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Файл:Элементы программирования при решении математических задач. Учебное пособие
.pdf
Министерств
Федеральное государственное бюджетное образовательное учреждение
«Московский педагогический государственный университет»
о образования и науки Российской Федерации
высшего профессионального образования
А. А. Привалов
ЭЛЕМЕНТЫ ПРОГРАММИРОВАНИЯ
ПРИ РЕШЕНИИ МАТЕМАТИЧЕСКИХ ЗАДАЧ
Учебное пособие
2-е издание, стереотипное
электронное
МПГУ
Москва • 2024
УДК 51
ББК 22.1я73
П752
Рецензенты:
И. И. Баврин, доктор физико-математических наук, академик РАО ,
профессор кафедры теоретической информатики
и дискретной математики,
Московский педагогический государственный университет
В. А. Стеценко, кандидат физико-математических наук, доцент
кафедры теоретической информатики и дискретной математики,
Московский педагогический государственный университет
Привалов, Алекса
П752
Элементы программирования при решении математических
ндр Андреевич.
задач. Учебное пособие / А. А. Привалов. – 2-е изд., стер. электрон.
– Москва: МПГУ, 2024. – 92 с. – Текст: электронный.
ISBN 978-5-4263-0186-3
Данное пособие содержит подробное и строгое изложение основ-
ных понятий комбинаторики, алгебры и вычислительной геометрии.
Предназначено для параллельного изучения математики и программирования. Пособие написано на основе лекций, прочитанных в течение нескольких последних лет студентам математического факультета
МПГУ.
УДК 51
ББК 22.1я73
ISBN 978-5-4263-0186-3
© МПГУ, 2014
© Привалов А. А., 2014
ɋɈȾȿɊɀȺɇɂȿ
ȼȼȿȾȿɇɂȿ .............................................................................................................. 4
ɈȻɈɁɇȺɑȿɇɂə ɂ ɈɉɊȿȾȿɅȿɇɂə ................................................................... 9
ɉɈɇəɌɂȿ ɆɈȾȿɅɂ ȼɕɑɂɋɅȿɇɂɃ .............................................................. 12
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ ................ 15
§ 1. ɇɚɱɚɥɶɧɵɟ ɫɜɟɞɟɧɢɹ ɢɡ ɤɨɦɛɢɧɚɬɨɪɢɤɢ ................................................... 15
§ 2. Ƚɪɭɩɩɵ ......................................................................................................... 25
§ 3. Ⱥɥɝɨɪɢɬɦ ɲɢɮɪɨɜɚɧɢɹ RSA ...................................................................... 31
§ 4. Ⱥɞɞɢɬɢɜɧɵɟ ɰɟɩɨɱɤɢ .................................................................................. 35
§ 5. Ɂɚɞɚɱɚ Ɏɪɨɛɟɧɢɭɫɚ ..................................................................................... 38
§ 6. ɐɟɩɧɵɟ ɞɪɨɛɢ .............................................................................................. 40
§ 7. ɉɟɪɟɫɬɚɧɨɜɤɢ ............................................................................................... 53
§ 8. ɑɢɫɥɚ ɋɬɢɪɥɢɧɝɚ ɜɬɨɪɨɝɨ ɪɨɞɚ .................................................................. 61
ɉɈɇəɌɂȿ Ɉ ȼɕɑɂɋɅɂɌȿɅɖɇɈɃ ȽȿɈɆȿɌɊɂɂ ......................................... 71
§ 1. Ɉɫɧɨɜɧɵɟ ɨɩɪɟɞɟɥɟɧɢɹ ............................................................................... 71
§ 2.
ɇɟɤɨɬɨɪɵɟ ɡɚɞɚɱɢ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɝɟɨɦɟɬɪɢɢ ....................................... 74
ɅɂɌȿɊȺɌɍɊȺ. ...................................................................................................... 91
3
ȼȼȿȾȿɇɂȿ
ɇɚɫɬɨɹɳɟɟ ɩɨɫɨɛɢɟ ɫɨɡɞɚɧɨ ɩɨ ɦɚɬɟɪɢɚɥɚɦ ɤɭɪɫɨɜ «ɂɧɮɨɪɦɚɰɢɨɧɧɵɟ
ɬɟɯɧɨɥɨɝɢɢ ɜ ɦɚɬɟɦɚɬɢɤɟ» ɢ «ɉɪɢɦɟɧɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɟɬɨɞɨɜ
ɜ ɢɧɮɨɪɦɚɬɢɤɟ», ɱɢɬɚɟɦɵɯ ɚɜɬɨɪɨɦ ɜ Ɇɨɫɤɨɜɫɤɨɦ ɩɟɞɚɝɨɝɢɱɟɫɤɨɦ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɦ ɭɧɢɜɟɪɫɢɬɟɬɟ.
Ʉɭɪɫɵ ɜɤɥɸɱɚɸɬ ɜ ɫɟɛɹ ɤɚɤ ɬɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ, ɬɚɤ ɢ ɷɥɟɦɟɧɬɵ
ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ. ɉɪɢ ɷɬɨɦ ɨɛɵɱɧɨ ɩɨɥɶɡɭɟɦɫɹ ɹɡɵɤɨɦ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ
JavaScript ɢ ɱɚɫɬɨ ɩɨɥɶɡɭɟɦɫɹ ɩɪɨɝɪɚɦɦɨɣ Graphics3Help.html, ɧɚɩɢɫɚɧɧɨɣ
ɫɩɟɰɢɚɥɶɧɨ ɞɥɹ ɥɭɱɲɟɝɨ
ɩɨɧɢɦɚɧɢɹ ɪɚɛɨɬɵ ɚɥɝɨɪɢɬɦɨɜ. ɉɪɨɝɪɚɦɦɚ ɩɪɨɫɬɨ
ɢɫɩɨɥɧɹɟɬ ɤɨɞɵ, ɧɚɩɢɫɚɧɧɵɟ ɜ ɨɞɧɨɦ ɨɤɧɟ, ɜɵɞɚɜɚɹ ɪɟɡɭɥɶɬɚɬɵ ɢɫɩɨɥɧɟɧɢɹ
ɜ ɞɪɭɝɨɦ ɨɤɧɟ. Ɍɚɤɠɟ ɩɪɨɝɪɚɦɦɚ ɥɟɝɤɨ ɫɬɪɨɢɬ ɝɪɚɮɢɤɢ ɢ ɦɨɠɟɬ ɢɡɦɟɧɹɬɶɫɹ
ɜ ɞɢɧɚɦɢɤɟ.
ɉɟɪɜɨɟ ɡɧɚɤɨɦɫɬɜɨ ɧɚɱɢɧɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦɢ ɡɚɞɚɱɚɦɢ.
