, m - m ( 3- ). - , . .
Ω n :
Ω = { x 1, x 2, , xn }.
( P) Ω :
xPy ↔ (xi ³ yi, i = 1, , m) ($ j: xj > yj). (1)
(.. y x ) , , . Ω (. 14-3); ΩP. - : 11-3 - ; 13-2.3 .
(1848 1923), , - , . . , . , (. 14-2.1).
, , ΩP, . - , ∕ . 4. . , , , -. ( ) ΩP. Ω ( , Ω). 3.
R 14-3.1. , R - xRy. , Ω Em. á x, y ñ, xy, - . .
m- Ω = { x 1, x 2, , xn } . ΩP.
1. .
1. . L = Ω, D =Æ, xb = x 1.
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2. xb L, x 1.
xb y :
) xbPy; y L; , 4; ! .! .! .! . xb - L;
) yPxb; xb ; , 4; xb = ; xb ( y) L;
) xb y b; xb L.
3. D = D { xb }, L = L { xb }, xb = 1- L. 2.
4. ΩP = D z, z L. 1 ■
1. ΩP Ω = {(3,7,0), (4,5,1), (2,3,0), (3,7,3), (2,6,3), (4,5,2)}. 1.
1. . L = Ω = {(3,7,0), (4,5,1), (2,3,0), (3,7,3), (2,6,3), (4,5,2)}, D =Æ, xb = x 1 = (3,7,0).
2. xb =(3,7,0) y =(4,5,1). ). y = (2,3,0).
2. xb = (3,7,0) y = (2,3,0). (3,7,0) P (2,3,0), ). (2,3,0) , L ={(3,7,0), (4,5,1), (3,7,3), (2,6,3), (4,5,2)}. - ( ) y = (3,7,3).
2. xb = (3,7,0) y = (3,7,3). (3,7,3) P (3,7,0), ). xb = (3,7,0) L ={(4,5,1), (3,7,3), (2,6,3), (4,5,2)}. xb = (3,7,3) y = (2,6,3).
2. xb = (3,7,3) y = (2,6,3). (3,7,3) P (2,6,3), ). (2,6,3) L ={(4,5,1), (3,7,3), (4,5,2)}. - ( ) y = (4,5,2)
2. xb = (3,7,3) y = (4,5,2) , ). - , 3.
3. D = Æ { xb } = {(3,7,3)}. (3,7,3) L, L ={(4,5,1), (4,5,2)}. xb = (4,5,1) 2.
2. xb = (4,5,1) y =(4,5,2). (4,5,2) P (4,5,1), ). xb = (4,5,1) xb = (4,5,2), L ={(4,5,2)}. L (4,5,2), 4.
4. D = D {(4,5,2)}= {(3,7,3), (4,5,2)}■
1. 1
01. Ω = {(3,7,5), (4,5,1), (2,3,0), (5,6,4), (1,4,3), (8,1,2)}
02. Ω ={(3,7,5), (4,5,-1), (2,3,0), (5,6,-4), (1,4,3), (8,1,2)}
03. Ω = {(3,7,5), (4,5,1), (2,3,0), (5,6,-4), (-1,4,3), (8,1,2)}
04. Ω = {(3,7,5), (4,5,-1), (2,-3,0), (5,6,4), (1,4,3), (8,1,2)}
05. Ω = {(3,7,5), (4,5,1), (2,3,0), (5,6,4), (1,4,-3), (8,1,2)}
06. Ω = {(3,7,5), (4,5,-1), (2,3,0), (-5,6,4), (1,4,3), (8,1,2)}
07. Ω = {(3,7,5), (4,5,1), (2,3,0), (5,6,-4), (-1,4,3), (8,1,2)}
08. Ω = {(3,7,5), (4,5,-1), (2,3,0), (5,6,4), (1,4,3), (8,1,-2)}
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09. Ω = {(3,7,5), (4,5,-1), (2,3,0), (5,6,-4), (1,4,3), (8,1,2)}
