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K Y B E R N E T I K A - V O L U M E 2 Í ( 1 9 9 2 ) , NUMBER 6, PAGES 4 2 5 - 4 4 3
C O M P A R I N G ALTERNATIVE D E F I N I T I O N S OF BOOLEAN-VALUED FUZZY SETS IVAN KRAMOSIL
Two definitions of fuzzy sets with Boolean-valued membership functions, introduced by Drossos and Markakis and called by them external and internal Boolean fuzzy sets, are compared with a third, classical definition descending more directly from the original Zadeh's and Goguen's ideas. Under some rather general conditions, internal and classical Boolean fuzzy sets are proved to be equivalent in the sense that there exists a one-to-one mapping to each other conserving the set theoretic operations. On the other side, the space of external Boolean fuzzy sets is richer, so that such a mapping exists only in some rather special cases. 1. THREE DEFINITIONS OF BOOLEAN-VALUED FUZZY SETS The two basic notions considered and combined together throughout this paper will be that of fuzzy set and that of Boolean algebra. Let us refer to [4] as far as the notion of fuzzy sets, their properties and basic results are concerned, let us refer to [5] for Boolean algebras. Fuzzy sets, in their classical setting with numerical real-valued membership functions, were conceived by Zadeh in 1965 [6] with the aim to develop a mathematical tool for uncertainty quantification and processing, alternative to that one represented by the classical probability theory. Hence, a fuzzy subset E of a nonempty basic space or universe A was defined by and identified with a function /% defined on A and taking its values in the unit interval (0,1) of real numbers. For a number of reasons, as soon as in 1967 Goguen presented the idea of fuzzy sets with non-numerical membership functions, cf. [3] for more details and motivation. Then, in 1985, [1] Drossos and Markakis argued in favour of taking profit of Boolean algebras when defining fuzzy sets, however, both the definitions of fuzzy sets, suggested by the same authors in [2], differ from the definitions resulting from direct applications of Goguen's ideas. So, the aim of this paper will be to compare the two definitions of Boolean-valued fuzzy sets from [2] with the third, classical or Goguen-like one. Let 1 = (2?,V,A,->,0 I ,lj) be a Boolean algebra over a nonempty support set B, hence, for each e, / € B, e V / is the supremum and e A / the infimum of e, / , ->e is
426
IVAN KRAMOSIL
the complement of e, OB is the zero and I s the unit (or: the minimal and the maximal) element of Iff. The partial ordering < on B will be defined in the usual way, i. e., for e, / € B, e < f holds iff e V / = / , or, what is the same, iff e A / = e, here = is the identity (relation) on B. There are numerous settings of the set of axioms which the operations and the distinguished elements of a Boolean algebra are to obey and we shall not repeat them here referring, e.g., to axioms A\ - A5 in [5]. In what follows, we shall always suppose that: i) The Boolean algebra B is complete, hence, for each I ^ C c B there exist such that (a) e < then (b) g < then (ii)
f,g£B
/ for each e G C and, if e < / , holds for some f\ G B and each e G C, / < f\\ such an / is denoted by Vegc e ani ^ called supremum of C. e for each e € C and, if #1 < e holds for some g\ G B and each e G C, {X(E)}EcA)
€ Dcp ( 1 B ) } .
(1.2)
(iii) External W-fuzzy subset of (the set) A or: external 1-fuzzy set over A is a classical (crisp) subset of the set A*. Hence, the set of all external I-fuzzy sets over A is the power-set V(A*) of the set A* of 1-fuzzy elements of A. Fact 1.1. In general, [V(A)}* is a proper subset of V(A*). Hence, each internal 1-fuzzy subset of A is also an external one, but, in general, not vice versa. P r o o f . Cf. [2] and references introduced there.
•
In the rest of this paper we shall investigate some relations between internal and classical 1-fuzzy sets over A and between external and classical B-fuzzy sets over A. In the extensional setting, we shall investigate the relations between [V(A)}* and A*, and between V(A*) and A*. 2. MUTUAL EMBEDDINGS OF INTERNAL AND CLASSICAL l-FUZZY SETS Let hie he a mapping defined on [V(A)\*, taking its values in A* and such that, for each internal 1-fuzzy set (over A) X <E [V(A)}*, hic(X), denoted also by X^*\ is the classical 1-fuzzy set (over A) defined by
hic(X)(a) = A-W(a) =
\J
X(E)
(2.1)
EcA,aeE
for all a € A; here ic abbreviates "internal to classical". It is evident that X'*' G A*, moreover, the mapping hic is one-to-one, as Theorem 2.1 proves. Theorem 2.1.
