195 |
P a g e
FUZZY
-CONTINUOUS MULTIFUNCTION
Anjana Bhattacharyya
1Department of Mathematics, Victoria Institution (College)
78 B, A.P.C. Road, Kolkata – 700009(India)
ABSTRACT
This paper deals with a new type of fuzzy multifunction, viz., fuzzy upper (lower) -continuous multifunction. Several characterizations and properties of this newly defined multifunction have been studied here. Also the mutual relationships of this fuzzy multifunction with fuzzy upper (lower) -precontinuous multifunctions [8] have been established here.
Keywords: Fuzzy nbd of a fuzzy set,
-nbd of a subset of a topological space,
-frontier of a subset
of a topological space, fuzzy compact space.
2000 Ams Subject Classification Code
Primary 54a40, Secondary 54c99
I
INTRODUCTION
A fuzzy mu ltifunction, introduced by Papageorgiou [21] in 1985 is a function from an ordinary topological space
to a fuzzy topological space (fts, for short) in the sense of Chang [10]. In this paper we use the definition of upper
inverse of fuzzy mu ltifunction given by Papageorgiou [21] and the lower inverse of fuzzy mult ifunction given by
Mukherjee and Mala kar [18].
Throughout this paper, or simply will stand for an ordinary topological space, while by or simp ly by
will always be denoted a fuzzy topological space. A fuzzy set [26] in is a function from into the unit
interval i.e ., . The support [26] of a fu zzy set in will be denoted by and is defined by
. The fu zzy point [22] with the singleton support and the value at
will be denoted by . For a set in (resp., a fuzzy set in ), and will respectively stand for closure
and interior o f in (resp., fu zzy closure and fuzzy interior of in [10]). For a fuzzy set in , the comp le ment
of in will be denoted by and is defined by , for each [26]. and will
respectively stand for the constant fuzzy sets taking values 0 and 1 on . For two fuzzy sets , in , we write
1
196 |
P a g e
[26] iff , for all while means is quasi-coincident [22] (q-coincident, for short)
with if there is some such that . The negation of the last two statements are written as
and respectively. A fuzzy set in is called a fuzzy neighbourhood (fuzzy nbd, for short) [22] of a
fuzzy set in if there is a fuzzy open set in such that .
A subset of (resp., a fuzzy set in ) is called se miopen[15], α -open [20], β-open [1], regular open [24],
preopen [16], -open [11] (resp., fu zzy semiopen [4], fu zzy α-open [9], fu zzy β-open [12], fu zzy regula r open [4],
fuzzy preopen [19], fuzzy -open) respectively if , ,
, , (resp., ,
, , , , ) respectively.
The comple ments of above mentioned sets in (resp., in ) are called semic losed [15], α-c losed [20], β-c losed [1],
preclosed [16], regular c losed [24], -closed [11] (resp., fuzzy semiclosed [4], fu zzy α-closed [9], fuzzy β-c losed
[12], fuzzy regular closed [4], fu zzy preclosed [19], fuzzy -closed ) respectively. The intersection of all semic losed, α-closed, β-closed, preclosed, -closed sets in (resp., fuzzy sets in ) containing are called semic losure [15],
α-closure [20], β-α-closure [1], p reα-closure [16], -closure [11] (resp., fu zzy semic losure [4], fuzzy α -closure [9], fu zzy
β-closure [12], fuzzy prec losure [19], fu zzy -closure) respectively and are denoted by , , , ,
respectively. The -interior [25] of a subset of is the union of all regular open sets of contained in and is
denoted by . A subset of is called -open if [25]. The comp le ment of a -open set in is
called -c losed. For a set in , . A subset of is called
-preopen [2] if . A fuzzy set is called a quasi-neighbourhood (q-nbd, for short) [21] of a fu zzy set
if there is a fuzzy open set in such that . If, in addit ion, is fuzzy regular open, then is ca lled fu zzy
regular open q-nbd of . A fuzzy point is said to be a fuzzy -c luster point of a fuzzy set in an fts if every
fuzzy regular open q-nbd of is q-coincident with [13]. The union of all fu zzy -cluster points of is called
the fuzzy -closure of and is denoted by [13]. A fu zzy set in an fts is called fuzzy -preopen [5] if
. The comple ment of a fuzzy -preopen set is called fuzzy -prec losed [5]. The intersection of all
fuzzy -prec losed sets containing a fuzzy set in an fts is called fuzzy -prec losure of and is denoted by
[5]. A fuzzy set in is -prec losed iff [5].