10. Ω = {(-3,7,5), (4,5,1), (2,3,0), (5,6,4), (1,4,3), (8,1,2)}
11. Ω ={(3,-7,5), (4,5,-1), (2,3,0), (5,6,-4), (1,4,3), (8,1,2)}
12. Ω = {(3,7,-5), (4,5,1), (2,3,0), (5,6,-4), (-1,4,3), (8,1,2)}
13. Ω = {(3,7,5), (-4,5,-1), (2,-3,0), (5,6,4), (1,4,3), (8,1,2)}
14. Ω = {(3,7,5), (4,-5,1), (2,3,0), (5,6,4), (1,4,-3), (8,1,2)}
15. Ω = {(3,7,5), (4,5,1), (2,3,0), (-5,6,4), (1,4,3), (8,1,2)}
16. Ω = {(3,7,5), (4,5,1), (-2,3,0), (5,6,-4), (-1,4,3), (8,1,2)}
17. Ω = {(3,7,5), (4,5,-1), (2,-3,0), (5,6,4), (1,4,3), (8,1,-2)}
18. Ω = {(3,7,5), (4,5,-1), (2,3,-0), (5,6,-4), (1,4,3), (8,1,2)}
19. Ω = {(3,7,5), (4,5,1), (2,3,0), (-5,6,4), (1,4,3), (8,1,2)}
20. Ω ={(3,7,5), (4,5,-1), (2,3,0), (5,-6,-4), (1,4,3), (8,1,2)}
21. Ω = {(3,7,5), (4,5,1), (2,3,0), (5,6,4), (-1,4,3), (8,1,2)}
22. Ω = {(3,7,5), (4,5,-1), (2,-3,0), (5,6,4), (-1,4,3), (8,1,2)}
23. Ω = {(3,7,5), (4,5,1), (2,3,0), (5,6,4), (1,-4,-3), (8,1,2)}
24. Ω = {(3,7,5), (4,5,-1), (2,3,0), (-5,6,4), (1,4,-3), (8,1,2)}
25. Ω = {(3,7,5), (4,5,1), (2,3,0), (5,6,-4), (-1,4,3), (-8,1,2)}
26. Ω = {(3,7,5), (4,5,-1), (2,3,0), (5,6,4), (1,4,3), (8,-1,-2)}
27. Ω = {(3,7,5), (4,5,-1), (2,3,0), (5,6,-4), (1,4,3), (8,1,-2)}
28. Ω = {(3,7,5), (4,5,1), (2,3,0), (5,6,4), (1,4,3), (8,-1,2)}
29. Ω ={(3,7,5), (4,5,-1), (2,3,0), (5,6,-4), (1,4,3), (-8,1,2)}
30. Ω = {(3,7,5), (4,5,1), (2,3,0), (5,6,-4), (-1,4,-3), (8,1,2)}
31. Ω = {(3,7,5), (4,5,-1), (2,-3,0), (5,6,4), (1,-4,3), (8,1,2)}
32. Ω = {(3,7,5), (4,5,1), (2,3,0), (5,6,4), (-1,4,-3), (8,1,2)}
33. Ω = {(3,7,5), (4,5,-1), (2,3,0), (-5,6,-4), (1,4,3), (8,1,2)}
34. Ω = {(3,7,5), (4,5,1), (2,3,0), (5,-6,-4), (-1,4,3), (8,1,2)}
35. Ω = {(3,7,5), (4,5,-1), (2,3,0), (-5,6,4), (1,4,3), (8,1,-2)}
36. Ω = {(3,7,5), (4,5,-1), (2,-3,0), (5,6,-4), (1,4,3), (8,1,2)}
37. Ω = {(3,7,5), (4,5,1), (-2,3,0), (5,6,4), (1,4,3), (8,1,2)}
38. Ω ={(3,7,5), (4,5,1), (2,3,0), (5,6,-4), (1,4,3), (8,1,2)}
39. Ω = {(3,7,5), (4,-5,1), (2,3,0), (5,6,-4), (-1,4,3), (8,1,2)}
40. Ω = {(3,7,5), (-4,5,-1), (2,-3,0), (5,6,4), (1,4,3), (8,1,2)}
