:
max f=3x1-4x2-2x3 䒺
5. - .
- 䒺 .
r =4, ,
4, x1, x6, x7, - x2, x3, x4, x5.
x1 = x6 = x7 = 0, x2=3; x3=4; x4=3; x5=2 - 䒺 . f x1, x6, x7,
, , :
1.
X1 | X2 | X3 | X4 | X5 | X6 | X7 | ³ | |
-1 | ||||||||
-1 | ||||||||
-1 | ||||||||
-3 | -20 |
; f 䒺 . 2.
X1 | X2 | X3 | X4 | X5 | X6 | X7 | ³ | |
X2 | -1 | |||||||
X3 | -1 | |||||||
X1 | ||||||||
X5 | ||||||||
f | -11 |
f 2 , :
max f = -11 (3,3,4,0,5,0,0) .
, max f = -11 (3,3,4).
6
.
1. , D .
2. , .
3. D.
4. .
5. D.
6. .
7. D.
8. .
9. D .
10. .
11. D .
12. .
13. , .
14. D.
15. - .
16. - D.
17. , | a | = n ak = 1, n k.
18. , | g | = n, " g Î G gk =1 , k n.
19. , | G | = pq, p, q G , G , | a | = pq.
20. , | G | = pq, p ¹ q , G .
|
|
21. , | G | = p 2, , .
22. C 1 C, I C. , C 1Ç I C 1.
23. , Z .
24. , j K 1 K 2 j(a b)= j(a)j(b).
25. , j K 1 K 2 j(a 1) = [j(a)]1 ( K 1 a 1).
26. , - I - C / I.
27. , I , I .
28. , - .
29. , - .
30. , Z [ i ] : 3; 2 + i.
31. , Z [ i ] : 3; 2 i.
32. , .
33. , .
1, 3, 5, 7, 9, 11, 14, 18, 22, 24, 26, 28, 31 | |
2, 4, 6, 8, 10, 12, 15, 17, 21, 23, 25, 29, 30 | |
1, 3, 7, 11, 13, 19, 22, 26, 29, 23, 32, 27 | |
2, 4, 6, 8, 11, 13, 15, 19, 23, 24, 27, 31, 28 | |
1, 3, 5, 7, 9, 14, 16, 23, 27, 30, 22, 29, 33 | |
2, 4, 6, 8, 10, 12, 15, 17, 20, 24, 27, 28, 31 | |
1, 3, 5, 7, 9, 14, 20, 22, 25, 26, 28, 30, 32 | |
2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 26, 30 | |
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 31 | |
2, 4, 6, 8, 10, 13, 18, 15, 20, 24, 26, 28, 32 |
6
1. ,
.
.
ϳ . , G:
1) (×);
2) ;
3) ;
4) .
(×) G.
S 1 | S 2 | S 3 | S 4 | S 5 | S 6 | |
S 1 | S 1 | S 2 | S 3 | S 4 | S 5 | S 6 |
S 2 | S 2 | S 3 | S 1 | S 6 | S 4 | S 5 |
S 3 | S 3 | S 1 | S 2 | S 5 | S 6 | S 4 |
S 4 | S 4 | S 5 | S 6 | S 1 | S 2 | S 3 |
S 5 | S 5 | S 6 | S 4 | S 3 | S 1 | S 2 |
S 6 | S 6 | S 4 | S 5 | S 2 | S 3 | S 1 |
(×) :
.
, (×).
, G . (S 3).
2) S 3.
.
S 3 6- . , , . , S 3 .
s 2, s 3 , (s 2)3 = (s 3)3 = s 1.
s 4, s 5, s 6 , (s 4)2 = (s 5)2 = (s 6)2 = s 1.
, S 3 3 2,
S 3 = { s 2, s 4} = { s 2, s 5} = { s 2, s 6} = { s 3, s 5} = { s 3, s 4} = { s 3, s 6}.
|
|
3) S 3.
.
. S 3 3 2 : E, S 3.
ϳ 3- S 3, < s 1, s 2, s 3> 2- S 3, < s 1, s 4>, < s 1, s 5>, < s 1, s 6>.
