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A generalization of the representation theorem
Fuzzy Sets and Systems 51 (1992) 309-311 North-Holland
309
A generalization of the representation theorem Dan A. Ralescu Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, USA Received August 1991 Revised October 1991
Abstract: We generalize the representation theorem for fuzzy sets. More specifically, we give necessary and sufficient conditions for a family of sets to represent a "permutation" of the levels of some fuzzy set.
Keywords: Fuzzy set; representation theorem.
level
set;
possibility
measure;
Introduction The representation theorem for fuzzy sets which first appeared in Negoita and Ralescu [7] (see also Negoita and Ralescu [6], pp. 53-57) proved to be a useful tool for the definition, analysis, and operation with fuzzy concepts (see Ralescu [10] for an early paper on this subject and H6hle [4] for a very recent historical perspective). The original idea was to compare the concept of fuzzy set [13] with a suitable extension of the concept of 'ensemble flou' [3]. The need for a representation of fuzzy sets was rediscovered: see Borillo and Fuentes [1], Zhang and Le [16], Dubois and Prade [2], pp. 136-137, Luo [5], Turksen [12], p. 9, and Swamy and Raju [11], pp. 191, among many others. In Negoita and Ralescu [8] we gave a simpler restatement of our result on representing fuzzy sets and indicated the need for such a representation as well as some extensions (some of them quite far reaching) that have appeared in the literature since 1974. The essence of the representation theorem has to do with the possibility of constructing a fuzzy set from a
family of sets. Clearly this can be done in many ways, but if we insist that the level sets of this fuzzy set be the items of the family we started from, some hypotheses are necessary. More specifically, the family of sets we start with must be decreasing and satisfy a continuity property. These conditions turn out to be necessary and sufficient for the existence of a (unique) fuzzy set with the desired property. Sometimes one starts with a family of sets which does not satisfy all the hypotheses; for instance, it is decreasing (but does not have continuity). Such a situation was considered in Ralescu [9] (quoted in [10]). All these results are relevant to the situation when we want our family of sets to represent exactly the levels of some fuzzy set. The present paper is concerned with a generalization of the representation theorem, by relaxing the above requirement. More precisely, we start with a family of sets and we want to construct a fuzzy set; however, this fuzzy set does not need to have exactly the prescribed levels. The levels may be 'shuffled' around or 'permuted' (this will be made more precise as we proceed). The rest of the paper is organized as follows: in Section 1 for completeness, we briefly describe the original representation theorem. In Section 2 we state and prove our generalization of the representation theorem.
1. Representation of fuzzy sets Let X be a set; all fuzzy subsets of X will be denoted by F X = {f I f :X--~ [0, 1]}. If f • FX, its c~-level is the set f~ = (x • X If(x)/> or}
Correspondence
to:
D.A. Ralescu, Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA.
(1)
where 0 <~ cr ~< 1. The fuzzy set f can be written in terms of its
0165-0114/92/$05.00 (~) 1992--Elsevier Science Publishers B.V. All rights reserved
310
D, Ralescu / Generalization of the representation theorem
levels (f~)~ as [14] f ( x ) = sup{or Ix
eft}.
(2)
Now, let us consider the converse problem: a family (A,~),, of subsets of X is given; does there exist a fuzzy set f • FX such that fo~ = A , for every a? Clearly, unless the family (A~)o~ satisfies some requirements, the answer is no. We must, at least, have
can take r/= sup or O = Lebesgue measure in [0, 1]. In either case, (3) gives a fuzzy set f whose levels f~ are equal to A~ for every o~. But notice that if (i) and (ii) are satisfied, then { c~ I x • A , } is an interval with left endpoint 0. Let us look, more specifically, at the choice r/(A)=supA,
Ac[0,1].
It is easy to show that r/is a possibility measure [15] in the sense that r / ( U Ai) = sup r/(Ai).
