, ; , .
X U Y . .
X . .
-
. | , | |
X Î ObS, < X, ~>, xÎX | - x, [ x ] | {yÎX | y~x} Í X |
X Î ObS, < X, ~> | - , X|~ | X|~ Í Ã(X), . X |
Xa, X, AÎObS Xa Í X "aÎA | . X, X = ëû Xa aÎA | {Xa} Í Ã(X) | X = È Xa aÎA & Xa ÇXb = Æ a ¹ b |
ObS - . | , [ X ] = [ Y ] | ObS: X ~ Y: = $ : X Y |
X ÎObS | . X, [ X ] | . X |
n Î N | [{1,2, , n}] | |
. | ||
. | , | |
, w | [N] | |
. | ||
, c. | [ [0,1] ] | |
. | ||
X, Y Î ObS | , [ X ] £ [ Y ]. | ObS: [ X ] £ [ Y ]: = X Y*ÍY |
-1 -. XÎ ObS. :
(~ - . X) Þ X = ëû [ x ].
[ x ] Î X|~
- X| ~ . X
-2
. . ObS
-3
3.1 Z
3.2 Q
3.3
3.4 a Î A Î ObS; Xa Î ObS "a Î A. :
([Xa] = w "aÎ A & [A] £ w) Þ [ È Xa ] = w
aÎA
3.5 XÎ ObS. :
X - . Þ $ X*Í X½ [ X* ] = w
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-4
4.1 a,b Î R & a < b. :
[ [0, 1] ] = [ [a,b] ] = [(a,b)] = [ R ] = [ R n] = [ [ a,b] ] = c
4.2 [ N ] = w & [[0,1]] =. > w
-5 ObS - .
: ([ X] £ [ Y]: = X Y*Í Y) Þ < ObS, £ > -
-6
" XÎ ObS: [ X] £ w Ú c £ [ X ]
,
-7
XÎ ObS. :
Ã(X) = {X*½X*ÍX} Þ [Ã(X)] > [X]
. X X
-8 . ., [X2] = [X].
-1 ~ - X. - X| ~:
1.1 () -, X( X2 )
1.2 [ x ], x Î X.
-2 XÎ ObS. [ X].
-1 X = R 2; <x1, y1> ~ <x2, y2>: = x12 + y12 = x22 + y22; x0 = <0,1>.
- X| ~.
[<0,1>]
-2. X= (0, 1). [ X].
. (0,1) Í [0,1] & [ 0.4, 0.5 ] Í (0,1) =([[ a, b ]] = c " a, b Î R ½ a < b)Þ [(0,1)] £ c & c £ [(0,1)] =( £ .)Þ [(0,1)] = c
4.2 [ N ] = w & [[0,1]] = . w < .
-. . [[0,1]] = w Ü( )Þ $ f: N [0,1] =(f , )Þ f(N) = {f(k) | kÎ N } = [0,1] (*). "k f(k) = Î [0,1], x = 0,x1x2xk Î[0,1], :
xk = , kÎ N
=( x)Þ f(k) ¹ x " kÎ N =(xÎ[0,1] & xÏf(N))Þ xÎ[0,1] \ f(N) (*).
1. X,Y Î ObS. .
1. XÍY Þ [X] £ [Y].
2. .
3. [ Z ] = w.
4. [[a,b]] = (a<b).
5. [(a,b)] = (a<b).
6. [ R ] = .
7. [D] = .
2. "" ( , , - 1). () (mod12) , .