ɁȺȾȺɑȺ 1. ɉɭɫɬɶ ɮɭɧɤɰɢɹ f(n) ɫɬɚɜɢɬ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɤɚɠɞɨɦɭ ɧɚɬɭ-
ɪɚɥɶɧɨɦɭ ɱɢɫɥɭ n ɫɭɦɦɭ ɤɭɛɨɜ ɰɢɮɪ, ɤɨɬɨɪɵɦɢ ɡɚɩɢɫɚɧɨ
ɷɬɨ ɱɢɫɥɨ. Ⱦɥɹ ɤɚɠ-
ɞɨɝɨ n ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɩɟɪɢɨɞ ɪɟɤɭɪɪɟɧɬɧɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ:
= n, a
a
0
= f(an), n = 1, 2, … .
n+1
ȼ ɱɚɫɬɧɨɫɬɢ, ɞɨɤɚɡɚɬɶ, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ ɱɢɫɥɚ, ɤɪɚɬɧɨɝɨ 3, ɷɬɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɛɭɞɟɬ ɢɦɟɬɶ ɩɟɪɢɨɞ 1 ɢ ɩɪɟɞɟɥ ɪɚɜɧɵɣ 153 (ɩɟɪɢɨɞɨɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ {a
ɧɚɱɢɧɚɹ ɫ ɧɟɤɨɬɨɪɨɝɨ ɧɨɦɟɪɚ N, a
} ɧɚɡɵɜɚɟɬɫɹ ɧɚɢɦɟɧɶɲɟɟ ɢɡ ɱɢɫɟɥ ɪ, ɟɫɥɢ ɬɚɤɨɟ ɫɭɳɟɫɬɜɭɟɬ, ɱɬɨ
k
= ak, k = N, N + 1, … ).
k+p
Ɋɟɲɟɧɢɟ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɨɥɶɡɭɹɫɶ «ɱɢɫɬɨ» ɦɚɬɟɦɚɬɢɱɟɫɤɢɦɢ ɦɟɬɨɞɚɦɢ
ɷɬɭ ɡɚɞɚɱɭ ɪɟɲɢɬɶ ɧɟ ɭɞɚɟɬɫɹ. Ɉɞɧɚɤɨ, ɥɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɞɥɹ ɥɸɛɨɣ {a
} ɧɚɣ-
n
ɞɟɬɫɹ ɧɨɦɟɪ N, ɧɚɱɢɧɚɹ ɫ ɤɨɬɨɪɨɝɨ ɱɥɟɧɵ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɛɭɞɭɬ ɧɟ ɛɨɥɟɟ
ɱɟɦ 4-ɡɧɚɱɧɵɦɢ ɱɢɫɥɚɦɢ. ɗɬɨ ɫɜɹɡɚɧɨ ɫ ɦɟɞɥɟɧɧɵɦ ɪɨɫɬɨɦ ɮɭɧɤɰɢɢ
lg
n
lim
lgn (
fo
n
ɰɢɮɪɚ 9). ɇɚɩɪɢɦɟɪ, ɟɫɥɢ a
ɢ, ɡɧɚɱɢɬ, f(a
0
, ɚ f(ɯ) < 729([lgn] + 1), ɝɞɟ [k] – ɰɟɥɚɹ ɱɚɫɬɶ k, ɚ ɫɚɦɚɹ ɛɨɥɶɲɚɹ
n
– 10-ɡɧɚɱɧɨɟ ɱɢɫɥɨ, ɬɨ ɨɧɨ ɦɟɧɶɲɟ, ɱɟɦ 1011,
) < 93ǜ10 = 7290 ɭɠɟ 4-ɡɧɚɱɧɨɟ ɱɢɫɥɨ. Ɍɟɩɟɪɶ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚ-
n
n
ɱɢ ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɫɦɨɬɪɟɬɶ ɜɫɟ ɱɢɫɥɚ ɞɨ 2195. ɋɥɟɞɭɸɳɚɹ ɩɪɨɝɪɚɦɦɚ ɞɨɤɚɡɵɜɚɟɬ ɜɬɨɪɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɡɚɞɚɱɢ:
Ⱥɥɝɨɪɢɬɦ 1.
t=''; z=new Date(); for(i=3; i<2195; i+=3){u=(''+i).split(''); t+=i+',';
while(1){s=u[0]*u[0]*u[0]; for(j=1; j<u.length; j++)s+=u[j]*u[j]*u[j];
if(s==153)break; u=(''+s).split(''); t+=s+','}t+='153\n'}
'Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ ɡɚ '+(new Date()-z)/1000+' ɫɟɤ.\n'+t
Ɂɞɟɫɶ ɦɟɬɨɞ split('') ɜɨɡɜɪɚɳɚɟɬ ɧɨɜɵɣ ɦɚɫɫɢɜ, ɪɚɡɛɢɜɚɹ ɫɬɪɨɤɭ ɩɨ
ɩɭɫɬɨɦɭ ɫɢɦɜɨɥɭ, ɬɨ ɟɫɬɶ ɱɢɫɥɨ i ɤɚɤ ɫɬɪɨɤɭ (''+i) ɧɚ ɰɢɮɪɵ. ȼ ɨɤɧɨ ɩɪɨɝɪɚɦ
ɦɵ Graphics3Help.html ɜɜɟɞɢɬɟ
ɷɬɨɬ ɬɟɤɫɬ, ɧɚɠɦɢɬɟ ɤɧɨɩɤɭ «calculator» ɢ ɜɨ
-
4
ȼȼȿȾȿɇɂȿ
ɜɬɨɪɨɦ ɨɤɧɟ ɩɨɥɭɱɢɬɟ ɪɟɡɭɥɶɬɚɬ: «Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ ɡɚ 0,082 ɫɟɤ.»
ɢ ɩɪɨɦɟɠɭɬɨɱɧɵɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ.
Ɂɞɟɫɶ ɢ ɧɢɠɟ ɩɪɢɜɨɞɹɬɫɹ ɪɟɡɭɥɶɬɚɬɵ, ɩɨɥɭɱɟɧɧɵɟ ɧɚ ɤɨɦɩɶɸɬɟɪɟ ɚɜɬɨɪɚ.
ɁȺȾȺɑȺ 2. Ɏɭɧɤɰɢɹ f ɤɚɠɞɨɦɭ ɧɚɬɭɪɚɥɶɧɨɦɭ ɱɢɫɥɭ n ɫɬɚɜɢɬ
ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɪɚɡɧɨɫɬɶ ɦɟɠɞɭ ɱɢɫɥɨɦ, ɡɚɩɢɫɚɧɧɵɦ ɰɢɮɪɚɦɢ ɱɢɫɥɚ n, ɪɚɫɩɨɥɨɠɟɧɧɵɦɢ ɜ ɩɨɪɹɞɤɟ ɭɛɵɜɚɧɢɹ ɢ ɱɢɫɥɨɦ, ɡɚɩɢɫɚɧɧɵɦ ɷɬɢɦɢ ɠɟ ɰɢɮɪɚɦɢ, ɧɨ
ɜ ɜɨɡɪɚɫɬɚɸɳɟɦ
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɡɚɞɚɧɚ ɪɟɤɭɪɪɟɧɬɧɨ: ɚ
ɩɨɪɹɞɤɟ. ɇɚɩɪɢɦɟɪ, f(957) = 396.
= n, ak = f(a
0
), k = 1, 2, … .
k–1
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɟɫɥɢ n ɫɨɫɬɨɢɬ ɢɡ ɨɞɢɧɚɤɨɜɵɯ ɰɢɮɪ, ɬɨ ɩɟɪɢɨɞ ɬɚɤɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɪɚɜɟɧ 1.
Ʉɚɤɢɟ ɩɟɪɢɨɞɵ ɦɨɝɭɬ ɛɵɬɶ ɭ ɞɜɭɡɧɚɱɧɵɯ, ɬɪɟɯɡɧɚɱɧɵɯ, ɱɟɬɵɪɟɯɡɧɚɱɧɵɯ ɢ ɩɹɬɢɡɧɚɱɧɵɯ ɱɢɫɟɥ?