The set of all semiopen, α-open, β-open, preopen, -open, -preopen sets in are denoted by , ,
, , , respectively. A semiopen (resp., α-open, β-open, preopen, -open, -preopen) set
in containing a point will be denoted by (resp., , , ,
197 |
P a g e
denoted by . A subset of a topological space is called a -nbd of a point [11] if there exists
such that .
II SOME WELL KNOWN DEFINITIONS, THEOREM AND LEMMAS
In this section we recall, some definitions, lemmas and theorem for ready references.
Definiti on 2.1 [21]. Let be an ordinary topological space and be an fts. We say that is a
fuzzy mu ltifunction if corresponding to each , is a unique fu zzy set in .
Henceforth by we shall mean a fuzzy mu ltifunction in the above sense.
Definiti on 2.2 [21, 18]. For a fuzzy mult ifunction , the upper inverse and the lower inverse are
defined as follo ws :
for any fu zzy set in , , .
The ore m 2.3 [18].For a fu zzy mult ifunction , we have , for any fu zzy set in .
Definiti on 2.4[10]. Let be a fuzzy set in an fts . A collection U of fu zzy sets in is called a fuzzy cover of if
U } = 1 for each . If, in addition, the me mbers of U are fu zzy open, then U is called a
fuzzy open cover of . In particular, if , we get the definition of fuzzy cover (resp., fuzzy open cover) of the
fts .
Definiti on 2.5 [14].A fu zzy cover U of a fu zzy set in an fts is said to have a finite subcover U 0 if U 0 is a
fin ite subcollection of U such that U 0 . Clearly, if , in particu lar, then the require ments on U 0 is U
0 .
Definiti on 2.6 [10].An fts is said to be fuzzy co mpact if every fuzzy open cover of has a finite subcover.
Definiti on 2.7 [18]. A fuzzy mu ltifunction is said to be fuzzy upper (lo wer) se mi-continuous if
(resp., ) is open in for every fu zzy open set in .
Definiti on 2.8 [8]. A fu zzy mult ifunction is said to be fuzzy
(i) upper -precontinuous (resp., upper quasi continuous) if for each and each fuzzy open set in
with , there e xists (resp., ) such that ,
(ii) lower -precontinuous (resp., lower quasi continuous) if for each and each fuzzy open set in
198 |
P a g e
(iii)upper (lowe r) -precontinuous (resp., upper (lo wer) quasi continuous) if it has this property at each point of
.
Le mma 2.9 [3, 11]. Let and be subsets of a topological space . If and , then
.
Le mma 2.10 [11]. Let , and , then .
III
FUZZY UPPER AND LOWER
-CONTINUOUS MULTIFUNCTIONS : SOME
CHARACTERIZATIONS
Now we define fuzzy upper (lower) -continuous multifunction and characterize these fuzzy mu ltifunctions in several ways.
Definiti on 3.1.A fu zzy mult ifunction is said to be fuzzy
(i) upper -continuous at a point if for each fuzzy open set in with , there exists
such that ,
(ii) lower -continuous at a point if fo r each fuzzy open set of with , there exists
such that , for a ll ,
(iii)upper (lowe r) -continuous if has this property at each point of .
The ore m 3.2. For a fu zzy mult ifunction , the follo wing are equivalent :
(a) is fu zzy upper -continuous,
(b) for any fu zzy open set of ,
(c) is -closed in for any fuzzy c losed set of ,
(d) , for any ,
(e) for each point and each fuzzy nbd of , is a -nbd of ,
(f) for each point and each fuzzy nbd of , there e xists a -nbd of such that ,
(g) , for any ,
(h) , for any .
Proof (a ) (b):Let be a fu zzy open set of and . Then by (a), there e xists such
199 |
P a g e
and so we have
.
(b) (c) :It fo llows fro m the fact that , for any .
(c) (d) :Let . Then is fu zzy closed in and so by (c), is -closed in and so
.
(d) (c) : Let be a fuzzy c losed set of . Then and so by (d),
is -closed in .
(b) (e) : Let and be a fuzzy nbd of . Then there e xists a fuzzy open set of such that
. Since (by (b)), is a γ-nbd of .