(since the levels f,, satisfy this property). Is this simple requirement sufficient? The answer is no (see the simple counterexample in Negoita and Ralescu, [7]). The necessary and sufficient conditions are stronger, as given in the following: Theorem 1. Let (A~)~ be a family of subsets of X. The necessary and sufficient conditions for the existence of a fuzzy set f • FX such that f~ = A , , 0<~ o: << - 1, are: (i) tr~
and c~,--->ol ~
iel
i~l
The main idea behind our generalization of the representation theorem is to consider any possibility measure r/, not just the particular one given by r/(A) = supA. Let then r/:P[0, 1]--->[0, 1] be any possibility measure, i.e., r/satisfies (4). But it is well known then that r/ is given by a 'possibility density'
q~:[0, 1]-, [0, 1]: r/(A) = sup qo(a).
(5)
a~m
We are then led to the formula f ( x ) = sup{q0(o 0 Ix • Ao~}
~-) A~ = Ao, n=l
The proof is given in Negoita and Ralescu [7] and will be omitted here.
(4)
(6)
(compare with (2) and (3)). The question now is, what are the level sets of f and how are they related to the original family (Ao,)~? The answer is given in the following theorem which is our generalization of the representation of fuzzy sets.
2. The generalization Assume that a family of sets (A,)~ is given; we want to use this family to construct a fuzzy set f but, this time, the levels of f need not be equal to A , for each c~ e [0, 1]. Clearly the set
{~lxeZ~}=_[O,
11
is important in connection with the fuzzy set we have in mind. The idea is to consider a set function r/ : P[0,1]---> [0,1] with certain properties, and to define the fuzzy set 'generated' by (A~)~ by the formula
f(x)=o({o~lxeA~}), xeX.
(3)
Many choices of r/ are possible; in the particular case when (A,~),~ satisfies hypotheses (i) and (ii) of the representation theorem, we
Theorem 2. Let q0:[0, 1]--->[0, 1] be given, let (A,)o~ be a family of subsets of X. necessary and sufficient conditions for existence of a fuzzy set f e F X such f~(~)=A~, 0<~ o~< 1, are (i) tp(o0~
and The the that
and (p(o~.)--->(p(a 0 :~, ~ - ~ A ~ = A ~ . n=l
Proof. Necessity is obvious. To prove sufficiency, define
f(x) = sup{~0(o0 Ix • A~} as in (6).
D. Ralescu / Generalization of the representation theorem
Let x e A s 0 . Then a o e { a ~ l x e A s } which implies f(x) = sup{tp(tr) I x E A s } i> (p(tr0) and hence x eft(s,,). To show the other inclusion, let x eft(s,,). Then
f(x)
=
Ix eA.}
(iv(a'o).
If strict inequality is true, then ::laq with x e A s , and such that qg(tx~)i> tp(O¢o). But then we have As, c As,, from (i) and thus x ~ A,,, as desired. Assume
f(x) = sup{(io(o~) I x eAs} = ~(ao). Then there exists a~, e [0, 1] with q0(ar~) T (p(tro). But then O ~ = j A s = A s , , from (ii). Since x e As. for each n, it follows that x e As,, and the proof is complete. Note. The function (iv such that A s =f~(s) acts as a 'permutation' or 'mixing' of the level sets of f. Obviously, if qg(tr)= tr, we get the representation theorem, Theorem 1. Examples. (1) Let q0(o~) = 1 - c~. Then condition
(ii) becomes:
n=l
This implies the existence of a fuzzy set f with
A.=f,-s
Yc~[O, 1].
(2) Let tp(cr) = tx:. In this case (ii) is, as usual,
trn'~ O: ~
(-) A.o=A.. n--I
This implies the existence of a fuzzy set f such that
A. =f~
Vo¢ e [0, 11.