Let Xu X2 € [V(A)}*, let Xx -- X2, then x{*] -. A f >•
P r o o f . First, let us prove a more general auxiliary assertion: let C C B, C e Dcp (1 B ), let F be a nonempty set, let 0 -£ U(A) C C for each a € F, then
(22)
A V e = V eaefeeW(a)
Write f)U(a)
instead of [)&"(*)•
Ve 6 n^(«) e ^ V e g W (a) e
fOT e a c h
e€f\a€F^)
* s f\U(a)
C U(a) for each aG
« e F , SO that Veen"*"'' ~
A
F, we obtain
^ F Ve€M(a) e"
Let
/
€
428
IVAN KRAMOSIL
C - f l u ( a ) , then / € C-U(af) for some af € F, hence, /A\/ e e U { a j ) e = VeeW(a / )(/ Ae ) = OB, as / 7^ e for each e 6 U(aj), consequently, / A e = 0 B . So, for / g C — p | u ( a ) e
A (/A V a6F \
) = / A ( A V e)=0*>
eeW(a) /
(2-3)
\ a 6 F e€U(a) )
so that
V
Ve W V
(/A A
/eC-nt/(a) \
a€f e£i/(a) /
V e )=°- (2-4)
/ W A
\/6C_p|W(a)
/
\a 6 Fe€W(a)
/
Hence, e
A V
H A V e)AlB=
agFeeWfa)
\a 6 Fe6W(a)
(2.5)
/
= fA V e W V /v V /) = \a 6 Fe6W(a) /
\}eC-f)U(a)
fef]^(a)
)
A
A V ' ) I V /IJvNA V - h i^aeFeeWfa) /
= 0BV
V /en^W
/=
\jeC-f)U{a) e
V
/ /
\\a6FeeW(a) /
V / \/en^W
'
een"(a)
as Ve e n"(°) e - ^a^F VeeW(a) e
is n o t h i n
A V
e
g
else t h a n
)A| V
a€Fe6W(a) /
e
)=
VeflWI") /
V
e
(2-6)
'
^fl"!")
hence, (2.2) is proved. Now, let us prove that, for each X G [V(A)}* and each F C A,
X(F) = A *M(a) A A a6F
-*W(«).
(2.7)
ag/4-F
Or, supposing that (2.7) holds, and considering X\, X2 6 ["P^)]* such that ^""'(a) = X^](a) for all a € A, we obtain that Xi(F) = A 2 ( F ) for all F C A. Hence, if there exists F C A such that ^ ( F ) ^ Xt(F), it must also exist aeA such that ^ " ' ( a ) ^ X W ( a ) . Due to (2.1), (2.7) yields that
A(F) = A
V
a€F EC-4, a£fi
X
(£)A
A ag/l-F
V
x £
£ 0 4 , ae£
( )-
(2-8)
429
Comparing Alternative Definitions of Boolean-Valued Fuzzy Sets As
fla6F'{£ : E O A, a e E} = {E : E C A, a € £ , for all a e F } = {E : F C E}, (2.2) yields that
/\
V
X(E) = \ / *(£)•
aeFEC/l.aeF
(2.9)
FDF
Moreover,
(J {E: EC A, aeE} = {E: E C A, E t-F
(2.17)
430
IVAN KRAMOSIL
We shall write also # ud XW(F) instead of h«(X) and hci(X)(F). Here, again, ci abbreviates "classical to internal", but in this time it is not evident that ;f(#> is in fP(.A)]* so that we have to prove it. The complete Boolean algebra IB is called completely set-isomorphic, if there exists a set S and a mapping H defined on B, taking its values in the power-set V(S) over S and such that, for each e £ B and each C C B, H(-,e) = S-H(e),
H (\J e\ = [j H(e). VegC / e£C
(2.18)
Consequently, also H(0B) = 0,
H(1B) = S,
H (f\ e) = f ] //(e) VegC / e€C
(2.19)
hold, as can be easily proved. When considering only finite operations, i.e., finite sets C in (2.18) and (2.19), each Boolean algebra is finitely set-isomorphic, due to the wellknown Stone representation theorem (cf., e.g., [5], § 8). For infinite operations the existence of an isomorphism between a complete Boolean algebra and the field of subsets of a set is a nontrivial property of the Boolean algebra in question, as the two facts introduced below demonstrate. An element 0 B ^ e € B is called an atom of the Boolean algebra B, if for each / € B such that / < e, either / = 0 B or / = e. Boolean algebra IB is called atomic, if for each Oj / / G B there exists an atom e £ B such that e < / . Complete Boolean algebra B is called completely distributive, if for each {ets}teT,ses C B,
