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Chapter 2.3 Functions and Operations
CHAPTER
2.3
F u n c t i o n s and O p e r a t i o n s
2.3.1
FUNCT I O N A L R E L A T I O N S
The r e l a t i o n R i s a Bunotiotr i f
(y,x) E R
and ( z , x ) E R , i m p l y
Thus, f u n c t i o n s R can be c h a r a c t e r i z e d by R 0 R - l
q = z .
ZID.
The f o l l o w i n g theorem, whose p r o o f i s l e f t t o t h e r e a d e r , g i v e s severI n t h i s book, R o R - l c I D a l equivalent possible d e f i n i t i o n s o f functions. i s used t o say t h a t R i s a f u n c t i o n . We a l s o assume i n t h i s Chapter t h a t R, S, T a r e r e l a t i o n s .
THEOREM,
2.3.1.1
(i) R0R-l cID(ii)
R O R - ~ 510-
R O R - ~
(v) R
OR-^
LIDc -D I
(vi) RoR-'cID-
( ~ v Ro) n R
=
0.
WSWT(SnT)oR = (SoR) n ( T O R ) .
( i i i ) R o R - lcID(iv)
( R o R - 1) n D v = 0 .
-
W S W T ( S ~ T ) =~ R ( s ~ R 2 ), (TOR).
wx
WY
R - ~ * ( xn Y ) = ( R - ~ * x )n ( R - ~ * Y ) .
V X W Y (R* X)
f-
Y = R*(XnR-'*Y).
I t i s c l e a r t h a t o f o u r c o n s t a n t r e l a t i o n s , I D and 0 a r e f u n c t i o n s , w h i l e V x V , E L , I N , and Dv a r e n o t .
The i n t e r s e c t i o n o f f u n c t i o n s i s a f u n c t i q n , b u t t h e u n i o n i s n o t , i n general, a function. Under some r e s t r i c t i o n s , i t i s : 2.3.1.2
THEOREM SCHEMA,
LeL
T
51
be a ,tm and @ a l;otvnu&;
then
ROLAND0 C H U A Q U I
52
The r e l a t i v e complement o f a f u n c t i o n i s never a f u n c t i o n . A subclass o f a f u n c t i o n i s a l s o a f u n c t i o n and t h e composition o f f u n c t i o n s i s a f u n c t i o n . The o t h e r r e l a t i o n a l o p e r a t i o n , t h e converse o r i n v e r s e , w i l l be discussed l a t e r . The f o l l o w i n g theorem, easy t o prove, summarizes these f a c t s .
2.3.1.4
DEFINITION,
R'x =
R*{x}
*
R ' x i s Lhc value o d x by R.
R*{x}
PROOF OF ( i ) . Suppose t h a t R i s a f u n c t i o n and = { q } f o r some y. Thus n R {XI = y.
x
E
D R . Then
L e t F be a unary o p e r a t i o n . The r e s t r i c t i o n o f F t o t h e c l a s s o f w i t h F(x) E V c a n be represented by t h e f u n c t i o n F
x
I t i s c l e a r t h a t F i s a f u n c t i o n , w i t h domain D F = {x : F ( x ) E V 1 , and F ' x = F(x) f o r x E D F . Thus F and F have t h e same v a l u e s f o r x w i t h
F(x) E
v.
However, t h e r e p r e s e n t a t i o n o f o p e r a t i o n s by r e l a t i o n s g i v e n i n 2 . 2 . 2 i s more general, because i t does n o t r e s t r i c t t h e domain t o those X w i t h
53
AXIOMATIC S E T T H E O R Y
But, when p o s s i b l e , t h e p r e s e n t r e p r e s e n t a t i o n by f u n c t i o n s i s E V. more convenient.
F(x)
The f o l l o w i n g d e f i n i t i o n f o r m a l i z e s t h e i n t r o d u c t i o n o f such f u n c t i o n s .
2.3.1.6
D E F I N I T I O N SCHEMA, ( 7
X
=
Let
T
:q5) = ( ( 7 , x ) :
be a t e r m and q5 a formula. Then
GI.
For i n s t a n c e , we have when F i s a unary o p e r a t i o n , { ( F ( x ) , x ) : F(x) E V )
.
(F(x): F(x)E V ) =
X
We can i n t r o d u c e f u n c t i o n s f o r a l l t h e o p e r a t i o n s a l r e a d y d e f i n e d . For example, f o r t h e image o p e r a t i o n ,
R* = ( R * x : x E
v)
.
I n t h i s case and o t h e r s , t h e same symbol w i l l be used f o r t h e operat i o n and t h e corresponding f u n c t i o n . With the value n o t a t i o n defined i n 2.3.1.4 can be g i v e n by, F*A = { F ' x : x E A } , and F-l*A have e a s i l y ,
tl x ( x
t h e image o f a f u n c t i o n F, = { x : F'xEA}. Also, we
2.3.1.7 THEOREM, R0R-l c CZD-I D A S0S-l E DR+ R ' x = S I X ) ) .
I n t h e r e s t o f t h i s s e c t i o n , t h e l e t t e r s F, G, H, s t r i c t e d t o functions.
2.3.1.8 (i) (ii)
= B n D F -1
F* F-'*B
.
F* F - ~ * B c -B . A n B =
o
-+
=
(F*A) n B .
(F-~*A) n (F-~*B) = 0 .
(v)
F-l*(AnB)
= (F-'*A)
n (F-l*B).
(vi)
F-l*(A%B)
= (F-'*A)
%
(vii) (viii) (ix)
6, g,
t--f
DR = D S A
and h. a r e r e -
THEOREM (PROPERTIES OF IMAGES OF FUNCTIONS)
(iii) F*(AnF-l*B) (iv)
(R = S
8
5 F*A
v B 3A
-+
3 C'(C'
B n D F-'
B n D F - ~ =F*
( X I F-~*A = F-~*B
%
-A
C
=
(F-'*&).
A B = F*C).
F*A.
~-l* % B .
+
A nDF = B nDF
.
I
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54
These p r o p e r t i e s o f images o f f u n c t i o n a r e n o t c h a r a c t e r i s t i c o f f u n c t i o n i n t h e sense t h a t t h e r e a r e o t h e r r e l a t i o n s w h i c h s a t i s f y them. B u t ; we have t h a t ( v i i i ) , ( i x ) , and ( x ) a r e e q u i v a l e n t and so a r e ( x i ) and ( x i i ) . PROOF,
2.2.1.10.
and ( i ) c a n be p r o v e d d i r e c t l y and t h e r e s t o b t a i n e d f r o m (i) We have, F 0 F - l
PROOF OF ( i ) .
I D I D ( F o F-') (ID IDF-')*B
=
I D I D F -I.
-ID.
By 2 . 3 . 1 . 3
C
T h e r e f o r e F*
F-'*E
( i i ) F o F-l =
o F-l)*
= (F
=
= B n DF-'.
PROOFOF ( i i ) :
By (i).
PROOF OF (iii). By 2 . 2 . 1 . 1 0
( v i i i ) , we h a v e (F*A) n B C - F*(AnF-'*B).
On t h e o t h e r hand, b y 2 . 2 . 1 . 1 0 ( i v ) , F*(AnF-'*E) U s i n g , now, ( i i ) we o b t a i n , F * ( A n F - l * B ) C - (F*A)
c ( F * A ) n ( F * F"*B).
17-8.
n B = 0. By ( i i i ) PROOF O F ( i v ) . Assume A n t 3 = 0. By ( i i ) (F* F-'*A) F*(f-l*AnF-'*B) = (F* F-'*A) n 8 = 0. T h e r e f o r e , F-l*A n F-l*B n D F = 0 . F-l*B C - D F . Thus, F-l*A n F-l*B = 0. B u t F-'*A
PROOFOF ( v ) .
By ( i v ) , we have ( F - 1 * ( A n 8 ) ) n ( F - 1 * ( 8 n J A ) )
= ( F - l * ( A n B ) n (F-'*(A%E))
( v ) , we o b t a i n , F-l*A
= (F-l*(A%B))
=
0 =
From 2 . 2 . 1 . 1 0
n (F-l*(B\A)).
(F-'*(AnB)) U (F-l*(A-E)) a n d F-'*B = T h e r e f o r e (Fql*A) n (F-'*E)=((( F - l * ( A n E ) ) U = (F-l*(AnB)) u (F-l*(B%A)). (F-'*(A
=
* 8 ) ) )) n (( (F-l*(A
PROOF OF ( v i ) .
n8 ) )
U
(F-'*( E % A ) ) ) = F-l*(A n B )
By 2 . 2 . 1 . 1 0
( v ) , F-l*A
t h e n , b y ( v ) , F-l*A = ( F - l * ( A % B ) ) U ( ( F - l * A )
(F-'*(A*B)) PROOF
n (F-l*B)
OF ( v i i ) .
= (F*A) n 8 = B.
= 0.
Therefore,
Suppose
Therefore, t a k e
E
= (F-l*(A%E))U(F-'*(ArlB));
(F-l*S)).
(F-l*A) \(F-'*E)
5 F*A. c'
Also, b y ( i v ) =
.
-DF-' c
= F*
,
F-'*(A%B).
Then, b y ( i i i ) , F * ( A n F - l * B )
= A n F-l*B
PROOF OF ( v i i i ) . We have, B n D F - ' i n g ( v i i ) , we o b t a i n ( v i i i ) .
.
=
V. Hence, a p p l y -
55
AXIOMATIC SET THEORY
PROOF OF ( i x ) .