311
Acknowledgements This paper was written while the author visited Japan in the Spring of 1991. Many thanks to Professors H. Inoue, M. Mizumoto, M. Mukaidono, R. Shimizu, and M. Sugeno for their kind hospitality. References [1] P.J. Borillo and R. Fuentes, A short note on representation of L-fuzzy sets by Moore's families, Stochastica 8 (1984) 291-295. [2] D. Dubois and H. Prade, Fuzzy sets, probability and measurement, European J. Oper. Res. 40 (1989) 135-154. [3] Y. Gentilhomme, Les ensembles flous en linguistique, Cahiers de Linguist. Thdorique et Appliqude V 47 (1968). [4] U. H6hle, Editorial, Fuzzy Sets and Systems 40 (1991) 253-256. [5] C.Z. Luo, Book announcement, Fuzzy Sets and Systems 39 (1991) 235-236. [6] C.V. Negoita and D.A. Ralescu, Multimi Vagi si Aplicatiile Lor (Ed. Technica, Bucharest, 1974). [7] C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis (Wiley, New York, 1975). [8] C.V. Negoita and D.A. Ralescu, Simulation, Knowledge-based Computing and Fuzzy Statistics (Van Nostrand, New York, 1987). [9] D.A. Ralescu, Inexact solutions for large-scale control problems, Proc. First Conf. on Math. at the Service of Man, Barcelona, Spain (1977). [10] D.A. Ralescu, A survey of the representation of fuzzy concepts and its applications, in: M. M. Gupta et al., Eds., Advances in Fuzzy Set Theory and Applications (North-Holland, Amsterdam, 1979) 77-91. [11] U.M. Swamy and D.V. Raju, Algebraic fuzzy systems, Fuzzy Sets and Systems 41 (1991) 187-194. [12] I.B. Turksen, Measurement of membership functions and their acquisition, Fuzzy Sets and Systems 40 (1991) 5-38. [13] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [14] L.A. Zadeh, Similarity relations and fuzzy orderings, Inform. Sciences 3 (1971) 177-200. [15] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28. [16] W. Zhang and H. Le, The structure of the norm system on fuzzy sets, J. Math. Anal. Applic. 127 (1987) 559-568.
309
A generalization of the representation theorem Dan A. Ralescu Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, USA Received August 1991 Revised October 1991
Abstract: We generalize the representation theorem for fuzzy sets. More specifically, we give necessary and sufficient conditions for a family of sets to represent a "permutation" of the levels of some fuzzy set.
Keywords: Fuzzy set; representation theorem.
level
set;
possibility
measure;
Introduction The representation theorem for fuzzy sets which first appeared in Negoita and Ralescu [7] (see also Negoita and Ralescu [6], pp. 53-57) proved to be a useful tool for the definition, analysis, and operation with fuzzy concepts (see Ralescu [10] for an early paper on this subject and H6hle [4] for a very recent historical perspective). The original idea was to compare the concept of fuzzy set [13] with a suitable extension of the concept of 'ensemble flou' [3]. The need for a representation of fuzzy sets was rediscovered: see Borillo and Fuentes [1], Zhang and Le [16], Dubois and Prade [2], pp. 136-137, Luo [5], Turksen [12], p. 9, and Swamy and Raju [11], pp. 191, among many others. In Negoita and Ralescu [8] we gave a simpler restatement of our result on representing fuzzy sets and indicated the need for such a representation as well as some extensions (some of them quite far reaching) that have appeared in the literature since 1974. The essence of the representation theorem has to do with the possibility of constructing a fuzzy set from a
family of sets. Clearly this can be done in many ways, but if we insist that the level sets of this fuzzy set be the items of the family we started from, some hypotheses are necessary. More specifically, the family of sets we start with must be decreasing and satisfy a continuity property. These conditions turn out to be necessary and sufficient for the existence of a (unique) fuzzy set with the desired property. Sometimes one starts with a family of sets which does not satisfy all the hypotheses; for instance, it is decreasing (but does not have continuity). Such a situation was considered in Ralescu [9] (quoted in [10]). All these results are relevant to the situation when we want our family of sets to represent exactly the levels of some fuzzy set. The present paper is concerned with a generalization of the representation theorem, by relaxing the above requirement. More precisely, we start with a family of sets and we want to construct a fuzzy set; however, this fuzzy set does not need to have exactly the prescribed levels. The levels may be 'shuffled' around or 'permuted' (this will be made more precise as we proceed). The rest of the paper is organized as follows: in Section 1 for completeness, we briefly describe the original representation theorem. In Section 2 we state and prove our generalization of the representation theorem.