(e) (f) : Let and be a fu zzy nbd of . Put . By (e ), is a -nbd of and .
(f) (a) :Let and be a fu zzy open set such that . Then is a fu zzy nbd of . By (f), there
e xists a -nbd of such that . There fore, there e xists such that and so
.
(c) (g) :Let . Then is fu zzy closed in and so by (c), is -closed in
.
(g) (h) :Let . Then . By (g),
=
.
(h) (b) :Let be a fu zzy open set of . By (h),
.
200 |
P a g e
(a) is fu zzy lo wer -continuous,
(b) , for any fu zzy open set of ,
(c) is -closed in for any fu zzy closed set of ,
(d) , for any ,
(e) , for any subset of ,
(f) , fo r any ,
(g ) , for any ,
(h ) for each and each fuzzy q-nbd of , is a -nbd of ,
(i) for each and each fuzzy q-nbd of , there e xists a -nbd of such that
, for all .
Proof (a) (b ): Let be a fuzzy open set of and . Then by (a), there e xists such that
, for all . Therefo re, we obtain
and so we have
.
(b) (c) :It fo llows fro m the fact that , for any .
(c) (d) :Let . Then is fu zzy closed in . By (c), is -closed in
.
(d) (c) : Let be a fuzzy c losed set of . Then . By (d),
is -closed in .
(c) (e) : Let be a subset of . Then is fu zzy closed in . By (c), is -closed in
201 |
P a g e
(e) (d) :Let . Then . By (e), .
(c) (f) :Let . Then is fuzzy c losed in . By (c), is -closed in
(f) (g) :Let . Then . By (f),
=
.
(g) (b) :Let be a fu zzy open set of . By (g),
.
(b) (h) :Let Let and be a fu zzy q-nbd of . Then there e xists a fuzzy open set of such that
. Since (by (b)), is a γ-nbd of .
(h) (i) : Let and be a fu zzy q-nbd of . Put . By (h), is a -nbd of and for a ll
.
(i) (a) :Let and be a fuzzy open set such that . Then is a fu zzy q-nbd of . By (i), there
e xists a -nbd of such that , for all . Therefo re, there e xists such that
and so , for all .
Definiti on 3.4 [18]. For a fu zzy mu ltifunction , the fuzzy graph mu ltifunction of is
defined as = the fu zzy set of , where is the fuzzy set in , whose value is 1 at and 0
other points of . We shall write for .
Le mma 3.5 [6]. The following hold for a fu zzy mult ifunction :
(a)
and(b)
for every subset of and every fuzzy set of .202 |
P a g e
Proof. Let be fu zzy lo wer -continuous. Let and be any fuzzy open set of such that .
Then there exists such that . Then . Let
. Then so that for so me open set in and
some fu zzy open set in with . Now . Since is fuzzy
lower -continuous, there exists such that , for all . Then by Le mma
2.5(b), . Moreover, and hence the proof.
Conversely, let be fuzzy lowe r -continuous. Let and be any fuzzy open set in such that . Then
there e xists such that . No w is open in such that
and so . Since is fuzzy lowe r -continuous, there e xists
such that , for a ll . By Le mma 2.5(b ), we obtain
. Hence , for all is fu zzy lo wer -continuous.
The ore m 3.7. Let be an α-open cover of a topological space . A fuzzy mu ltifunction is
fuzzy upper -continuous iff the restrict ion is fu zzy upper -continuous.
Proof. Let . Then there e xists such that . Let be a fuzzy open set of such that
. Since is fuzzy upper -continuous, there e xists such that . Put
. Then by Le mma 1.9, and . There fore, is fu zzy upper
-continuous.
Conversely, let and , a fu zzy open set of such that . Since is an α-open cover of ,
there e xists such that . Since is fuzzy upper -continuous and , there
e xists such that . Then by Lemma 1.10 (as α-open sets are -open),
and is fuzzy upper -continuous.
The ore m 3.8. Let be an α-open cover of a topological space . A fuzzy mu ltifunction is
fuzzy lower -continuous iff the restrict ion is fu zzy lower -continuous.
Proof. Let . Then there e xists such that . Let be a fu zzy open set of such that
203 |
P a g e
. Put . Then by Lemma 1.9, and , for all . Therefore,
is fu zzy lo wer -continuous.