Using 2.2.1.10
By ( v i i i ) , we have B n DF-'
( B n D F - ' ) = B n DF-'. But, by 2.2.1.10 ( F - ' * % 8 ) u (F-'* x D 0 F - l ) = F - l * % B . (xi,
=
F*A
f o r some A .
F* % F-'* % ( v ) , F - l * % ( 8 n OF-') = = F*A,
( x i ) , we o b t a i n , F*%F-'*%F*A
i.e.
( x i ) and f x i i ) a r e ' l e f t t o t h e r e a d e r .
We a l s o have d i s t r i b u t i v i t y o f i n v e r s e image o f f u n c t i o n s w i t h generalized intersection.
2.3.1.9
THEOREM SCHEMA, L& r be a t m and 6 a 6o/un&.
On t h e o t h e r hand, i f x $ implies that
f o r a l l such r . IT
x
6
F-'*r
E
nx
, i.e. y
Therefore
FIX
E
DEFINITION
BA i s t h e & a n
06
F'x
=
:$I,
t h e n f o r a l l Xo...
f o r some y
nx
0.. .x n-1
:$I. 2.3.1.10
r
0"' Xn-1
{r
:$I,
Then,
E 7,
i.e.
Xn-l,
and hence F'xE r
x c ~ - l *n
'0'
*
'Xn-1
I
6uncLionn w L t h domain 8 and m n g e included i n A.
Since i n G we almost never can prove t h a t f u n c t i o n which i s a s e t w i t h domain 8, 'A 8
BA
f
0, i.e.
t h a t t h e r e is a
w i l l n o t be much used i n P a r t 2.
On t h e o t h e r hand A ( ? ) i s t h e no;tion t h a t says t h a t f 12 a 6unOtiun w a h domain 8 m d tange i n c h d e d in A. W i t h t h i s n o t i o n , we can express t h a t ? i s a f u n c t i o n by D F V ( F ) . T h i s w i l l o f t e n be used.
2.3.1.11 DEFINITION SCHEMA (GENERALIZED CARTESIAN PRODUCT). Let be a term and $ a formula. Then n ( 7 :q,) = Ed : 6 & x V A
~ ; O ~ - ' C J D A DI x~: = $1 A
Vx($
+
6kE
r)).
X
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56
2.3.1.12
For instance, we have
F, by F ( 0 ) = A xEA A
"6 = nX (6'~:x
DEFINITION,
gEBl.
nX
( A : xEB) =
E
06).
G
A.
I f we d e f i n e t h e o p e r a t i o n
XIx (F(x) : X E 2 ) = (I( x,O) , ( y , 1 ) I : I ( x,O) , ( g , l ) l i s a p o s s i b l e d e f i n i t i o n o f t h e o r -
and F ( 1 ) = 8, then The s e t
2
dered p a i r o f x and g. I t i s t h e f u n c t i o n 1; E {x,gl, w i t h 6'0= x a n d 4'1 = y . I f we ' i d e n t i f y ' t h i s 6 w i t h t h e ordered p a i r ( X , L J ) , t h e n t h e Ilx (F(x): x E 2 ) i s ' i d e n t i f i e d ' w i t h A x B .
I n some c o n t e x t , ( x , g ) i s b e t t e r as ordered p a i r than {C x,O), ( y , l ) } ; i n o t h e r s t h e o p p o s i t e i s t h e case. The same i s t r u e f o r t h e t w o operat i o n s of C a r t e s i a n Droduct. T h i s m u l t i p l i c i t y o f d i f f e r e n t p o s s i b l e d e f i n i t i o n s f o r t h e same conc e p t i s an i n e l e g a n t c h a r a c t e r i s t i c o f a l l s e t t h e o r i e s and i t seems unavoidable. We say t h a t F i s a biunLquc
2.3.1.13
(ii)
one-one i j u n c t i o n i f F and F - l a r e func-
i f D F ~ ~ - Al D ( F~- l) D F (F-').
t i o n s , i.e.
(i)
ofi
-
DEFINITIONa
!A(F)
! A = (1;:
.
A ~ ( A~ A) ~ ( ~ - l ) A
A(ij)l,
! A ( F ) says t h a t F i s a pe,trnLLtCLtion o f A , i.e. F i s a b i u n i q u e funct i o n from A o n t o A. !A i s t h e c l a s s o f a l l permutations o f A.
PROBLEMS
Prove 2.3.1.1
1. 2. 3.
Characterize a l l r e l a t i o n s R t h a t s a t i s f y R = R o R - l o R
4.
Show t h a t F i s a b i u n i q u e f u n c t i o n i f and o n l y i f D F * = V A DF-'"
Prove 2.3.1.3
= PDF-'
A (F*
I
.
P D F i s a biunique function).
-
5.
Prove 2.3.1.8
6.
F i n d a r e l a t i o n R which i s n o t a f u n c t i o n b u t s a t i s f i e s
7.
Characterize t h e r e l a t i o n s
(x)
(xii).
W B 3 A ( B n D R - ~= R*A).
R
that satisfy
R* R'l*
R*A
=
R*A
.
=
A X I O M A T I C SET THEORY 2.3.2
57
MONOTONE O P E R A T I O N S ,
z
A unary o p e r a t i o n F i s Y,Z-monutone i f i t s a t i s f i e s : Y LA c B C Y c F [ A ) C_ F ( B ) 5 2 . We say t h a t F i s rnoncdone i f i t i s 0, V-KonoTone. S i m i l a r l y , f o r f u n c t i o n s we have: 2.3.2.1
F ' x C- F ' y ) .
DEFINITION,
Mo ( F )
- DbF
(F) Avxt'q(x,y
E
DFA
x
--f
5q
-+
The f o l l o w i n g theorem g i v e s c o n d i t i o n s e q u i v a l e n t t o monotony. - 1. 2.3.2.2 THEOREM SCHEMA, L e R F be a unmy opehCLtion and Y c Then t h e doflowing CvnditioMn CVLQ e.qLvaRen2:
( i ) W A WB(Y c - A c- 8 c- Z
( i v ) W A WB(Y ( v ) W A WB(Y
c - A, B c- Z
- A, C
B C -Z
+
--t
+
Y c - F(A) c - F ( B ) c- Z ) .
Y c - F(A) u F ( B ) C_F(AUB)C - Z).
-
Y C - F ( A n B ) C F(A) n F ( B )
5 Z).
Suppose t h a t V AW B ( Y C A C B C Z Y C F(A) C F ( B ) 2 Z ) . PROOF, Therefore, f o r e v e r y X t h a t s a t i s f i e s - @ A-Y C-X c Z , we have-Y z F ( X ) 5 F( u(X : 4 A Y 5 X c Z}) c Y c - X C- Z } ) C_F(X)& Z . - Z and Y & F ( n{X : @ +
n
Thus,
( i ) i m p l i e s ( i i ) and ( i i i ) .
I t i s c l e a r t h a t ( i i ) i m p l i e s ( i v ) and i t i s enough t o show t h a t ( i v ) o r ( v ) i m p l y i m p l i e s ( i ) . Suppose Y S A 5 8 C Z. Then, Thus, Z ? F ( B ) 2 F(A) 2 2 F(A) u F ( B ) 3 Y .
-
( i i i ) i m p l i e s ( v ) . Therefore, ( i ) .I s h a l l prove t h a t ( i v ) by ( i v ) , Z > F ( B ) = F(AUB) Y.
The i m p l i c a t i o n o f ( v ) t o ( i ) i s proved s i m i l a r l y . I n a s i m i l a r way, t h e f o l l o w i n g can be proved.
.
58
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(iv)
t!xtlq F(x) u F ( q ) gF(xuq),
(v)
W x w q F ( x n q ) 5F(x) " F ( Y ) .
We now begin t h e s t u d y o f f i x e d p o i n t s o f monotone o p e r a t i o n s . T h i s study i s based on T a r s k i 1955. The theorems proved here w i l l be useful i n several chapters o f t h e book. We say t h a t a c l a s s X i s a hixed p a i d of a unary o p e r a t i o n F, i f We have t h a t Y, Z-monotone o p e r a t i o n s always have f i x e d p o i n t s . F(X) = X. F i r s t , t h e theorem f o r 0, 2-monotone o p e r a t i o n s . THEOREM SCHEMA, L e t F be a unmq opehation.
2.3.2.4
W A W B ( A-c B c Z +
F ( A )CF(B)LZ)+
n {X : X c - Z A F(X) = X ] A U =
LJ
Then
~C~U(F(C)=CAF(U)=DAC=
{ X :X C - Z A F(X) = A } ) .
F can be considered as a monotone o p e r a t i o n from subclasses o f Z t o subclasses o f Z .C i s t h e l e a s t f i x e d p o i n t and Q i s t h e g r e a t e s t f i x e d p o i n t . Thus, t h e c o n c l u s i o n can a l s o be w r i t t e n , WA W B ( A - c B-c Z + F ( A ) & F ( B ) 5 Z ) + F ( n { X : X-C Z A F ( X )= X } ) =
= n
{x : x
A F(X) =
c -
z
A F ( x ) = X I A F( u { x : x c -
z
A F(x)=
XI.
