Математика для юристов - Д.А. Ловцова
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141
ПРИЛОЖЕНИЕ
ȼ ɫɥɭɱɚɟ ɡɥɨɫɬɧɨɝɨ ɭɤɥɨɧɟɧɢɹ ɨɬ ɨɬɛɵɜɚɧɢɹ ɧɚɤɚɡɚɧɢɹ ɥɢɰɨɦ, ɨɫɭɠɞɟɧɧɵɦ ɤ ɢɫɩɪɚɜɢɬɟɥɶɧɵɦ ɪɚɛɨɬɚɦ, ɫɭɞ ɦɨɠɟɬ ɡɚɦɟɧɢɬɶ ɧɟɨɬɛɵɬɨɟ ɧɚɤɚɡɚɧɢɟ ɨɝɪɚɧɢɱɟɧɢɟɦ ɫɜɨɛɨɞɵ, ɚɪɟɫɬɨɦ ɢɥɢ ɥɢɲɟɧɢɟɦ ɫɜɨɛɨɞɵ ɢɡ ɪɚɫɱɟɬɚ ɨɞɢɧ ɞɟɧɶ ɨɝɪɚɧɢɱɟɧɢɹ ɫɜɨɛɨɞɵ ɡɚ ɨɞɢɧ ɞɟɧɶ ɢɫɩɪɚɜɢɬɟɥɶɧɵɯ ɪɚɛɨɬ, ɨɞɢɧ ɞɟɧɶ ɚɪɟɫɬɚ ɡɚ ɞɜɚ ɞɧɹ ɢɫɩɪɚɜɢɬɟɥɶɧɵɯ ɪɚɛɨɬ ɢɥɢ ɨɞɢɧ ɞɟɧɶ ɥɢɲɟɧɢɹ ɫɜɨɛɨɞɵ ɡɚ ɬɪɢ ɞɧɹ ɢɫɩɪɚɜɢɬɟɥɶɧɵɯ
ɪɚɛɨɬ.
(ɍɄ ɊɎ, ɋɬ. 50, ɩ. 4)
ТЕСТЫ
Ɍɟɫɬ [ɚɧɝɥ. test – ɤɨɧɬɪɨɥɶɧɚɹ ɪɚɛɨɬɚ, ɩɪɨɜɟɪɤɚ, ɷɤɡɚɦɟɧ] – ɡɚɞɚɱɚ, ɤɨɬɨɪɭɸ ɩɪɟɞɥɚɝɚɟɬɫɹ ɪɟɲɢɬɶ ɨɛɭɱɚɟɦɨɦɭ ɞɥɹ ɨɰɟɧɤɢ ɟɝɨ ɡɧɚɧɢɣ ɢ ɭɦɟɧɢɣ ɩɭɬɟɦ ɫɪɚɜɧɟɧɢɹ ɩɨɥɭɱɟɧɧɵɯ ɢɦ ɪɟɡɭɥɶɬɚɬɨɜ ɫ ɢɡɜɟɫɬɧɵɦ ɪɟɲɟɧɢɟɦ (ɫ ɷɬɚɥɨɧɨɦ).
Ʉɚɠɞɵɣ ɬɟɫɬ ɩɨ ɞɚɧɧɨɣ ɞɢɫɰɢɩɥɢɧɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɸ ɤɨɧɬɪɨɥɶɧɨɟ ɞɨɦɚɲɧɟɟ ɡɚɞɚɧɢɟ ɄȾɁ, ɤɨɬɨɪɨɟ ɫɬɭɞɟɧɬ ɜɵɩɨɥɧɹɟɬ ɜ ɱɚɫɵ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɣ ɪɚɛɨɬɵ ɩɨɫɥɟ ɢɡɭɱɟɧɢɹ ɨɱɟɪɟɞɧɨɣ ɬɟɦɵ ɭɱɟɛɧɨɣ ɩɪɨɝɪɚɦɦɵ ɩɨ ɞɢɫɰɢɩɥɢɧɟ.
ɋɬɭɞɟɧɬ ɜɵɩɨɥɧɹɟɬ ɜɚɪɢɚɧɬ ɧɨɦɟɪ W ɤɚɠɞɨɝɨ ɬɟɫɬɚ:
W (ɇȽ V)mod 20,
ɇȽ – ɧɨɦɟɪ ɝɪɭɩɩɵ ɫɬɭɞɟɧɬɚ,
V – ɧɨɦɟɪ ɪɚɛɨɱɟɝɨ ɦɟɫɬɚ ɫɬɭɞɟɧɬɚ ɨɱɧɨɣ ɮɨɪɦɵ ɨɛɭɱɟɧɢɹ ɜ ɤɨɦɩɶɸɬɟɪɧɨɦ ɤɥɚɫɫɟ ɢɥɢ ɞɜɟ ɦɥɚɞɲɢɟ ɰɢɮɪɵ ɧɨɦɟɪɚ ɫɬɭɞɟɧɱɟɫɤɨɝɨ ɛɢɥɟɬɚ ɫɬɭɞɟɧɬɚ ɞɪɭɝɢɯ ɮɨɪɦ ɨɛɭɱɟɧɢɹ.