Ɉɤɚɡɵɜɚɟɬɫɹ, ɞɥɹ 4-ɡɧɚɱɧɵɯ ɱɢɫɟɥ ɷɬɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɩɪɢɯɨɞɢɬ
ɤ ɱɢɫɥɭ 6174. Ɉɩɹɬɶ ɠɟ ɧɟɩɨɧɹɬɧɨ, ɤɚɤ ɞɨɤɚɡɚɬɶ ɷɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢ, ɧɨ ɥɟɝɤɨ ɩɪɨɜɟɪɢɬɶ ɟɝɨ ɫ
ɩɨɦɨɳɶɸ ɩɪɨɝɪɚɦɦɵ:
Ⱥɥɝɨɪɢɬɦ 2.
z=new Date(); for(i=1000; i<9999; i++){p=i;
while(p!=6174){u=(''+p).split('').sort();
g=''; s=''; for(j=0; j<4; j++) {if(u[j]!=0)g+=u[j]; s+=u[3-j]}p=s-g;
if(isNaN(p))break; }}
'Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ ɡɚ ɜɪɟɦɹ t='+(new Date()-z)/1000+' sec.\n',
ɝɞɟ ɮɭɧɤɰɢɹ isNaN(p) ɜɨɡɜɪɚɳɚɟɬ true, ɟɫɥɢ ɪ ɹɜɥɹɟɬɫɹ ɫɬɪɨɤɨɣ (ɧɟ ɱɢɫɥɨɦ)
ɢ false – ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ. ɋ ɩɨɦɨɳɶɸ ɩɪɨɝɪɚɦɦɵ Graphics3Help.html ɦɵ
ɩɨɥɭɱɢɦ ɨɬɜɟɬ: «Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ ɡɚ ɜɪɟɦɹ t = 1.004 ɫɟɤ.». ɗɬɚ ɩɪɨɝɪɚɦɦɚ
ɩɪɨɫɬɨ ɩɪɨɜɟɪɹɟɬ ɜɫɟ 4-ɡɧɚɱɧɵɟ ɱɢɫɥɚ. ȼɨɡɧɢɤɚɟɬ ɜɨɩɪɨɫ, ɚ ɦɨɠɧɨ ɥɢ ɭɦɟɧɶ-
ɭɥɭɱɲɢɜ ɷɬɨɬ ɚɥɝɨɪɢɬɦ? Ɉɤɚɡɵɜɚɟɬɫɹ, ȾȺ,
ɲɢɬɶ ɜɪɟɦɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ
,
ɢ ɫɥɟɞɭɸɳɢɣ ɤɨɞ ɩɨɞɬɜɟɪɠɞɚɟɬ ɷɬɨ:
Ⱥɥɝɨɪɢɬɦ 3.
z=new Date(); for(i=1; i<10; i++){for(k=0; k<i; k++){p=999*i+90*k;
while(p!=6174){u=(''+p).split('').sort(); g=''; s=''; for(j=0; j<4;
j++){if(u[j]!=0)g+=u[j]; s+=u[3-j]}p=s-g; if(isNaN(p))break; }}}
'Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ ɡɚ ɜɪɟɦɹ t='+(new Date()-z)/1000+' sec.'
Ɉɬɜɟɬɨɦ ɤɨɞɚ ɛɭɞɟɬ: «Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ ɡɚ ɜɪɟɦɹ t=0.004 ɫɟɤ.». Ⱦɥɹ ɪɟɲɟɧɢɹ
ɦɵ ɩɪɢɦɟɧɢɥɢ ɧɟɫɥɨɠɧɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɪɚɫɫɭɠɞɟɧɢɹ. ɉɭɫɬɶ ɧɚɲɟ ɱɢɫɥɨ n
ɡɚɩɢɫɚɧɨ ɰɢɮɪɚɦɢ x, y, z, t ɢ x y z t
. Ɍɨɝɞɚ
f(n) = 1000x + 100y + 10z + t – (1000t + 100z + 10y + x) = 999(x – t) + 90(y – z),
ɢ ɜ ɷɬɨɦ ɜɵɪɚɠɟɧɢɢ ɜɟɥɢɱɢɧɚ (x – t) ɦɟɧɹɟɬɫɹ ɨɬ 1 ɞɨ 9, ɜɟɥɢɱɢɧɚ (y – z) ɜɫɟɝɞɚ
ɦɟɧɶɲɟ, ɱɟɦ (x – t) ɢ ɧɟ ɦɟɧɶɲɟ 0. Ɂɧɚɱɢɬ, ɧɚɞɨ ɫɞɟɥɚɬɶ ɜɫɟɝɨ 45 ɩɪɨɫɦɨɬɪɨɜ
5
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
(ɩɪɨɜɟɪɨɤ). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫ ɩɨɦɨɳɶɸ ɧɟɫɥɨɠɧɨɝɨ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɪɚɫɫɭɠ-
ɞɟɧɢɹ ɜɪɟɦɹ ɢɫɩɨɥɧɟɧɢɹ ɭɦɟɧɶɲɢɥɨɫɶ ɩɨɱɬɢ ɜ 1000 ɪɚɡ.
ɁȺȾȺɑȺ Ɉ ɏȺɇɈɃɋɄɈɃ ȻȺɒɇȿ. ɗɬɭ ɦɚɥɟɧɶɤɭɸ ɢɡɹɳɧɭɸ ɝɨɥɨɜɨɥɨɦɤɭ ɩɪɢɞɭɦɚɥ ɮɪɚɧɰɭɡɫɤɢɣ ɦɚɬɟɦɚɬɢɤ ɗɞɭɚɪɞ Ʌɸɤɚ ɜ 1883 ɝɨɞɭ. Ȼɚɲɧɹ
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɜɨɫɟɦɶ ɞɢɫɤɨɜ, ɧɚɧɢɡɚɧɧɵɯ ɜ ɩɨɪɹɞɤɟ ɭɦɟɧɶɲɟɧɢɹ ɪɚɡɦɟɪɨɜ ɧɚ ɨɞɢɧ ɢɡ ɬɪɟɯ ɤɨɥɵɲɤɨɜ:
Ɂɚɞɚɱɚ ɫɨɫɬɨɢɬ
ɜ ɬɨɦ, ɱɬɨɛɵ ɩɟɪɟɦɟɫɬɢɬɶ ɜɫɸ ɛɚɲɧɸ ɧɚ ɨɞɢɧ ɢɡ ɤɨɥɵɲɤɨɜ, ɩɟɪɟɧɨɫɹ ɤɚɠɞɵɣ ɪɚɡ ɬɨɥɶɤɨ ɨɞɢɧ ɞɢɫɤ ɢ ɧɟ ɩɨɦɟɳɚɹ ɛɨɥɶɲɢɣ ɞɢɫɤ
ɧɚ ɦɟɧɶɲɢɣ.