41. Ω = {(3,7,-5), (4,5,1), (2,3,0), (5,6,4), (1,4,-3), (8,1,2)}
42. Ω = {(3,-7,5), (4,5,-1), (2,3,0), (-5,6,4), (1,4,3), (8,1,2)}
43. Ω = {(-3,7,5), (4,5,1), (2,3,0), (5,6,-4), (-1,4,3), (8,1,2)}
44. Ω = {(-3,7,5), (-4,5,-1), (2,3,0), (5,6,4), (1,4,3), (8,1,-2)}
45. Ω = {(3,7,5), (-4,5,-1), (-2,3,0), (5,6,-4), (1,4,3), (8,1,2)}
46. Ω = {(3,7,5), (4,5,-1), (-2,3,0), (-5,6,4), (1,4,3), (8,1,-2)}
47. Ω = {(3,7,5), (4,5,-1), (2,3,0), (-5,6,-4), (1,4,3), (8,1,2)}
48. Ω = {(3,7,5), (4,5,-1), (2,3,0), (5,6,4), (-1,4,3), (8,1,-2)}
49. Ω = {(3,7,5), (4,5,-1), (2,3,0), (5,6,-4), (-1,4,3), (-8,1,2)}
50. Ω = {(-3,7,5), (4,5,-1), (2,3,0), (5,6,4), (1,4,3), (-8,1,-2)}■
, ( 1- ) -, .. . .
2. Ω = {(3,7,0), (4,5,1), (2,3,0), (3,7,3), (2,6,3), (4,5,2)} ΩP = {(3,7,3), (4,5,2)} (c 1). , Ω 1= Ω ΩP = {(3,7,0), (4,5,1), (2,3,0), (2,6,3)}. 1 Ω 1 , = {(3,7,0), (4,5,1), (2,3,0), (2,6,3)} P = {(3,7,0), (4,5,1), (2,6,3)}. Ω 2= Ω 1 = {(2,3,0)}, , : {(3,7,3), (4,5,2)}; {(3,7,0), (4,5,1), (2,6,3)} {(2,3,0)}■
2. 7- , .
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, , , - . , - - , (). - - . , 30 .. ( ), - . - .
, (, , , - ..), . - . , () . , -: - . , , , (). , , -, , , , . - . , , , . -- . , , (, , ) 2,5 , -. (. 1.1).
, ( ), -, .. . - (. 1, 08 2), , , - .
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7. |
3.1. . . i Kj i j, Ki Kj − i ≈ j. Θ (). ( , / ) , , .. i ≈ j i j. , , , -, Θ = {1 2, 2 ≈ 3}.
- ., , . - ( ) - , , (.. , , , − ), , -. , , , - , , -, . , , , .
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. xij - , xi xj. , x = (5,4,3,4), 14 = (4,4,3,5) 23 = (5,3,4,4).
. Ki Kj , -, xij . Ki - Kj, , xi > xj,, xij. , i ≈ j , -, =xij, I i≈j. , i > j , , xi > xj, =xij, Pi > j. , (5,4,3,4) P 1>2(4,5,3,4), , - ( ) 5 − ó 4 5. - , . (5,4,3,4) I 2≈3(5,3,4,4). , , . . (5,4,3,4) I 2≈3(5,3,5,4) -, 2- 1- 2- 3- -.
. , . , , , , - . - (, (5,4,3,3) (4,5,3,4)), P: - . . .
3.2. . Θ - , . , Θ , -, . - . , Θ 1 2,2 3, 3 1, - , , 1 1. , , Θ . , 3 1, 1 3. , . , Θ -.
3. . , , .. Θ = {1 2}. , (5,3) (2,4) (5,3) P 1>2(2,4) , 2- 1- . , (5,3) P (2,4), 2- , : 3 < 4. :
(5,3) P 1>2(3,5) (3,5) P (2,4).
