Дискретная математика. Учебное пособие-1
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ȾɂɋɄɊȿɌɇȺə ɆȺɌȿɆȺɌɂɄȺ
ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ
Ɇɨɫɤɜɚ ȼɨɥɨɝɞɚ
©ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹª
2024
1
ɍȾɄ 51
ȻȻɄ 22.144 Ʉ19
Ɋɟɰɟɧɡɟɧɬɵ ɤɚɧɞɢɞɚɬ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ ɞɨɰɟɧɬ ɤɚɮɟɞɪɵ ɫɢɫɬɟɦ ɚɜɬɨɦɚɬɢɱɟɫɤɨɝɨ
ɭɩɪɚɜɥɟɧɢɹ Ʉɚɥɭɠɫɤɨɝɨ ɮɢɥɢɚɥɚ ɆȽɌɍ ɢɦ ɇ ɗ Ȼɚɭɦɚɧɚ ɋɟɪɟɝɢɧɚ ȿɥɟɧɚ ȼɥɚɞɢɦɢɪɨɜɧɚ ɤɚɧɞɢɞɚɬ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ ɞɨɰɟɧɬ ɤɚɮɟɞɪɵ ɮɢɡɢɤɢ ɢ ɦɚɬɟɦɚɬɢɤɢ ɎȽȻɈɍ ȼɈ ©Ʉɚɥɭɠɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ ɭɧɢɜɟɪɫɢɬɟɬ ɢɦ Ʉ ɗ ɐɢɨɥɤɨɜɫɤɨɝɨª ɋɚɜɨɫɶɤɢɧɚ ɂɪɢɧɚ ɂɜɚɧɨɜɧɚ
Ʉɚɧɚɪɟɣɤɢɧ, Ⱥ ɂ Ʉ19 Ⱦɢɫɤɪɟɬɧɚɹ ɦɚɬɟɦɚɬɢɤɚ ɭɱɟɛɧɨɟ ɩɨɫɨɛɢɟ Ⱥ ɂ Ʉɚɧɚɪɟɣɤɢɧ – Ɇɨɫɤɜɚ ;
ȼɨɥɨɝɞɚ ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ 4. – 100 ɫ ɢɥ ɬɚɛɥ
ISBN 978-5-9729-1739-6
ɂɡɥɨɠɟɧɵ ɨɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɬɟɨɪɢɢ ɦɧɨɠɟɫɬɜ ɨɛɳɢɟ ɫɜɨɣɫɬɜɚ ɩɨɫɬɪɨɟɧɧɵɯ ɧɚ ɟɟ ɛɚɡɟ ɞɢɫɤɪɟɬɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɫɬɪɭɤɬɭɪ Ɋɚɫɫɦɨɬɪɟɧɵ ɤɨɧɤɪɟɬɧɵɟ ɩɪɢɦɟɪɵ ɫɬɪɭɤɬɭɪ Ƚɚɥɭɚ ɝɪɭɩɩ ɤɨɥɟɰ ɢ ɩɨɥɟɣ ɜ ɬɨɦ ɱɢɫɥɟ ɛɭɥɟɜɨɣ ɚɥɝɟɛɪɵ ɧɚɢɛɨɥɟɟ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɟɦɵɟ ɜ ɩɪɢɥɨɠɟɧɢɹɯ ɞɢɫɤɪɟɬɧɨɣ ɦɚɬɟɦɚɬɢɤɢ ɤɨɧɟɱɧɨɣ ɝɟɨɦɟɬɪɢɢ ɬɟɨɪɢɢ ɤɨɞɢɪɨɜɚɧɢɹ ɫɟɬɟɣ ɢ ɤɨɧɟɱɧɵɯ ɚɜɬɨɦɚɬɨɜ
Ⱦɥɹ ɫɬɭɞɟɧɬɨɜ ɜɫɟɯ ɧɚɩɪɚɜɥɟɧɢɣ ɢ ɫɩɟɰɢɚɥɶɧɨɫɬɟɣ ɩɨ ɞɢɫɰɢɩɥɢɧɟ ©Ⱦɢɫɤɪɟɬɧɚɹ ɦɚɬɟɦɚɬɢɤɚª
ɍȾɄ ȻȻɄ
ISBN 978-5-9729-1739-6 |
Ʉɚɧɚɪɟɣɤɢɧ Ⱥ ɂ. 24 |
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ɂɡɞɚɬɟɥɶɫɬɜɨ ©ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹª 24 |
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Ɉɮɨɪɦɥɟɧɢɟ ɂɡɞɚɬɟɥɶɫɬɜɨ ©ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹª 24 |
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2 |
ʝʒʚʏʑʚʔʜʗʔ |
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ȼȼȿȾȿɇɂȿ................................................................................................................. |
4 |
ɊȺɁȾȿɅ , ɆɇɈɀȿɋɌȼȺ ɎɍɇɄɐɂɂ ɈɌɇɈɒȿɇɂə................................. |
6 |
Ʌɟɤɰɢɹ ʋ Ɇɧɨɠɟɫɬɜɚ ɢ ɨɩɟɪɚɰɢɢ ɧɚɞ ɧɢɦɢ..................................................... |
