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Файл:Элементы программирования при решении математических задач. Учебное пособие
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ɈȻɈɁɇȺɑȿɇɂə ɂ ɈɉɊȿȾȿɅȿɇɂə
Ⱥɧɚɥɨɝɢɱɧɨ, ɞɥɹ Yȼ ɦɧɨɠɟɫɬɜɨ f
-–1
(Y) = {aA: f(a) = b} – ɩɪɨɨɛɪɚɡ Y
ɮɭɧɤɰɢɢ f.
ȿɫɥɢ f(Ⱥ) = ȼ, ɬɨ ɬɚɤɨɟ ɨɬɨɛɪɚɠɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɮɭɧɤɰɢɟɣ ɢɡ Ⱥ ɧɚ ȼ ɢɥɢ
ɫɸɪɴɟɤɬɢɜɧɵɦ ɨɬɨɛɪɚɠɟɧɢɟɦ ɢɥɢ ɫɸɪɴɟɤɬɢɜɧɨɣ ɮɭɧɤɰɢɟɣ.
ȿɫɥɢ ɞɥɹ ɥɸɛɵɯ a ɢ bA
a z b f(a) z f(b),
o
ɬɨ ɬɚɤɚɹ ɮɭɧɤɰɢɹ f: Ⱥ
ȼ ɧɚɡɵɜɚɟɬɫɹ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɨɣ.
11
ɉɈɇəɌɂȿ ɆɈȾȿɅɂ ȼɕɑɂɋɅȿɇɂɃ
Ƚɥɚɜɧɵɣ ɦɟɬɨɞɨɥɨɝɢɱɟɫɤɢɣ ɜɨɩɪɨɫ, ɫ ɤɨɬɨɪɵɦ ɧɟɢɡɛɟɠɧɨ ɫɬɚɥɤɢɜɚɟɲɶɫɹ ɩɪɟɠɞɟ ɜɫɟɝɨ ɩɪɢ ɥɸɛɨɦ ɢɫɫɥɟɞɨɜɚɧɢɢ ɚɥɝɨɪɢɬɦɨɜ, ɡɚɤɥɸɱɚɟɬɫɹ
ɜ ɬɳɚɬɟɥɶɧɨɦ ɨɩɢɫɚɧɢɢ ɩɪɢɧɹɬɨɣ ɦɨɞɟɥɢ ɜɵɱɢɫɥɟɧɢɣ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɚɥɝɨɪɢɬɦ, ɩɪɟɞɥɨɠɟɧɧɵɣ ɞɥɹ ɪɟɲɟɧɢɹ ɨɩɪɟɞɟɥɟɧɧɨɣ ɡɚɞɚɱɢ, ɞɨɥɠɟɧ ɛɵɬɶ ɨɰɟɧɟɧ
ɜ ɬɟɪɦɢɧɚɯ ɟɝɨ ɫɬɨɢɦɨɫɬɢ ɤɚɤ ɧɟɤɨɣ ɮɭɧɤɰɢɢ ɨɬ ɪɚɡɦɟɪɚ ɢɧɞɢɜɢɞɭɚɥɶɧɨɝɨ ɷɤɡɟɦɩɥɹɪɚ ɷɬɨɣ ɡɚɞɚɱɢ. ɉɪɢɧɰɢɩɢɚɥɶɧɚɹ
ɜɚɠɧɨɫɬɶ ɦɨɞɟɥɢ ɜɵɱɢɫɥɟɧɢɣ ɨɩɪɟ-
ɞɟɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: Ɇɨɞɟɥɶ ɜɵɱɢɫɥɟɧɢɣ ɨɩɪɟɞɟɥɹɟɬ ɧɚɛɨɪ ɞɨɩɭɫ-
ɬɢɦɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɨɩɟɪɚɰɢɣ ɢ ɫɬɨɢɦɨɫɬɢ ɷɬɢɯ ɨɩɟɪɚɰɢɣ.
ɗɥɟɦɟɧɬɚɪɧɵɟ ɨɩɟɪɚɰɢɢ – ɷɬɨ ɬɚɤɢɟ ɨɩɟɪɚɰɢɢ, ɞɥɹ ɤɚɠɞɨɣ ɢɡ ɤɨɬɨɪɵɯ
ɦɵ ɧɚɡɧɚɱɚɟɦ ɮɢɤɫɢɪɨɜɚɧɧɭɸ ɫɬɨɢɦɨɫɬɶ, ɯɨɬɹ ɷɬɚ ɫɬɨɢɦɨɫɬɶ ɧɟɨɞɢɧɚɤɨɜɚ ɞɥɹ
ɪɚɡɧɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɨɩɟɪɚɰɢɣ. ɇɚɩɪɢɦɟɪ, ɟɫɥɢ ɷɥɟɦɟɧɬɚɪɧɵɟ ɨɩɟɪɚɰɢɢ ɨɛɪɚɛɚɬɵɜɚɸɬ ɨɬɞɟɥɶɧɵɟ ɰɢɮɪɵ ɜ ɱɢɫɥɚɯ (
ɤɚɤ ɜ ɛɭɥɟɜɫɤɢɯ ɮɭɧɤɰɢɹɯ ɫ ɞɜɭɦɹ
ɞɜɨɢɱɧɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ, ɜɵɱɢɫɥɹɟɦɵɦɢ ɧɚ ɦɨɞɟɥɢ ɦɚɲɢɧɵ Ɍɶɸɪɢɧɝɚ), ɬɨ
ɫɬɨɢɦɨɫɬɶ ɨɩɟɪɚɰɢɢ ɫɥɨɠɟɧɢɹ ɞɜɭɯ ɰɟɥɵɯ ɱɢɫɟɥ ɜɨɡɪɚɫɬɚɟɬ ɫ ɪɨɫɬɨɦ ɞɥɢɧɵ
ɨɩɟɪɚɧɞɨɜ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɫɬɨɢɦɨɫɬɶ ɷɬɨɣ ɨɩɟɪɚɰɢɢ ɩɨɫɬɨɹɧɧɚ ɞɥɹ ɦɨɞɟɥɢ,
ɜ ɤɨɬɨɪɨɣ ɨɩɟɪɚɧɞɵ ɢɦɟɸɬ ɮɢɤɫɢɪɨɜɚɧɧɭɸ ɞɥɢɧɭ (ɤɚɤ ɜ ɥɸɛɨɣ ɦɨɞɟɥɢ, ɨɪɢɟɧɬɢɪɨɜɚɧɧɨɣ ɧɚ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɪɟɚɥɶɧɵɯ ɰɢɮɪɨɜɵɯ
ɤɨɦɩɶɸɬɟɪɨɜ). ɉɪɢ ɜɵɛɨɪɟ ɦɨɞɟɥɢ ɩɪɢɯɨɞɢɬɫɹ ɢɞɬɢ ɧɚ ɤɨɦɩɪɨɦɢɫɫɵ ɦɟɠɞɭ ɪɟɚɥɶɧɨɫɬɶɸ
ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɫɬɪɨɝɨɫɬɶɸ, ɜɵɛɢɪɚɹ ɫɯɟɦɭ, ɤɨɬɨɪɚɹ ɨɬɪɚɠɚɟɬ ɨɫɧɨɜɧɵɟ
ɱɟɪɬɵ ɢɫɩɨɥɶɡɭɟɦɵɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɟɬɨɞɨɜ ɢ ɜɦɟɫɬɟ ɫ ɬɟɦ ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɫɬɚ, ɱɬɨɛɵ ɩɪɨɜɟɫɬɢ ɬɳɚɬɟɥɶɧɵɣ ɚɧɚɥɢɡ. ɇɚɩɪɢɦɟɪ, ɱɚɫɬɨ, ɩɪɢ ɪɟɲɟɧɢɢ ɦɧɨɝɢɯ ɡɚɞɚɱ ɫɥɟɞɭɸɳɢɟ ɨɩɟɪɚɰɢɢ ɫɱɢɬɚɸɬɫɹ ɷɥɟɦɟɧɬɚɪɧɵɦɢ ɢ ɨɛɥɚɞɚɸɳɢɦɢ
ɟɞɢɧɢɱɧɨɣ ɫɬɨɢɦɨɫɬɶɸ (ɟɞɢɧɢɱɧɵɦ
ɜɪɟɦɟɧɟɦ ɜɵɩɨɥɧɟɧɢɹ):
1. Ⱥɪɢɮɦɟɬɢɱɟɫɤɢɟ ɨɩɟɪɚɰɢɢ (+, –, ×, /).