Ʌɸɤɚ ɫɜɹɡɵɜɚɥ ɫɜɨɸ ɢɝɪɭɲɤɭ ɫ ɦɢɮɢɱɟɫɤɨɣ ɥɟɝɟɧɞɨɣ ɨ ɡɧɚɱɢɬɟɥɶɧɨ
ɛɨɥɶɲɟɣ ɛɚɲɧɟ Ȼɪɚɦɵ, ɤɨɬɨɪɚɹ, ɤɚɤ ɭɬɜɟɪɠɞɚɟɬɫɹ, ɫɨɫɬɨɢɬ ɢɡ 64 ɞɢɫɤɨɜ ɱɢɫɬɨɝɨ ɡɨɥɨɬɚ, ɚ ɤɨɥɵɲɤɢ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɬɪɢ
ɚɥɦɚɡɧɵɯ ɲɩɢɥɹ. ɉɪɢ ɫɨɬɜɨɪɟɧɢɢ ɦɢɪɚ ȼɫɟɜɵɲɧɢɣ ɩɨɦɟɫɬɢɥ ɞɢɫɤɢ ɧɚ ɩɟɪɜɵɣ ɲɩɢɥɶ ɢ ɩɨɜɟɥɟɥ, ɱɬɨɛɵ
ɠɪɟɰɵ ɩɟɪɟɦɟɫɬɢɥɢ ɢɯ ɧɚ ɬɪɟɬɢɣ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɟɞɩɢɫɚɧɧɵɦɢ ɩɪɚɜɢɥɚɦɢ. ɉɨ ɢɦɟɸɳɢɦɫɹ ɫɜɟɞɟɧɢɹɦ, ɠɪɟɰɵ ɬɪɭɞɹɬɫɹ ɧɚɞ ɷɬɨɣ ɡɚɞɚɱɟɣ ɞɟɧɧɨ
ɢ ɧɨɳɧɨ – ɤɚɤ ɬɨɥɶɤɨ ɨɧɢ ɡɚɤɨɧɱɚɬ, ɛɚɲɧɹ ɪɚɫɫɵɩɥɟɬɫɹ ɜ ɩɪɚɯ ɢ ɧɚɫɬɭɩɢɬ ɤɨɧɟɰ ɫɜɟɬɚ.
ɬɨɬɱɚɫ ɨɱɟɜɢɞɧɨ, ɱɬɨ ɡɚɞɚɱɚ ɪɚɡɪɟɲɢɦɚ, ɧɨ ɩɨ ɤɪɚɬɤɨɦ ɪɚɡɦɵɲɥɟɧɢɢ
ɇɟ
ɭɛɟɠɞɚɟɦɫɹ, ɱɬɨ ɷɬɨ ɬɚɤ. Ɍɨɝɞɚ ɜɨɡɧɢɤɚɟɬ ɫɥɟɞɭɸɳɢɣ ɜɨɩɪɨɫ: ɤɚɤɨɣ ɫɩɨɫɨɛ
ɫɚɦɵɣ ɨɩɬɢɦɚɥɶɧɵɣ? Ɍɨ ɟɫɬɶ ɤɚɤɨɟ ɤɨɥɢɱɟɫɬɜɨ ɩɟɪɟɦɟɳɟɧɢɣ ɞɢɫɤɨɜ ɹɜɥɹɟɬɫɹ
ɧɟɨɛɯɨɞɢɦɵɦ ɢ ɞɨɫɬɚɬɨɱɧɵɦ ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɩɨɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ.
Ɉɛɨɛɳɚɹ ɡɚɞɚɱɭ, ɨɛɨɡɧɚɱɢɦ f(n) – ɦɢɧɢɦɚɥɶɧɨɟ ɱɢɫɥɨ ɩɟɪɟɤɥɚɞɵɜɚɧɢɣ,
ɧɟɨɛɯɨɞɢɦɵɯ ɞɥɹ ɩɟɪɟɦɟɳɟɧɢɹ n ɞɢɫɤɨɜ ɫ ɨɞɧɨɝɨ
ɤɨɥɵɲɤɚ ɧɚ ɞɪɭɝɨɣ ɩɨ ɩɪɚɜɢɥɚɦ Ʌɸɤɚ. Ɍɨɝɞɚ, ɨɱɟɜɢɞɧɨ, ɱɬɨ f(0) = 0, f(1) = 1, ɚ f(2) = 3. Ɍɟɩɟɪɶ ɡɚɦɟɬɢɦ,
ɱɬɨ ɞɥɹ ɩɟɪɟɦɟɳɟɧɢɹ n (n > 2) ɞɢɫɤɨɜ ɧɚ ɬɪɟɬɢɣ ɤɨɥɵɲɟɤ ɦɵ ɦɨɠɟɦ ɩɟɪɟɦɟɫɬɢɬɶ (n – 1) ɜɟɪɯɧɢɯ ɞɢɫɤɨɜ ɧɚ ɜɬɨɪɨɣ ɤɨɥɵɲɟɤ ɡɚ f(n – 1) ɩɟɪɟɤɥɚɞɵɜɚɧɢɣ,
ɡɚɬɟɦ ɛɨɥɶɲɢɣ ɞɢɫɤ ɩɨɦɟɫɬɢɬɶ ɧɚ ɬɪɟɬɢɣ ɤɨɥɵɲɟɤ (ɨɞɧɨ ɩɟɪɟɤɥɚɞɵɜɚɧɢɟ) ɢ
– 1) ɩɟɪɟɤɥɚɞɵɜɚɧɢɣ ɩɟɪɟɦɟɫɬɢɬɶ ɞɢɫɤɢ ɫɨ ɜɬɨɪɨɝɨ ɤɨɥɵɲɤɚ ɧɚ ɬɪɟɬɢɣ (ɧɚ
f(n
ɛɨɥɶɲɢɣ ɞɢɫɤ). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, n (n > 0) ɞɢɫɤɨɜ ɦɨɠɧɨ ɩɟɪɟɦɟɫɬɢɬɶ ɫɚɦɨɟ
ɛɨɥɶɲɨɟ ɡɚ (2f(n – 1) + 1) ɩɟɪɟɤɥɚɞɵɜɚɧɢɣ:
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ȼȼȿȾȿɇɂȿ
f(n) 2f(n – 1) + 1 ɩɪɢ n > 0.
ȼ ɷɬɨɣ ɮɨɪɦɭɥɟ ɮɢɝɭɪɢɪɭɟɬ ɡɧɚɤ «» ɜɦɟɫɬɨ «=», ɩɨɫɤɨɥɶɤɭ ɧɚɲɟ ɩɨɫɬɪɨɟɧɢɟ ɩɨɤɚɡɵɜɚɟɬ ɬɨɥɶɤɨ, ɱɬɨ ɞɨɫɬɚɬɨɱɧɨ (2f(n – 1) + 1) ɩɟɪɟɤɥɚɞɵɜɚɧɢɣ;
ɦɵ ɧɟ ɞɨɤɚɡɚɥɢ, ɱɬɨ ɧɟɨɛɯɨɞɢɦɨ (2f(n – 1) + 1) ɩɟɪɟɤɥɚɞɵɜɚɧɢɣ!
Ɉɤɚɡɵɜɚɟɬɫɹ, ɱɬɨ ɛɨɥɟɟ ɤɨɪɨɬɤɨɝɨ ɩɭɬɢ ɧɟɬ. ȼ ɫɚɦɨɦ ɞɟɥɟ, ɧɚ ɧɟɤɨɬɨɪɨɦ
ɷɬɚɩɟ ɦɵ ɞɨɥɠɧɵ ɩɟɪɟɦɟɫɬɢɬɶ ɫɚɦɵɣ ɛɨɥɶɲɨɣ
ɞɢɫɤ. Ʉɨɝɞɚ ɦɵ ɷɬɨ ɞɟɥɚɟɦ,
(n – 1) ɦɟɧɶɲɢɯ ɞɢɫɤɨɜ ɞɨɥɠɧɵ ɧɚɯɨɞɢɬɶɫɹ ɧɚ ɨɞɧɨɦ ɤɨɥɵɲɤɟ, ɚ ɞɥɹ ɬɨɝɨ,
ɱɬɨɛɵ ɫɨɛɪɚɬɶ ɢɯ ɬɚɦ ɜɦɟɫɬɟ, ɩɨɬɪɟɛɭɟɬɫɹ, ɩɨ ɦɟɧɶɲɟɣ ɦɟɪɟ, f(n – 1) ɩɟɪɟɤɥɚɞɵɜɚɧɢɣ. ɉɨɫɥɟ ɩɟɪɟɦɟɳɟɧɢɹ ɫɚɦɨɝɨ ɛɨɥɶɲɨɝɨ ɞɢɫɤɚ ɜ ɩɨɫɥɟɞɧɢɣ ɪɚɡ ɦɵ
ɨɛɹɡɚɧɵ ɩɟɪɟɦɟɫɬɢɬɶ (n – 1) ɦɟɧɶɲɢɯ ɞɢɫɤɨɜ (ɤɨɬɨɪɵɟ ɞɨɥɠɧɵ ɧɚɯɨɞɢɬɶɫɹ ɧɚ
ɨɞɧɨɦ ɤɨɥɵɲɤɟ) ɨɛɪɚɬɧɨ
ɧɚ ɧɚɢɛɨɥɶɲɢɣ ɞɢɫɤ, ɱɬɨ ɬɚɤɠɟ ɩɨɬɪɟɛɭɟɬ f(n – 1)
ɩɟɪɟɤɥɚɞɵɜɚɧɢɣ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
f(n) 2f(n – 1) + 1 ɩɪɢ n > 0.