PROOF, Assume F(A) c F(B) C Z). L e t that C Z. Suppose, By t h e assumption, we we g e t t h a t F(C) 5 X,
c
X I )= u { x : x c -z
A
t h a t F i s a unary o p e r a t i o n such t h a t , V A W B ( A c-B C-Z + C = n { X : F ( X ) 2 X C Z l . Since F ( Z ) 5 Z, we have now t h a t X i s such t h a t F ( X ) 5 X 5 Z. Then C C X have F(C) L F ( X ) . S i n c e we assumed t h a t F(X)-z X , f o r a l l X w i t h F(X) 5 X E Z. T h e r e f o r e F(C) 5 C.
.
On t h e o t h e r hand, from F(C) c C
5 Z,
by t h e monotony o f F , we deduce
F (F(C))E F ( C )5 Z. Thus F ( C ) i s o n e o f t h e c l a s s e s whose i n t e r s e c t i o n i s - F(C),and, hence, F(C) = C. C. Therefore, C c
c
=
Also, C = n { X : F ( X ) c c Z} c -X - n { X :F(X) = X n { x : F ( X )= x c z).
5 Z } C- C.
Therefore
.
I n o r d e r t o p r o v e t h e r e s t o f t h e theorem, t a k e U = LJ { X :X c Z A X c F ( X ) } . The p r o o f t h a t U = F ( U ) and D = u { X : X = F ( X ) 5 2 1 i s s i m i l a r t o t h e above.
2.3.2.5
THEOREM SCHEMA,
L e t F be a unmq opetration. Then
Y C - F ( A ) C F ( B ) 5 Z) F ( n{x :Y c x = F ( x ) c z } ) = n { x : Y c- X = F(X) 5 Z } A -
Y c c A c - Z A vAWB(Y - 8 c- Z
F ( u { X :Y
-X C
=
F(X) C Z})
+
= U {X :Y
+
5X
= F(X)
5 Z}.
AXIOMATIC S E T T H E O R Y
59
The proof i s s i m i l a r t o t h a t of 2.3.2.4.
PROOF,
By 2.3.2.2,
we have
F ( n {X : @ A Y c X = F(X) c C X = F(X) - Z } ) -c n { F ( X ) : @ A Y -
.
5Zl
=
n {X:@ A Y c - X = F(X) 5 Z ) . S i m i l a r l y f o r unions.
n
{x : u
PROOF,
{ V :Q A Y c - V = F(V)
By 2.3.2.6
and
5 Z}
C_X = F ( X
CZ)
.
2.3.2.5.
* 2 . 3 . 2 . 8 E X A M P L E , In a topological space X, l e t F ( A ) be t h e c l a s s o f accumulation points of A f o r A C X . Let Z be a closed subset of X. T h u s , we have A C 8 C Z + F ( A ) CF(i3) Z. Therefore, by 2.3.2.4, t h e r e i s a l a r g e s t D such t h a t D = F ( D ) . Then D i s p e r f e c t and Z % D i s s c a t t e r e d (Theorem o f Cantor-Bendixon. ) From t h e theorems proved, we now deduce theorems f o r two unary operations. 2.3.2.9
THEOREM SCHEMA,
W XW Y ( ( X c -
Let F and G be u w y ope&onb.
Yc A -,F ( X ) C F(Y))
A (X
5Y 5 8
+
G(X)
G(Y)))
3 A 1 3 B1(A1 Z G ( 8 ) A B1 c F ( A ) A F(d%A1)= B1 A G(B%B1) =
Then, *
All.
60
ROLAND0
CHUAQUI
PROOF, Assume t h a t F and C a r e a u n a r y o p e r a t i o n s such t h a t W X W Y F(X) C F(Y))A X 5 Y 2 B G(X) c G(Y))). D e f i n e t h e operat i o n H,-by
((X
5Y c A
-
+
+
Then 8 Suppose X 2 Y c F ( A ) . and, hence A % G ( B % X ) 5 A%LG(B%Y)
A p p l y i n g 2.3.2.4
1 B%X
5 A.
2 &%Y, t h u s , G(B%X) > G ( B Therefore,
Y)
t o H a n d A, we o b t a i n a B1 such t h a t H(B1) = B1.
We
have, B1 = H(B1) = F ( A % G ( B % B 1 ) ) c F ( A ) . Let
.
A1 = G ( B % B 1 ) .
8 1 = H(B1) = F ( A % A 1 ) . 2.3.2.10
S i n c e 8 % B , C 8 , we have A1 z G ( 8 ) .
Finally,
1 -
THEOREM SCHEMA,
Le.L F a n d G be unarry 0perrcc;tiunn.
Then
W X W Y ( ( X-c Y c-A - + F ( X ) cF(Y))A(XcY5B+G(X) cG(Y)))A F(A) c 8 A G(B) L A A nA = 0 = 1 2
B 1 nB 2
+
3 A1 3 A 2
3B1
3B2 (A = A U A A 1 2
A F ( A 2 ) = B1 A G ( B 2 ) = A1)
.
B
=
B1UB2 A
L e t F and G be monotone o p e r a t i o n s f o r s u b c l a s s e s o f A and PROOF, r e s p e c t i v e l y and suppose F ( A ) 2 B and G ( B ) 5 A. By 2.3.2.5, there a r e A1, B1 such t h a t A1 E G ( B ) , B1 c F ( A ) , F ( A % A 1 ) = B1, and G ( B % B l ) =A1.
B,
L e t A2 = A % A 1 and
B2
=
8%Bl.
We have, A1 L G ( B )
5A
t h u s A1 C A and B1 5 B. T h e r e f o r e A = A1 u A2 and B = rn c l e a r t h a t A1 n A2 = 0 = 8 n 8 1 2'
and B1 s F ( A )
B 1 LJ
B2.
5
It i s also
PROBLEMS
1.
Prove:
DFDF(F)A CLO(R)ADF=DR A
WaWb(aRb+F'aRF'b) R 3 c 3 d ( F ' c = c A F ' d = d A c = A E x : F ' x = x} A d = V { x : F ' x = x } ) .
R
8;
+
61
A X I O M A T I C S E T THEORY
2.
Prove:
CLO(R)AW6Wg(6,gES-+6ED6D6 A D 6 = D g = D R A 6 o g = g o d ) - + n
*3.
A p p l y 2.3.2.9
2.3.3
t o r e a l numbers, r e p l a c i n g
-
by s u b s t r a c t i o n .
ADDITIVE AND MULTIPLICATIVE OPERATIONS,
An o p e r a t i o n F i s compLdAQeii a d d t t i v e , i f f o r a l l t e r m s i and f o r m u l a s t h e g e n e r a l i z e d u n i o n i s c o m p l e t e l y a d d i t i v e . Also, b y 2.2.1.12 ( i i ) R* i s c o m p l e t e l y a d d i t i v e and, by 2.2.1.12 ( i v ) , i f f o r a c l a s s A we d e f i n e t h e o p e r a t i o n A b y A ( X ) = X*A , so i s t h i s A .
adAn a p p a r e n t l y weaker n o t i o n i s t h a t o f c l a s s a d d i t i v e . F i s C&b d i t i v e , i f f o r a l l c l a s s e s A, F U A = u C F ( y ) : Y E A } . I t terms o u t t h a t t h e s e two n o t i o n s a r e e q u i v a l e n t :
( i i i ) W A F(A) = u I F ( I y I ) :
YEA}
.
(iv) IRWAF(A) =R*A. I ( i ) C l e a r l y i m p l i e s ( i i ) , and ( i i ) i m p l i e s ( i i i ) , because CCyl : y E A l
PROOF
A =
U
.
The i m p l i c a t i o n o f ( i i i ) t o ( i v ) i s proved, as f o l l o w s : F(A) = u {F(Iy})
: Y E A ) f o r a l l A.
R
=
Suppose
D e f i n e R, by
[F({x})
: xEV].
By 2.2.2.1 and Def. 2.2.2.2, R*CxI = F ( I x 1 ) f o r a l l x E V . Therefore, by 2.2.1.12 ( i v ) , R*A = u {R*{xI : X E A I = u { F ( C x I ) : x E A 1 = F ( A ) .
ROLAND0 CHUAQUI
62
F(UCr
The i m p l i c a t i o n from ( i v ) t o ( i ) i s o b t a i n e d from 2.2.1.12 : $ I ) = R*(U{P : $1) = UER* 7 : $j = UCF(r) : $1.
( i v ) , i.e.
A n o t i o n weaker than complete a d d i t i v i t y i s s e t a d d i t i v i t y . An ope r a t i o n F i s neA a d d i t i v e i f f o r a l l s e t s x we have F(u x) = uCF(y) : EX}. T h i s i s d e f i n i t e l y weaker as i s shown b y t h e o p e r a t i o n F d e f i n e d by,
[uA,
if
A E V ,
This F i s set a d d i t i v e but n o t completely a d d i t i v e . i t i s p o s s i b l e t o prove. 2.3.3.1, 2.3.3.2 THEOREM SCHEMA, i n c o n d i t i a a a.te e q u i v d e n i .
LeA F be a n opemuXon.
(i)
w
x F ( u x ) = u I F ( y ) : y ~ x 1,
(ii)
w
xF(x)
(iii)
=
Similarly
as
Then ,the d0Uvw-
u W((g1):y~xl,
3 R W x F ( x ) = R*x
.
A s l i g h t g e n e r a l i z a t i o n of these theorems i s t h e f o l l o w i n g : 2.3.3.3
THEOREM SCHEMA,
ing c o n d i t i o n b me equivaLent.
L e A F be an opmaaXon.
( i ) F a t e v m y t m r and ~vhmvnuRa9
PROOF,
Take
G ( A ) = F(A)
%
Then t h e dvUow-
,
F ( 0 ) and a p p l y t h e p r e v i o u s theorems..