2. ɋɪɚɜɧɟɧɢɹ ɞɜɭɯ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ (<, >, =, , , ).
3. Ʉɨɫɜɟɧɧɚɹ ɚɞɪɟɫɚɰɢɹ ɩɚɦɹɬɢ (ɞɨɩɭɫɬɢɦɵ ɬɨɥɶɤɨ ɰɟɥɨɱɢɫɥɟɧɧɵɟ ɚɞɪɟɫɚ).
4. Ʉɨɪɧɢ k-ɣ ɫɬɟɩɟɧɢ, ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ, EXP, LOG (ɜɨɨɛɳɟ
ɚɧɚɥɢɬɢɱɟɫɤɢɟ ɜɵɪɚɠɟɧɢɹ).
Ɍɚɤɢɟ ɦɨɞɟɥɢ ɧɚɡɵɜɚɸɬ ɜɟɳɟɫɬɜɟɧɧɨɡɧɚɱɧɵɦɢ. Ɉɧɢ ɯɨɪɨɲɨ ɨɬɪɚɠɚɸɬ
ɬɢɩɵ ɩɪɨɝɪɚɦɦ, ɤɨɬɨɪɵɟ ɨɛɵɱɧɨ ɩɢɲɭɬɫɹ ɧɚ ɚɥɝɨɪɢɬɦɢɱɟɫɤɢɯ ɹɡɵɤɚɯ ɜɵɫɨɤɨɝɨ ɭɪɨɜɧɹ, ɝɞɟ ɩɪɢɧɹɬɨ
ɫɱɢɬɚɬɶ, ɱɬɨ ɩɟɪɟɦɟɧɧɵɟ ɬɢɩɚ REAL ɢɦɟɸɬ ɧɟɨɝɪɚ-
ɧɢɱɟɧɧɭɸ ɬɨɱɧɨɫɬɶ.
Ɉɞɧɚɤɨ ɫɬɨɢɦɨɫɬɶ ɨɩɟɪɚɰɢɣ ɦɨɠɧɨ ɧɚɡɧɚɱɚɬɶ. Ⱦɥɹ ɩɪɢɦɟɪɚ ɪɚɫɫɦɨɬɪɢɦ
ɡɚɞɚɱɭ. ɉɭɫɬɶ ɜ ɫɥɟɞɭɸɳɟɣ ɩɪɨɝɪɚɦɦɟ, ɤɨɬɨɪɚɹ ɧɚɯɨɞɢɬ ɦɢɧɢɦɚɥɶɧɵɣ ɷɥɟɦɟɧɬ n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ {a
, …, an}={1, 2, …, n}:
1
Ⱥɥɝɨɪɢɬɦ 6.
m=a[1]; for(i=2; i<=n; i++) if(a[i]<m)m=a[i]; return m
ɨɩɟɪɚɰɢɹ ɩɪɢɫɜɚɢɜɚɧɢɹ (=) ɢɦɟɟɬ ɟɞɢɧɢɱɧɭɸ ɫɬɨɢɦɨɫɬɶ. Ɍɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɱɢɫɥɨ N
ɬɚɤɢɯ ɦɧɨɠɟɫɬɜ, ɫɬɨɢɦɨɫɬɶ ɨɛɪɚɛɨɬɤɢ ɤɨɬɨɪɵɯ ɫɨɫɬɚɜɥɹɟɬ k
k
(k = 1, 2, …, n) ɟɞɢɧɢɰ ɢ ɫɪɟɞɧɸɸ ɫɬɨɢɦɨɫɬɶ ɷɬɨɣ ɩɪɨɝɪɚɦɦɵ, ɬɨ ɟɫɬɶ ɦɚɬɟɦɚ-
12
ɉɈɇəɌɂȿ ɆɈȾȿɅɂ ȼɕɑɂɋɅȿɇɂɃ
n
kSkPMS
ɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ
¦
k
1
ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɦɧɨɠɟɫɬɜɚ {a
N
k
kSP
.
!)(n
)(
, ɝɞɟ P(S = k) – ɜɟɪɨɹɬɧɨɫɬɶ ɨɛɪɚɛɨɬɤɢ
, …, an} ɢɦɟɟɬ ɫɬɨɢɦɨɫɬɶ S = k, ɬɨ ɟɫɬɶ
1
Ɋɟɲɟɧɢɟ ɷɬɨɣ ɡɚɞɚɱɢ ɛɭɞɟɬ ɩɪɢɜɟɞɟɧɨ ɜ ɪɚɡɞɟɥɟ «Ⱥɥɝɟɛɪɚ».
ɂɧɵɦɢ ɫɥɨɜɚɦɢ, ɨɫɧɨɜɧɵɦ ɩɚɪɚɦɟɬɪɨɦ ɚɥɝɨɪɢɬɦɚ, ɤɨɬɨɪɵɣ ɧɚɫ ɢɧɬɟɪɟɫɭɟɬ, ɹɜɥɹɟɬɫɹ ɟɝɨ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɫɥɨɠɧɨɫɬɶ (ɢɥɢ ɩɪɨɫɬɨ ɫɥɨɠɧɨɫɬɶ),
ɬɨ ɟɫɬɶ ɱɢɫɥɨ ɲɚɝɨɜ, ɜɵɩɨɥɧɹɟɦɵɯ ɚɥɝɨɪɢɬɦɨɦ ɜ ɯɭɞɲɟɦ ɫɥɭɱɚɟ ɤɚɤ ɮɭɧɤɰɢɹ
ɪɚɡɦɟɪɧɨɫɬɢ ɡɚɞɚɱɢ, ɩɪɟɞɫɬɚɜɥɟɧɧɨɣ ɜɯɨɞɧɵɦɢ ɞɚɧɧɵɦɢ. ɇɚɩɪɢɦɟɪ, ɟɫɥɢ ɚɥɝɨɪɢɬɦ ɩɪɢɧɢɦɚɟɬ ɤɚɤ ɞɚɧɧɵɟ ɩɪɨɢɡɜɨɥɶɧɵɣ ɝɪɚɮ <V,
E>, ɬɨ ɩɨɞ ɪɚɡɦɟɪɧɨɫɬɶɸ ɡɚɞɚɱɢ ɦɨɠɧɨ ɩɪɢɧɢɦɚɬɶ ɱɢɫɥɨ ɟɝɨ ɜɟɪɲɢɧ |V |. ɋɥɨɠɧɨɫɬɶ ɚɥɝɨɪɢɬɦɚ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɝɞɚ ɤɚɤ ɮɭɧɤɰɢɹ f , ɬɚɤɚɹ, ɱɬɨ f(n) ɪɚɜɧɨ ɧɚɢɛɨɥɶɲɟɦɭ ɱɢɫɥɭ
ɲɚɝɨɜ ɚɥɝɨɪɢɬɦɚ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɝɪɚɮɚ ɫ n ɜɟɪɲɢɧɚɦɢ. Ɇɨɠɧɨ ɬɚɤɠɟ ɫɱɢɬɚɬɶ ɪɚɡɦɟɪɧɨɫɬɶɸ ɡɚɞɚɱɢ ɩɚɪɭ <|V|, |E|> – ɬɨɝɞɚ ɫɥɨɠɧɨɫɬɶɸ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ ɢ
f(n,m) ɪɚɜɧɨ ɧɚɢɛɨɥɶɲɟɦɭ ɱɢɫɥɭ ɲɚɝɨɜ, ɜɵɩɨɥɧɹɟɦɵɯ
ɚɥɝɨɪɢɬɦɨɦ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɝɪɚɮɚ ɫ n ɜɟɪɲɢɧɚɦɢ ɢ m ɪɟɛɪɚɦɢ.