Conversely, let and , a fuzzy open set of such that . Since is an α-open cover of ,
there e xists such that . Since is fu zzy lower -continuous and , there e xists
such that , for all . Then by Lemma 1.10 (as α-open sets are -open),
and , for a ll is fu zzy lo wer -continuous.
Definiti on 3.9. For a fuzzy multifunction , fuzzy mult ifunction , ,
, [6], , [6], [8] are defined by
, , , ,
, , , fo r a ll .
Le mma 3.10. Let be a fuzzy mu ltifunction. Then we have , for each fuzzy open
set of .
Proof. Let be a fuzzy open set of . Let . Then [Indeed, if ,
then , for each ) = (since is fuzzy
open is fuzzy -open in ) , a contradiction.] .
Similarly we can prove that
Le mma 3.11. Let be a fu zzy mult ifunction. Then we have , ,
, , , , for each fuzzy open set
of .
The ore m 3.12. For a fuzzy mu ltifunction , the following are equiva lent :
(a) is fu zzy lo wer -continuous.
(b) is fu zzy lo wer -continuous.
(c) is fu zzy lo wer -continuous.
(d) is fu zzy lo wer -continuous.
(e) is fu zzy lo wer -continuous.
204 |
P a g e
(g) is fu zzy lo wer -continuous.
(h) is fu zzy lo wer -continuous.
Proof. The proof follows fro m Le mma 2.10 and Le mma 2.1 1.
IV SOME RELATIONSHIP
In this section it has been shown that fuzzy upper (lowe r) -continuous multifunction may not be fuzzy upper
(lowe r) semi-continuous, fuzzy upper (lo wer) quasi continuous, fuzzy upper (lo wer) -precontinuous mult ifunctions.
We first reca ll some theore ms for ready references.
The ore m 4.1 [7]. A fu zzy mult ifunction is fuzzy upper (lower) quasi continuous iff (resp.,
) is semiopen in for every fuzzy open set of .
The ore m 4.2 [18]. A fuzzy multifunction is fuzzy upper (lowe r) semi-continuous iff (resp.,
) is open in for every fuzzy open set of .
The ore m 4.3 [8]. A fu zzy mult ifunction is fuzzy upper (lowe r) -precontinuous iff (resp.,
) is -preopen in for every fuzzy open set of .
Re mark 4.4. Since open set, semiopen set and -preopen set are -open, it is clear fro m Theore m 3.1, Theore m 3.2 and Theorem 3.3 that fuzzy upper (lowe r) quasi continuous, fuzzy upper (lower) se mi -continuous and fuzzy upper
(lowe r) -precontinuous multifunctions are fuzzy upper (lower) -continuous. But the converses are not true, in
general, as seen from the following e xa mp les.
Example 4.5. fuzzy upper -continuity fuzzy upper se mi-continuity
Let , , , where , for all . Then
and ) are ordinary topological space and an fts respectively. Let be a fuzzy
mu ltifunction defined by where , for all . Now
and so is not fuzzy upper semi-continuous.
But is -open in .
Again is
-open in which shows that is -open in for a ll fu zzy open set of is fuzzy upper -continuous.
205 |
P a g e
Let , , , where , for all . Then
and ) are ordinary topological space and an fts respectively. Let be a fuzzy
mu ltifunction defined by where , for all . Now
and , is fu zzy lo wer -continuous. But
and so is not fuzzy lower semi-continuous.
Example 4.7. fuzzy upper -continuity fuzzy upper quasi-continuity
Consider Exa mple 3.5. Here is not fuzzy upper quasi-continuous, though
is upper -continuous.
Example 4.8. fuzzy lower -continuity fuzzy lowe r quasi-continuity
Consider Exa mple 3.6. He re is not semiopen in is not fuzzy
lower quasi-continuous, though it is fuzzy lower -continuous.
Example 4.9. fuzzy upper -continuity fuzzy upper -precontinuity
Consider Exa mple 3.5. Here . Now and
, is not -p reopen in is not fuzzy upper -precontinuous,
though is fu zzy upper -continuous.
Example 4.10. fu zzy lo wer -continuity fuzzy lower -precontinuity
Consider Exa mp le 3.6. He re . No w is fuzzy lowe r
-precontinuous, though is fuzzy lowe r -continuous.