nu e >
teTses
ves
(2.20)
teT
or, what is the same due" to the fact that de Morgan rules are valid also for infinite operations, if for each {e t .}< e r,ses C B,
u n e - = n Ue1
= H ( \/ *(#)(F)) = VFC>1
b u t a so
l
So
>
for
(2.24)
/
= U I (~)H (*(*)) n f ] (S - H(X(a)))) = S. FcA Va6F
a6>l-F
/
However, 1B is the only element of B which is mapped onto S by H. Hence, it follows VFCA Xm(F) = 1B, so that {X{*\F)}FcA € Dcp(l B ). The assertion is proved. D
Theorem 2.3. Let IB be completely set-isomorphic, let Xx, X2 G A*, let Xx ^ X2,
then X[*] ± Xf\
P r o o f . Let us prove that, for each X £ A* and each a0 G A,
X(a0)= V X{*\F)= \/ ( f\X(a)A F9a 0
F3a 0 VaeF
f\ ^X(a)Y a&A-F
(2.25)
J
Supposing that (2.25) holds and that xi*\F) = X{*\F) for each FcA, then X1(a0) = A'2(a0) for each a0 G A, hence, X](a) ^ X2(a) for some a G A implies that X$*\P) ^ X{*\F) for some F C A.
432
IVAN KRAMOSIL
Take a0 £ A arbitrarily and set A0 = A - {a0}. Define a 1-fuzzy subset X0 of A0 (classical), setting X0(a) = X(a) for each a £ A0, so that X0 £ A*0 = BA°. Set, for each ECA0, 4*\E) = f\ X0(a) A / \ -^X0(a). (2.26) agF
ae^o-E
Applying Theorem 2.2 to A0 and X0 we obtain that
V (A^a)A
V ^ ) = FC/»o
FC/io VaEF
A -V'6T
/
v
(A*« ( # W) = :
{E 1 } lerC P(/l),n i€T t "1-£),П 1 е г Е 1 =в'бТ \ a e F °
Л
-ад).
аб(Л-Е)-Е°
/
so t h a t t h e only we h a v e t o prove is t h a t
A -(A'V'(«))= A (V-A'(«))= aeA-E
\ieT
V
)
aeA-E \teT
(413)
I
A -*«(«))•
A(A^(«)A 0
{E?} i e rCP(4-E),n i s r E, =(9<eT \ a 6 F o
a£(/l-E)-F°
/
S u p p o s i n g t h a t IB is completely set-isomorphic we m a y also s u p p o s e , w i t h o u t any loss of g e n e r a l i t y a n d in order t o simplify our n o t a t i o n , t h a t Xt(a) S for each t £T,
is a subset of a basic s p a c e
a 6 A, a n d t h a t (4.13) converts i n t o
H
* = n u^'-'WH
(4.14)
ae>4-E(€T
u
n(n<wn n (*-w)
{F?} 1 £ TCP(4-E),n i 6 T E t =B'6T \ a € F ° =
a6(>t
_B)_^
H2,
w h e r e H\ a n d Hi d e n o t e a b b r e v i a t e l y t h e corresponding sets.
439 Comparing Alternative Definitions of Boolean-Valued Fuzzy Sets
Let s G S, s G # 1 , set, for each t € T, at(s) = {a G A, s G Xt(a)} ,
F°(s) = at(s) - E-
(4.15)
Obviously, f]at(s)=
i a e A : s € f) *.(a) 1 =
(€T
I
16T
(4-16)
J
= J a G A : s G 5 - (J(5 - *.(«)) 1 C £, as for each a G A - E, s € # i yields that s € U i e T ^ - Xt(a)). Hence, flteT^ 0 = f|i6T (at(s) -E) = 9, so that {F°(s)} ( € r is one of the sequences over which the union operation in # 2 is taken. Moreover, F?(s) C at(s), so that s G Xt(a) for each a G F°(s) and each t G T, hence, s G flaeF?
