13.2 süzgeçler n kar“ıla“

advertisement
!"#$ %&' (#)*+,"+-
!"
!"# $%&'()*(+,- ./+01*/021+1*3/$1
!"#"
$%&'()*()%
# $%&'( )*+ ,-./ -0/+*&1/,* )-2-& .-.,-& 3-04/5'/+* )('(&(0#
!"#$% 6+&/7*& X = {a, b, c, d, e} %'3(&# 8( ,-./ -0/+*&1/
9:; S1 = {X}
9); Sa = {{a}, X}< Sb = {{b}, X}< Sc = {{c}, X}< Sd = {{d}, X}<
Se = {{e}, X}
9=; Sab = {{a, b}, X}< Sac = {{a, c}, X}< Sad = {{a, d}, X}< Sae =
{{a, e}, X}< Sbc = {{b, c}, X}< # # #
91; Sabc = {{a, b, c}, X}< Sacd = {{a, c, d}, X}< Sace = {{a, c, e}, X}<
Sbcd = {{b, c, d}, X}< Sbce = {{b, c, e}, X}< # # #
9/; Sabcd = {{a, b, c, d}, X}< Sabce = {{a, b, c, e}, X}< Sbcde = {{b, c, d, e}, X}<
Sabde = {{a, b, d, e}, X}< Sacde = {{a, c, d, e}, X}< # # #
> ?7/'* )*+ ,-./1/ @(,:+A1:,* B C:+,'A 3-04/=* 2:&A.':.AD %'1(,#
!# $%&3(0 )*+ X ,-./3* -0/+*&1/,* )*+ A 3-04/=*&*& )-2-& ,-./'/+*&*& :+:,/E
3*2* )%D *3/< )(&(& X -0/+*&1/,* 3%&'( 2-.'/@/&'/+ 3-04/=*&1/& 1:F: *&=/
%'1(7(&( 4?32/+*&*0#
!"#$% $%&'( 2-.'/@/&'/+ 3-04/=*&* S *'/ 4?32/+/'*.# S ⊂ A %'1(7(&(
4?32/+./'*@*0# G'.:@:&: H+4* I?&2/.*&* ,('':&:'A.# 8*+ S ∈ S ,-./3*&*&
A @: :*2 %'.:1A7A&A J:+3:@:'A.# 8( 1(+(.1: S ,-./3* A @: :*2 F*5 )*+
,-./&*& -32 ,-./3* %':.:0# 8(+:1:& F/+ A ∈ A *5*& A ∩ S ′ 6= ∅ %'1(7(
3%&(=( 5A,:+# 8( *3/
\
\
∅=
6
(A ∩ S ′ ) ⊂
A=∅
A∈A
A∈A
%'(+1(# 8( 5/'*D,* %':.:@:=:7A&: 4?+/< ,:)('-.-0 @:&'AD2A+K @:&* F/+ S ∈
S ,-./3* A 3-04/=*&/ :*22*+# L%':@A3A@':< S ⊂ A %'(+#
B# S J/ H )*+ X ,-./3* -0/+*&1/ *,* 3-04/5 %'3(&# X -0/+*&1/,* 3-04/5'/+
:+:3A&1: )( *,*3*&*& )*+ /& ,-5-, -32 3A&A+A J:+3:< )(&(&
D = {S ∩ H
%'1(7(&( 4?32/+*&*0#
|
S ∈ S, H ∈ H }
!"#$% 6&/+./ B#!#> 4/+/7*&=/< 3?0 ,%&(3( /& ,-5-, -32 3A&A+A& 93(M;
%'.:3A *5*& 4/+/,'* J/ @/2/+'* ,%D(' F/+ S ∈ S J/ F/+ H ∈ H *5*& S ∩H 6= ∅
%'.:3A1A+# 8( ,%D(' :'2A&1: D &*& )*+ 3-04/5 %'1(7(&( 4?32/+/)*'*+*0N
9:; ∅ ∈
/ D 1*+#
!"#" $%&'()*(+,- ./+01*/021+1*3/$1
!"