ȼ ɞɚɥɶɧɟɣɲɟɦ ɱɚɳɟ ɧɚɫ ɧɟ ɛɭɞɟɬ ɢɧɬɟɪɟɫɨɜɚɬɶ ɬɨɱɧɚɹ ɫɥɨɠɧɨɫɬɶ ɚɥɝɨɪɢɬɦɚ, ɚ ɬɨɥɶɤɨ ɟɝɨ ɚɫɢɦɩɬɨɬɢɱɟɫɤɚɹ ɫɥɨɠɧɨɫɬɶ, ɬɨ ɟɫɬɶ ɚɫɢɦɩɬɨɬɢɱɟɫɤɚɹ
ɫɤɨɪɨɫɬɶ ɭɜɟɥɢɱɟɧɢɹ ɲɚɝɨɜ ɚɥɝɨɪɢɬɦɚ, ɤɨɝɞɚ ɪɚɡɦɟɪɧɨɫɬɶ ɡɚɞɚɱɢ ɧɟɨɝɪɚɧɢɱɟɧɧɨ ɪɚɫɬɟɬ (ɱɬɨɛɵ ɦɨɠɧɨ ɛɵɥɨ
ɝɨɜɨɪɢɬɶ ɨ ɬɚɤɨɣ ɫɤɨɪɨɫɬɢ ɪɨɫɬɚ, ɩɪɟɞɩɨɥɚɝɚɟɦ, ɱɬɨ ɨɛɴɟɦ ɩɚɦɹɬɢ ɧɚɲɟɝɨ ɤɨɦɩɶɸɬɟɪɚ ɧɟɨɝɪɚɧɢɱɟɧɧɵɣ, ɚ ɬɚɤɠɟ, ɱɬɨ
ɤɚɠɞɚɹ ɹɱɟɣɤɚ ɩɚɦɹɬɢ ɦɨɠɟɬ ɫɨɞɟɪɠɚɬɶ ɩɪɨɢɡɜɨɥɶɧɨ ɛɨɥɶɲɨɟ ɰɟɥɨɟ ɱɢɫɥɨ).
ɉɪɢ ɫɪɚɜɧɟɧɢɢ ɫɤɨɪɨɫɬɢ ɪɨɫɬɚ ɞɜɭɯ ɮɭɧɤɰɢɣ f(n) ɢ g(n)
(ɫ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ) ɨɱɟɧɶ ɭɞɨɛɧɵ ɫɥɟɞɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ:
f(n) = Ɉ(g(n))
ɋ ɢ N (ɤɨɧɫɬɚɧɬɵ) ɬɚɤɢɟ, ɱɬɨ f(n) ɋg(n), n N,
f(n) = ȍ(g(n)) ɋ ɢ N (ɤɨɧɫɬɚɧɬɵ) ɬɚɤɢɟ, ɱɬɨ f(n) ɋg(n), n N.
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɟɫɥɢ f(n)=Ɉ(g(n)), ɬɨ g(n)=ȍ(f(n
)). ɋɢɦɜɨɥɵ Ɉ(g(n))
ɢ ȍ(f(n)) ɱɢɬɚɸɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ: «ɩɨɪɹɞɤɚ ɧɟ ɛɨɥɟɟ ɱɟɦ g(n)» ɢ «ɩɨɪɹɞɤɚ ɧɟ
ɦɟɧɟɟ ɱɟɦ f(n)». ȿɫɥɢ ɫɥɨɠɧɨɫɬɶ ɤɚɤɨɝɨ-ɥɢɛɨ ɚɥɝɨɪɢɬɦɚ ɟɫɬɶ Ɉ(g(n)), ɬɨ ɝɨɜɨɪɢɦ, ɱɬɨ ɷɬɨɬ ɚɥɝɨɪɢɬɦ «ɡɚɬɪɚɱɢɜɚɟɬ ɩɨɪɹɞɤɚ Ɉ(g(n)) ɜɪɟɦɟɧɢ».
ȼ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɦ ɪɚɡɞɟɥɟ ɪɚɫɫɦɨɬɪɢɦ ɦɨɞɟɥɢ, ɤɨɬɨɪɵɟ
ɩɨɞɯɨɞɹɬ
ɞɥɹ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɩɪɢɥɨɠɟɧɢɣ. ȼ ɱɚɫɬɧɨɫɬɢ, ɧɟɤɨɬɨɪɵɟ ɡɚɞɚɱɢ, ɨɬɧɨɫɹɳɢɟɫɹ ɤ ɞɢɫɰɢɩɥɢɧɟ, ɢɡɜɟɫɬɧɨɣ ɧɵɧɟ ɤɚɤ ɜɵɱɢɫɥɢɬɟɥɶɧɚɹ ɝɟɨɦɟɬɪɢɹ.
ɍ ɤɨɥɵɛɟɥɢ ɷɬɨɣ ɞɢɫɰɢɩɥɢɧɵ ɫɬɨɹɥɨ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɩɪɢɥɨɠɟɧɢɣ, ɢɛɨ
ɨɧɢ ɩɨɪɨɠɞɚɸɬ ɫɜɨɣɫɬɜɟɧɧɵɟ ɢɦ ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ ɡɚɞɚɱɢ, ɞɥɹ ɤɨɬɨɪɵɯ
ɞɨɥɠɧɵ ɫɨɡɞɚɜɚɬɶɫɹ ɷɮɮɟɤɬɢɜɧɵɟ ɚɥɝɨɪɢɬɦɵ. Ʉ ɱɢɫɥɭ ɬɚɤɢɯ ɡɚɞɚɱ ɨɬɧɨɫɹɬɫɹ: ɡɚɞɚɱɚ ɤɨɦɦɢɜɨɹɠɟɪɚ ɧɚ ɟɜɤɥɢɞɨɜɨɣ ɦɟɬɪɢɤɟ, ɩɨɫɬɪɨɟɧɢɟ ɦɢɧɢ
ɦɚɥɶɧɨɝɨ ɨɫɬɨɜɧɨɝɨ ɞɟɪɟɜɚ, ɭɞɚɥɟɧɢɟ ɧɟɜɢɞɢɦɵɯ ɥɢɧɢɣ, ɥɢɧɟɣɧɨɟ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɟ ɢ ɦɧɨɝɢɟ ɞɪɭɝɢɟ.
-
13
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
Ɂɚɞɚɱɢ, ɜɨɡɧɢɤɚɸɳɢɟ ɜ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɝɟɨɦɟɬɪɢɢ, ɦɨɠɧɨ ɫɝɪɭɩɩɢɪɨ-
ɜɚɬɶ ɜ ɬɪɢ ɫɥɟɞɭɸɳɢɯ ɤɥɚɫɫɚ:
1. ɉɨɢɫɤ ɩɨɞɦɧɨɠɟɫɬɜɚ. ȼ ɡɚɞɚɱɚɯ ɷɬɨɝɨ ɪɨɞɚ ɡɚɞɚɧ ɧɚɛɨɪ ɨɛɴɟɤɬɨɜ
ɢ ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɧɟɤɨɟ ɩɨɞɦɧɨɠɟɫɬɜɨ, ɤɨɬɨɪɨɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɨɩɪɟɞɟɥɟɧɧɨɦɭ ɭɫɥɨɜɢɸ. ɇɚɩɪɢɦɟɪ, ɩɨɢɫɤ ɛɥɢɠɚɣɲɟɣ ɩɚɪɵ ɬɨɱɟɤ ɧɚ ɩɥɨɫɤɨɫɬɢ ɢɡ N ɬɨɱɟɤ ɢɥɢ ɩɨɢɫɤ ɜɟɪɲɢɧ ɜɵɩɭɤɥɨɣ ɨɛɨɥɨɱɤɢ ɦɧɨɠɟɫɬɜɚ ɢ ɬ
ȼɚɠɧɚɹ ɱɟɪɬɚ ɡɚɞɚɱ ɨ ɜɵɛɨɪɟ ɩɨɞɦɧɨɠɟɫɬɜɚ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɧɟ
ɧɭɠɧɨ ɫɨɡɞɚɜɚɬɶ ɧɢɤɚɤɢɯ ɧɨɜɵɯ ɨɛɴɟɤɬɨɜ; ɪɟɲɟɧɢɟ ɩɨɥɧɨɫɬɶɸ ɫɨɫɬɨɢɬ
ɢɡ ɷɥɟɦɟɧɬɨɜ, ɤɨɬɨɪɵɟ ɡɚɞɚɧɵ ɧɚ ɜɯɨɞɟ.
2. ȼɵɱɢɫɥɟɧɢɟ. Ⱦɚɧɨ ɦɧɨɠɟɫɬɜɨ ɨɛɴɟɤɬɨɜ, ɧɭɠɧɨ ɜɵɱɢɫɥɢɬɶ ɜɟɥɢɱɢɧɭ ɧɟɤɨɟɝɨ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ ɩɚɪɚɦɟɬɪɚ ɧɚ ɷɬɨɦ ɦɧɨɠɟɫɬɜɟ. Ⱦɨɩɭɫɬɢɦɵɟ ɷɥɟɦɟɧɬɚɪɧɵɟ ɨɩɟɪɚɰɢɢ ɜ ɦɨɞɟɥɢ ɞɨɥɠɧɵ ɛɵɬɶ ɞɨɫɬɚɬɨɱɧɨ
ɦɨɳɧɵɦɢ, ɱɬɨɛɵ ɪɟɚɥɢɡɨɜɚɬɶ ɷɬɨ ɜɵɱɢɫɥɟɧɢɟ, ɧɚɩɪɢɦɟɪ, ɜɵɱɢɫɥɟɧɢɹ ɢɪɪɚɰɢɨɧɚɥɶɧɵɯ ɱɢɫɟɥ ɢ ɡɧɚɱɟɧɢɣ ɚɧɚɥɢɬɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ.