Ɉɬɫɸɞɚ ɢɦɟɟɦ:
f(0) = 0, f(n) = 2f(n – 1) + 1 ɩɪɢ n > 0.
Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɹɜɧɨɣ ɮɨɪɦɭɥɵ ɞɥɹ f(n) ɪɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ
g(n) = f(n) + 1.
Ɍɨɝɞɚ, g(0) = 1, g(n) = 2f(n – 1) + 2 = 2(g(n – 1) – 1) + 2 = 2g(
ɩɪɢ n > 0. Ɂɧɚɱɢɬ, g(n) = 2
n
, ɚ
f(n) = 2
n
– 1.
n – 1) ɩɪɢ
ɂɡ ɩɪɢɜɟɞɟɧɧɵɯ ɪɚɫɫɭɠɞɟɧɢɣ ɥɟɝɤɨ ɧɚɩɢɫɚɬɶ ɩɪɨɝɪɚɦɦɭ, ɪɟɲɚɸɳɭɸ
ɷɬɭ ɡɚɞɚɱɭ. ɉɨɩɵɬɚɣɬɟɫɶ ɫɞɟɥɚɬɶ ɷɬɨ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ. Ⱥ ɦɵ ɜɨɫɩɨɥɶɡɭɟɦɫɹ
ɬɟɦ ɨɛɫɬɨɹɬɟɥɶɫɬɜɨɦ, ɱɬɨ ɧɚɲ ɤɨɦɩɶɸɬɟɪ ɦɨɠɟɬ ɧɟ ɬɨɥɶɤɨ ɜɵɩɨɥɧɹɬɶ ɜɵɱɢɫɥɟɧɢɹ, ɧɨ ɢ ɡɚɩɨɦɢɧɚɬɶ ɩɪɟɞɥɨɠɟɧɧɵɟ ɞɟɣɫɬɜɢɹ, ɪɚɫɫɬɚɜɥɹɹ ɢɯ ɜ ɨɱɟɪɟɞɶ.
ɉɪɨɝɪɚɦɦɵ, ɢɫɩɨɥɶɡɭɸɳɢɟ ɷɬɨ ɫɜɨɣɫɬɜɨ, ɧɚɡɵɜɚɸɬɫɹ ɪɟɤɭɪɫɢɜɧɵɦɢ. ɉɪɟɞɫɬɚɜɢɦ ɬɚɤɭɸ ɜɟɪɫɢɸ ɧɚɲɟɣ ɩɪɨɝɪɚɦɦɵ. Ⱦɥɹ ɷɬɨɝɨ
ɨɛɨɡɧɚɱɢɦ ɧɚɲɢ ɤɨɥɵɲɤɢ
ɤɚɤ 1, 2, 3 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɚ «--> » – ɩɟɪɟɤɥɚɞɵɜɚɧɢɹ. ɇɚɩɪɢɦɟɪ, «1-->3» ɛɭɞɟɬ ɨɡɧɚɱɚɬɶ ɩɟɪɟɤɥɚɞɵɜɚɧɢɟ ɫ ɩɟɪɜɨɝɨ ɤɨɥɵɲɤɚ ɧɚ ɬɪɟɬɢɣ. ɂɬɚɤ, ɧɚɲ ɤɨɞ:
Ⱥɥɝɨɪɢɬɦ 4 (ɏɚɧɨɣɫɤɢɟ ɛɚɲɧɢ).
function f(n,x,y,z){if(n==1)g+=x+'-->'+z+' ';
else {f(n–1,x,z,y); g+=x+'-->'+z+' '; f(n–1,y,x,z)}}
g=''; f(n=parseInt(prompt('ɑɢɫɥɨ ɞɢɫɤɨɜ','3')),1,2,3); g
ɉɪɢ n = 3 ɩɨɥɭɱɢɦ ɨɬɜɟɬ: 1-->3 1-->2 3-->2 1-->3 2-->1 2-->3 1-->3. Ɋɚɡ-
ɛɟɪɢɬɟɫɶ ɩɨ ɲɚɝɚɦ ɜ ɷɬɨɣ ɩɪɨɝɪɚɦɦɟ!
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ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ȼ ɤɪɢɩɬɨɝɪɚɮɢɢ ɞɥɹ ɲɢɮɪɨɜɚɧɢɹ ɬɟɤɫɬɨɜ ɩɨɥɶɡɭɸɬɫɹ ɞɥɢɧɧɵɦɢ ɱɢɫɥɚɦɢ, ɞɥɹ ɡɚɩɢɫɢ ɤɨɬɨɪɵɯ ɬɪɟɛɭɸɬɫɹ ɞɟɫɹɬɤɢ ɢ ɫɨɬɧɢ ɰɢɮɪ. Ɉɫɨɛɵɣ ɢɧɬɟɪɟɫ
ɩɪɟɞɫɬɚɜɥɹɸɬ ɱɢɫɥɚ, ɹɜɥɹɸɳɢɟɫɹ ɩɪɨɫɬɵɦɢ. Ɏɨɪɦɭɥɚ ɩɪɨɫɬɨɝɨ ɱɢɫɥɚ ɞɨ ɫɢɯ
ɩɨɪ ɧɟ ɧɚɣɞɟɧɚ, ɢ ɧɟ ɧɚɣɞɟɧ ɬɚɤɠɟ ɷɮɮɟɤɬɢɜɧɵɣ ɚɥɝɨɪɢɬɦ ɪɚɡɥɨɠɟɧɢɹ ɱɢɫɥɚ
ɧɚ ɩɪɨɫɬɵɟ ɦɧɨɠɢɬɟɥɢ. Ⱦɥɢɧɧɵɟ ɩɪɨɫɬɵɟ ɱɢɫɥɚ ɢɫɩɨɥɶɡɭɸɬɫɹ ɚɥɝɨɪɢɬɦɨɦ
ɲɢɮɪɨɜɚɧɢɹ RSA (ɚɦɟɪɢɤɚɧɫɤɢɟ ɦɚɬɟɦɚɬɢɤɢ Ɋɢɜɟɫɬ
, ɒɚɦɢɪ ɢ Ⱥɞɥɟɦɚɧ