ɇɚɩɨɦɧɢɦ, ɱɬɨ (A)mod M – ɨɫɬɚɬɨɤ ɨɬ ɞɟɥɟɧɢɹ ɱɢɫɥɚ Ⱥ ɧɚ ɦɨɞɭɥɶ M ɧɚɰɟɥɨ.
Ɉɬɱɟɬ ɩɨ ɄȾɁ ɨɮɨɪɦɥɹɟɬɫɹ ɩɨ ɨɛɪɚɡɰɭ, ɤɨɬɨɪɵɣ ɩɪɢɜɟɞɟɧ ɧɢɠɟ.
Ɉɬɱɟɬ ɞɨɥɠɟɧ ɫɨɞɟɪɠɚɬɶ:
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ɪɚɫɱɟɬ ɡɧɚɱɟɧɢɹ W, ɭɫɥɨɜɢɟ ɡɚɞɚɱɢ, ɪɟɲɟɧɢɟ.
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɞɨɥɠɧɨ ɫɨɩɪɨɜɨɠɞɚɬɶɫɹ ɧɟɨɛɯɨɞɢɦɵɦɢ ɩɨɹɫɧɟɧɢɹɦɢ, ɢɥɥɸɫɬɪɚɰɢɹɦɢ, ɫɫɵɥɤɚɦɢ ɧɚ ɮɨɪɦɭɥɵ, ɚɤɫɢɨɦɵ, ɬɟɨɪɟɦɵ ɢ ɬ.ɩ. ɢɡ ɭɱɟɛɧɢɤɚ, ɤɨɧɫɩɟɤɬɚ ɥɟɤɰɢɣ.
Ⱥɤɤɭɪɚɬɧɨ ɨɮɨɪɦɥɟɧɧɵɣ (ɩɭɫɬɶ ɢ ɨɬ ɪɭɤɢ) ɨɬɱɟɬ ɩɨ ɄȾɁ ɩɪɟɞɨɫɬɚɜɥɹɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɸ ɞɥɹ ɨɰɟɧɤɢ.
143
Ɉ Ȼ Ɋ Ⱥ Ɂ ȿ ɐ
Ʉɚɮɟɞɪɚ ɩɪɚɜɨɜɨɣ ɢɧɮɨɪɦɚɬɢɤɢ, ɢɧɮɨɪɦɚɰɢɨɧɧɨɝɨ ɩɪɚɜɚ ɢ ɦɚɬɟɦɚɬɢɤɢ
ɄɈɇɌɊɈɅɖɇɈȿ ȾɈɆȺɒɇȿȿ ɁȺȾȺɇɂȿ
ɩɨ ɞɢɫɰɢɩɥɢɧɟ
ɂɇɎɈɊɆȺɌɂɄȺ ɂ ɆȺɌȿɆȺɌɂɄȺ
Ɍɟɫɬ 4. ɎɍɇɄɐɂɂ
Ⱥ. Ⱥɩɩɪɨɤɫɢɦɚɰɢɹ ɬɚɛɥɢɱɧɵɯ ɮɭɧɤɰɢɣ
ȼɵɩɨɥɧɢɥ ɫɬɭɞɟɧɬ ɝɪɭɩɩɵ 9
ȼ. ɑɚɩɚɟɜ
« 15 » ɨɤɬɹɛɪɹ 2008 ɝ.
144
ɉɄ 28.
Ɂ ɚ ɞ ɚ ɱ ɚ
ɉɨ ɡɚɞɚɧɧɨɦɭ ɜɵɪɚɠɟɧɢɸ f(x) ɩɨɫɬɪɨɢɬɶ ɧɚ ɨɬɪɟɡɤɟ [a,b] ɬɚɛɥɢɰɭ ɨɛɴɟɦɨɦ n.