3. Ɋɚɫɩɨɡɧɚɜɚɧɢɟ. Ɂɚɞɚɱɚ ɪɚɫɩɨɡɧɚɜɚɧɢɹ ɟɫɬɟɫɬɜɟɧɧɵɦ ɨɛɪɚɡɨɦ ɫɜɹɡɚɧɚ
ɫ ɡɚɞɚɱɚɦɢ «ɉɨɢɫɤ ɩɨɞɦɧɨɠɟɫɬɜɚ» ɢ «ȼɵɱɢɫɥɟɧɢɟ». ȼ ɱɚɫɬɧɨɫɬɢ,
(1) ȿɫɥɢ ɜ ɡɚɞɚɱɟ S ɨ ɜɵɱɢɫɥɟɧɢɢ ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɜɟɥɢɱɢɧɭ ɩɚɪɚɦɟɬɪɚ s,
ɬɨ ɜ ɫɜɹɡɚɧɧɨɣ ɫ ɧɟɣ ɡɚɞɚɱɟ ɪɚɫɩɨɡɧɚɜɚɧɢɹ D(S
ȾȺ/ɇȿɌ ɧɚ ɜɨɩɪɨɫ ɬɢɩɚ: «ȼɟɪɧɨ ɥɢ, ɱɬɨ s s
) ɬɪɟɛɭɟɬɫɹ ɞɚɬɶ ɨɬɜɟɬ:
», ɝɞɟ s0 – ɤɨɧɫɬɚɧɬɚ.
0
(2) ȿɫɥɢ ɜ ɡɚɞɚɱɟ S ɨ ɩɨɢɫɤɟ ɩɨɞɦɧɨɠɟɫɬɜɚ ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɩɨɞɦɧɨɠɟ-
ɫɬɜɨ ɡɚɞɚɧɧɨɝɨ ɦɧɨɠɟɫɬɜɚ Ɇ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɨɩɪɟɞɟɥɟɧɧɨɦɭ
ɫɜɨɣɫɬɜɭ Ɋ, ɬɨ ɜ ɡɚɞɚɱɟ ɪɚɫɩɨɡɧɚɜɚɧɢɹ D(S) ɬɪɟɛɭɟɬɫɹ ɞɚɬɶ ɨɬɜɟɬ:
ȾȺ/ɇȿɌ ɧɚ ɜɨɩɪɨɫ ɬɢɩɚ: «ȼɟɪɧɨ ɥɢ, ɱɬɨ Ɇ
ɩɨɞɦɧɨɠɟɫɬɜɨ Ɇ.
Ɇ
1 –
ɭɞɨɜɥɟɬɜɨɪɹɟɬ Ɋ», ɝɞɟ
1
.ɩ.
14
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
§ 1. ɇɚɱɚɥɶɧɵɟ ɫɜɟɞɟɧɢɹ ɢɡ ɤɨɦɛɢɧɚɬɨɪɢɤɢ
ȼ ɷɬɨɦ ɪɚɡɞɟɥɟ ɧɚɩɨɦɧɢɦ ɨɩɪɟɞɟɥɟɧɢɹ, ɤɚɫɚɸɳɢɟɫɹ ɭɤɚɡɚɧɧɵɯ ɦɚɬɟɦɚ-
ɬɢɱɟɫɤɢɯ ɞɢɫɰɢɩɥɢɧ. ɇɚɱɧɟɦ ɫ ɤɨɦɛɢɧɚɬɨɪɢɤɢ.
Ʉɨɦɛɢɧɚɬɨɪɢɤɚ – ɨɞɢɧ ɢɡ ɪɚɡɞɟɥɨɜ ɞɢɫɤɪɟɬɧɨɣ ɦɚɬɟɦɚɬɢɤɢ, ɢɦɟɟɬ ɜɚɠ-
ɧɨɟ ɡɧɚɱɟɧɢɟ ɜ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ, ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɥɨɝɢɤɟ, ɬɟɨɪɢɢ ɱɢɫɟɥ,
ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɬɟɯɧɢɤɟ, ɤɢɛɟɪɧɟɬɢɤɟ. ɋ ɤɨɦɛɢɧɚɬɨɪɧɵɦɢ ɡɚɞɚɱɚɦɢ ɩɪɢɯɨɞɢɬɫɹ ɢɦɟɬɶ ɞɟɥɨ
ɮɢɡɢɤɚɦ, ɯɢɦɢɤɚɦ, ɛɢɨɥɨɝɚɦ, ɥɢɧɝɜɢɫɬɚɦ, ɫɩɟɰɢɚɥɢɫɬɚɦ ɩɨ
ɬɟɨɪɢɢ ɤɨɞɨɜ. ȼ ɨɫɧɨɜɧɨɦ ɷɬɨ ɧɚɭɤɚ ɨ ɤɨɧɟɱɧɵɯ ɦɧɨɠɟɫɬɜɚɯ.
Ʉɨɦɛɢɧɚɬɨɪɢɤɚ ɧɚɱɢɧɚɟɬɫɹ ɫ ɨɞɧɨɝɨ ɩɪɚɜɢɥɚ, ɤɨɬɨɪɨɟ ɧɚɡɵɜɚɸɬ ɨɫɧɨɜ-
ɧɵɦ ɩɪɚɜɢɥɨɦ ɤɨɦɛɢɧɚɬɨɪɢɤɢ. Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ: «ɢɡ ɩɭɧɤɬɚ Ɉ ɞɨ ɩɭɧɤɬɚ
Ⱥ ɦɨɠɧɨ ɞɨɛɪɚɬɶɫɹ ɬɪɟɦɹ ɫɩɨɫɨɛɚɦɢ: ɫɚɦɨɥɟɬɨɦ, ɩɨɟɡɞɨɦ ɢɥɢ ɚɜɬɨɦɨɛɢɥɟɦ,
ɚ ɢɡ Ⱥ ɜ ɩɭɧɤɬ ȼ – ɞɜɭɦɹ ɫɩɨɫɨɛɚɦɢ:
ɜɟɪɬɨɥɟɬɨɦ ɢɥɢ ɤɚɬɟɪɨɦ. ɋɤɨɥɶɤɨ ɫɭɳɟɫɬɜɭɟɬ ɦɚɪɲɪɭɬɨɜ ɢɡ Ɉ ɜ ȼ?». Ɉɬɜɟɬ ɧɚ ɷɬɭ ɡɚɞɚɱɭ ɨɱɟɜɢɞɟɧ: «ɲɟɫɬɶɸ ɫɩɨɫɨɛɚɦɢ». Ⱥ ɬɟɩɟɪɶ ɫɮɨɪɦɭɥɢɪɭɟɦ ɨɫɧɨɜɧɨɟ ɩɪɚɜɢɥɨ ɤɨɦɛɢɧɚɬɨɪɢɤɢ, ɤɨɬɨɪɨɟ
ɬɚɤɠɟ ɧɚɡɵɜɚɸɬ ɩɪɚɜɢɥɨɦ ɭɦɧɨɠɟɧɢɹ.
ȿɫɥɢ ɜɵɛɨɪ Ⱥ ɦɨɠɧɨ ɨɫɭɳɟɫɬɜɢɬɶ n ɫɩɨɫɨɛɚɦɢ, ɞɥɹ ɤɚɠɞɨɝɨ ɢɡ ɷɬɢɯ
ɫɩɨɫɨɛɨɜ ɞɪɭɝɨɣ ɜɵɛɨɪ ȼ ɦɨɠɧɨ ɨɫɭɳɟɫɬɜɢɬɶ k ɫɩɨɫɨɛɚɦɢ, ɬɨ ɜɵɛɨɪ
ȼ (ɜ ɭɤɚɡɚɧɧɨɦ ɩɨɪɹɞɤɟ) ɦɨɠɧɨ ɨɫɭɳɟɫɬɜɢɬɶ n×k ɫɩɨɫɨɛɚɦɢ.