ɨɩɭɛɥɢɤɨɜɚɥɢ ɷɬɨɬ ɚɥɝɨɪɢɬɦ ɜ 1977 ɝɨɞɭ). ɉɨɷɬɨɦɭ ɩɨɥɭɱɟɧɢɟ ɞɥɢɧɧɵɯ ɩɪɨɫɬɵɯ ɱɢɫɟɥ ɹɜɥɹɟɬɫɹ ɬɚɤɠɟ ɧɟɪɟɲɟɧɧɨɣ ɩɪɨɛɥɟɦɨɣ, ɬɨ ɟɫɬɶ ɞɨ ɫɢɯ ɩɨɪ ɧɟ ɢɡɜɟɫɬɟɧ ɛɨɥɟɟ-ɦɟɧɟɟ ɷɮɮɟɤɬɢɜɧɵɣ ɚɥɝɨɪɢɬɦ ɢɯ ɩɨɥɭɱɟɧɢɹ. Ɂɞɟɫɶ ɫɬɨɢɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɩɨɩɵɬɤɢ ɪɟɲɢɬɶ ɷɬɭ ɡɚɞɚɱɭ ɢɦɟɸɬ ɝɥɭɛɨɤɢɟ ɤɨɪɧɢ. ɇɚɩɪɢɦɟɪ, ɉɶɟɪ
Ɏɟɪɦɚ ɛɵɥ ɭɜɟɪɟɧ, ɱɬɨ
ɜɫɟ ɱɢɫɥɚ ɜɢɞɚ
n
F
n
1 22
ɩɪɨɫɬɵɟ (ɱɢɫɥɚ ɬɚɤɨɝɨ ɜɢ-
ɞɚ ɧɚɡɵɜɚɸɬɫɹ ɱɢɫɥɚɦɢ Ɏɟɪɦɚ), ɨɞɧɚɤɨ Ʌɟɨɧɚɪɞ ɗɣɥɟɪ ɩɨɤɚɡɚɥ, ɱɬɨ ɭɠɟ ɱɢɫ-
= 4294967297 = 641·6700417, ɬɨ ɟɫɬɶ ɧɟ ɹɜɥɹɟɬɫɹ ɩɪɨɫɬɵɦ. ɂɡɜɟɫɬɧɨ, ɱɬɨ
ɥɨ F
5
ɫɪɟɞɢ ɱɢɫɟɥ ɜɢɞɚ 2
ɪ
– 1, ɝɞɟ ɪ ɩɪɨɫɬɨɟ ɱɢɫɥɨ, ɛɟɫɤɨɧɟɱɧɨ ɦɧɨɝɨ ɩɪɨɫɬɵɯ. ɑɢɫ-
ɥɚ ɬɚɤɨɝɨ ɜɢɞɚ ɧɚɡɵɜɚɸɬɫɹ ɱɢɫɥɚɦɢ Ɇɟɪɫɟɧɧɚ.
ɇɚ ɫɟɧɬɹɛɪɶ 2013 ɝɨɞɚ ɫɚɦɵɦ ɛɨɥɶɲɢɦ ɩɪɨɫɬɵɦ ɱɢɫɥɨɦ ɹɜɥɹɟɬɫɹ ɱɢɫɥɨ
Ɇɟɪɫɟɧɧɚ Ɇ
57885161
ɩɟɪɨɦ. Ⱦɟɫɹɬɢɱɧɚɹ ɡɚɩɢɫɶ ɱɢɫɥɚ Ɇ
= 2
57885161
– 1, ɧɚɣɞɟɧɧɨɟ 25 ɹɧɜɚɪɹ 2013 ɝɨɞɚ Ʉɟɪɬɢɫɨɦ Ʉɭ-
ɫɨɞɟɪɠɢɬ 17 425 170 ɰɢɮɪ.
57885161
ɂɧɨɝɞɚ ɧɚ ɨɥɢɦɩɢɚɞɚɯ ɩɨ ɢɧɮɨɪɦɚɬɢɤɟ ɩɪɟɞɥɚɝɚɸɬ ɡɚɞɚɱɢ, ɫɜɹɡɚɧɧɵɟ
ɫ ɱɢɫɥɚɦɢ ɬɢɩɚ Ɇɟɪɫɟɧɧɚ. ɇɚɩɪɢɦɟɪ,
ɁȺȾȺɑȺ 4. ɇɚ ɩɟɪɜɭɸ ɤɥɟɬɤɭ ɲɚɯɦɚɬɧɨɣ ɞɨɫɤɢ N×N (N < 100) ɤɥɚɞɟɬɫɹ ɨɞɧɨ ɡɟɪɧɵɲɤɨ ɢ ɧɚ ɤɚɠɞɭɸ ɫɥɟɞɭɸɳɭɸ – ɜ ɞɜɚ ɪɚɡɚ ɛɨɥɶɲɟ, ɱɟɦ ɧɚ ɩɪɟɞɵɞɭɳɭɸ. ȼɵɜɟɞɢɬɟ ɧɚ ɞɢɫɩɥɟɣ ɱɢɫɥɨ (ɧɟ ɮɨɪɦɭɥɭ!) ɩɨɥɨɠɟɧɧɵɯ ɡɟɪɧɵɲɟɤ.
ɉɪɟɞɫɬɚɜɥɟɧɧɵɣ ɧɢɠɟ ɤɨɞ
ɜɵɩɢɫɵɜɚɟɬ ɱɢɫɥɚ ɜɢɞɚ 2n – 1:
Ⱥɥɝɨɪɢɬɦ 5.
u=new Array(); u[0]=2; p=1000; v=new Array(); z=new Date(); n=10000;
for(j=1; j<n; j++){r=0; for(i=0; i<u.length; i++){x=u[i]*2+r; u[i]=x%p; r=(x–
u[i])/p;
while((''+u[i]).length<3)u[i]='0'+u[i]}if(r)u[u.length]=r; }
u[0]--; while((''+u[0]).length<3)u[0]='0'+u[0];
s=''; for(i=u.length–1; i>=0; i--)s+=u[i];
while(s.substr(0,1)=='0')s=s.substr(1,s.length–1);
'Ⱦɥɢɧɚ ɱɢɫɥɚ='+s.length+', ȼɪɟɦɹ='+(new Date()–z)/1000+'ɫɟɤ.\nɑɢɫɥɨ='+s
Ɂɞɟɫɶ ɱɢɫɥɨ 2
ɫɱɢɫɥɟɧɢɹ. ȼ ɞɚɧɧɨɦ ɤɨɧɤɪɟɬɧɨɦ ɫɥɭɱɚɟ ɪ = 1000, n = 10000 ɩɪɨɝɪɚɦɦɚ ɪɚɛɨɬɚɟɬ 24 ɫɟɤɭɧɞɵ ɢ ɜɵɩɢɫɵɜɚɟɬ 3011-ɡɧɚɱɧɨɟ ɱɢɫɥɨ (2
n
– 1 ɡɚɩɢɫɵɜɚɟɬɫɹ ɤɚɤ ɦɚɫɫɢɜ ɰɢɮɪ ɜ ɪ-ɱɧɨɣ ɫɢɫɬɟɦɟ
10000
– 1).
ȼ ɡɚɤɥɸɱɟɧɢɟ ɨɬɦɟɬɢɦ, ɱɬɨ ɫɭɳɟɫɬɜɭɸɬ ɹɡɵɤɢ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ, ɧɚɩɪɢɦɟɪ “Python”, ɩɨɡɜɨɥɹɸɳɢɟ ɪɚɛɨɬɚɬɶ ɫ ɰɟɥɵɦɢ ɱɢɫɥɚɦɢ ɥɸɛɨɣ, ɞɨɫɬɭɩɧɨɣ
ɩɚɦɹɬɢ ɤɨɦɩɶɸɬɟɪɚ, ɞɥɢɧɵ.