An o p e r a t i o n F i s c a l l e d compLetdy &pLica.LLue r a n d e v e r y f o r m u l a 4 we have
i f f o r every t e r m
AXIOMATIC S E T T H E O R Y
f o r every s e t x , F ( n . x ) = n { F ( y ) : y E x ) . m u l t i p l i c a t i v e operations. 2.3.3.4 THEOREM SCHEMA, ing condLtiorzcs me e q u i w d e n t
63
We have s i m i l a r theorems for
L e t F be a n a p e h a t i o n .
Then t h e 6 a U a w -
( i ) Fvk e v e h y tm r and 6vzmvnLLea @,
( i i ) WA(A
f
0
+
F(nA)
=
n IF(y) : YEA})
,
( i i i ) 3R3CWAF(A) = C % R * $ A . &ned
.tax
The t h e e c v n d i t i a a &emmain equivalent id t h e ned-OukZion AA eRiminated dhvm (i)and (L], and C = V p u t in &a c o n d i t i o a m e equivaLent kepeacing A by x.
t a nonempty
(AX). Tha
The proof i s l e f t t o the reader. The theorems we have s t a t e d show t h a t completely a d d i t i v e operations can be represented by c l a s s e s ; i.e. by r e l a t i o n s . An a r b i t r a r y operation, on t h e other hand, can be represented by a r e l a t i o n , only r e s t r i c t e d t o
sets.
An operation F i s AiniteLy a d d i t i v e , i f F ( A u 8 ) = F(A) U F ( B ) , f o r a l l A , B ; ~ i n i t d yn e t a d d i t i v e i f t h i s condition i s t r u e with x, y rei f F(An8) = F(A) n F ( B ) placing A , 8. F i s &kLteLy ( n e t ) muLt.Lp&cat.ive, ( F ( x n y ) = F(x) n F ( y ) ) f o r a l l A , B ( f o r a l l x, y ) . I t i s c l e a r by 2.1.3.5 ( i ) ( o r ( i i ) ) t h a t completely a d d i t i v e ( o r m u l t i p l i c a t i v e ) operations a r e f i n i t e l y a d d i t i v e ( o r m u l t i p l i c a t i v e ) . S i m i l a r l y , by 2.1.1.15, s e t a d d i t i v i t y ( o r m u l t i p l i c a t i v i t y ) implies f i n i t e s e t a d d i t i v i t y ( o r multiplicativity).
PROOF, The implication from ( i i ) t o ( i ) i s obtained from 2.3.1.8 ( v ) and 2.2.1.12 ( i i ) .
In order t o prove t h e converse implication, assume t h a t F i s complete-
By 2 . 3 . 3 . 1 , t h e r e i s an R n ( V x V ), and suppose t h a t y G x and
l y a d d i t i v e and f i n i t e l y s e t m u l t i p l i c a t i v e .
Take
such t h a t R*A = F ( A ) .
G-l =
R
.
z G x . Then x E ( G - ' * C y 1 ) n (G-'*{z>) = F ( C y 1 ) n F ( C z ) ) = F ( C q I n ( Z 1 ) + 0. But; s i n c e F i s c o m p l e t e l y a d d i t i v e , F ( 0 ) = 0. T h e r e f o r e , C g ) n C z 1 f 0 , and hence y = z. Thus, we have p r o v e d t h a t G i s a f u n c t i o n .
A v a r i a n t o f 2 . 2 . 3 . 5 w h i c h i s easy t o p r o o f i s : L e R F be a n opehation.
THEOREM SCHEMA,
2.3.3.6
ing atre equivalent.
(i)W A ( A f 0 + F ( u A ) = u{F(q) : q f A } ) g v z (F(qnz) = F(q) n F(z))
T h u e btatemevdh atre & o
A
,
3 G ~ c ( ~ ~ v A( GA )F ( A ) = ( G - ' * A )
(ii)
Then Lhe 6oMow-
equivaLevLt w d h
u
c)
x hephcing
A thhoughoLLt.
The n e x t theorem g i v e s a c h a r a c t e r i z a t i o n o f o p e r a t i o n s t h a t a r e f i n i t e l y a d d i t i v e and m u l t i p l i c a t i v e . 2.3.3.7
L e t F be a n openation.
THEOREM SCHEMA,
Then
W A V B (F(AuB) = F ( A ) uF(B) AF(AnB) = F ( A ) n F ( B ) ) A
F(0) = 0-
T k i n hXaXement
WAWBF(A%B) = F ( A ) %F(8). heYnain6
D ~ u ewhen x,g atre buhbaXuZed t h t a u g k t doh A,B.
The p r o o f i s l e f t t o t h e r e a d e r . O p e r a t i o n s w h i c h a r e c o m p l e t e l y a d d i t i v e and f i n i t e l y s e t m u l t i p l i c a t i v e , are a l s o completely m u l t i p l i c a t i v e . S i m i l a r l y , complete m u l t i p l i c a t i v i t y and f i n i t e a d d i t i v i t y i m p l y c o m p l e t e a d d i t i v i t y . 2.3.3.8
L e t F be a n opehation.
THEOREM SCHEMA,
ing c o n d i t i o n 6 atre e q u i v d e n t : (i) (ii)
W A ( A f 0 -+ F ( u A ) = u { F ( y ) : g E A 1 A F ( n A ) = n { F ( y ) : Y E A ) ) W A(A # 0
+
F(uA)=
u
F(Y) "F(z)
(iii)
Then t h e doUow-
W A(A # 0
+
{ F ( y ) : y € A } ) A W y Wz F ( g n z ) = 3
F ( n A ) = n { F ( g ) : Y E A ) ) A W A WB
F(AUB) = F(A)
U
F(B)
.
T h u e thhee c a n d i t i o n h hemain equivaLerCt when x and z m e h u b h U u Z e d
doh A and 8.
65
AXIOMATIC SET THEORY
I t i s c l e a r t h a t ( i ) i m p l i e s ( i i ) and ( i i i ) . and 2.3.1.10.
PROOF,
( i )by 2.3.3.6
( i i ) implies
I n o r d e r t o prove t h e r e m a i n i n g i m p l i c a t i o n , assume ( i i i ) and suppose F ( A % 8 ) = F ( A ) % F ( B ) f o r a l l A , 8. A l s o , f i r s t t h a t F ( 0 ) = 0. By 2.3.3.7, f i n i t e s e t a d d i t i v i t y i m p l i e s , by 2.3.2.3, monotony, and t h i s , a g a i n by 2.3.2.3, implies W F ( y ) : Y E A } C F ( U A ) . Now ,
F(uA)
u W y )
-.
YEA} = n {F(uA) %F(y) : YEA} Y =
n {F(UA Y
=
Fin { U A % y : y E A } }
.
But, i f u E U A , t h e n u E y E A f o r a c e r t a i n y . Thus, n { u A % y : Y E A } = 0 . Y Therefore, s i n c e we assumed F ( 0 ) = 0
F ( u A ) ~u Thus,
,
y) : YEA}
Y
Y
Hence, u
4
u A%y.
,
{ F ( y ) : Y E A } = F ( 0 ) = 0.
F ( u A ) L u {F(y): Y E A ) , and complete a d d i t i v i t y i s proved.
Now we prove (iii)i m p l i e s ( i ) w i t h o u t t h e assumption F(0) =. 0. L e t t h e o p e r a t i o n G be d e f i n e d f o r a l l A , by G ( A ) = F ( A ) % F ( O ) . We have, G ( 0 ) = 0. Also, i f A Z 0,
G ( n A ) = F ( n A ) %F(O) = n{F(y) : Y E A } %F(O) = n
{F(y)%F(O): Y E A }
,
= n {C(y) : YEA]
S i m i l a r l y we can show, G ( A u 8 ) = G ( A ) U G(8). Applying, t h e case proved above, we have f o r A f 0, G ( u A ) = U CG(y) : Y E A } ; i.e. F ( u A ) F ( 0 ) = u { F ( y ) % F ( O ) : Y E A } . Since f i n i t e a d d i t i v i t y i m p l i e s , monotony, we have f o r a l l 8, F ( 0 ) 5 F ( 8 ) . Hence,
F(uA) = (F(uA)%F(O)) u F(0) = = u
U
{F(y) % F ( O ) : Y E A }
U
.
F(O)
{ ( F ( y ) s F ( 0 ) ) u F ( 0 ) : Y E A ] = u EF(y) : Y E A } .
The requirement t h a t F s a t i s f y t h e c o n d i t i o n s f o r a l l c l a s s e s or a l l s e t s i s e s s e n t i a l i n these theorems. For i n s t a n c e , t h e r e a r e o p e r a t i o n s which a r e c o m p l e t e l y m u l t i p l i c a t i v e and f i n i t e l y a d d i t i v e ( t h e t o p o l o g i c a l c l o s u r e ) i n some c o l l e c t i o n , b u t a r e n o t c o m p l e t e l y a d d i t i v e i n t h e same collection. W i t h t h e axiom o f c h o i c e i t i s p o s s i b l e t o f i n d a f i n i t e l y a d d i t i v e and m u l t i p l i c a t i v e o p e r a t i o n t h a t i s n o t c o m p l e t e l y a d d i t i v e . Without A x C , t h i s problem seems t o be open.
66
ROLAND0 CHUAQUI
PROBLEMS
1.
P r o v e 2.3.3.4
2.