Ⱥ ɢ
ȼ ɛɨɥɟɟ ɨɛɳɟɦ ɜɢɞɟ ɷɬɨ ɩɪɚɜɢɥɨ ɡɜɭɱɢɬ ɬɚɤ:
ɉɭɫɬɶ ɬɪɟɛɭɟɬɫɹ ɜɵɩɨɥɧɢɬɶ ɨɞɧɨ ɡɚ ɞɪɭɝɢɦ k ɞɟɣɫɬɜɢɣ. ȿɫɥɢ ɩɟɪɜɨɟ
ɞɟɣɫɬɜɢɟ ɦɨɠɧɨ ɜɵɩɨɥɧɢɬɶ n
ɞɟɣɫɬɜɢɟ – n
ɩɨɥɧɢɬɶ n
ɫɩɨɫɨɛɚɦɢ ɢ ɬɚɤ ɞɚɥɟɟ ɞɨ k-ɝɨ ɞɟɣɫɬɜɢɹ, ɤɨɬɨɪɨɟ ɦɨɠɧɨ ɜɵ-
3
ɫɩɨɫɨɛɚɦɢ, ɬɨ ɜɫɟ k ɞɟɣɫɬɜɢɣ ɜɦɟɫɬɟ ɦɨɝɭɬ ɛɵɬɶ ɜɵɩɨɥɧɟɧɵ
k
ɫɩɨɫɨɛɚɦɢ, ɜɬɨɪɨɟ – n2 ɫɩɨɫɨɛɚɦɢ, ɬɪɟɬɶɟ
1
u
n
u
. . . u nk
n
1
2
ɫɩɨɫɨɛɚɦɢ.
ɉɪɢɜɟɞɟɦ ɞɥɹ ɩɪɢɦɟɪɚ ɧɟɫɤɨɥɶɤɨ ɩɪɨɫɬɵɯ ɡɚɞɚɱ, ɞɨɫɬɭɩɧɵɯ ɲɤɨɥɶɧɢɤɭ
ɫɪɟɞɧɢɯ ɤɥɚɫɫɨɜ, ɡɧɚɤɨɦɨɦɭ ɫ ɨɫɧɨɜɧɵɦ ɩɪɚɜɢɥɨɦ ɤɨɦɛɢɧɚɬɨɪɢɤɢ.
ɁȺȾȺɑɂ
1. ɋɤɨɥɶɤɨ ɬɪɟɯɡɧɚɱɧɵɯ ɱɢɫɟɥ ɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ ɢɡ ɰɢɮɪ 1, 2, 3, 4, 5?
2. ɋɤɨɥɶɤɨ ɬɪɟɯɡɧɚɱɧɵɯ ɱɢɫɟɥ ɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ ɢɡ ɰɢɮɪ 1, 2, 3, 4, 5, ɟɫɥɢ
ɤɚɠɞɭɸ ɢɡ ɷɬɢɯ ɰɢɮɪ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɧɟ ɛɨɥɟɟ ɨɞɧɨɝɨ ɪɚɡɚ?
3. ɋɤɨɥɶɤɨ ɢɦɟɟɬɫɹ ɩɹɬɢɡɧɚɱɧɵɯ ɱɢɫɟɥ, ɤɨɬɨɪɵɟ
ɞɟɥɹɬɫɹ ɧɚ 5?
4. ɇɚ ɨɞɧɨɣ ɢɡ ɛɨɤɨɜɵɯ ɫɬɨɪɨɧ ɬɪɟɭɝɨɥɶɧɢɤɚ ɜɡɹɬɨ n ɬɨɱɟɤ, ɧɚ ɞɪɭɝɨɣ – m
ɬɨɱɟɤ. Ʉɚɠɞɚɹ ɢɡ ɜɟɪɲɢɧ ɩɪɢ ɨɫɧɨɜɚɧɢɢ ɬɪɟɭɝɨɥɶɧɢɤɚ ɫɨɟɞɢɧɟɧɚ ɩɪɹɦɵɦɢ ɫ ɬɨɱɤɚɦɢ, ɜɡɹɬɵɦɢ ɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɣ ɫɬɨɪɨɧɟ.
ɚ) ɋɤɨɥɶɤɨ ɬɨɱɟɤ ɩɟɪɟɫɟɱɟɧɢɹ ɷɬɢɯ ɩɪɹɦɵɯ ɨɛɪɚɡɭɟɬɫɹ ɜɧɭɬɪɢ
ɬɪɟɭɝɨɥɶɧɢɤɚ?
ɛ) ɇɚ ɫɤɨɥɶɤɨ ɱɚɫɬɟɣ ɞɟɥɹɬ ɬɪɟɭɝɨɥɶɧɢɤ ɷɬɢ ɩɪɹɦɵɟ?
ɋɤɨɥɶɤɨ ɟɫɬɶ ɩɹɬɢɡɧɚɱɧɵɯ ɱɢɫɟɥ, ɤɨɬɨɪɵɟ ɨɞɢɧɚɤɨɜɨ ɱɢɬɚɸɬɫɹ ɫɥɟɜɚ
5.
ɧɚɩɪɚɜɨ ɢ ɫɩɪɚɜɚ ɧɚɥɟɜɨ (ɧɚɩɪɢɦɟɪ 23 732, 45 654, ... )?
15
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
A
A
A
6. 5 ɦɚɥɶɱɢɤɨɜ ɢ 5 ɞɟɜɨɱɟɤ ɫɚɞɹɬɫɹ ɜ ɪɹɞ ɧɚ 10 ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɩɨɞɪɹɞ
ɫɬɭɥɶɟɜ, ɩɪɢɱɟɦ ɦɚɥɶɱɢɤɢ ɫɚɞɹɬɫɹ ɧɚ ɦɟɫɬɚ ɫ ɧɟɱɟɬɧɵɦɢ ɧɨɦɟɪɚɦɢ,
ɚ ɞɟɜɨɱɤɢ – ɧɚ ɦɟɫɬɚ ɫ ɱɟɬɧɵɦɢ ɧɨɦɟɪɚɦɢ. ɋɤɨɥɶɤɢɦɢ ɫɩɨɫɨɛɚɦɢ ɷɬɨ
ɦɨɠɧɨ ɫɞɟɥɚɬɶ?
7. ȼ ɫɟɥɟɧɢɢ ɠɢɜɭɬ 1500 ɠɢɬɟɥɟɣ. Ⱦɨɤɚɡɚɬɶ, ɱɬɨ, ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ, ɞɜɨɟ ɢɡ
ɧɢɯ ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɵɟ ɢɧɢɰɢɚɥɵ.
8. ɚ) ɋɤɨɥɶɤɨ
ɛ) ɉɭɫɬɶ p
ɪɚɡɧɵɯ ɞɟɥɢɬɟɥɟɣ ɢɦɟɟɬ ɱɢɫɥɨ 35 u 54?
, p2, . . . , pn – ɪɚɡɥɢɱɧɵɟ ɩɪɨɫɬɵɟ ɱɢɫɥɚ. ɋɤɨɥɶɤɨ
1
ɞɟɥɢɬɟɥɟɣ ɢɦɟɟɬ ɱɢɫɥɨ
mm
21
21
m
n
pppm ...
n
,
, m2 , . . . , mn – ɧɟɤɨɬɨɪɵɟ ɧɚɬɭɪɚɥɶɧɵɟ ɱɢɫɥɚ?
ɝɞɟ m
1
9. Ɉɬ Ⱥ ɞɨ ȼ 999 ɤɦ. ȼɞɨɥɶ ɞɨɪɨɝɢ ɫɬɨɹɬ ɫɬɨɥɛɵ, ɧɚ ɤɨɬɨɪɵɯ ɭɤɚɡɚɧɵ ɪɚɫɫɬɨɹɧɢɹ ɞɨ Ⱥ ɢ ɞɨ ȼ:
0; 999 1; 998 2; 997 . . . 998; 1 999; 0
ɋɤɨɥɶɤɨ ɫɪɟɞɢ ɧɢɯ ɬɚɤɢɯ, ɧɚ ɤɨɬɨɪɵɯ ɢɦɟɸɬɫɹ ɬɨɥɶɤɨ 2 ɪɚɡɥɢɱɧɵɟ
ɰɢɮɪɵ?
ȼ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɬɚɛɥɢɰɟ ɢɡ m ɫɬɪɨɤ ɢ n ɫɬɨɥɛɰɨɜ ɡɚɩɢɫɚɧɵ ɱɢɫɥɚ +1
ɤɚɠɞɨɣ ɫɬɪɨɤɟ ɢ ɤɚɠɞɨɦ ɫɬɨɥɛɰɟ ɪɚɜɧɨ 1.