1. Representation of fuzzy sets Let X be a set; all fuzzy subsets of X will be denoted by F X = {f I f :X--~ [0, 1]}. If f • FX, its c~-level is the set f~ = (x • X If(x)/> or}
Correspondence
to:
D.A. Ralescu, Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA.
(1)
where 0 <~ cr ~< 1. The fuzzy set f can be written in terms of its
0165-0114/92/$05.00 (~) 1992--Elsevier Science Publishers B.V. All rights reserved
310
D, Ralescu / Generalization of the representation theorem
levels (f~)~ as [14] f ( x ) = sup{or Ix
eft}.
(2)
Now, let us consider the converse problem: a family (A,~),, of subsets of X is given; does there exist a fuzzy set f • FX such that fo~ = A , for every a? Clearly, unless the family (A~)o~ satisfies some requirements, the answer is no. We must, at least, have
can take r/= sup or O = Lebesgue measure in [0, 1]. In either case, (3) gives a fuzzy set f whose levels f~ are equal to A~ for every o~. But notice that if (i) and (ii) are satisfied, then { c~ I x • A , } is an interval with left endpoint 0. Let us look, more specifically, at the choice r/(A)=supA,
Ac[0,1].
It is easy to show that r/is a possibility measure [15] in the sense that r / ( U Ai) = sup r/(Ai).
(since the levels f,, satisfy this property). Is this simple requirement sufficient? The answer is no (see the simple counterexample in Negoita and Ralescu, [7]). The necessary and sufficient conditions are stronger, as given in the following: Theorem 1. Let (A~)~ be a family of subsets of X. The necessary and sufficient conditions for the existence of a fuzzy set f • FX such that f~ = A , , 0<~ o: << - 1, are: (i) tr~
and c~,--->ol ~
iel
i~l
The main idea behind our generalization of the representation theorem is to consider any possibility measure r/, not just the particular one given by r/(A) = supA. Let then r/:P[0, 1]--->[0, 1] be any possibility measure, i.e., r/satisfies (4). But it is well known then that r/ is given by a 'possibility density'
q~:[0, 1]-, [0, 1]: r/(A) = sup qo(a).
(5)
a~m
We are then led to the formula f ( x ) = sup{q0(o 0 Ix • Ao~}
~-) A~ = Ao, n=l
The proof is given in Negoita and Ralescu [7] and will be omitted here.
(4)
(6)
(compare with (2) and (3)). The question now is, what are the level sets of f and how are they related to the original family (Ao,)~? The answer is given in the following theorem which is our generalization of the representation of fuzzy sets.
2. The generalization Assume that a family of sets (A,)~ is given; we want to use this family to construct a fuzzy set f but, this time, the levels of f need not be equal to A , for each c~ e [0, 1]. Clearly the set
{~lxeZ~}=_[O,
11
is important in connection with the fuzzy set we have in mind. The idea is to consider a set function r/ : P[0,1]---> [0,1] with certain properties, and to define the fuzzy set 'generated' by (A~)~ by the formula
f(x)=o({o~lxeA~}), xeX.
(3)
Many choices of r/ are possible; in the particular case when (A,~),~ satisfies hypotheses (i) and (ii) of the representation theorem, we
Theorem 2. Let q0:[0, 1]--->[0, 1] be given, let (A,)o~ be a family of subsets of X. necessary and sufficient conditions for existence of a fuzzy set f e F X such f~(~)=A~, 0<~ o~< 1, are (i) tp(o0~
and The the that
and (p(o~.)--->(p(a 0 :~, ~ - ~ A ~ = A ~ . n=l
Proof. Necessity is obvious. To prove sufficiency, define
f(x) = sup{~0(o0 Ix • A~} as in (6).