P r o v e 2.3.3.7
2.3
F u n c t i o n s and O p e r a t i o n s
2.3.1
FUNCT I O N A L R E L A T I O N S
The r e l a t i o n R i s a Bunotiotr i f
(y,x) E R
and ( z , x ) E R , i m p l y
Thus, f u n c t i o n s R can be c h a r a c t e r i z e d by R 0 R - l
q = z .
ZID.
The f o l l o w i n g theorem, whose p r o o f i s l e f t t o t h e r e a d e r , g i v e s severI n t h i s book, R o R - l c I D a l equivalent possible d e f i n i t i o n s o f functions. i s used t o say t h a t R i s a f u n c t i o n . We a l s o assume i n t h i s Chapter t h a t R, S, T a r e r e l a t i o n s .
THEOREM,
2.3.1.1
(i) R0R-l cID(ii)
R O R - ~ 510-
R O R - ~
(v) R
OR-^
LIDc -D I
(vi) RoR-'cID-
( ~ v Ro) n R
=
0.
WSWT(SnT)oR = (SoR) n ( T O R ) .
( i i i ) R o R - lcID(iv)
( R o R - 1) n D v = 0 .
-
W S W T ( S ~ T ) =~ R ( s ~ R 2 ), (TOR).
wx
WY
R - ~ * ( xn Y ) = ( R - ~ * x )n ( R - ~ * Y ) .
V X W Y (R* X)
f-
Y = R*(XnR-'*Y).
I t i s c l e a r t h a t o f o u r c o n s t a n t r e l a t i o n s , I D and 0 a r e f u n c t i o n s , w h i l e V x V , E L , I N , and Dv a r e n o t .
The i n t e r s e c t i o n o f f u n c t i o n s i s a f u n c t i q n , b u t t h e u n i o n i s n o t , i n general, a function. Under some r e s t r i c t i o n s , i t i s : 2.3.1.2
THEOREM SCHEMA,
LeL
T
51
be a ,tm and @ a l;otvnu&;
then
ROLAND0 C H U A Q U I
52
The r e l a t i v e complement o f a f u n c t i o n i s never a f u n c t i o n . A subclass o f a f u n c t i o n i s a l s o a f u n c t i o n and t h e composition o f f u n c t i o n s i s a f u n c t i o n . The o t h e r r e l a t i o n a l o p e r a t i o n , t h e converse o r i n v e r s e , w i l l be discussed l a t e r . The f o l l o w i n g theorem, easy t o prove, summarizes these f a c t s .
2.3.1.4
DEFINITION,
R'x =
R*{x}
*
R ' x i s Lhc value o d x by R.
R*{x}
PROOF OF ( i ) . Suppose t h a t R i s a f u n c t i o n and = { q } f o r some y. Thus n R {XI = y.
x
E
D R . Then
L e t F be a unary o p e r a t i o n . The r e s t r i c t i o n o f F t o t h e c l a s s o f w i t h F(x) E V c a n be represented by t h e f u n c t i o n F
x
I t i s c l e a r t h a t F i s a f u n c t i o n , w i t h domain D F = {x : F ( x ) E V 1 , and F ' x = F(x) f o r x E D F . Thus F and F have t h e same v a l u e s f o r x w i t h
F(x) E
v.
However, t h e r e p r e s e n t a t i o n o f o p e r a t i o n s by r e l a t i o n s g i v e n i n 2 . 2 . 2 i s more general, because i t does n o t r e s t r i c t t h e domain t o those X w i t h
53
AXIOMATIC S E T T H E O R Y
But, when p o s s i b l e , t h e p r e s e n t r e p r e s e n t a t i o n by f u n c t i o n s i s E V. more convenient.
F(x)
The f o l l o w i n g d e f i n i t i o n f o r m a l i z e s t h e i n t r o d u c t i o n o f such f u n c t i o n s .
2.3.1.6
D E F I N I T I O N SCHEMA, ( 7
X
=
Let
T
:q5) = ( ( 7 , x ) :
be a t e r m and q5 a formula. Then
GI.
For i n s t a n c e , we have when F i s a unary o p e r a t i o n , { ( F ( x ) , x ) : F(x) E V )
.
(F(x): F(x)E V ) =
X
We can i n t r o d u c e f u n c t i o n s f o r a l l t h e o p e r a t i o n s a l r e a d y d e f i n e d . For example, f o r t h e image o p e r a t i o n ,
R* = ( R * x : x E
v)
.
I n t h i s case and o t h e r s , t h e same symbol w i l l be used f o r t h e operat i o n and t h e corresponding f u n c t i o n . With the value n o t a t i o n defined i n 2.3.1.4 can be g i v e n by, F*A = { F ' x : x E A } , and F-l*A have e a s i l y ,
tl x ( x
t h e image o f a f u n c t i o n F, = { x : F'xEA}. Also, we
2.3.1.7 THEOREM, R0R-l c CZD-I D A S0S-l E DR+ R ' x = S I X ) ) .
I n t h e r e s t o f t h i s s e c t i o n , t h e l e t t e r s F, G, H, s t r i c t e d t o functions.
2.3.1.8 (i) (ii)
= B n D F -1
F* F-'*B
.
F* F - ~ * B c -B . A n B =
o
-+
=
(F*A) n B .
(F-~*A) n (F-~*B) = 0 .
(v)
F-l*(AnB)
= (F-'*A)
n (F-l*B).
(vi)
F-l*(A%B)
= (F-'*A)
%
(vii) (viii) (ix)
6, g,
t--f
DR = D S A
and h. a r e r e -
THEOREM (PROPERTIES OF IMAGES OF FUNCTIONS)
(iii) F*(AnF-l*B) (iv)
(R = S
8
5 F*A
v B 3A
-+
3 C'(C'
B n D F-'
B n D F - ~ =F*
( X I F-~*A = F-~*B
%
-A
C
=
(F-'*&).
A B = F*C).
F*A.
~-l* % B .
+
A nDF = B nDF
.
I
ROLAND0 CHUAQUI
54
These p r o p e r t i e s o f images o f f u n c t i o n a r e n o t c h a r a c t e r i s t i c o f f u n c t i o n i n t h e sense t h a t t h e r e a r e o t h e r r e l a t i o n s w h i c h s a t i s f y them. B u t ; we have t h a t ( v i i i ) , ( i x ) , and ( x ) a r e e q u i v a l e n t and so a r e ( x i ) and ( x i i ) . PROOF,
2.2.1.10.
and ( i ) c a n be p r o v e d d i r e c t l y and t h e r e s t o b t a i n e d f r o m (i) We have, F 0 F - l
PROOF OF ( i ) .
I D I D ( F o F-') (ID IDF-')*B
=
I D I D F -I.
-ID.
By 2 . 3 . 1 . 3
C
T h e r e f o r e F*
F-'*E
( i i ) F o F-l =
o F-l)*
= (F
=
= B n DF-'.
PROOFOF ( i i ) :
By (i).
PROOF OF (iii). By 2 . 2 . 1 . 1 0
( v i i i ) , we h a v e (F*A) n B C - F*(AnF-'*B).
On t h e o t h e r hand, b y 2 . 2 . 1 . 1 0 ( i v ) , F*(AnF-'*E) U s i n g , now, ( i i ) we o b t a i n , F * ( A n F - l * B ) C - (F*A)
c ( F * A ) n ( F * F"*B).
17-8.
n B = 0. By ( i i i ) PROOF O F ( i v ) . Assume A n t 3 = 0. By ( i i ) (F* F-'*A) F*(f-l*AnF-'*B) = (F* F-'*A) n 8 = 0. T h e r e f o r e , F-l*A n F-l*B n D F = 0 . F-l*B C - D F . Thus, F-l*A n F-l*B = 0. B u t F-'*A
PROOFOF ( v ) .
By ( i v ) , we have ( F - 1 * ( A n 8 ) ) n ( F - 1 * ( 8 n J A ) )
= ( F - l * ( A n B ) n (F-'*(A%E))
( v ) , we o b t a i n , F-l*A
= (F-l*(A%B))
=
0 =
From 2 . 2 . 1 . 1 0
n (F-l*(B\A)).
(F-'*(AnB)) U (F-l*(A-E)) a n d F-'*B = T h e r e f o r e (Fql*A) n (F-'*E)=((( F - l * ( A n E ) ) U = (F-l*(AnB)) u (F-l*(B%A)). (F-'*(A
=
* 8 ) ) )) n (( (F-l*(A
PROOF OF ( v i ) .
n8 ) )
U
(F-'*( E % A ) ) ) = F-l*(A n B )
By 2 . 2 . 1 . 1 0
( v ) , F-l*A
t h e n , b y ( v ) , F-l*A = ( F - l * ( A % B ) ) U ( ( F - l * A )
(F-'*(A*B)) PROOF
n (F-l*B)
OF ( v i i ) .
= (F*A) n 8 = B.
= 0.
Therefore,
Suppose
Therefore, t a k e
E
= (F-l*(A%E))U(F-'*(ArlB));
(F-l*S)).
(F-l*A) \(F-'*E)
5 F*A. c'
Also, b y ( i v ) =
.
-DF-' c
= F*
,
F-'*(A%B).
Then, b y ( i i i ) , F * ( A n F - l * B )
= A n F-l*B
PROOF OF ( v i i i ) . We have, B n D F - ' i n g ( v i i ) , we o b t a i n ( v i i i ) .
.
=
V. Hence, a p p l y -
55
AXIOMATIC SET THEORY
PROOF OF ( i x ) .