, (5,3) ( Θ - ) (2,4). PΘ ■
. , - Θ = {1 2} (5,3) (4,5). , 1 , (5,3) P 1>2(3,5). , , (4,5) P (3,5). , : - (3,5), - . .
1. , , , , P , .
2. , , , , .
, , -, . , 1 2 , -. , -. . , .
4. . 5- 6- 1: (5,4,3,5) (3,5,4,4). , : Θ = {1≈2, 2 3,3≈4}. .
(5,4,3,5) P 1>2(5,3,4,5) I 1≈2(3,5,4,5) I 3≈4(3,5,5,4) P (3,5,4,4).
1- 2- 3- , 2- 1- 2- , 3- - 3- 4- , 4- - ■
4 . -, x , Θ. , - , , , . - , . - . , - .
, , - . - .
2- 7- 1: (3,4,4,3) (3,5,3,3). . , ( 1). ( 5, ), 1 2 . , Θ.
, Θ . 5- 6- 1: (5,4,3,5) (3,5,4,4). , : Θ = {1≈2, 3≈4},.. 1 2 3 4. . , , (5,4,3,5), 3,5 5,3. (3,5,4,4) 4,4. 3,5, 5,3 4,4, .
.
1. , . - , .. ( ). ↓. ↓. (PΘ) , ↓ P ↓.
2. (Θ =Æ). P Ø = P (.. , ).
3.3. . x, y Θ . , . - :
) S (x) = S (y) ↓ = y ↓;
) S (x) > S (y).
, , -.
) ) .
). S (x) = S (y) ↓ = y ↓, , - x . - 2.
2. .
0. (0-) x.
i (i >0). i - , - , (i− 1)- , -. (, s t, Θ s t, , s t, s - , t -). , 0-, : (i− 1)- , ( Ps > t I s≈t), ( ).
:
) 0- i -, ;
) y, .
) (i− 1)- i - -, ■
2 .
2 ), , - . , Pi > j, , y PΘ. Pi > j, , x y PΘ y x PΘ. .
, ) - y x . y x (.. y).
5. Θ = {3 2, 1 ≈ 3}. : x= (5,3,4,5) y = (4,5,3,5). S (x) = S (y) ↓ = y ↓, 2.
( )
1. (5,3,4,5)
2. (5,3,4,5) I 1≈3(4,3,5,5), (5,3,4,5) P 3>2(5,4,3,5).
3. (4,3,5,5) P 3>2(4,5,3,5).
, (4,5,3,5) = y. :
x = (5,3,4,5) I 1≈3(4,3,5,5) P 3>2(4,5,3,5) = y.
P 3>2, - x y PΘ, y x ■
6. Θ = {3 2, 1≈ 3}. x = (5,3,4,5) y = (3,5,5,4). S (x) = S (y) ↓ = y ↓, 2. ( )
1. (5,3,4,5)
2. (5,3,4,5) I 1≈3 (4,3,5,5), (5,3,4,5) P 3>2 (5,4,3,5).
3. (4,3,5,5) P 3>2 (4,5,3,5), (5,4,3,5) I 1≈3 (3,4,5,5).
4. (4,5,3,5) I 1≈3 (3,5,4,5) ( (3,4,5,5) P 3>2 (3,5,4,5)).
, (3,5,4,5) : P 3>2 , 3- 4, 2- 5 4 < 5. I 1≈3 , . . , (5,3,4,5) (3,5,5,4) PΘ ■
7. , .. (3,5,5,4). -
1. (3,5,5,4).
2. (3,5,5,4) I 1≈3 (5,5,3,4).
, P 3>2 , I 1≈3 , . (3,5,5,4) (5,3,4,5)
PΘ. (. 6), (5,3,4,5) (3,5,5,4), , ■
8. Θ = {3 2, 1≈3, 4 1}. x = (5,3,4,5) y = (3,5,5,4). , ó Θ. 2
1. (5,3,4,5)
2. (5,3,4,5) I 1≈3(4,3,5,5), (5,3,4,5) P 3>2 (5,4,3,5).
