965, vol. 8, iss. 3, pp. 338-353.

2. Zadeh L. A. The concept of a linguistic variable and its application to approximate reasoning. Information Sciences, 1975, vol. 8, iss. 3, pp. 199-249.

3. Liang Q., Mendel J. M. MPEG VBR video traffic modeling and classification using fuzzy techniques. IEEE Transactions on Fuzzy Systems, 2001, vol. 9, iss. 1, pp. 183-193.

4. Hisdal E. The IF THEN ELSE statement and interval-valued fuzzy sets of higher type. International Journal of Man-Machine Studies, 1981, vol. 15, pp. 385-455.

5. Mizumoto M., Tanaka K. Some properties of fuzzy sets of type-2. Fuzzy Sets and Systems, 1976, vol. 31, pp. 312-340.

6. Noghin V. D. Pareto set reduction. An axiomatic approach. Cham, Springer Publ., 2018, 232 p.

7. Karnik N. N., Mendel J. M. Operations on type-2 fuzzy sets. Fuzzy Sets and Systems, 2001, vol. 122, pp. 327-348.

Received: December 02, 2020. Accepted: April 05, 2021.

Author's information:

Oleg V. Baskov — PhD in Physics and Mathematics, Associate Professor; o.baskov@spbu.ru

*

О. В. Басков

Санкт-Петербургский государственный университет, Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7^9

Для цитирования: Baskov О. V. Join and meet of infinite number of type-2 fuzzy sets // Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления. 2021. Т. 17. Вып. 2. С. 108-119. https://doi.org/10.21638/11701/spbul0.2021.201

Изучаются свойства соединения и слияния бесконечного количества нечетких множеств типа 2. Также исследуются тесно связанные с ними операции инфимума и супремума над нечеткими числами. Показано, что классы нечетких множеств типа 2 с нормальными или выпуклыми степенями принадлежности замкнуты относительно соединения и слияния, однако полунепрерывность сверху при этом может не сохраняться. Установлены условия, при которых справедливы такие основные свойства соединения и слияния как идемпотентность, коммутативность и ассоциативность. Доказана корректность применения инфимума и супремума к обеим частям неравенств нормальных выпуклых полунепрерывных сверху нечетких чисел. Наконец, приведена формула для нахождения функций принадлежности соединения и слияния бесконечного количества нечетких множеств типа 2.

Ключевые слова: нечеткие множества типа 2, соединение, слияние, нечеткие числа, ин-фимум, супремум.

Контактная информация:

Басков Олег Владимирович — канд. физ.-мат. наук, доц.; o.baskov@spbu.ru

* Работа выполнена при финансовой поддержке Российского фонда фундаментальных исследований (грант № 20-07-00298-а).