Using 2.2.1.10
By ( v i i i ) , we have B n DF-'
( B n D F - ' ) = B n DF-'. But, by 2.2.1.10 ( F - ' * % 8 ) u (F-'* x D 0 F - l ) = F - l * % B . (xi,
=
F*A
f o r some A .
F* % F-'* % ( v ) , F - l * % ( 8 n OF-') = = F*A,
( x i ) , we o b t a i n , F*%F-'*%F*A
i.e.
( x i ) and f x i i ) a r e ' l e f t t o t h e r e a d e r .
We a l s o have d i s t r i b u t i v i t y o f i n v e r s e image o f f u n c t i o n s w i t h generalized intersection.
2.3.1.9
THEOREM SCHEMA, L& r be a t m and 6 a 6o/un&.
On t h e o t h e r hand, i f x $ implies that
f o r a l l such r . IT
x
6
F-'*r
E
nx
, i.e. y
Therefore
FIX
E
DEFINITION
BA i s t h e & a n
06
F'x
=
:$I,
t h e n f o r a l l Xo...
f o r some y
nx
0.. .x n-1
:$I. 2.3.1.10
r
0"' Xn-1
{r
:$I,
Then,
E 7,
i.e.
Xn-l,
and hence F'xE r
x c ~ - l *n
'0'
*
'Xn-1
I
6uncLionn w L t h domain 8 and m n g e included i n A.
Since i n G we almost never can prove t h a t f u n c t i o n which i s a s e t w i t h domain 8, 'A 8
BA
f
0, i.e.
t h a t t h e r e is a
w i l l n o t be much used i n P a r t 2.
On t h e o t h e r hand A ( ? ) i s t h e no;tion t h a t says t h a t f 12 a 6unOtiun w a h domain 8 m d tange i n c h d e d in A. W i t h t h i s n o t i o n , we can express t h a t ? i s a f u n c t i o n by D F V ( F ) . T h i s w i l l o f t e n be used.
2.3.1.11 DEFINITION SCHEMA (GENERALIZED CARTESIAN PRODUCT). Let be a term and $ a formula. Then n ( 7 :q,) = Ed : 6 & x V A
~ ; O ~ - ' C J D A DI x~: = $1 A
Vx($
+
6kE
r)).
X
ROLAND0 CHUAQUI
56
2.3.1.12
For instance, we have
F, by F ( 0 ) = A xEA A
"6 = nX (6'~:x
DEFINITION,
gEBl.
nX
( A : xEB) =
E
06).
G
A.
I f we d e f i n e t h e o p e r a t i o n
XIx (F(x) : X E 2 ) = (I( x,O) , ( y , 1 ) I : I ( x,O) , ( g , l ) l i s a p o s s i b l e d e f i n i t i o n o f t h e o r -
and F ( 1 ) = 8, then The s e t
2
dered p a i r o f x and g. I t i s t h e f u n c t i o n 1; E {x,gl, w i t h 6'0= x a n d 4'1 = y . I f we ' i d e n t i f y ' t h i s 6 w i t h t h e ordered p a i r ( X , L J ) , t h e n t h e Ilx (F(x): x E 2 ) i s ' i d e n t i f i e d ' w i t h A x B .
I n some c o n t e x t , ( x , g ) i s b e t t e r as ordered p a i r than {C x,O), ( y , l ) } ; i n o t h e r s t h e o p p o s i t e i s t h e case. The same i s t r u e f o r t h e t w o operat i o n s of C a r t e s i a n Droduct. T h i s m u l t i p l i c i t y o f d i f f e r e n t p o s s i b l e d e f i n i t i o n s f o r t h e same conc e p t i s an i n e l e g a n t c h a r a c t e r i s t i c o f a l l s e t t h e o r i e s and i t seems unavoidable. We say t h a t F i s a biunLquc
2.3.1.13
(ii)
one-one i j u n c t i o n i f F and F - l a r e func-
i f D F ~ ~ - Al D ( F~- l) D F (F-').
t i o n s , i.e.
(i)
ofi
-
DEFINITIONa
!A(F)
! A = (1;:
.
A ~ ( A~ A) ~ ( ~ - l ) A
A(ij)l,
! A ( F ) says t h a t F i s a pe,trnLLtCLtion o f A , i.e. F i s a b i u n i q u e funct i o n from A o n t o A. !A i s t h e c l a s s o f a l l permutations o f A.
PROBLEMS
Prove 2.3.1.1
1. 2. 3.
Characterize a l l r e l a t i o n s R t h a t s a t i s f y R = R o R - l o R
4.
Show t h a t F i s a b i u n i q u e f u n c t i o n i f and o n l y i f D F * = V A DF-'"
Prove 2.3.1.3
= PDF-'
A (F*
I
.
P D F i s a biunique function).
-
5.
Prove 2.3.1.8
6.
F i n d a r e l a t i o n R which i s n o t a f u n c t i o n b u t s a t i s f i e s
7.
Characterize t h e r e l a t i o n s
(x)
(xii).
W B 3 A ( B n D R - ~= R*A).
R
that satisfy
R* R'l*
R*A
=
R*A
.
=
A X I O M A T I C SET THEORY 2.3.2
57
MONOTONE O P E R A T I O N S ,
z
A unary o p e r a t i o n F i s Y,Z-monutone i f i t s a t i s f i e s : Y LA c B C Y c F [ A ) C_ F ( B ) 5 2 . We say t h a t F i s rnoncdone i f i t i s 0, V-KonoTone. S i m i l a r l y , f o r f u n c t i o n s we have: 2.3.2.1
F ' x C- F ' y ) .
DEFINITION,
Mo ( F )
- DbF
(F) Avxt'q(x,y
E
DFA
x
--f
5q
-+
The f o l l o w i n g theorem g i v e s c o n d i t i o n s e q u i v a l e n t t o monotony. - 1. 2.3.2.2 THEOREM SCHEMA, L e R F be a unmy opehCLtion and Y c Then t h e doflowing CvnditioMn CVLQ e.qLvaRen2:
( i ) W A WB(Y c - A c- 8 c- Z
( i v ) W A WB(Y ( v ) W A WB(Y
c - A, B c- Z
- A, C
B C -Z
+
--t
+
Y c - F(A) c - F ( B ) c- Z ) .
Y c - F(A) u F ( B ) C_F(AUB)C - Z).
-
Y C - F ( A n B ) C F(A) n F ( B )
5 Z).
Suppose t h a t V AW B ( Y C A C B C Z Y C F(A) C F ( B ) 2 Z ) . PROOF, Therefore, f o r e v e r y X t h a t s a t i s f i e s - @ A-Y C-X c Z , we have-Y z F ( X ) 5 F( u(X : 4 A Y 5 X c Z}) c Y c - X C- Z } ) C_F(X)& Z . - Z and Y & F ( n{X : @ +
n
Thus,
( i ) i m p l i e s ( i i ) and ( i i i ) .
I t i s c l e a r t h a t ( i i ) i m p l i e s ( i v ) and i t i s enough t o show t h a t ( i v ) o r ( v ) i m p l y i m p l i e s ( i ) . Suppose Y S A 5 8 C Z. Then, Thus, Z ? F ( B ) 2 F(A) 2 2 F(A) u F ( B ) 3 Y .
-
( i i i ) i m p l i e s ( v ) . Therefore, ( i ) .I s h a l l prove t h a t ( i v ) by ( i v ) , Z > F ( B ) = F(AUB) Y.
The i m p l i c a t i o n o f ( v ) t o ( i ) i s proved s i m i l a r l y . I n a s i m i l a r way, t h e f o l l o w i n g can be proved.
.
58
ROLAND0 CHUAQUI
(iv)
t!xtlq F(x) u F ( q ) gF(xuq),
(v)
W x w q F ( x n q ) 5F(x) " F ( Y ) .
We now begin t h e s t u d y o f f i x e d p o i n t s o f monotone o p e r a t i o n s . T h i s study i s based on T a r s k i 1955. The theorems proved here w i l l be useful i n several chapters o f t h e book. We say t h a t a c l a s s X i s a hixed p a i d of a unary o p e r a t i o n F, i f We have t h a t Y, Z-monotone o p e r a t i o n s always have f i x e d p o i n t s . F(X) = X. F i r s t , t h e theorem f o r 0, 2-monotone o p e r a t i o n s . THEOREM SCHEMA, L e t F be a unmq opehation.
2.3.2.4
W A W B ( A-c B c Z +
F ( A )CF(B)LZ)+
n {X : X c - Z A F(X) = X ] A U =
LJ
Then
~C~U(F(C)=CAF(U)=DAC=
{ X :X C - Z A F(X) = A } ) .
F can be considered as a monotone o p e r a t i o n from subclasses o f Z t o subclasses o f Z .C i s t h e l e a s t f i x e d p o i n t and Q i s t h e g r e a t e s t f i x e d p o i n t . Thus, t h e c o n c l u s i o n can a l s o be w r i t t e n , WA W B ( A - c B-c Z + F ( A ) & F ( B ) 5 Z ) + F ( n { X : X-C Z A F ( X )= X } ) =
= n
{x : x
A F(X) =
c -
z
A F ( x ) = X I A F( u { x : x c -
z
A F(x)=
XI.