ɢ –1 ɬɚɤ, ɱɬɨ ɩɪɨɢɡɜɟɞɟɧɢɟ ɱɢɫɟɥ ɜ
ɋɤɨɥɶɤɢɦɢ ɫɩɨɫɨɛɚɦɢ ɷɬɨ ɦɨɠɧɨ ɫɞɟɥɚɬɶ?
ɗɬɢ ɡɚɞɚɱɢ ɥɟɝɤɨ ɩɪɨɜɟɪɹɸɬɫɹ ɤɨɦɩɶɸɬɟɪɨɦ, ɧɚɩɪɢɦɟɪ, ɩɪɢɦɟɧɢɜ
ɜ ɡɚɞɚɱɟ 9 ɦɟɬɨɞ split ɤ ɫɬɪɨɤɚɦ g – ɧɚɛɨɪɚɦ ɰɢɮɪ ɪɚɫɫɬɨɹɧɢɣ ɧɚ ɫɬɨɥɛɚɯ:
(''+g.split(''+i)). split(''+(9–i)).
ɉɟɪɟɣɞɟɦ ɬɟɩɟɪɶ ɤ ɩɪɟɞɦɟɬɭ, ɢɡɭɱɚɟɦɨɦɭ ɤɨɦɛɢɧɚɬɨɪɢɤɨɣ, – ɦɧɨɠɟɫɬɜɚɦ.
Ɉɩɪɟɞɟɥɟɧɧɵɟ ɜɵɲɟ ɨɩɟɪɚɰɢɢ ɧɚɞ ɦɧɨɠɟɫɬɜɚɦɢ ɨɛɥɚɞɚɸɬ ɫɥɟɞɭɸɳɢ-
ɦɢ ɫɜɨɣɫɬɜɚɦɢ, ɤɨɬɨɪɵɟ ɜɚɦ ɧɚɞɨ ɨɛɹɡɚɬɟɥɶɧɨ ɞɨɤɚɡɚɬɶ.
ɉɨɥɨɠɢɦ
: – ɨɛɴɟ-
ɞɢɧɟɧɢɟ ɜɫɟɯ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɦɧɨɠɟɫɬɜ Ⱥ, ȼ, ɋ, … (ɭɧɢɜɟɪɫɚɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ). Ȼɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ
ɟɫɬɶ ɷɬɨ ɬɚɤɨɟ ɦɧɨɠɟɫɬɜɨ, ɱɬɨ
ɞɨɩɨɥɧɟɧɢɟ ɦɧɨɠɟɫɬɜɚ Ⱥ ɞɨ ɦɧɨɠɟɫɬɜɚ :, ɬɨ
:
ɢɥɢ
def
AAaaA
\}:{ : :
,
Ⱥ\ȼ – ɪɚɡɧɨɫɬɶ ɦɧɨɠɟɫɬɜ:
ɢ Ⱥ¨ȼ – ɫɢɦɦɟɬɪɢɱɟɫɤɚɹ ɪɚɡɧɨɫɬɶ ɦɧɨɠɟɫɬɜ:
}:{\ BaAaBABA
)\()\( ABBABA '
.
16
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ABA
ABA
Ⱦɨɤɚɠɢɬɟ ɫɥɟɞɭɸɳɢɟ ɫɜɨɣɫɬɜɚ:
1. Ʉɨɦɦɭɬɚɬɢɜɧɨɫɬɶ ɨɛɴɟɞɢɧɟɧɢɹ: AB = BA.
2. Ʉɨɦɦɭɬɚɬɢɜɧɨɫɬɶ ɩɟɪɟɫɟɱɟɧɢɹ: BA = AB.
3. Ⱥɫɫɨɰɢɚɬɢɜɧɨɫɬɶ ɨɛɴɟɞɢɧɟɧɢɹ: A(BC) = (AB)C.
4. Ⱥɫɫɨɰɢɚɬɢɜɧɨɫɬɶ ɩɟɪɟɫɟɱɟɧɢɹ: Ⱥ(BC) = (AB)C.
5. Ⱦɢɫɬɪɢɛɭɬɢɜɧɨɫɬɶ ɨɛɴɟɞɢɧɟɧɢɹ: (AB)C = (AC)(BC)
.
6. Ⱦɢɫɬɪɢɛɭɬɢɜɧɨɫɬɶ ɩɟɪɟɫɟɱɟɧɢɹ: A(BC) = (AB)(AC).
7. ɋɜɨɣɫɬɜɚ ɩɭɫɬɨɝɨ ɦɧɨɠɟɫɬɜɚ: A = A; A = .
8. ɋɜɨɣɫɬɜɚ ɭɧɢɜɟɪɫɚɥɶɧɨɝɨ ɦɧɨɠɟɫɬɜɚ: A: = :; A: = A.
9. Ɂɚɤɨɧɵ ɞɟ Ɇɨɪɝɚɧɚ:
B
B
.
.
:: : ,,,, AAAAAA
10.
Ʉɪɨɦɟ ɬɨɝɨ, ɧɚɞɨ ɭɦɟɬɶ ɞɨɤɚɡɵɜɚɬɶ ɫɥɟɞɭɸɳɢɟ ɭɬɜɟɪɠɞɟɧɢɹ:
ɚ) A(AB) = A; A(AB) = A; AA = AA = A;
ɛ) (AB)\B = A\B; A(CB) = A\(A\B)(A\C);
ɜ) (A\B)C = (AB) \ (BC); ABC = A \ (
A\(BC));
ɝ) (AC) \ B = (AC) \ (BC); (A \ B)(A \ C) = A \ (BC);
ɞ) (A \ B = ) (AB = A);
ɟ) A \ B = AB;
f) Ⱥ¨ȼ = B¨A = (AB)\(AB).
Ⱦɨɤɚɠɟɦ ɞɥɹ ɩɪɢɦɟɪɚ ɬɨɠɞɟɫɬɜɨ 5 (ɞɢɫɬɪɢɛɭɬɢɜɧɨɫɬɶ ɩɟɪɟɫɟɱɟɧɢɹ).
ɉɭɫɬɶ ɯA(
BC), ɬɨɝɞɚ ɯ ɞɨɥɠɟɧ ɹɜɥɹɬɶɫɹ ɷɥɟɦɟɧɬɨɦ Ⱥ ɢ ɨɞɧɨɜɪɟɦɟɧɧɨ
ɛɵɬɶ ɷɥɟɦɟɧɬɨɦ BC, ɬɨ ɟɫɬɶ ɩɪɢɧɚɞɥɟɠɚɬɶ ȼ ɢɥɢ ɋ. Ɂɧɚɱɢɬ, ɯAB ɢɥɢ
ɯAC, ɬɨ ɟɫɬɶ ɯ(AB)(AC). ɋɟɣɱɚɫ ɦɵ ɞɨɤɚɡɚɥɢ, ɱɬɨ
A(BC)(AB)(AC). Ɉɛɪɚɬɧɨ
, ɩɭɫɬɶ ɯ(AB)(AC), ɬɨɝɞɚ ɯAB ɢɥɢ
ɯAC, ɬɨ ɟɫɬɶ ɯ ɹɜɥɹɟɬɫɹ ɨɛɳɢɦ ɷɥɟɦɟɧɬɨɦ Ⱥ ɢ ȼ ɢɥɢ ɨɛɳɢɦ ɷɥɟɦɟɧɬɨɦ
Ⱥ ɢ ɋ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɯA ɢ ɩɪɢɧɚɞɥɟɠɢɬ ɨɞɧɨɦɭ ɢɡ ɦɧɨɠɟɫɬɜ ȼ ɢɥɢ ɋ, ɬɨ
ɟɫɬɶ ɯA(BC) ɢ (A
B)(AC)A(BC).
ɋɭɳɟɫɬɜɭɸɬ ɢ ɞɪɭɝɢɟ ɫɩɨɫɨɛɵ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɩɨɞɨɛɧɵɯ ɫɨɨɬɧɨɲɟɧɢɣ.
Ɉɞɧɢɦ ɢɡ ɜɚɠɧɵɯ ɩɪɢɧɰɢɩɨɜ ɜ ɬɟɨɪɢɢ ɦɧɨɠɟɫɬɜ ɹɜɥɹɟɬɫɹ ɩɪɢɧɰɢɩ
ɜɤɥɸɱɟɧɢɹ ɢ ɢɫɤɥɸɱɟɧɢɹ. ȿɝɨ ɦɨɠɧɨ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɜ ɜɢɞɟ ɪɚɜɟɧɫɬɜɚ:
| AB| = |A| + |B| – |AB|. (1)
Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɪɚɜɟɧɫɬɜɚ ɫɧɚɱɚɥɚ ɡɚɦɟɬɢɦ, ɱɬɨ ɟɫɥɢ Aȼ = ,
ɬɨ
| AB| = |A| + |B|. (2)
17
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
A
A
BA
Ɋɚɫɫɦɨɬɪɢɦ ɦɧɨɠɟɫɬɜɚ:
ɢɦɟɟɦ
Ⱥɧɚɥɨɝɢɱɧɨ
Ɉɬɫɸɞɚ
BA
Ɇɧɨɠɟɫɬɜɚ:
ɢɥɢ
Ⱥɧɚɥɨɝɢɱɧɨ
ɢ
Ɋɚɜɟɧɫɬɜɨ (1) ɞɨɤɚɡɚɧɨ.