ȼɵɩɨɥɧɢɬɶ ɥɢɧɟɣɧɭɸ ɚɩɩɪɨɤɫɢɦɚɰɢɸ ɷɬɨɣ ɬɚɛɥɢɱɧɨɣ ɮɭɧɤɰɢɢ ɢ ɧɚɣɬɢ ɩɪɢɛɥɢɠɟɧɧɵɟ ɡɧɚɱɟɧɢɹ y ɩɨ ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɢɦ ɮɨɪɦɭɥɚɦ ɢ
ɬɨɱɧɵɟ ɡɧɚɱɟɧɢɹ yT ɞɥɹ x |
{x1,x2}. |
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Ɉɰɟɧɢɬɶ ɩɨɝɪɟɲɧɨɫɬɶ ɚɩɩɪɨɤɫɢɦɚɰɢɢ. |
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(ɬɚɛɥ.1): |
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Ɂɚɩɢɫɵɜɚɟɦ ɮɨɪɦɭɥɵ ɥɢɧɟɣɧɨɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɬɚɛɥɢɱɧɨɣ |
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ɮɭɧɤɰɢɢ: |
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ɋɬɪɨɢɦ ɝɪɚɮɢɤɢ ɬɚɛɥɢɱɧɨɣ ɮɭɧɤɰɢɢ {xti,yti} ɢ ɡɚɞɚɧɧɨɣ ɮɭɧɤɰɢɢ f(x). Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɝɪɚɮɢɤɚ f(x) ɞɨɩɨɥɧɢɬɟɥɶɧɨ ɜɵɱɢɫɥɢɦ ɟɟ ɡɧɚɱɟ-
ɧɢɹ ɞɥɹ x {1, 3, 5, 7 9}: f(1) 16, f(3) 12, f(5) 5, f(7) 1.75, f(9) 9 #0.56. 16
Ɉɬɦɟɱɚɟɦ ɷɬɢ ɡɧɚɱɟɧɢɹ ɧɚ ɝɪɚɮɢɤɟ. ɋɨɟɞɢɧɹɟɦ ɷɬɢ ɬɨɱɤɢ ɢ ɬɨɱɤɢ, ɡɚɞɚɧɧɵɟ ɬɚɛɥ. 1, ɩɥɚɜɧɨɣ ɩɭɧɤɬɢɪɧɨɣ ɥɢɧɢɟɣ. Ɍɨɱɤɢ ɬɚɛɥɢɱɧɨɣ ɮɭɧɤɰɢɢ {xti,yti} ɫɨɟɞɢɧɹɟɦ ɨɬɪɟɡɤɚɦɢ ɩɪɹɦɵɯ (4.2). Ɉɬɦɟɱɚɟɦ ɤɪɢɜɭɸ
ɧɚɞɩɢɫɶɸ f(x), ɚ ɨɬɪɟɡɤɢ ɩɪɹɦɵɯ – ɧɚɞɩɢɫɹɦɢ Mk(x), k 0,3 . Ɋɟɡɭɥɶɬɚɬ – ɧɚ ɪɢɫ. 1.
145
ɉɨɥɚɝɚɟɦ x x1 1. Ʉɚɤ ɜɢɞɢɦ (ɫɦ. ɪɢɫ.1 ɢɥɢ ɬɚɛɥ. 1), xt0 x1 xt1. |
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Ɂɧɚɱɢɬ, ɩɨ ɮɨɪɦɭɥɟ (4.3) k1 0 ɢ ɩɭɬɟɦ ɢɧɬɟɪɩɨɥɹɰɢɢ ɧɚɯɨɞɢɦ ɩɪɢ- |
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ɛɥɢɠɟɧɧɨɟ ɡɧɚɱɟɧɢɟ y1 ɞɥɹ x1: |
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y1T f(x1) xu2 x1 5 32u1u2 1 16. |
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ɇɚɧɨɫɢɦ ɬɨɱɤɭ y1 ɧɚ ɩɪɹɦɭɸ M0(x), ɚ ɬɨɱɤɭ y1T ɧɚ ɤɪɢɜɭɸ f(x) |
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(ɪɢɫ. 1). |
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ɉɨɝɪɟɲɧɨɫɬɶ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɫɨɫɬɚɜɢɬ |
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y1T y1 u100% |
16 8 u100% |
50%. |
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y1T |
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ɬɟɦ, ɱɬɨ ɧɚ ɨɬɪɟɡɤɟ [0,2] ɩɪɹɦɚɹ M0(x) ɫɢɥɶɧɨ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɤɪɢɜɨɣ |
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f(x), ɢ ɩɪɢ x x1 1 ɪɚɡɧɨɫɬɶ f(x1) M0(x1) y1T y1 ɩɨɱɬɢ ɦɚɤɫɢɦɚɥɶɧɚ.