PROOF, Assume F(A) c F(B) C Z). L e t that C Z. Suppose, By t h e assumption, we we g e t t h a t F(C) 5 X,
c
X I )= u { x : x c -z
A
t h a t F i s a unary o p e r a t i o n such t h a t , V A W B ( A c-B C-Z + C = n { X : F ( X ) 2 X C Z l . Since F ( Z ) 5 Z, we have now t h a t X i s such t h a t F ( X ) 5 X 5 Z. Then C C X have F(C) L F ( X ) . S i n c e we assumed t h a t F(X)-z X , f o r a l l X w i t h F(X) 5 X E Z. T h e r e f o r e F(C) 5 C.
.
On t h e o t h e r hand, from F(C) c C
5 Z,
by t h e monotony o f F , we deduce
F (F(C))E F ( C )5 Z. Thus F ( C ) i s o n e o f t h e c l a s s e s whose i n t e r s e c t i o n i s - F(C),and, hence, F(C) = C. C. Therefore, C c
c
=
Also, C = n { X : F ( X ) c c Z} c -X - n { X :F(X) = X n { x : F ( X )= x c z).
5 Z } C- C.
Therefore
.
I n o r d e r t o p r o v e t h e r e s t o f t h e theorem, t a k e U = LJ { X :X c Z A X c F ( X ) } . The p r o o f t h a t U = F ( U ) and D = u { X : X = F ( X ) 5 2 1 i s s i m i l a r t o t h e above.
2.3.2.5
THEOREM SCHEMA,
L e t F be a unmq opetration. Then
Y C - F ( A ) C F ( B ) 5 Z) F ( n{x :Y c x = F ( x ) c z } ) = n { x : Y c- X = F(X) 5 Z } A -
Y c c A c - Z A vAWB(Y - 8 c- Z
F ( u { X :Y
-X C
=
F(X) C Z})
+
= U {X :Y
+
5X
= F(X)
5 Z}.
AXIOMATIC S E T T H E O R Y
59
The proof i s s i m i l a r t o t h a t of 2.3.2.4.
PROOF,
By 2.3.2.2,
we have
F ( n {X : @ A Y c X = F(X) c C X = F(X) - Z } ) -c n { F ( X ) : @ A Y -
.
5Zl
=
n {X:@ A Y c - X = F(X) 5 Z ) . S i m i l a r l y f o r unions.
n
{x : u
PROOF,
{ V :Q A Y c - V = F(V)
By 2.3.2.6
and
5 Z}
C_X = F ( X
CZ)
.
2.3.2.5.
* 2 . 3 . 2 . 8 E X A M P L E , In a topological space X, l e t F ( A ) be t h e c l a s s o f accumulation points of A f o r A C X . Let Z be a closed subset of X. T h u s , we have A C 8 C Z + F ( A ) CF(i3) Z. Therefore, by 2.3.2.4, t h e r e i s a l a r g e s t D such t h a t D = F ( D ) . Then D i s p e r f e c t and Z % D i s s c a t t e r e d (Theorem o f Cantor-Bendixon. ) From t h e theorems proved, we now deduce theorems f o r two unary operations. 2.3.2.9
THEOREM SCHEMA,
W XW Y ( ( X c -
Let F and G be u w y ope&onb.
Yc A -,F ( X ) C F(Y))
A (X
5Y 5 8
+
G(X)
G(Y)))
3 A 1 3 B1(A1 Z G ( 8 ) A B1 c F ( A ) A F(d%A1)= B1 A G(B%B1) =
Then, *
All.
60
ROLAND0
CHUAQUI
PROOF, Assume t h a t F and C a r e a u n a r y o p e r a t i o n s such t h a t W X W Y F(X) C F(Y))A X 5 Y 2 B G(X) c G(Y))). D e f i n e t h e operat i o n H,-by
((X
5Y c A
-
+
+
Then 8 Suppose X 2 Y c F ( A ) . and, hence A % G ( B % X ) 5 A%LG(B%Y)
A p p l y i n g 2.3.2.4
1 B%X
5 A.
2 &%Y, t h u s , G(B%X) > G ( B Therefore,
Y)
t o H a n d A, we o b t a i n a B1 such t h a t H(B1) = B1.
We
have, B1 = H(B1) = F ( A % G ( B % B 1 ) ) c F ( A ) . Let
.
A1 = G ( B % B 1 ) .
8 1 = H(B1) = F ( A % A 1 ) . 2.3.2.10
S i n c e 8 % B , C 8 , we have A1 z G ( 8 ) .
Finally,
1 -
THEOREM SCHEMA,
Le.L F a n d G be unarry 0perrcc;tiunn.
Then
W X W Y ( ( X-c Y c-A - + F ( X ) cF(Y))A(XcY5B+G(X) cG(Y)))A F(A) c 8 A G(B) L A A nA = 0 = 1 2
B 1 nB 2
+
3 A1 3 A 2
3B1
3B2 (A = A U A A 1 2
A F ( A 2 ) = B1 A G ( B 2 ) = A1)
.
B
=
B1UB2 A
L e t F and G be monotone o p e r a t i o n s f o r s u b c l a s s e s o f A and PROOF, r e s p e c t i v e l y and suppose F ( A ) 2 B and G ( B ) 5 A. By 2.3.2.5, there a r e A1, B1 such t h a t A1 E G ( B ) , B1 c F ( A ) , F ( A % A 1 ) = B1, and G ( B % B l ) =A1.
B,
L e t A2 = A % A 1 and
B2
=
8%Bl.
We have, A1 L G ( B )
5A
t h u s A1 C A and B1 5 B. T h e r e f o r e A = A1 u A2 and B = rn c l e a r t h a t A1 n A2 = 0 = 8 n 8 1 2'
and B1 s F ( A )
B 1 LJ
B2.
5
It i s also
PROBLEMS
1.
Prove:
DFDF(F)A CLO(R)ADF=DR A
WaWb(aRb+F'aRF'b) R 3 c 3 d ( F ' c = c A F ' d = d A c = A E x : F ' x = x} A d = V { x : F ' x = x } ) .
R
8;
+
61
A X I O M A T I C S E T THEORY
2.
Prove:
CLO(R)AW6Wg(6,gES-+6ED6D6 A D 6 = D g = D R A 6 o g = g o d ) - + n
*3.
A p p l y 2.3.2.9
2.3.3
t o r e a l numbers, r e p l a c i n g
-
by s u b s t r a c t i o n .
ADDITIVE AND MULTIPLICATIVE OPERATIONS,
An o p e r a t i o n F i s compLdAQeii a d d t t i v e , i f f o r a l l t e r m s i and f o r m u l a s t h e g e n e r a l i z e d u n i o n i s c o m p l e t e l y a d d i t i v e . Also, b y 2.2.1.12 ( i i ) R* i s c o m p l e t e l y a d d i t i v e and, by 2.2.1.12 ( i v ) , i f f o r a c l a s s A we d e f i n e t h e o p e r a t i o n A b y A ( X ) = X*A , so i s t h i s A .
adAn a p p a r e n t l y weaker n o t i o n i s t h a t o f c l a s s a d d i t i v e . F i s C&b d i t i v e , i f f o r a l l c l a s s e s A, F U A = u C F ( y ) : Y E A } . I t terms o u t t h a t t h e s e two n o t i o n s a r e e q u i v a l e n t :
( i i i ) W A F(A) = u I F ( I y I ) :
YEA}
.
(iv) IRWAF(A) =R*A. I ( i ) C l e a r l y i m p l i e s ( i i ) , and ( i i ) i m p l i e s ( i i i ) , because CCyl : y E A l
PROOF
A =
U
.
The i m p l i c a t i o n o f ( i i i ) t o ( i v ) i s proved, as f o l l o w s : F(A) = u {F(Iy})
: Y E A ) f o r a l l A.
R
=
Suppose
D e f i n e R, by
[F({x})
: xEV].
By 2.2.2.1 and Def. 2.2.2.2, R*CxI = F ( I x 1 ) f o r a l l x E V . Therefore, by 2.2.1.12 ( i v ) , R*A = u {R*{xI : X E A I = u { F ( C x I ) : x E A 1 = F ( A ) .
ROLAND0 CHUAQUI
62
F(UCr
The i m p l i c a t i o n from ( i v ) t o ( i ) i s o b t a i n e d from 2.2.1.12 : $ I ) = R*(U{P : $1) = UER* 7 : $j = UCF(r) : $1.
( i v ) , i.e.
A n o t i o n weaker than complete a d d i t i v i t y i s s e t a d d i t i v i t y . An ope r a t i o n F i s neA a d d i t i v e i f f o r a l l s e t s x we have F(u x) = uCF(y) : EX}. T h i s i s d e f i n i t e l y weaker as i s shown b y t h e o p e r a t i o n F d e f i n e d by,
[uA,
if
A E V ,
This F i s set a d d i t i v e but n o t completely a d d i t i v e . i t i s p o s s i b l e t o prove. 2.3.3.1, 2.3.3.2 THEOREM SCHEMA, i n c o n d i t i a a a.te e q u i v d e n i .
LeA F be a n opemuXon.
(i)
w
x F ( u x ) = u I F ( y ) : y ~ x 1,
(ii)
w
xF(x)
(iii)
=
Similarly
as
Then ,the d0Uvw-
u W((g1):y~xl,
3 R W x F ( x ) = R*x
.
A s l i g h t g e n e r a l i z a t i o n of these theorems i s t h e f o l l o w i n g : 2.3.3.3
THEOREM SCHEMA,
ing c o n d i t i o n b me equivaLent.
L e A F be an opmaaXon.
( i ) F a t e v m y t m r and ~vhmvnuRa9
PROOF,
Take
G ( A ) = F(A)
%
Then t h e dvUow-
,
F ( 0 ) and a p p l y t h e p r e v i o u s theorems..