C O M P A R I N G ALTERNATIVE D E F I N I T I O N S OF BOOLEAN-VALUED FUZZY SETS IVAN KRAMOSIL
Two definitions of fuzzy sets with Boolean-valued membership functions, introduced by Drossos and Markakis and called by them external and internal Boolean fuzzy sets, are compared with a third, classical definition descending more directly from the original Zadeh's and Goguen's ideas. Under some rather general conditions, internal and classical Boolean fuzzy sets are proved to be equivalent in the sense that there exists a one-to-one mapping to each other conserving the set theoretic operations. On the other side, the space of external Boolean fuzzy sets is richer, so that such a mapping exists only in some rather special cases. 1. THREE DEFINITIONS OF BOOLEAN-VALUED FUZZY SETS The two basic notions considered and combined together throughout this paper will be that of fuzzy set and that of Boolean algebra. Let us refer to [4] as far as the notion of fuzzy sets, their properties and basic results are concerned, let us refer to [5] for Boolean algebras. Fuzzy sets, in their classical setting with numerical real-valued membership functions, were conceived by Zadeh in 1965 [6] with the aim to develop a mathematical tool for uncertainty quantification and processing, alternative to that one represented by the classical probability theory. Hence, a fuzzy subset E of a nonempty basic space or universe A was defined by and identified with a function /% defined on A and taking its values in the unit interval (0,1) of real numbers. For a number of reasons, as soon as in 1967 Goguen presented the idea of fuzzy sets with non-numerical membership functions, cf. [3] for more details and motivation. Then, in 1985, [1] Drossos and Markakis argued in favour of taking profit of Boolean algebras when defining fuzzy sets, however, both the definitions of fuzzy sets, suggested by the same authors in [2], differ from the definitions resulting from direct applications of Goguen's ideas. So, the aim of this paper will be to compare the two definitions of Boolean-valued fuzzy sets from [2] with the third, classical or Goguen-like one. Let 1 = (2?,V,A,->,0 I ,lj) be a Boolean algebra over a nonempty support set B, hence, for each e, / € B, e V / is the supremum and e A / the infimum of e, / , ->e is
426
IVAN KRAMOSIL
the complement of e, OB is the zero and I s the unit (or: the minimal and the maximal) element of Iff. The partial ordering < on B will be defined in the usual way, i. e., for e, / € B, e < f holds iff e V / = / , or, what is the same, iff e A / = e, here = is the identity (relation) on B. There are numerous settings of the set of axioms which the operations and the distinguished elements of a Boolean algebra are to obey and we shall not repeat them here referring, e.g., to axioms A\ - A5 in [5]. In what follows, we shall always suppose that: i) The Boolean algebra B is complete, hence, for each I ^ C c B there exist such that (a) e < then (b) g < then (ii)
f,g£B
/ for each e G C and, if e < / , holds for some f\ G B and each e G C, / < f\\ such an / is denoted by Vegc e ani ^ called supremum of C. e for each e € C and, if #1 < e holds for some g\ G B and each e G C, {X(E)}EcA)
€ Dcp ( 1 B ) } .
(1.2)
(iii) External W-fuzzy subset of (the set) A or: external 1-fuzzy set over A is a classical (crisp) subset of the set A*. Hence, the set of all external I-fuzzy sets over A is the power-set V(A*) of the set A* of 1-fuzzy elements of A. Fact 1.1. In general, [V(A)}* is a proper subset of V(A*). Hence, each internal 1-fuzzy subset of A is also an external one, but, in general, not vice versa. P r o o f . Cf. [2] and references introduced there.
•
In the rest of this paper we shall investigate some relations between internal and classical 1-fuzzy sets over A and between external and classical B-fuzzy sets over A. In the extensional setting, we shall investigate the relations between [V(A)}* and A*, and between V(A*) and A*. 2. MUTUAL EMBEDDINGS OF INTERNAL AND CLASSICAL l-FUZZY SETS Let hie he a mapping defined on [V(A)\*, taking its values in A* and such that, for each internal 1-fuzzy set (over A) X <E [V(A)}*, hic(X), denoted also by X^*\ is the classical 1-fuzzy set (over A) defined by
hic(X)(a) = A-W(a) =
\J
X(E)
(2.1)
EcA,aeE
for all a € A; here ic abbreviates "internal to classical". It is evident that X'*' G A*, moreover, the mapping hic is one-to-one, as Theorem 2.1 proves. Theorem 2.1.