#$% S1 ∩ H1 ∈ D &' S2 ∩ H2 ∈ D ()'
(S1 ∩ H1 ) ∩ (S2 ∩ H2 ) = (S1 ∩ S2 ) ∩ (H1 ∩ H2 ) ∈ D
*(+,
#-% S1 , S2 ∈ S &' H1 , H2 ∈ H ./012 34'+' S1 ∩ H1 ⊂ S2 ∩ H2 ()'
S1 ⊂ S2 &' H1 ⊂ H2 ./1-1567*17 S2 ∩ H2 ∈ D ./8+,
9, S∞ 1(/')( R 7(7 (a, ∞), a ∈ R :'2/(7*' $(+ 1+1/65676 (;'+'7 $3<37 1/<=
230'/'+(7*'7 ./8:)87, >8 1(/'7(7 R 34'+(7*' $(+ )34?'; ./8:<8+1-15676
?@)<'+(7(4,
!"#$% A1760
B, , 2.:8//1+6767 )15/17*65676 ?@)<'+'-'5(4,
#1% C'+ S ∈ S∞ 230')( $(+ (a, ∞) 1+1/65676 (;'+*(5(7*'7 D(; $(+()( $.:
./1014,
#$% A, B ∈ S∞ ()' (a, ∞) ⊂ A &' (b, ∞) ⊂ B ./1-12 $(;(0*' a, b ?'+;'/
)1E6/1+6 &1+*6+, c = max{a, b} *(E'/(0, (c, ∞) ⊂ A ∩ B ./*8587*17
A ∩ B ∈ S∞ ./1-12<6+,
#-% S ∈ S∞ &' S ⊂ A ./)87, (a, ∞) ⊂ S ()' (a, ∞) ⊂ A ./01/6*6+,
F./1E6)6E/1 S ∈ S∞ ./8+,
", S 1(/')( X 34'+(7*' $(+ )34?'; ./)87 &' $(+ A ⊂ X 1/<=230')( &'+(/)(7, S
7(7 A 34'+(7*'2( SA (4(7(7 #<+1-'% $(+ )34?'; ./01)6 (;(7 ?'+'2/( &' E'<'+/(
2.:8/ S 7(7 D'+ 230')(7(7 A (/' 2')(:0')(*(+, G@)<'+(7(4,
!"#$%
SA = {A ∩ S| S ∈ S
1(/')(7(7 A 34'+(7*' $(+ )34?'; ./*85878 ?@)<'+'-'5(4,
#1% S 7(7 D'+ 230')( A (/' 2')(:(E.+)1 A ∩ S 6= ∅ ./8+, H D1/*' ∅ ∈
/ SA
*(+,
#$% A ∩ S1 ∈ SA &' A ∩ S2 ∈ SA ()'
(A ∩ S1 ) ∩ (A ∩ S2 ) = A ∩ (S1 ∩ S2 ) ∈ SA
./8+,
#-% (A ∩ S1 ) ⊂ (A ∩ S2 ) ()' S1 ⊂ S2 ./1-1567*17 S2 ∈ S *(+, H D1/*'
(A ∩ S2 ) ∈ SA *6+,
I, F 1(/')( X 230')( 34'+(7*' $(+ 1:267 )34?'; #8/<+1 J/<'+% ./)87, K5'+
A1 ∪ A2 ∪ . . . ∪ An ∈ F
()'L {A1 , A2 , . . . , An } 230'/'+(7*'7 '7 14 $(+ <17')( F 1:267 )34?'-(7'
1(<<(+, G@)<'+(7(4,
!"#$%
!"#$ %&' (#)*+,"+-
!"
!./0 123245 #$%& '()*+&,- -.- ./,& -0-$ 012&+-,3 A ∪ B ∈ F )+4565$47
87 A ∈ F 87 47 B ∈ F )+4565$5 91:;&(&+-,3
<+,787$7 =(9- >1$;&,-$- .5++7$7+?,3 A ∈
/ F @& B ∈
/ F )+4565$5 @7(:78A
7+?,3
G = {C | ∅ 6= C ⊂ X, A ∪ C ∈ F }
7-+&:-$- ;7$?,+787+?,3 B5 7-+&$-$ X ./,&:- /2&(-$4& *-( :/29&0 )+4565 .)A
+78%7 91(/+/(3 C8(?%7 F ⊂ G @& B ∈ G 4-(3 <8:7 F 7D.?$ E5+;(7F :/29&0;-(3
#8+&8:& F = G )+,7+?4?(3 < G7+4& B ∈ F 4-(3 B5 -:& .7*5+/,/2/$ 0&+-D.87(7;;?6?$? 91:;&(-(3 H&,&. .-I A ∪ B ∈ F )+4565$47 87 A ∈ F 87 47
B ∈ F )+,7+?4?(3
*/6/0 123245
Ai ∈
/ F -:&I #$ J()*+&, E%F 587(?$%7I Ai ∈ F I (n =
1, 2, . . . , n) )+7%7.;?(3 B5$+7(?$ :)$+5 7(7.&:-;- 4& :/29&%& 7-; )+,7+?4?(K
87$!′
n
n
\
[
′
Ai =
Ai ∈ F
′
i=1
i=1
)+7%7.;?(3 <8:7 *5 -0&(,& )+7,72K 0/$./
n
[
Ai ∈ F
i=1
-0&(,&:- @7(4?(3 < G7+4& .7*5+/,/2 87$+?D;?(3 =$ 72 *-( Ai ∈ F )+,7+?4?(3
Download