Ⱦɥɹ x x2 10 ɢɦɟɟɬ ɦɟɫɬɨ x2!xt3, ɢ k2 3. ɗɤɫɬɪɚɩɨɥɹɰɢɹ ɜɩɟɪɟɞ:
y2 M3(x2) yt3 yt4 yt3 u(x2 xt3) xt4 xt3
3 1 3 u(10 6) 1.
8 6
Ɍɨɱɧɨɟ ɡɧɚɱɟɧɢɟ y2T ɞɥɹ x2 ɪɚɜɧɨ
y2T f(x2) xu2 x2 5 32u10u2 10 320 0.31. 1024
ɇɚɧɨɫɢɦ ɬɨɱɤɭ y2 ɧɚ ɩɪɹɦɭɸ M3(x), ɚ ɬɨɱɤɭ y2T ɧɚ ɤɪɢɜɭɸ f(x) (ɪɢɫ. 1).
ɉɨɝɪɟɲɧɨɫɬɶ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɫɨɫɬɚɜɢɬ
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Ʉɚɤ ɜɢɞɢɦ, ɷɤɫɬɪɚɩɨɥɹɰɢɹ ɞɚɥɟɤɨ ɡɚ ɩɪɟɞɟɥɵ ɬɚɛɥɢɰɵ ɧɟɰɟɥɟɫɨɨɛɪɚɡɧɚ ɢɡ-ɡɚ ɨɝɪɨɦɧɵɯ ɩɨɝɪɟɲɧɨɫɬɟɣ.
Ⱦɥɹ ɭɦɟɧɶɲɟɧɢɹ ɩɨɝɪɟɲɧɨɫɬɟɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɧɭɠɧɨ ɭɜɟɥɢɱɢɜɚɬɶ ɨɛɴɟɦ ɬɚɛɥɢɰɵ n, ɬɨ ɟɫɬɶ ɭɦɟɧɶɲɚɬɶ ɟɟ ɲɚɝ h.
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Тест 1. Элементы теории множеств
Ɂ ɚ ɞ ɚ ɱ ɚ . Ɂɚɞɚɧɵ ɦɧɨɠɟɫɬɜɚ Ⱥ, ȼ, ɋ, : (ɪɢɫ. 1). Ɂɚɲɬɪɢɯɨɜɚɬɶ ɨɛɥɚɫɬɶ Q, ɡɚɞɚɧɧɭɸ ɮɨɪɦɭɥɨɣ ɚɥɝɟɛɪɵ ɦɧɨɠɟɫɬɜ ɧɨɦɟɪ W.
0.Q º(A B C) ((A B) (A C) (B C))º(A B C).
1.Q º(A B C)º(Aº(B C)).
2. |
Q º(A B C)º((C B)ºA). |
3. |
Q (A C B) (ºAºBºC). |
4. |
Q º(Bº(A C)º(AºB C). |
5. |
Q º(A C)º(B C) (A B C). |
6. |
C |
Q º(B C)º(Aº(B C)). |
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7. |
Q º((A C)ºB)º(ºC B). |
8. |
Q º(Aº(B C))º(ºA (B C)). |
9. |
Q º(Aº(B C))º(Bº(A C))º(C |
º(B A)). |
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10. |
Q º(A B C)º(Bº(A C)). |
11. |
Q º(A B C)º((A C)ºB). |
12. |
Q (A B C) (ºAºBºC). |
13. |
Q º(Cº(A B)º(A BºC). |
14. |
Q º(A B)º(B C) (A B C). |
15. |
Q º(A C)º(Bº(A C)). |
16. |
Q º(ºA B)º(ºA C B)). |
17. |
Q º(Bº(A C))º(ºB (A C)). |
18. |
Q º(A B C)ºA B C. |
19. |
Q (A B C)º(A B)º(A C)º(B C). |
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Ɋɢɫ. 1 |
Ɇɟɬɨɞɢɱɟɫɤɢɟ ɭɤɚɡɚɧɢɹ.
Ɋɟɤɨɦɟɧɞɭɟɦ ɫɧɚɱɚɥɚ, ɢɫɩɨɥɶɡɭɹ ɬɨɠɞɟɫɬɜɚ 10 ɢ 10’, ɚ ɬɚɤɠɟ ɮɨɪɦɭɥɭ (1.1), ɜ ɡɚɞɚɧɧɨɦ ɜɵɪɚɠɟɧɢɢ ɞɥɹ Q ɜɵɩɨɥɧɢɬɶ ɡɚɦɟɧɵ ɜɢɞɚ
ºRºS º(R S), ºRºS º(R S), RºS R\S,
ɬɨ ɟɫɬɶ ɩɪɟɨɛɪɚɡɨɜɚɬɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ Q ɤ ɜɢɞɭ, ɛɨɥɟɟ ɭɞɨɛɧɨɦɭ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɢɫɤɨɦɨɝɨ ɪɟɡɭɥɶɬɚɬɚ.