1. -1
- x, [ x ] | - , X|~ | , < X, ~> | ||
. | ||||
, w | [N] | |||
, , | , [ X ] £ [ Y ]. | |||
{1,2,3} | Ã({1,2}) | . | ||
[0,1] | Ã([0,1]) | . | ||
Ã({1,2,3}) | {1,2,3,4,5,6,7} | . | ||
x Î [x] | [x] Í X | . , < X, ~>, x,y,zÎX | ||
XÍY | [X] £ [Y] | . , X,YÎObS | ||
-. - . {y = - x + c| cÎR} | -. - . {x = y + c} | , <x1,y1>~<x2,y2> Û x1 - y1 = x2 - y2 | ||
-. - . {y = c| cÎR} | -. - . {x = c} | , <x1,y1>~<x2,y2> Û x1 = x2 | ||
[<1,1>] = {<x,y>| x = 1} | [<1,1>] = {<x,y>| y = 1} | , <x1,y1>~<x2,y2> Û x1 = x2 | ||
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2. -2
X|~ Í Ã(X), . X | - , X|~ | , < X, ~> | ||||
[{1,2, , n}] | . X, [ X ] | |||||
, w | . | |||||
ObS [X]£[Y]: = X Y*ÍY | , [ X ] £ [ Y ]. | |||||
N | Q | . | ||||
[0,1] | Rn | . | ||||
Æ | {1} | . | ||||
[x]Ç[y]¹Æ | [x] = [y] | . , < X, ~>, x,y,zÎX | ||||
$ 1X :XY* = X Í Y | [X] £ [Y] | . , X,YÎObS | ||||
-. - . {x = - y2 + c} | -. - . {y = - x2 + c} | , <x1,y1>~<x2,y2> Û x1 + y12 = x2 + y22 | ||||
-. - . {x = c| cÎR} | -. - . {y = c} | , <x1,y1>~<x2,y2> Û y1 = y2 | ||||
[<1,1>] = {<x,y>| x = 1} | [<1,1>] = {<x,y>| y = 1} | , <x1,y1>~<x2,y2> Û y1 = y2 | ||||
3. -3
. X, X = ëû Xa aÎA | {Xa} Í Ã(X) | X = È Xa aÎA & Xa ÇXb = Æ a ¹ b | , < X, ~> | ||
. X | - x, [ x ] | |||
. | ||||
. | ||||
R | Z | . | ||
[0,1]´[0,3] | [-1,1]´[-3,3] | . | ||
R2 | Z3 | . | ||
z ~ x & z ~ y | x ~ y | . , < X, ~>, x,y,zÎX | ||
-. - . {x = - y2 + c} | -. - . {y = - x2 + c} | , <x1,y1>~<x2,y2> Û x1 + y12 = x2 + y22 | ||
-. - . {y = c| cÎR} | -. - . {x = c} | , <x1,y1>~<x2,y2> Û x1 = x2 | ||
[<1,1>] = {<x,y>| x = - y2+2} | [<1,1>] = {<x,y> | x = - y2 + 1} | , <x1,y1>~<x2,y2>Ûx1+y12=x2 + y22 | ||
[<1,1>] = {<x,y>| x = 1} | [<1,1>] = {<x,y>| y = 1} | , <x1,y1>~<x2,y2> Û x1 = x2 | ||
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4. -4
ObS: X ~ Y: = $ : X Y | , | |||
. | . | |||
, c | ||||
[N] | [ [0,1] ] | |||
[0, + ¥) | [0,1] | . | ||
[0,1]´{0, 3} | {0, 1}´{0, 3} | . | ||
[0,1]´{1,2} | Ã({1,2,3}) | . | ||
[x] Í [y] | x ~ y | . , < X, ~>, x,y,zÎX | ||
-. - . {y = - x + c| cÎR} | -. - . {x = y + c} | , <x1,y1>~<x2,y2> Û x1 - y1 = x2 - y2 | ||
-. - . {x = c| cÎR} | -. - . {y = c} | , <x1,y1>~<x2,y2> Û y1 = y2 | ||
[<1,2>] = {<x,y>| y = - x + 1} | [<1,2>] = {<x,y>| x = y - 1} | , <x1,y1>~<x2,y2> Û x1 - y1 = x2 - y2 | ||
[<1,1>] = {<x,y>| x = 1} | [<1,1>] = {<x,y>| y = 1} | , <x1,y1>~<x2,y2> Û y1 = y2 | ||
ã .. 2005. m_oe.doc