Let Xu X2 € [V(A)}*, let Xx -- X2, then x{*] -. A f >•
P r o o f . First, let us prove a more general auxiliary assertion: let C C B, C e Dcp (1 B ), let F be a nonempty set, let 0 -£ U(A) C C for each a € F, then
(22)
A V e = V eaefeeW(a)
Write f)U(a)
instead of [)&"(*)•
Ve 6 n^(«) e ^ V e g W (a) e
fOT e a c h
e€f\a€F^)
* s f\U(a)
C U(a) for each aG
« e F , SO that Veen"*"'' ~
A
F, we obtain
^ F Ve€M(a) e"
Let
/
€
428
IVAN KRAMOSIL
C - f l u ( a ) , then / € C-U(af) for some af € F, hence, /A\/ e e U { a j ) e = VeeW(a / )(/ Ae ) = OB, as / 7^ e for each e 6 U(aj), consequently, / A e = 0 B . So, for / g C — p | u ( a ) e
A (/A V a6F \
) = / A ( A V e)=0*>
eeW(a) /
(2-3)
\ a 6 F e€U(a) )
so that
V
Ve W V
(/A A
/eC-nt/(a) \
a€f e£i/(a) /
V e )=°- (2-4)
/ W A
\/6C_p|W(a)
/
\a 6 Fe€W(a)
/
Hence, e
A V
H A V e)AlB=
agFeeWfa)
\a 6 Fe6W(a)
(2.5)
/
= fA V e W V /v V /) = \a 6 Fe6W(a) /
\}eC-f)U(a)
fef]^(a)
)
A
A V ' ) I V /IJvNA V - h i^aeFeeWfa) /
= 0BV
V /en^W
/=
\jeC-f)U{a) e
V
/ /
\\a6FeeW(a) /
V / \/en^W
'
een"(a)
as Ve e n"(°) e - ^a^F VeeW(a) e
is n o t h i n
A V
e
g
else t h a n
)A| V
a€Fe6W(a) /
e
)=
VeflWI") /
V
e
(2-6)
'
^fl"!")
hence, (2.2) is proved. Now, let us prove that, for each X G [V(A)}* and each F C A,
X(F) = A *M(a) A A a6F
-*W(«).
(2.7)
ag/4-F
Or, supposing that (2.7) holds, and considering X\, X2 6 ["P^)]* such that ^""'(a) = X^](a) for all a € A, we obtain that Xi(F) = A 2 ( F ) for all F C A. Hence, if there exists F C A such that ^ ( F ) ^ Xt(F), it must also exist aeA such that ^ " ' ( a ) ^ X W ( a ) . Due to (2.1), (2.7) yields that
A(F) = A
V
a€F EC-4, a£fi
X
(£)A
A ag/l-F
V
x £
£ 0 4 , ae£
( )-
(2-8)
429
Comparing Alternative Definitions of Boolean-Valued Fuzzy Sets As
fla6F'{£ : E O A, a e E} = {E : E C A, a € £ , for all a e F } = {E : F C E}, (2.2) yields that
/\
V
X(E) = \ / *(£)•
aeFEC/l.aeF
(2.9)
FDF
Moreover,
(J {E: EC A, aeE} = {E: E C A, E t-F
(2.17)
430
IVAN KRAMOSIL
We shall write also # ud XW(F) instead of h«(X) and hci(X)(F). Here, again, ci abbreviates "classical to internal", but in this time it is not evident that ;f(#> is in fP(.A)]* so that we have to prove it. The complete Boolean algebra IB is called completely set-isomorphic, if there exists a set S and a mapping H defined on B, taking its values in the power-set V(S) over S and such that, for each e £ B and each C C B, H(-,e) = S-H(e),
H (\J e\ = [j H(e). VegC / e£C
(2.18)
Consequently, also H(0B) = 0,
H(1B) = S,
H (f\ e) = f ] //(e) VegC / e€C
(2.19)
hold, as can be easily proved. When considering only finite operations, i.e., finite sets C in (2.18) and (2.19), each Boolean algebra is finitely set-isomorphic, due to the wellknown Stone representation theorem (cf., e.g., [5], § 8). For infinite operations the existence of an isomorphism between a complete Boolean algebra and the field of subsets of a set is a nontrivial property of the Boolean algebra in question, as the two facts introduced below demonstrate. An element 0 B ^ e € B is called an atom of the Boolean algebra B, if for each / € B such that / < e, either / = 0 B or / = e. Boolean algebra IB is called atomic, if for each Oj / / G B there exists an atom e £ B such that e < / . Complete Boolean algebra B is called completely distributive, if for each {ets}teT,ses C B,