An o p e r a t i o n F i s c a l l e d compLetdy &pLica.LLue r a n d e v e r y f o r m u l a 4 we have
i f f o r every t e r m
AXIOMATIC S E T T H E O R Y
f o r every s e t x , F ( n . x ) = n { F ( y ) : y E x ) . m u l t i p l i c a t i v e operations. 2.3.3.4 THEOREM SCHEMA, ing condLtiorzcs me e q u i w d e n t
63
We have s i m i l a r theorems for
L e t F be a n a p e h a t i o n .
Then t h e 6 a U a w -
( i ) Fvk e v e h y tm r and 6vzmvnLLea @,
( i i ) WA(A
f
0
+
F(nA)
=
n IF(y) : YEA})
,
( i i i ) 3R3CWAF(A) = C % R * $ A . &ned
.tax
The t h e e c v n d i t i a a &emmain equivalent id t h e ned-OukZion AA eRiminated dhvm (i)and (L], and C = V p u t in &a c o n d i t i o a m e equivaLent kepeacing A by x.
t a nonempty
(AX). Tha
The proof i s l e f t t o the reader. The theorems we have s t a t e d show t h a t completely a d d i t i v e operations can be represented by c l a s s e s ; i.e. by r e l a t i o n s . An a r b i t r a r y operation, on t h e other hand, can be represented by a r e l a t i o n , only r e s t r i c t e d t o
sets.
An operation F i s AiniteLy a d d i t i v e , i f F ( A u 8 ) = F(A) U F ( B ) , f o r a l l A , B ; ~ i n i t d yn e t a d d i t i v e i f t h i s condition i s t r u e with x, y rei f F(An8) = F(A) n F ( B ) placing A , 8. F i s &kLteLy ( n e t ) muLt.Lp&cat.ive, ( F ( x n y ) = F(x) n F ( y ) ) f o r a l l A , B ( f o r a l l x, y ) . I t i s c l e a r by 2.1.3.5 ( i ) ( o r ( i i ) ) t h a t completely a d d i t i v e ( o r m u l t i p l i c a t i v e ) operations a r e f i n i t e l y a d d i t i v e ( o r m u l t i p l i c a t i v e ) . S i m i l a r l y , by 2.1.1.15, s e t a d d i t i v i t y ( o r m u l t i p l i c a t i v i t y ) implies f i n i t e s e t a d d i t i v i t y ( o r multiplicativity).
PROOF, The implication from ( i i ) t o ( i ) i s obtained from 2.3.1.8 ( v ) and 2.2.1.12 ( i i ) .
In order t o prove t h e converse implication, assume t h a t F i s complete-
By 2 . 3 . 3 . 1 , t h e r e i s an R n ( V x V ), and suppose t h a t y G x and
l y a d d i t i v e and f i n i t e l y s e t m u l t i p l i c a t i v e .
Take
such t h a t R*A = F ( A ) .
G-l =
R
.
z G x . Then x E ( G - ' * C y 1 ) n (G-'*{z>) = F ( C y 1 ) n F ( C z ) ) = F ( C q I n ( Z 1 ) + 0. But; s i n c e F i s c o m p l e t e l y a d d i t i v e , F ( 0 ) = 0. T h e r e f o r e , C g ) n C z 1 f 0 , and hence y = z. Thus, we have p r o v e d t h a t G i s a f u n c t i o n .
A v a r i a n t o f 2 . 2 . 3 . 5 w h i c h i s easy t o p r o o f i s : L e R F be a n opehation.
THEOREM SCHEMA,
2.3.3.6
ing atre equivalent.
(i)W A ( A f 0 + F ( u A ) = u{F(q) : q f A } ) g v z (F(qnz) = F(q) n F(z))
T h u e btatemevdh atre & o
A
,
3 G ~ c ( ~ ~ v A( GA )F ( A ) = ( G - ' * A )
(ii)
Then Lhe 6oMow-
equivaLevLt w d h
u
c)
x hephcing
A thhoughoLLt.
The n e x t theorem g i v e s a c h a r a c t e r i z a t i o n o f o p e r a t i o n s t h a t a r e f i n i t e l y a d d i t i v e and m u l t i p l i c a t i v e . 2.3.3.7
L e t F be a n openation.
THEOREM SCHEMA,
Then
W A V B (F(AuB) = F ( A ) uF(B) AF(AnB) = F ( A ) n F ( B ) ) A
F(0) = 0-
T k i n hXaXement
WAWBF(A%B) = F ( A ) %F(8). heYnain6
D ~ u ewhen x,g atre buhbaXuZed t h t a u g k t doh A,B.
The p r o o f i s l e f t t o t h e r e a d e r . O p e r a t i o n s w h i c h a r e c o m p l e t e l y a d d i t i v e and f i n i t e l y s e t m u l t i p l i c a t i v e , are a l s o completely m u l t i p l i c a t i v e . S i m i l a r l y , complete m u l t i p l i c a t i v i t y and f i n i t e a d d i t i v i t y i m p l y c o m p l e t e a d d i t i v i t y . 2.3.3.8
L e t F be a n opehation.
THEOREM SCHEMA,
ing c o n d i t i o n 6 atre e q u i v d e n t : (i) (ii)
W A ( A f 0 -+ F ( u A ) = u { F ( y ) : g E A 1 A F ( n A ) = n { F ( y ) : Y E A ) ) W A(A # 0
+
F(uA)=
u
F(Y) "F(z)
(iii)
Then t h e doUow-
W A(A # 0
+
{ F ( y ) : y € A } ) A W y Wz F ( g n z ) = 3
F ( n A ) = n { F ( g ) : Y E A ) ) A W A WB
F(AUB) = F(A)
U
F(B)
.
T h u e thhee c a n d i t i o n h hemain equivaLerCt when x and z m e h u b h U u Z e d
doh A and 8.
65
AXIOMATIC SET THEORY
I t i s c l e a r t h a t ( i ) i m p l i e s ( i i ) and ( i i i ) . and 2.3.1.10.
PROOF,
( i )by 2.3.3.6
( i i ) implies
I n o r d e r t o prove t h e r e m a i n i n g i m p l i c a t i o n , assume ( i i i ) and suppose F ( A % 8 ) = F ( A ) % F ( B ) f o r a l l A , 8. A l s o , f i r s t t h a t F ( 0 ) = 0. By 2.3.3.7, f i n i t e s e t a d d i t i v i t y i m p l i e s , by 2.3.2.3, monotony, and t h i s , a g a i n by 2.3.2.3, implies W F ( y ) : Y E A } C F ( U A ) . Now ,
F(uA)
u W y )
-.
YEA} = n {F(uA) %F(y) : YEA} Y =
n {F(UA Y
=
Fin { U A % y : y E A } }
.
But, i f u E U A , t h e n u E y E A f o r a c e r t a i n y . Thus, n { u A % y : Y E A } = 0 . Y Therefore, s i n c e we assumed F ( 0 ) = 0
F ( u A ) ~u Thus,
,
y) : YEA}
Y
Y
Hence, u
4
u A%y.
,
{ F ( y ) : Y E A } = F ( 0 ) = 0.
F ( u A ) L u {F(y): Y E A ) , and complete a d d i t i v i t y i s proved.
Now we prove (iii)i m p l i e s ( i ) w i t h o u t t h e assumption F(0) =. 0. L e t t h e o p e r a t i o n G be d e f i n e d f o r a l l A , by G ( A ) = F ( A ) % F ( O ) . We have, G ( 0 ) = 0. Also, i f A Z 0,
G ( n A ) = F ( n A ) %F(O) = n{F(y) : Y E A } %F(O) = n
{F(y)%F(O): Y E A }
,
= n {C(y) : YEA]
S i m i l a r l y we can show, G ( A u 8 ) = G ( A ) U G(8). Applying, t h e case proved above, we have f o r A f 0, G ( u A ) = U CG(y) : Y E A } ; i.e. F ( u A ) F ( 0 ) = u { F ( y ) % F ( O ) : Y E A } . Since f i n i t e a d d i t i v i t y i m p l i e s , monotony, we have f o r a l l 8, F ( 0 ) 5 F ( 8 ) . Hence,
F(uA) = (F(uA)%F(O)) u F(0) = = u
U
{F(y) % F ( O ) : Y E A }
U
.
F(O)
{ ( F ( y ) s F ( 0 ) ) u F ( 0 ) : Y E A ] = u EF(y) : Y E A } .
The requirement t h a t F s a t i s f y t h e c o n d i t i o n s f o r a l l c l a s s e s or a l l s e t s i s e s s e n t i a l i n these theorems. For i n s t a n c e , t h e r e a r e o p e r a t i o n s which a r e c o m p l e t e l y m u l t i p l i c a t i v e and f i n i t e l y a d d i t i v e ( t h e t o p o l o g i c a l c l o s u r e ) i n some c o l l e c t i o n , b u t a r e n o t c o m p l e t e l y a d d i t i v e i n t h e same collection. W i t h t h e axiom o f c h o i c e i t i s p o s s i b l e t o f i n d a f i n i t e l y a d d i t i v e and m u l t i p l i c a t i v e o p e r a t i o n t h a t i s n o t c o m p l e t e l y a d d i t i v e . Without A x C , t h i s problem seems t o be open.
66
ROLAND0 CHUAQUI
PROBLEMS
1.
P r o v e 2.3.3.4
2.
P r o v e 2.3.3.7