ɉɪɨɞɨɥɠɢɦ ɧɚɲɢ ɪɚɫɫɭɠɞɟɧɢɹ. Ɋɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ ɬɪɟɯ ɦɧɨɠɟɫɬɜ Ⱥ,
ȼ ɢ ɋ. Ʉɚɤ ɦɵ ɡɧɚɟɦ
B
,
B
,
ɢ AB ɧɟ ɩɟɪɟɫɟɤɚɸɬɫɹ, ɩɨɷɬɨɦɭ ɜ ɫɢɥɭ (2)
ɢ AB. ȼ ɫɢɥɭ ɫɜɨɣɫɬɜ 3, 10 ɢ ɚ)
BBAAB )()(
.
|||| |)()(| BABABABA
.
|||| |||)()(| || BAABABABABA
|||||||)()(||| BABBABAABAB
.|||||| ||||||||||
BABABABABBAA
,CCBBAA :
ɡɧɚɱɢɬ,
.ABCABCACBACBBCA
ABBAABABBAABABA )(())(()()()(
.
BABAABBABAABBA )())()(()()()(
.
|||||| |)()()(| ||
BAABBABAABBABA
BCACBACBACCBBAA
:
18
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
Ɉɬɫɸɞɚ
||||||||||
BCACBACBAABC
:
.||||||||
ABCACBACBBCA
ȼɵɪɚɡɢɦ ɞɨɩɨɥɧɟɧɢɹ
CBA ,,
, ɧɚɩɪɢɦɟɪ,
|||||)\(||| CBACBCBACBA :
.
Ɍɨɝɞɚ ɩɨɥɭɱɢɦ ɟɳɟ ɨɞɧɭ ɮɨɪɦɭɥɭ ɜɤɥɸɱɟɧɢɹ ɢ ɢɫɤɥɸɱɟɧɢɹ:
|||||||||||||||||| CBACBCABACBAABC :
(3)
Ⱦɨɤɚɠɢɬɟ ɷɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ, ɚ ɡɚɨɞɧɨ ɢ ɟɳɟ ɨɞɧɭ ɩɪɨ-
ɫɬɭɸ, ɧɨ ɩɨɥɟɡɧɭɸ ɮɨɪɦɭɥɭ:
)( DCBADCBA :
.
ɉɪɢɜɟɞɟɦ ɡɞɟɫɶ ɬɚɤɠɟ ɧɟɫɤɨɥɶɤɨ ɩɪɨɫɬɵɯ ɡɚɞɚɱ ɩɨ ɷɬɨɣ ɬɟɦɟ.
ɁȺȾȺɑɂ
11. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ
|ABˁ|=|A|+|B|+|C|–|AB|–|AC|–|CB|+|ABC|.
12. Ⱦɨɤɚɡɚɬɶ, ɱɬɨ
ɚ) |AB| + |
Aɋ| + |ɋB| |A| + |ȼ| + |ɋ| – |Aȼɋ|;
b) |AB| + |Aɋ| – |ɋB| |A|.
13. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɟɫɥɢ ɩɪɢɡɧɚɤ Ⱥ ɜɫɬɪɟɱɚɟɬɫɹ ɱɚɳɟ, ɤɨɝɞɚ ɟɫɬɶ ɩɪɢɡɧɚɤ ȼ,
ɱɟɦ ɤɨɝɞɚ ɟɝɨ ɧɟɬ, ɬɨ ȼ ɜɫɬɪɟɬɢɬɫɹ ɫɨɜɦɟɫɬɧɨ ɫ Ⱥ ɜ ɛɨɥɶɲɟɦ ɱɢɫɥɟ ɫɥɭɱɚ-
||||||
ɟɜ, ɱɟɦ ɛɟɡ ɧɟɝɨ. Ɍɨ
ɟɫɬɶ ɞɨɤɚɡɚɬɶ, ɱɬɨ ɟɫɥɢ
BA
A
AB
!
||
A
||||||
.
14. Ⱦɨɤɚɡɚɬɶ, ɱɬɨ ɟɫɥɢ 2|A| = 2|ȼ| = |Aȼɋ|, ɬɨ
15. Ⱦɨɤɚɡɚɬɶ, ɱɬɨ ɟɫɥɢ 2|A| = 2|ȼ| = 2|ɋ| = |A + ȼ + ɋ| ɢ
BA
B
BA
!
, ɬɨ
||
B
.
|||| BABA
|| || CBACBA
ɬɨ 4|AȼC| = 2(|Aȼ| + |AC| + |Cȼ|) – |Aȼɋ|.
16. ȼɟɪɧɨ ɥɢ, ɱɬɨ ɟɫɥɢ |Aȼɋ| = 1000, |Aȼ| = 100, |Aɋ| = 40, |A| = 300,
|ȼ| = 500, |ɋ| = 400 ɢ |AȼC| = 50, ɬɨ |ȼɋ| = 50?
17. Ɂɚɞɚɱɚ-ɲɭɬɤɚ (Ʌɶɸɢɫ Ʉɷɪɪɨɥɥ (Ⱦɨɞɠɫɨɧ), «Ɂɚɩɭɬɚɧɧɚɹ ɫɤɚɡɤɚ», 1881).
ȼ ɨɠɟɫɬɨɱɟɧɧɨɦ ɛɨɸ ɧɟ ɦɟɧɟɟ 70% ɛɨɣɰɨɜ ɩɨɬɟɪɹɥɢ
ɨɞɢɧ ɝɥɚɡ, ɧɟ ɦɟɧɟɟ
75% – ɨɞɧɨ ɭɯɨ, ɧɟ ɦɟɧɟɟ 80% – ɨɞɧɭ ɪɭɤɭ ɢ ɧɟ ɦɟɧɟɟ 85% – ɨɞɧɭ ɧɨɝɭ.
Ʉɚɤɨɜɨ ɦɢɧɢɦɚɥɶɧɨɟ ɱɢɫɥɨ ɩɨɬɟɪɹɜɲɢɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɝɥɚɡ, ɭɯɨ, ɪɭɤɭ
ɢ ɧɨɝɭ?
18. Ⱦɨɤɚɡɚɬɶ, ɱɬɨ ɟɫɥɢ |A| = Nx, |B| =
2Nx, |C| = 3Nx ɢ |Aȼ| = |AC| = |Cȼ| = Ny,
ɝɞɟ N = |Aȼɋ|, ɬɨ ɡɧɚɱɟɧɢɹ x ɢ y ɧɟ ɛɨɥɶɲɟ 1/4.
19. ɂɡ 1000 ɨɛɫɥɟɞɨɜɚɧɧɵɯ ɞɟɜɨɱɟɤ 70 ɢɦɟɥɢ ɧɟɞɨɫɬɚɬɤɢ ɮɢɡɢɱɟɫɤɨɝɨ ɪɚɡ-
ɜɢɬɢɹ (Ⱥ), 90 – ɩɪɢɡɧɚɤɢ ɧɟɪɜɧɨɫɬɢ (ȼ) ɢ 75 – ɭɦɫɬɜɟɧɧɭɸ ɜɹɥɨɫɬɶ (ɋ),
,
19
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ɩɪɢ ɷɬɨɦ |Aȼ| = 25, ɚ |ɋȼ| = 40. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɧɟɤɨɬɨɪɵɟ ɮɢɡɢɱɟɫɤɢ
ɧɟɪɚɡɜɢɬɵɟ ɞɟɜɨɱɤɢ ɧɟ ɹɜɥɹɸɬɫɹ ɭɦɫɬɜɟɧɧɨ ɜɹɥɵɦɢ ɢ ɧɚɣɬɢ ɢɯ ɦɢɧɢɦɚɥɶɧɨɟ ɱɢɫɥɨ.