3. (4,3,5,5) P 3>2(4,5,3,5), (4,3,5,5) P 4>1(5,3,5,4), (5,4,3,5) I 1≈3 (3,4,5,5).
4. (4,5,3,5) P 4>1(5,5,3,4), (4,5,3,5) I 1≈3(3,5,4,5), (3,4,5,5) P 4>1(5,4,5,3).
5. (5,5,3,4) I 1≈3(3,5,5,4),
(3,5,5,4) 5- . , x = (5,3,4,5) I 1≈3 (4,3,5,5) P 3>2 (4,5,3,5) P 4>1 (5,5,3,4) I 1≈3(3,5,5,4) = y.
, Θ = {3 2, 1≈3, 4 1} - , (5,3,4,5) PΘ (3,5,5,4), .. (5,3,4,5) -, (3,5,5,4). , ■
. : S (x) > S (y). : xPΘy x y . 2, - 2.
2.
0. (0-) x.
i (i >0). i - , - , (i− 1)- , -. (, s t, Θ s t, , s t, s - , t -). , 0-, : (i− 1)- , (Ps > t I s≈t), ( ).
:
) 0- i -, ;
) y, - .
) (i− 1)- i - -, ■
2 .
2 ), , - . 1, x y PΘ y x PΘ. - .
, 2 ), 2, x y: S (x) > S (y) y x.
9. Θ = {3 2, 1 ≈ 3}. : x = (5,3,4,5) y = (4,5,3,4). S (x) > S (y), 2.
( )
1. (5,3,4,5)
2. (5,3,4,5) I 1≈3(4,3,5,5), (5,3,4,5) P 3>2(5,4,3,5).
3. (4,3,5,5) P 3>2(4,5,3,5).
, (4,5,3,5) P (4,5,3,4) = y. -:
x = (5,3,4,5) I 1≈3 (4,3,5,5) P 3>2 (4,5,3,5) P (4,5,3,4)= y.
2 x y PΘ, y x ■
10. Θ = {3 2, 1 ≈ 3}. x = (5,3,4,5) y = (3,5,5,3). S (x) > S (y), 2.
( )
1. (5,3,4,5)
2. (5,3,4,5) I 1≈3 (4,3,5,5), (5,3,4,5) P 3>2 (5,4,3,5).
3. (4,3,5,5) P 3>2 (4,5,3,5), (5,4,3,5) I 1≈3 (3,4,5,5).
4. (4,5,3,5) I 1≈3 (3,5,4,5).
, (3,5,4,5) : P 3>2 , 3- 4, 2- 5 4 < 5. I 1≈3 , .
2 . , (5,3,4,5) (3,5,5,3) PΘ ■
11. Θ = {3 2, 1≈3, 4 1}. x = (5,3,4,5) y = (3,5,5,3). , ó Ω. 2
1. (5,3,4,5)
2. (5,3,4,5) I 1≈3(4,3,5,5), (5,3,4,5) P 3>2 (5,4,3,5).
3. (4,3,5,5) P 3>2(4,5,3,5), (4,3,5,5) P 4>1(5,3,5,4), (5,4,3,5) I 1≈3 (3,4,5,5).
4. (4,5,3,5) P 4>1(5,5,3,4), (4,5,3,5) I 1≈3(3,5,4,5), (3,4,5,5) P 4>1(5,4,5,3).
5. (5,5,3,4) I 1≈3(3,5,5,4),
, 5- (3,5,5,4) (3,5,5,3) = y. :
x = (5,3,4,5) I1≈3 (4,3,5,5) P3>2 (4,5,3,5) P4>1 (5,5,3,4) I1≈3(3,5,5,4) P(3,5,5,3) = y.
2 x y PΘ, y x ■
3. Θ = {3 2, 1≈3, 4 3} 2 ■
,
,
,
,
,
,
1.
2.
3.
4.