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ɉ ɪ ɢ ɦ ɟ ɪ . Ɂɚɞɚɧɵ ɦɧɨɠɟɫɬɜɚ Ⱥ, ȼ, ɋ, : (ɪɢɫ. 1). Ɂɚɲɬɪɢɯɨɜɚɬɶ ɨɛɥɚɫɬɶ Q, ɡɚɞɚɧɧɭɸ ɮɨɪɦɭɥɨɣ ɚɥɝɟɛɪɵ ɦɧɨɠɟɫɬɜ ɧɨɦɟɪ W 20.
20.Q º(Bº(A C))º(Cº(A B))º(A B C) ¢10, (1.1)² º(B\(A C) (C\(A B)) (A B C).
A B C |
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B\(A C) C\(A B)
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Тест 2. Элементы математической логики
Ɂ ɚ ɞ ɚ ɱ ɚ . ɂɫɩɨɥɶɡɭɹ ɬɨɠɞɟɫɬɜɚ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɥɨɝɢɤɢ, ɞɨɤɚɡɚɬɶ ɢɫɬɢɧɧɨɫɬɶ ɜɵɫɤɚɡɵɜɚɧɢɹ S ɧɨɦɟɪ W.
0.S ((A B)o(BoA))o((A B)o(A ºB)).
1.S (º((AoB)o(A B)) (BoA)oº(A B)).
2.S (º(A ºB) ºA)o(ºA B).
3.S ((A B ºA ºB) (ºA B) (A ºB)).
4.S ((ºB (ºA B))oºA).
5.((BoA)o(A B) (AoB) (A B)).
6.S (AoB)o((A ºB)oºA).
7.S ((A B)o(A B) º((AoB)o(ºA B))).
8.S º(AoB) º(ºBoºA) (ºA B).
9.S (º((A B)oº(BoA)) (AoB)o(A B)).
10.S (º(ºA ºB) º(A B)) (BoA).
11.S (º((BoA)o(A B)) (AoB)oº(A B)).
12.S ((A ºB)oB)o(AoB).
13.S (º(A B) (AoºB)).
14.S (ºA B)o((A ºB)oB).
15.S (º(ºAoB) (AoºB)).
16.S (BoA)o(A B) (AoB)oº(A B).
17.S º((AoB)o(A B)) º((BoA)o(ºA ºB)).
18.S ((A B)oº(BoA) º((AoB)o(A B))).
19.S ((AoB) º(A B))oº((BoA) (A B)).
Ɇɟɬɨɞɢɱɟɫɤɢɟ ɭɤɚɡɚɧɢɹ.
ɋɧɚɱɚɥɚ ɪɟɤɨɦɟɧɞɭɟɦ ɩɨɫɬɪɨɢɬɶ ɬɚɛɥɢɰɭ ɢɫɬɢɧɧɨɫɬɢ ɞɥɹ ɜɵɫɤɚɡɵɜɚɧɢɹ S ɧɨɦɟɪ W ɢ ɭɛɟɞɢɬɶɫɹ ɜ ɬɨɦ, ɱɬɨ ɩɪɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ ɜɵɫɤɚɡɵɜɚɧɢɣ A ɢ B ɡɧɚɱɟɧɢɟ ɢɫɯɨɞɧɨɝɨ ɭɬɜɟɪɠɞɟɧɢɹ S ɢɫɬɢɧɧɨ (ɪɚɜɧɨ ɟɞɢɧɢɰɟ). Ɍɨɝɞɚ ɢɫɬɢɧɧɨɫɬɶ ɜɵɫɤɚɡɵɜɚɧɢɹ S ɦɨɠɧɨ ɞɨɤɚɡɚɬɶ ɚɧɚɥɢɬɢɱɟɫɤɢ.
Ⱥɧɚɥɢɬɢɱɟɫɤɨɟ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ ɢɫɬɢɧɧɨɫɬɢ ɜɵɫɤɚɡɵɜɚɧɢɹ S ɧɨɦɟɪ W ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨɛɵ, ɢɫɩɨɥɶɡɭɹ ɬɨɠɞɟɫɬɜɚ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɥɨɝɢɤɢ, ɭɩɪɨɫɬɢɬɶ ɢɫɯɨɞɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɨ ɟɞɢɧɢɰɵ. Ⱥ ɞɥɹ ɷɬɨɝɨ ɫɥɟ-
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