8
ɈȻɈɁɇȺɑȿɇɂə ɂ ɈɉɊȿȾȿɅȿɇɂə
ȼɜɟɞɟɦ ɧɟɨɛɯɨɞɢɦɵɟ ɨɛɨɡɧɚɱɟɧɢɹ. Ʉɚɤ ɨɛɵɱɧɨ,
ɡɧɚɤ … – ɨɡɧɚɱɚɟɬ ɢ ɬɚɤ ɞɚɥɟɟ;
– : (ɢɥɢ Ň ɢ °) – ɬɚɤɨɟ (ɬɚɤɢɟ), ɱɬɨ, ɝɞɟ, … ;
– – ɫɭɳɟɫɬɜɭɟɬ, ɧɚɣɞɟɬɫɹ, … ;
– – ɥɸɛɨɣ, ɜɫɹɤɢɣ, ɞɥɹ ɥɸɛɨɝɨ, … ;
– { … } – ɩɪɢɦɟɧɹɟɬɫɹ ɞɥɹ ɨɛɨɡɧɚɱɟɧɢɹ ɦɧɨɠɟɫɬɜ. ɇɚɩɪɢɦɟɪ, {3, 7, 9, a}
ɟɫɬɶ ɦɧɨɠɟɫɬɜɨ, ɫɨɫɬɨɹɳɟɟ ɢɡ ɱɟɬɵɪɟɯ ɷɥɟɦɟɧɬɨɜ: 3, 7, 9, a;
– , – ɡɧɚɤɢ ɩɪɢɧɚɞɥɟɠɧɨɫɬɢ ɢ ɜɤɥɸɱɟɧɢɹ, ɧɚɩɪɢɦɟɪ,
7{3, 7}, {3, 7}{3, 7, 9}, {1, 7, 9, 5}{5, 7}, 3{2, 8},
{3}{9, 11, 3, 6}{9, 7, 3, 2},
ɬɨ ɟɫɬɶ ɩɟɪɜɵɣ ɡɧɚɤ ɨɡɧɚɱɚɟɬ ɩɪɢɧɚɞɥɟɠɧɨɫɬɶ ɦɧɨɠɟɫɬɜɭ ɨɞɧɨɝɨ ɷɥɟɦɟɧɬɚ,
ɜɬɨɪɨɣ ɡɧɚɤ – ɤɨɝɞɚ ɪɟɱɶ ɢɞɟɬ ɨ ɜɤɥɸɱɟɧɢɢ ɨɞɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɜ ɞɪɭɝɨɟ;
– , , \ – ɡɧɚɤɢ ɩɟɪɟɫɟɱɟɧɢɹ, ɨɛɴɟɞɢɧɟɧɢɹ ɢ ɪɚɡɧɨɫɬɢ ɦɧɨɠɟɫɬɜ. ɇɚɩɪɢɦɟɪ,
{2, 3, 7}{3, 7, 9} = {3, 7}, {2, 3, 7}{3, 7, 9} = {2, 3, 7, 9},
{2, 1, 3, 7}\{3, 7, 9} = {2, 1}.
Ɇɧɨɠɟɫɬɜɚ ɱɚɫɬɨ ɨɛɨɡɧɚɱɚɸɬ ɡɚɝɥɚɜɧɵɦɢ ɥɚɬɢɧɫɤɢɦɢ ɢɥɢ ɞɪɭɝɢɦɢ ɛɭ-
Ⱥ, ȼ, , , R … ɨɛɹɡɚɬɟɥɶɧɨ ɨɬɞɟɥɹɹ ɢɯ ɨɛɨɡɧɚɱɟɧɢɹ ɨɬ ɨɛɨɡɧɚɱɟɧɢɣ
ɤɜɚɦɢ
ɷɥɟɦɟɧɬɨɜ ɦɧɨɠɟɫɬɜ.
ȿɫɥɢ ɦɧɨɠɟɫɬɜɨ Ⱥ ɤɨɧɟɱɧɨ, ɬɨ ɟɫɬɶ ɫɨɫɬɨɢɬ ɢɡ ɤɨɧɟɱɧɨɝɨ ɱɢɫɥɚ ɷɥɟɦɟɧɬɨɜ,
ɬɨ |Ⱥ| – ɨɡɧɚɱɚɟɬ ɱɢɫɥɨ ɟɝɨ ɷɥɟɦɟɧɬɨɜ ɢ ɧɚɡɵɜɚɟɬɫɹ ɦɨɳɧɨɫɬɶɸ ɦɧɨɠɟɫɬɜɚ Ⱥ.
Ɉɛɨɡɧɚɱɟɧɢɟ Ⱥ
ɜɚ , ɧɚɡɵɜɚɟɦɨɝɨ ɭɧɢɜɟɪɫɚɥɶɧɵɦ, ɢ ɤɨɬɨɪɨɟ ɜɤɥɸɱɚɟɬ ɜ ɫɟɛɹ Ⱥ ɢ ɞɪɭɝɢɟ
ɦɧɨɠɟɫɬɜɚ, ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɜ ɤɨɧɤɪɟɬɧɨɣ ɡɚɞɚɱɟ. Ɍɨɝɞɚ Ⱥ
ɫ
ɨɛɵɱɧɨ ɩɪɢɦɟɧɹɟɬɫɹ ɩɪɢ ɧɚɥɢɱɢɢ ɧɟɤɨɬɨɪɨɝɨ ɦɧɨɠɟɫɬ-
ɫ
= \ Ⱥ.
Ⱥ×ȼ = {(a, b) | aA, bB}. Ɍɚɤɨɟ ɦɧɨɠɟɫɬɜɨ ɩɚɪ ɧɚɡɵɜɚɸɬɫɹ ɞɟɤɚɪɬɨɜɵɦ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ.
ɉɪɢɜɟɞɟɦ ɧɟɤɨɬɨɪɵɟ ɨɛɨɡɧɚɱɟɧɢɹ, ɱɚɫɬɨ ɜɫɬɪɟɱɚɸɳɢɯɫɹ, ɦɧɨɠɟɫɬɜ:
N = {1, 2, 3, … } – ɦɧɨɠɟɫɬɜɨ ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɟɥ,
Z = {0, ±1, ±2, … } – ɦɧɨɠɟɫɬɜɨ ɰɟɥɵɯ ɱɢɫɟɥ,
Q = {m/nŇnN, mZ} – ɦɧɨɠɟɫɬɜɨ ɪɚɰɢɨɧɚɥɶɧɵɯ ɱɢɫɟɥ.
Ɉɱɟɜɢɞɧɨ, ɱɬɨ NZQ
ɥɚɦɢ. Ⱥ ɜɫɟ ɷɬɢ ɱɢɫɥɚ ɧɚɡɵɜɚɸɬɫɹ ɞɟɣɫɬɜɢɬɟɥɶɧɵɦɢ ɱɢɫɥɚɦɢ. ɗɬɨ ɦɧɨɠɟɫɬɜɨ
ɱɚɫɬɨ ɨɛɨɡɧɚɱɚɸɬ ɛɭɤɜɨɣ R = (–f, f).
. ɑɢɫɥɚ xQ ɧɚɡɵɜɚɸɬɫɹ ɢɪɪɚɰɢɨɧɚɥɶɧɵɦɢ ɱɢɫ-
Z+ = N {0} = {0, 1, 2, … } – ɦɧɨɠɟɫɬɜɨ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɯ ɰɟɥɵɯ ɱɢɫɟɥ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, NZ
ZQR.