ϳ G 1 = < s 1, s 2, s 3> = { s 2} = { s 3}; G 2 = < s 1, s 3> = { s 3};
G 3 = < s 1, s 5> = { s 5}; G 4 = < s 1, s 6> = { s 6}.
, - S 3 , () .
3. S 3 .
.
s 1 S 3; s 2, s 3 3 ,
si 1× s 3× si = s 2, si 1× s 2× si = s 3;
s 4, s 5, s 6 2 ,
s 5 s 4 s 5 = s 6, s 6 s 5 s 6 = s 4, s 4 s 5 s 4 = s 6,
. ,
S 3 = < s 1> + < s 2, s 3> + < s 4, s 5, s 6 >.
4. S 3 G 2 = < s 1, s 6>.
.
˳ S 3 G 2 : G 2, s 2 G 2 = s 4 G 2 = { s 2, s 4}, s 5 G 2 = s 6 G 2 = { s 4, s 6}.
S 3 = G 2 È{ s 2, s 4}È{ s 4, s 6}.
5. S 3.
.
, , . , S 3 G 1 = < s 1, s 2, s 3> , S 3 G 1 : : G 1 i < s 4, s 5, s 6>,
S 3 = G 1È G 1 s 4 = s 4 G 1 = { s 4, s 5, s 6}.
6. - S 3 G 1.
.
S 3 G 1 , - S 3 G 1 (S 3/ G 1).
siG 1× sjs 1 = si × sjs 1, si, sj Î S 3, s 1 .
, .
7. S 3:
(siG 1× sjs 1)× skG 1= (sisj) s 1× skG 1 = (sisj)× skG 1= si (sj × sk) G 1 =
= siG 1×(sj × sk) G 1= siG 1×(sjG 1× skG 1)
8. G 1:
G 1× siG 1= s 1 G 1 ×siG 1 = s 1 siG 1 = siG 1,
siG 1 ×G 1 = siG 1 ×s 1 G 1 = sis 1 G 1 = siG 1.
9. siG 1 si 1 G 1,
siG 1× si 1 G 1 = si × si 1 G 1 = eG 1 = G 1,
si 1 G 1× siG 1 = si 1× siG 1 = eG 1 = G 1.
S 3/ G 1 - S 3 G 1.
S 3/ G 1 = < G 1, s 4 G 1>.
7. , | a | = n ak = 1, n k.
.
G n. , n , an = 1. k - , k n,
k = nq + r, 0 £ r < n,
ak = (an) q × ar = ar.
, n ak = 1, n k.
8. , | G | = pq, p ¹ q , G .
.
p, q ,. p - q - G, , . { a }, { b } p - q- . G 1 + kq pq, q - { b } . , G, .
|
|
9. 1 , . , 1Ç I .
.
D = 1Ç I. , , - , - . ¹0, . , = 0 I. 0Î , 0Î , 0Î D, D ¹Æ.
, b Î D, , b Î , b Î . , .
Î D, b Î 1. , ab ba D. D Í , , - Î D Í - b Î 1 Í K , ab, ba Î .
, ab, ba Î 1Ç I = D. D = 1Ç I 1.
10. , j K 1 K 2 j(a b) = j(a) j(b).
.
j K 1 K 2.
" a, ( b)Î 1
j(a + ( b) = j(a) + j( b), j( b) = j(b).
, j(a + ( b) = j(a b) = j(a) j(b), .
11. , , .
.
, . me ¹0 m, .
me =0 m , , m . m ¹1, ¹0. m , .
m . s t , 1 < s, t < m m = st. K ,
0 = me = (st) × e = (se) × (te).
, m s < m, t < m, se ¹0, te ¹0, (se) × (te) = me ¹0, , . , .
12. , 4 .
.
. , , a, b, c, d Î Z.
()() = 1.
. (*)
Nr () = .
г (*) , . (**) г (**), , .
, : 1, 1.
, 4 : .
, 2, , 2, 2, . , , 2, . , , .
4 : 4=2×2 =1×4.
, b 2 < 1, b =0. a 2 = 2, a. , .
, .
, . , 2 . , , . , 4 .
13. , , .
.
p = st, s < p, t < p.
(se)×(te) = 0.
, (se)×(te) = 0 , se = 0, te = 0, , . , , , .
|
|