D. Ralescu / Generalization of the representation theorem
Let x e A s 0 . Then a o e { a ~ l x e A s } which implies f(x) = sup{tp(tr) I x E A s } i> (p(tr0) and hence x eft(s,,). To show the other inclusion, let x eft(s,,). Then
f(x)
=
Ix eA.}
(iv(a'o).
If strict inequality is true, then ::laq with x e A s , and such that qg(tx~)i> tp(O¢o). But then we have As, c As,, from (i) and thus x ~ A,,, as desired. Assume
f(x) = sup{(io(o~) I x eAs} = ~(ao). Then there exists a~, e [0, 1] with q0(ar~) T (p(tro). But then O ~ = j A s = A s , , from (ii). Since x e As. for each n, it follows that x e As,, and the proof is complete. Note. The function (iv such that A s =f~(s) acts as a 'permutation' or 'mixing' of the level sets of f. Obviously, if qg(tr)= tr, we get the representation theorem, Theorem 1. Examples. (1) Let q0(o~) = 1 - c~. Then condition
(ii) becomes:
n=l
This implies the existence of a fuzzy set f with
A.=f,-s
Yc~[O, 1].
(2) Let tp(cr) = tx:. In this case (ii) is, as usual,
trn'~ O: ~
(-) A.o=A.. n--I
This implies the existence of a fuzzy set f such that
A. =f~
Vo¢ e [0, 11.
311
Acknowledgements This paper was written while the author visited Japan in the Spring of 1991. Many thanks to Professors H. Inoue, M. Mizumoto, M. Mukaidono, R. Shimizu, and M. Sugeno for their kind hospitality. References [1] P.J. Borillo and R. Fuentes, A short note on representation of L-fuzzy sets by Moore's families, Stochastica 8 (1984) 291-295. [2] D. Dubois and H. Prade, Fuzzy sets, probability and measurement, European J. Oper. Res. 40 (1989) 135-154. [3] Y. Gentilhomme, Les ensembles flous en linguistique, Cahiers de Linguist. Thdorique et Appliqude V 47 (1968). [4] U. H6hle, Editorial, Fuzzy Sets and Systems 40 (1991) 253-256. [5] C.Z. Luo, Book announcement, Fuzzy Sets and Systems 39 (1991) 235-236. [6] C.V. Negoita and D.A. Ralescu, Multimi Vagi si Aplicatiile Lor (Ed. Technica, Bucharest, 1974). [7] C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis (Wiley, New York, 1975). [8] C.V. Negoita and D.A. Ralescu, Simulation, Knowledge-based Computing and Fuzzy Statistics (Van Nostrand, New York, 1987). [9] D.A. Ralescu, Inexact solutions for large-scale control problems, Proc. First Conf. on Math. at the Service of Man, Barcelona, Spain (1977). [10] D.A. Ralescu, A survey of the representation of fuzzy concepts and its applications, in: M. M. Gupta et al., Eds., Advances in Fuzzy Set Theory and Applications (North-Holland, Amsterdam, 1979) 77-91. [11] U.M. Swamy and D.V. Raju, Algebraic fuzzy systems, Fuzzy Sets and Systems 41 (1991) 187-194. [12] I.B. Turksen, Measurement of membership functions and their acquisition, Fuzzy Sets and Systems 40 (1991) 5-38. [13] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [14] L.A. Zadeh, Similarity relations and fuzzy orderings, Inform. Sciences 3 (1971) 177-200. [15] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28. [16] W. Zhang and H. Le, The structure of the norm system on fuzzy sets, J. Math. Anal. Applic. 127 (1987) 559-568.