, .
, | ||
<X, £ >, AÍ X | A | xÎX ½ a £ x " aÎA |
<X, £ >, AÍ X | A | AÎ A |
<X, £ >, AÍ X | A | xÎX ½ x £ a " aÎA |
<X, £ >, AÍ X | A | AÎ A |
<X, £ >, AÍ X | A, supA | a A, |
<X, £ >, AÍ X | A, infA | a A, |
<X, £ >, AÍ X | A, maxA | maxA Î A½ aÎA & a ³ maxA Þ a = maxA |
<X, £ >, AÍ X | A, minA | minA Î A½ aÎA & a £ minA Þ a = minA |
<X, £ >, AÍ X | A, | |
<X, £ >, AÍ X | A, | |
<X, £ >, AÍ X | A, (.. <A, £ > - ) | |
<X, £ >, AÍ X | C | A, : $ xÎX½ a < x "aÎA |
<X, £ > - | <X, £ > ½ ƹA Í X Þ A | |
X,YÎ ObS | (..) | F: X Ã(Y) |
F: X Ã(Y) - . . | O . . F | f: X Y½ xÎX Þ fx Î Fx |
.
.
in
(/)
/
/ / .
. ..
-1
B
-2 < X, £ > - . :
(" X ) Þ X $ maxX
< X, £ > , X
-3 F: X Ã(Y) - . . :
(Fx ¹ Æ " xÎX) Û F
. .
-4
|
|
-5 < X, £ > - . A:X Pr - ½1) X'1 - X Þ A(1) ;
2) (1¹yÎX & A(x) "x < y) Þ A(y) . A(x) "xÎX.
-1 A Í < X, £ > - . ():
1.1 A;
1.2 A ( $);
1.3 supA, infA ( $);
1.4 A ( $);
1.5 maxA minA ( $);
1.6 A, 2 ( $);
1.7 C X;
1.8 A ( $).
-1 X = < R2, £ >; < x1, x2> £ < y1, y2> Û x1 £ y1 & x2 £ y2 ;
A = {<0, 1>, <1, 0>}
<0, 1> -
<1, 1> = supA
<1, 0>
. 1
-5 < X, £ > - . A:X Pr - ½1) X'1 - X Þ A(1) ;
2) (1¹yÎX & A(x) "x < y) Þ A(y) . A(x) "xÎX.
-. . X ={xÎX | A(x) }¹Æ =(X )Þ $ y X; A(y) =(1))Þ y¹1 =( )Þ "x < y¹1 A(x) =(2))Þ A(y) . ¨
1. < X, £ > - , AÍX, BÍX. .
1. .
2. .
3. supA.
4. infA.
5. supA, .
6. infA, .
7. maxA, .
8. minA, .
9. $ supA $ supB AÍB Þ supA £ supB.
10. $ infA $ infB AÍB Þ infA ³ infB.
(5 ).
b =(?)Þ b = supA.
-. b Ü( )Þ
b = & bÎA =( : a£m "aÎA)Þ b = & b£m " m Ü( )Þ b = , Ü( supA)Þ b = supA ª
. < R, £ >, A = (0,1). SupA = 1, . , supA .
2. "" ( , , - 1). () (mod12) , .
|
|
1. -1
A | A | , <X, £ >, AÍ X | ||||||
A, supA | a A | , <X, £ >, AÍ X | ||||||
A, infA | AÎ A | , <X, £ >, AÍ X | ||||||
A, | , <X, £ >, AÍ X | |||||||
A, : $ xÎX½a < x "aÎA | , <X, £ >, AÍ X | |||||||
. {{2}, {2,3,4}} | sup{{1}, {2,3,4}} | , Ã(N) | ||||||
x - . | x = supA | , <X,£>, AÍX | ||||||
a £ x "aÎA & xÎA | x = supA | , <X,£>, AÍX | ||||||
x - . | x £ a "aÎA & xÎA | , <X,£>, AÍX | ||||||
$ | - -, <X,£> - | |||||||
, | minA | , X,YÎ ObS | ||||||
, . | maxA | , X,YÎ ObS | ||||||
2. -2
AÎ A | xÎX ½ a £ x "aÎA | , <X, £ >, AÍ X | |||
A | AÎ A | , <X, £ >, AÍ X | |||
A | A, infA | , <X, £ >, AÍ X | |||
. | minA Î A½ aÎA & a £ minA Þ a=minA | , <X, £ >, AÍ X | |||
A, | C | , <X, £ >, AÍ X | |||
S(X, Ã(Y)) | , X,YÎ ObS | ||||
{min{{1}, {2,3}}} | {max{{1}, {2,3}}} | , Ã(N) | |||
a £ x "aÎA & xÎA | x £ z "z - | , <X,£>, AÍX | |||
x = supA | x £ z "z - | , <X,£>, AÍX | |||
x1 £ x2 & x2 £ x1 | x1, x2 - . | , <X,£>, AÍX | |||
F:XÃ(Y) | Fx¹Æ "xÎX | $ f:XY | fxÎFx "xÎX | - -, <X,£> - | |||
supA | maxA | , X,YÎ ObS | |||
3. -3
AÎ A | A | , <X, £ >, AÍ X | ||
a A | xÎX ½ x £ a " aÎA | , <X, £ >, AÍ X | ||
A | A, minA | , <X, £ >, AÍ X | ||
F:XÃ(Y) | S(X,Y) | , X,YÎ ObS | ||
f: X Y½ xÎX Þ fx Î Fx | XÈY | , X,YÎ ObS | ||
sup{Æ, {2}, {3,4}} | sup{Æ, {2}, {3}, {4}} | , Ã(N) | ||
inf{{2,3}, {1,2,3}, {3,4}} | sup{{2,5}, {1,2,5}} | , Ã(N) | ||
{{2}, Æ, {2,3}} - | {{2}, Æ, {2,3}} - | , Ã({1,2,3}) | ||
x = infA | x - . | , <X,£>, AÍX | ||
x = minA | x - . | , <X,£>, AÍX | ||
supA | infA | , <X,£>, AÍX | ||
infA | minA | , X,YÎ ObS | ||
|
|
4. -4
a A | A | , <X, £ >, AÍ X | |||||
maxA Î A½ aÎA & a³maxAÞa=maxA | A | , <X, £ >, AÍ X | |||||
A, | , <X, £ >, AÍ X | ||||||
A, | , <X, £ >, AÍ X | ||||||
O . . F | f: X Y½ xÎX Þ fx Î Fx | , X,YÎ ObS | |||||
. {{1}, {2}, {1,2}} | . . {{1}, {1,3,5}} | , Ã(N) | |||||
x = maxA | x - . | , <X,£>, AÍX | |||||
x1, x2 - . | x2 £ x1 & x1 £ x2 | , <X,£>, AÍX | |||||
- <X,£> | $yÎX | x£y " xÎC | - -, <X,£> - , " | |||||
" | $ . maxX | - -, <X,£> - | |||||
$ | $yÎX | xÎX & x ³ y Þ x = y | - -, <X,£> - | |||||
. yÎA | aÎA & a £ y Þ a = y | infA | , X,YÎ ObS | |||||
ã .. 2003. m_ochp.doc