20. ȼ ɥɢɰɟɟ 87 ɭɱɚɳɢɯɫɹ. ȼ ɜɨɫɤɪɟɫɟɧɶɟ ɜɫɟ ɨɧɢ ɫɯɨɞɢɥɢ ɜ ɤɢɧɨ ɢɥɢ ɧɚ ɮɭɬ-
ɛɨɥ ɢɥɢ ɩɨɫɟɬɢɥɢ ɨɛɚ ɦɟɪɨɩɪɢɹɬɢɹ. ɂɡɜɟɫɬɧɨ, ɱɬɨ
3
ɞɢɜɲɢɯ ɧɚ ɮɭɬɛɨɥ, ɞɟɜɨɱɤɢ, ɢ
ɜɫɟɯ ɭɱɟɧɢɤɨɜ, ɫɯɨɞɢɜɲɢɯ ɜ ɤɢɧɨ, –
7
2
ɜɫɟɯ ɭɱɟɧɢɤɨɜ, ɫɯɨ-
5
ɦɚɥɶɱɢɤɢ. Ʉɚɤɨɟ ɦɢɧɢɦɚɥɶɧɨɟ ɱɢɫɥɨ ɞɟɜɨɱɟɤ ɦɨɠɟɬ ɛɵɬɶ ɭɱɚɳɢɦɢɫɹ
ɥɢɰɟɹ?
21. ɇɚɣɬɢ ɦɧɨɠɟɫɬɜɨ X, ɟɫɥɢ ɢɡɜɟɫɬɧɵ ɦɧɨɠɟɫɬɜɚ A, B ɢ ɋ, ɝɞɟ
ɚ) B = A X, C = AX;
b) B = A \ X, C = AX.
*
ɇɚɩɢɫɚɬɶ ɤɨɦɩɶɸɬɟɪɧɭɸ ɢɝɪɭ ɩɨ ɦɨɬɢɜɚɦ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɢ.
22.
ɉɪɢ ɩɪɢɦɟɧɟɧɢɢ ɜɵɱɢɫɥɢɬɟɥɶɧɵɯ ɭɫɬɪɨɣɫɬɜ ɞɥɹ ɪɟɲɟɧɢɹ ɬɚɤɢɯ ɡɚɞɚɱ,
ɫɧɚɱɚɥɚ ɧɟɨɛɯɨɞɢɦɨ ɨɩɢɫɚɬɶ ɭɧɢɜɟɪɫɚɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ : (ɨɛɴɟɞɢɧɟɧɢɟ ɜɫɟɯ
ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɦɧɨɠɟɫɬɜ). ɉɭɫɬɶ : ɫɨɫɬɨɢɬ ɢɡ n ɷɥɟɦɟɧɬɨɜ (|:| = n), ɬɨɝɞɚ
ɦɨɠɧɨ ɩɪɨɧɭɦɟɪɨɜɚɬɶ ɷɥɟɦɟɧɬɵ : ɢ ɪɚɫɫɦɚɬɪɢɜɚɬɶ : ɤɚɤ ɨɬɪɟɡɨɤ ɪɹɞɚ ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɟɥ {1, 2, ... , n}, ɚ ɩɨɞɦɧɨɠɟɫɬɜɚ
: – ɤɚɤ ɩɨɞɦɧɨɠɟɫɬɜɚ ɦɧɨɠɟɫɬɜɚ
{1, 2, ..., n}. ɑɢɫɥɨ ɜɫɟɯ ɩɨɞɦɧɨɠɟɫɬɜ : ɧɚɣɬɢ ɧɟɫɥɨɠɧɨ – ɞɨɫɬɚɬɨɱɧɨ ɡɚɦɟɬɢɬɶ,
ɱɬɨ ɜɫɟ ɩɨɞɦɧɨɠɟɫɬɜɚ ɦɧɨɠɟɫɬɜɚ {1, 2, ..., n} ɦɨɠɧɨ ɪɚɡɛɢɬɶ ɧɚ ɞɜɚ ɧɟɩɟɪɟɫɟ-
ɤɚɸɳɢɯɫɹ ɪɚɜɧɨɦɨɳɧɵɯ ɤɥɚɫɫɚ: ɩɨɞɦɧɨɠɟɫɬɜɚ, ɧɟ ɫɨɞɟɪɠɚɳɢɟ ɷɥɟɦɟɧɬ n
ɢ ɩɨɞɦɧɨɠɟɫɬɜɚ, ɫɨɞɟɪɠɚɳɢɟ ɟɝɨ. ɉɨɷɬɨɦɭ ɱɢɫɥɨ ɜɫɟɯ ɩɨɞɦɧɨɠɟɫɬɜ ɦɧɨɠɟɫɬɜɚ
{1, 2, ..., n} ɪɨɜɧɨ ɜ ɞɜɚ ɪɚɡɚ ɛɨɥɶɲɟ ɱɢɫɥɚ ɜɫɟɯ ɩɨɞɦɧɨɠɟɫɬɜ ɦɧɨɠɟɫɬɜɚ
n – 1} ɢ, ɡɧɚɱɢɬ, ɪɚɜɧɨ 2
ɩɨɞɦɧɨɠɟɫɬɜɚ , 1-ɷɥɟɦɟɧɬɨɟ – ɢɡ ɞɜɭɯ (ɩɭɫɬɨɝɨ ɢ {1}), ɬɨ ɟɫɬɶ 2
ɪɢɪɭɟɦ (ɧɚɣɞɟɦ) ɜɫɟ 2
Ɍɚɤ ɤɚɤ ɜ ɛɨɥɶɲɢɧɫɬɜɟ ɤɨɦɩɶɸɬɟɪɨɜ ɰɟɥɵɟ ɱɢɫɥɚ ɩɪɟɞɫɬɚɜɥɹɸɬɫɹ ɤɨɞɚɦɢ ɜ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɟ, ɩɪɢɱɟɦ ɱɢɫɥɨ (2
ɠɚɳɢɦ n ɟɞɢɧɢɰ, ɚ ɜɫɟ ɱɢɫɥɚ ɦɟɧɶɲɢɟ (2
n
(0-ɷɥɟɦɟɧɬɨɟ ɦɧɨɠɟɫɬɜɨ ɫɨɫɬɨɢɬ ɢɡ ɨɞɧɨɝɨ ɩɭɫɬɨɝɨ
n
ɩɨɞɦɧɨɠɟɫɬɜ ɦɧɨɠɟɫɬɜɚ {1, 2, ..., n}.
n –
1) ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɤɨɞɨɦ, ɫɨɞɟɪ-
n –
1), ɩɪɟɞɫɬɚɜɥɹɸɬɫɹ ɤɨɞɚɦɢ,
{1, 2, ...,
1
ɢ ɬ.ɞ.). ɋɝɟɧɟ-
ɢɦɟɸɳɢɦɢ ɧɟ ɛɨɥɟɟ n ɧɟɧɭɥɟɜɵɯ ɪɚɡɪɹɞɨɜ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɤɨɞ ɱɢɫɥɚ 0 ɹɜɥɹɟɬɫɹ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɩɭɫɬɨɝɨ ɦɧɨɠɟɫɬɜɚ , ɤɨɞ ɱɢɫɥɚ 1 ɹɜɥɹɟɬɫɹ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɩɨɞɦɧɨɠɟɫɬɜɚ, ɫɨɫɬɨɹɳɟɝɨ ɢɡ ɩɟɪɜɨɝɨ ɷɥɟɦɟɧɬɚ, ɢ ɬ.ɞ., ɤɨɞ ɱɢɫɥɚ (2
n –
1)
ɹɜɥɹɟɬɫɹ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɜɫɟɝɨ ɦɧɨɠɟɫɬɜɚ {1, 2, ..., n} (ɢɥɢ :). ɗɬɨ ɧɚɛɥɸɞɟɧɢɟ ɩɪɢɜɨɞɢɬ ɧɚɫ ɤ ɫɥɟɞɭɸɳɟɦɭ ɬɪɢɜɢɚɥɶɧɨɦɭ ɚɥɝɨɪɢɬɦɭ:
Ⱥɥɝɨɪɢɬɦ 7 (ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɜɫɟɯ ɩɨɞɦɧɨɠɟɫɬɜ ɦɧɨɠɟɫɬɜɚ).
n=+prompt('ȼɜɟɞɢ ɦɨɳɧɨɫɬɶ ɦɧɨɠɟɫɬɜɚ n','2'); m=1; g=''; for(i=0; i<n;
i++){g+='0'; m*=2}t=g;
for(i=1; i<m; i++){j=i; p='';
while(j){if(j%2){p+=1; j--}else p+=0; j/=2}p+=g.substr(0,n-p.length); t+='\n'+p}t
20
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