+
9
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
– ɩɭɫɬɨɟ ɦɧɨɠɟɫɬɜɨ, ɬɨ ɟɫɬɶ ɦɧɨɠɟɫɬɜɨ, ɧɟ ɫɨɞɟɪɠɚɳɟɟ ɧɢɤɚɤɢɯ
ɷɥɟɦɟɧɬɨɜ. Ɉɧɨ ɜɜɟɞɟɧɨ ɞɥɹ ɭɞɨɛɫɬɜɚ ɢ ɢɝɪɚɟɬ ɬɚɤɭɸ ɠɟ ɪɨɥɶ ɜ ɬɟɨɪɢɢ ɦɧɨɠɟɫɬɜ, ɤɚɤ ɱɢɫɥɨ ɧɨɥɶ ɜ ɚɪɢɮɦɟɬɢɤɟ. ɇɚɩɪɢɦɟɪ,
M = , M = M, M\ = M, \M = , [1,2](2,f) = ,
ɝɞɟ M – ɩɪɨɢɡɜɨɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ. ɉɪɢɧɹɬɨ ɫɱɢɬɚɬɶ, ɱɬɨ
ɹɜɥɹɟɬɫɹ ɱɚɫɬɶɸ
(ɩɨɞɦɧɨɠɟɫɬɜɨɦ) ɥɸɛɨɝɨ ɦɧɨɠɟɫɬɜɚ.
Ⱥ ɜɨɬ ɡɧɚɤɢ, ɤɨɬɨɪɵɟ ɦɵ ɛɭɞɟɦ ɭɩɨɬɪɟɛɥɹɬɶ ɩɪɢ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚɯ ɭɬɜɟɪɠɞɟɧɢɣ ɢ ɪɟɲɟɧɢɹɯ ɡɚɞɚɱ. ɗɬɨ ɡɧɚɤɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɥɨɝɢɤɢ:
, – ɢɦɩɥɢɤɚɰɢɢ (implicatio – ɫɩɥɟɬɟɧɢɟ ɢɥɢ imply – ɜɥɟɱɶ ɡɚ ɫɨɛɨɣ):
x = 2 x{3, 2, 4} (ɟɫɥɢ x = 2, ɬɨ x{3, 2, 4}); x > 3 x > 4 (ɧɟɪɚɜɟɧɫɬɜɨ x > 4
ɜɥɟɱɟɬ (ɢɦɩɥɢɰɢɪɭɟɬ) x > 3);
– ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɶ
(aequus – ɪɚɜɧɵɣ ɢ valeo – ɡɧɚɱɟɧɢɟ, ɰɟɧɚ):
(x – 1)(x – 2) = 0 x{1, 2} ((x – 1)(x – 2) = 0 ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ (ɜ ɬɨɦ
ɢ ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ), ɤɨɝɞɚ x ɪɚɜɟɧ 1 ɢɥɢ 2, ɬɨ ɟɫɬɶ {1, 2} ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ
ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ);
– ɞɢɡɴɸɧɤɰɢɹ (disjunctio – ɪɚɡɨɛɳɟɧɢɟ, ɪɚɡɥɢɱɢɟ, ɱɢɬɚɟɬɫɹ: «ɢɥɢ»):
(x – 1)(x – 2) = 0 x = 1 x = 2 (x = 1 ɢɥɢ
x = 2 ɢ ɬɨɥɶɤɨ ɨɧɢ ɹɜɥɹɸɬɫɹ ɪɟɲɟ-
ɧɢɹɦɢ ɭɪɚɜɧɟɧɢɹ (x – 1)(x – 2) = 0);
, & – ɤɨɧɴɸɧɤɰɢɹ (conjunctio – ɫɨɸɡ, ɫɜɹɡɶ, ɱɢɬɚɟɬɫɹ: «ɢ»):
(x > 1 x < 3) (xN) x=2 (ɧɚɬɭɪɚɥɶɧɨɟ ɱɢɫɥɨ
x, ɛɨɥɶɲɟɟ 1
ɢ ɦɟɧɶɲɟɟ ɬɪɟɯ, ɟɫɬɶ x = 2); (x + 1)(x – 1½)(x – 4) = 0 xZ x = –1 x = 4
x{–1, 4} x{2, 5, 1, 3, 4}.
ɑɬɨɛɵ ɧɟ ɛɵɥɨ ɪɚɡɧɨɱɬɟɧɢɣ, ɞɟɫɹɬɢɱɧɵɟ ɞɪɨɛɢ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɬɚɤ:
3½ = 3.5, ¾ = 0.75 = .75, … – ɨɬɞɟɥɹɹ ɞɪɨɛɧɭɸ ɱɚɫɬɶ ɱɢɫɥɚ ɨɬ ɟɝɨ ɰɟɥɨɣ ɱɚɫɬɢ
ɬɨɱɤɨɣ ɢ ɢɧɨɝɞɚ ɡɚɩɹɬɨɣ.
Ɉɞɧɢɦ ɢɡ ɨɫɧɨɜɧɵɯ ɩɨɧɹɬɢɣ ɦɚɬɟɦɚɬɢɤɢ ɹɜɥɹɟɬɫɹ ɩɨɧɹɬɢɟ
ɮɭɧɤɰɢɢ.
ɉɭɫɬɶ Ⱥ ɢ ȼ – ɦɧɨɠɟɫɬɜɚ ɨɛɴɟɤɬɨɜ ɥɸɛɨɣ ɩɪɢɪɨɞɵ ɢ ɩɭɫɬɶ ɩɨ ɧɟɤɨɬɨɪɨɦɭ ɡɚɤɨɧɭ f ɤɚɠɞɨɦɭ ɷɥɟɦɟɧɬɭ ɚȺ ɫɬɚɜɢɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɷɥɟɦɟɧɬ bB.
Ɍɨɝɞɚ ɝɨɜɨɪɹɬ, ɱɬɨ ɧɚ ɦɧɨɠɟɫɬɜɟ Ⱥ ɡɚɞɚɧɚ ɮɭɧɤɰɢɹ f ɢɡ Ⱥ ɜ ȼ (f: Ⱥ o ȼ). ɉɪɢ
ɷɬɨɦ ɦɧɨɠɟɫɬɜɨ Ⱥ ɧɚɡɵɜɚɸɬ ɨɛɥɚɫɬɶɸ ɨɩɪɟɞɟɥɟɧɢɹ
ɮɭɧɤɰɢɢ f ; ɷɥɟɦɟɧɬ bB,
ɩɨɫɬɚɜɥɟɧɧɵɣ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɷɥɟɦɟɧɬɭ ɚȺ, ɨɛɨɡɧɚɱɚɟɬɫɹ ɤɚɤ f(a) (b = f(a))
ɢ ɧɚɡɵɜɚɟɬɫɹ ɨɛɪɚɡɨɦ ɷɥɟɦɟɧɬɚ ɚȺ ɮɭɧɤɰɢɢ f ; ɞɥɹ ɩɨɞɦɧɨɠɟɫɬɜɚ ɏȺ
ɦɧɨɠɟɫɬɜɨ
f(ɏ) = {bB | aɏ: f(a) = b}
ɧɚɡɵɜɚɟɬɫɹ ɨɛɪɚɡɨɦ ɦɧɨɠɟɫɬɜɚ ɏ ɮɭɧɤɰɢɢ f; ɞɥɹ ɷɥɟɦɟɧɬɚ b
B, ɦɧɨɠɟɫɬɜɨ
ɚȺ ɬɚɤɢɯ, ɱɬɨ b=f(a) ɧɚɡɵɜɚɟɬɫɹ ɩɪɨɨɛɪɚɡɨɦ bB ɮɭɧɤɰɢɢ f ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ:
-–1
f
(b) = {aA: f(a) = b} f –1(b) = {ɚȺ : f(a) = b}.
10
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