V FUZZY UPPER (LOWER)
-CONTINUOUS MULTIFUNCTION: SOME
CHARACTERIZATIONS AND APPLICATIONS
In this section fuzzy upper (lo wer) -continuous mult ifunction is characterized by fuzzy upper (lowe r) nbd of a
fuzzy set and also some applications of these fuzzy mu ltifunctions have been given.
Definiti on 5.1. A fuzzy set in an fts is said to be a fuzzy lowe r (upper) nbd of a fuzzy set in if there e xists a
fuzzy open set of such that (resp., ) and .
The ore m 5.2. A fu zzy mult ifunction is fu zzy upper -continuous on iff for each point and each
206 |
P a g e
Proof. Let be fuzzy upper -continuous. Then for any and for any fuzzy upper nbd of , there
e xists a fuzzy open set of such that and . Since is fuzzy upper -continuous,
there exists such that . Therefore,
is a -nbd of .
Conversely, let for any and any fuzzy open set of with , we have . Therefore, is a
fuzzy upper nbd of . Then by hypothesis, is a -nbd of . Then there exists such that
is fuzzy upper -continuous.
The ore m 5.3. A fu zzy mult ifunction is fu zzy lower -continuous on iff for each point and each
fuzzy lower nbd of , is a -nbd of .
Proof. Let be fuzzy lo wer -continuous on . Then for any and for any fuzzy lowe r nbd of , there
e xists a fuzzy open set of such that and . Since is fuzzy lower -continuous,
there e xists such that . Therefore, is a -nbd of .
Conversely, let for any and any fuzzy open set of with , we have . Therefore, is a
fuzzy lower nbd of . Then by hypothesis, is a -nbd of . Then there exists such that
is fuzzy lower -continuous.
Definiti on 5.4 [11]. A topological space is said to be -compact if for every covering of by -open sets in
has a finite subcovering.
The ore m 5.5. Let be a fuzzy upper -continuous surjective mu ltifunction and be a fuzzy co mpact
set in for each If is -compact, then is fu zzy co mpact.
Proof. Let A = be a fuzzy open cover of . Now for each , is fuzzy compact in and so
there is a finite subset of such that . Let . Then
where is a fuzzy open set of . Since is fuzzy upper -continuous, there exists such that
. Then U = { is a cover of by -open sets of . Since is -compact, there e xists finite ly
many points in such that . As is surjective,
is fuzzy co mpact.
Definiti on 5.6 [17]. An fts is said to be an -space if every fuzzy regular open cover of has a finite
207 |
P a g e
Re mark 5.7. As every fu zzy regular open set is fuzzy open, we set the following theorem.The ore m 5.8. Let be a fuzzy upper -continuous surjective mu ltifunction and be a fuzzy co mpact
set in for each . If is -compact, then is -space.
The ore m 5.9. When is product related to , a fu zzy mu ltifunction is fu zzy upper -continuous if its
fuzzy graph mult ifunction is fuzzy upper -continuous.
Proof. Let be fuzzy upper -continuous. Let and be any fuzzy open set in such that . Then
and is easily seen to be open in . By hypothesis, there exists such that
. Now for any and for any , , i.e.,
, for all , for any is fuzzy upper -continuous.
Definiti on 5.10 [2]. The -frontier of a subset of a topological space , denoted by , is defined by
.
The ore m 5.11. Let be a fuzzy mult ifunction. Let : is not fuzzy upper -continuous at ,
and is fu zzy open in . Then .
Proof. Let be such that is not fuzzy upper -continuous at . Then there exists a fuzzy open set of with
such that , for all
, but . Therefore,
.
Conversely, let and be a fuzzy open set of with such that . If possible, let
be fuzzy upper -continuous at . Then there exists such that . Then
, a contradiction and hence is not
fuzzy upper -continuous at .
The ore m 5.12. Let be a fuzzy mult ifunction. Let : is not fuzzy lowe r -continuous at ,
and is fu zzy open set of . Then .
Proof. Let be such that is not fuzzy lower -continuous at . Then there exists a fuzzy open set of with
such that i.e ., , for all
, but . Therefore,
208 |
P a g e
Conversely, let and be a fuzzy open set of with such that . If possible, let be
fuzzy lowe r -continuous at . Then there exists such that . Then
, a contradiction and hence is not
fuzzy lower -continuous at .