nu e >
teTses
ves
(2.20)
teT
or, what is the same due" to the fact that de Morgan rules are valid also for infinite operations, if for each {e t .}< e r,ses C B,
u n e - = n Ue1
= H ( \/ *(#)(F)) = VFC>1
b u t a so
l
So
>
for
(2.24)
/
= U I (~)H (*(*)) n f ] (S - H(X(a)))) = S. FcA Va6F
a6>l-F
/
However, 1B is the only element of B which is mapped onto S by H. Hence, it follows VFCA Xm(F) = 1B, so that {X{*\F)}FcA € Dcp(l B ). The assertion is proved. D
Theorem 2.3. Let IB be completely set-isomorphic, let Xx, X2 G A*, let Xx ^ X2,
then X[*] ± Xf\
P r o o f . Let us prove that, for each X £ A* and each a0 G A,
X(a0)= V X{*\F)= \/ ( f\X(a)A F9a 0
F3a 0 VaeF
f\ ^X(a)Y a&A-F
(2.25)
J
Supposing that (2.25) holds and that xi*\F) = X{*\F) for each FcA, then X1(a0) = A'2(a0) for each a0 G A, hence, X](a) ^ X2(a) for some a G A implies that X$*\P) ^ X{*\F) for some F C A.
432
IVAN KRAMOSIL
Take a0 £ A arbitrarily and set A0 = A - {a0}. Define a 1-fuzzy subset X0 of A0 (classical), setting X0(a) = X(a) for each a £ A0, so that X0 £ A*0 = BA°. Set, for each ECA0, 4*\E) = f\ X0(a) A / \ -^X0(a). (2.26) agF
ae^o-E
Applying Theorem 2.2 to A0 and X0 we obtain that
V (A^a)A
V ^ ) = FC/»o
FC/io VaEF
A -V'6T
/
v
(A*« ( # W) = :
{E 1 } lerC P(/l),n i€T t "1-£),П 1 е г Е 1 =в'бТ \ a e F °
Л
-ад).
аб(Л-Е)-Е°
/
so t h a t t h e only we h a v e t o prove is t h a t
A -(A'V'(«))= A (V-A'(«))= aeA-E
\ieT
V
)
aeA-E \teT
(413)
I
A -*«(«))•
A(A^(«)A 0
{E?} i e rCP(4-E),n i s r E, =(9<eT \ a 6 F o
a£(/l-E)-F°
/
S u p p o s i n g t h a t IB is completely set-isomorphic we m a y also s u p p o s e , w i t h o u t any loss of g e n e r a l i t y a n d in order t o simplify our n o t a t i o n , t h a t Xt(a) S for each t £T,
is a subset of a basic s p a c e
a 6 A, a n d t h a t (4.13) converts i n t o
H
* = n u^'-'WH
(4.14)
ae>4-E(€T
u
n(n<wn n (*-w)
{F?} 1 £ TCP(4-E),n i 6 T E t =B'6T \ a € F ° =
a6(>t
_B)_^
H2,
w h e r e H\ a n d Hi d e n o t e a b b r e v i a t e l y t h e corresponding sets.
439 Comparing Alternative Definitions of Boolean-Valued Fuzzy Sets
Let s G S, s G # 1 , set, for each t € T, at(s) = {a G A, s G Xt(a)} ,
F°(s) = at(s) - E-
(4.15)
Obviously, f]at(s)=
i a e A : s € f) *.(a) 1 =
(€T
I
16T
(4-16)
J
= J a G A : s G 5 - (J(5 - *.(«)) 1 C £, as for each a G A - E, s € # i yields that s € U i e T ^ - Xt(a)). Hence, flteT^ 0 = f|i6T (at(s) -E) = 9, so that {F°(s)} ( € r is one of the sequences over which the union operation in # 2 is taken. Moreover, F?(s) C at(s), so that s G Xt(a) for each a G F°(s) and each t G T, hence, s G flaeF?