REFERENCES
[1] Abd El-Monsef, M.E., El-Deeb, S.N. and Mahmoud, R.A.; β-open sets and β-continuous mappings, Bull. Fac. Sci. Assiut Univ. 12 (1983), 77 – 90.
[2] Abd El-Monsef, M.E. and Nasef, A.A.; On mu ltifunctions, Chaos, Solitons and Fractals 12 (2001), 2387-2394.
[3] Andrijevič, D. ; On -open sets, Mat. Bech. 48 (1996), 59-64.
[4] Azad, K.K. ; On fuzzy semicontinuity, fuzzy almost continuity and fuzzy wea kly continuity, J. Math. Anal. Appl. 82 (1981), 14-32.
[5] Bhattacharyya, Anjana and Mukherjee, M.N. ; On fuzzy -almost continuous and *-almost continuous functions, J. Tripura Math. Soc. 2 (2000), 45-57.
[6] Bhattacharyya, Anjana ; Concerning almost quasi continuous fuzzy mu ltifunctions, Universitatea Din Bacău Studii Şi Cercetări Ştiinţifice, Seria : Mate matică Nr. 11 (2001), 35-48.
[7] Bhattacharyya, Anjana : Upper and lower weakly quasi continuous fuzzy mu ltifunctions, Analele Universităţii Oradea, Fasc. Matematica, Tom XX (2013), Issue No. 8, 5-17.
[8] Bhattacharyya, Anjana : On upper and lower -precontinuous fuzzy multifunctions, International Journal of Advancements in Research and Technology Vol. 2, Issue 7 (2013), 328-333
.
[9] Bin Shahna, A.S. ; On fuzzy strong semicontinuity and fuzzy precontinuity, Fuzzy Sets and Systems 44 (1991), 303-308.
[10] Chang, C.L. ; Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182-190.
[11] El-Atik, A.A. ; A study of some types of mappings on topological spaces, Master’s Thesis, Faculty of
Science, Tanta University, Tanta, Egypt, 1997.
[12] Fath Alla, M.A. ; On fuzzy topological spaces, Ph.D. Thesis, Assiut Univ., Sohag, Egypt (1984)
.
[13] Ganguly, S. and Saha, S. ; A note on -continuity and -connected sets in fuzzy set theory, Simon Stevin 62 (1988), 127-141.
[14] Ganguly, S. and Saha, S. ; A note on compactness in fuzzy setting, Fuzzy Sets and Systems 34 (1990), 117-124.
209 |
P a g e
[16] Mashhour, A.S., Abd El-Monsef, M.E. and El-Deeb, S.N. ; On precontinuous and weak precontinuous mappings, Proc. Phys. Soc. Egypt 53 (1982), 47-53.
[17] Mukherjee, M.N. and Ghosh, B. ; On nearly compact and -rig id fuzzy sets in fuzzy topological spaces,
Fuzzy Sets and Systems 43 (1991), 57-68.
[18] Mukherjee, M .N. and Mala kar, S. ; On almost continuous and weakly continuous fuzzy multifunctions,
Fuzzy Sets and Systems 41 (1991), 113-125.
[19] Nanda, S. ; Strongly co mpact fuzzy topological spaces, Fuzzy Sets and Systems 42 (1991), 259-262. [20] Njåstad, O. ; On some c lasses of nearly open sets, Pacific J. Math. 15 (1965), 961-970.
[21] Papageorgiou, N.S. ; Fu zzy topology and fuzzy mu ltifunctions, J. Math. Anal. Appl. 109 (1985), 397-425. [22] Pu, Pao Ming and Liu, Ying Ming ; Fuzzy topology I, Neighbourhood structure of a fuzzy point and
Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980), 571-599.
[23] Raychaudhuri, S. ; Concerning -almost continuity and -preopen sets, Bull. Inst. Math. Acad. Sinica 21(4) (1993), 357-366.
[24] Stone, M.H. ; Applications of the theory of Boolean ring to general topology, TAMS 41 (1937), 375. [25] Veličko, N.V. ; H-closed topological spaces, Amer. Math. Soc. Transl. 78 (1968), 103-118.
[26] Zadeh, L.A. ; Fu zzy sets, Inform. Control 8 (1965), 338-353