6 |
Ʌɟɤɰɢɹ ʋ 2. ɋɨɨɬɜɟɬɫɬɜɢɹ ɢ ɮɭɧɤɰɢɢ................................................................. |
13 |
Ʌɟɤɰɢɹ ʋ 3. Ɉɬɧɨɲɟɧɢɹ ɢ ɢɯ ɫɜɨɣɫɬɜɚ ............................................................... |
18 |
Ʌɟɤɰɢɹ ʋ 4. Ɉɫɧɨɜɧɵɟ ɜɢɞɵ ɨɬɧɨɲɟɧɢɣ............................................................ |
21 |
ɊȺɁȾȿɅ ,, ȼȼȿȾȿɇɂȿ ȼ ɈȻɓɍɘ ȺɅȽȿȻɊɍ.............................................. |
25 |
Ʌɟɤɰɢɹ ʋ 5. ɗɥɟɦɟɧɬɵ ɨɛɳɟɣ ɚɥɝɟɛɪɵ............................................................... |
25 |
Ʌɟɤɰɢɹ ʋ 6. Ɋɚɡɥɢɱɧɵɟ ɜɢɞɵ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɫɬɪɭɤɬɭɪ................................... |
29 |
ɊȺɁȾȿɅ ,,, ȼȼȿȾȿɇɂȿ ȼ ɅɈȽɂɄɍ................................................................. |
35 |
Ʌɟɤɰɢɹ ʋ 7. ɗɥɟɦɟɧɬɵ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɥɨɝɢɤɢ ................................................ |
35 |
Ʌɟɤɰɢɹ ʋ 8 Ʌɨɝɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ ....................................................................... |
39 |
Ʌɟɤɰɢɹ ʋ 9. Ȼɭɥɟɜɵ ɚɥɝɟɛɪɵ................................................................................ |
44 |
Ʌɟɤɰɢɹ ʋ 10. Ȼɭɥɟɜɵ ɚɥɝɟɛɪɵ ɢ ɬɟɨɪɢɹ ɦɧɨɠɟɫɬɜ ............................................ |
50 |
Ʌɟɤɰɢɹ ʋ 11. ɉɨɥɧɨɬɚ ɢ ɡɚɦɤɧɭɬɨɫɬɶ.................................................................. |
54 |
Ʌɟɤɰɢɹ ʋ 12. əɡɵɤ ɥɨɝɢɤɢ ɩɪɟɞɢɤɚɬɨɜ............................................................... |
60 |
Ʌɟɤɰɢɹ ʋ 13. Ʉɨɦɛɢɧɚɬɨɪɢɤɚ............................................................................... |
65 |
ɊȺɁȾȿɅ ,V ɌȿɈɊɂə ȽɊȺɎɈȼ........................................................................... |
71 |
Ʌɟɤɰɢɹ ʋ 14. Ƚɪɚɮɵ ɨɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɢ ɨɩɟɪɚɰɢɢ........................................ |
71 |
Ʌɟɤɰɢɹ ʋ 15. Ɇɚɪɲɪɭɬɵ ɰɟɩɢ ɢ ɰɢɤɥɵ............................................................. |
78 |
Ʌɟɤɰɢɹ ʋ 16. ɇɟɤɨɬɨɪɵɟ ɤɥɚɫɫɵ ɝɪɚɮɨɜ ɢ ɢɯ ɱɚɫɬɟɣ........................................ |
84 |
ɉɊȺɄɌɂɑȿɋɄɂȿ ɁȺȾȺɇɂə ȾɅə ɋȺɆɈɄɈɇɌɊɈɅə............................... |
89 |
ɋɉɂɋɈɄ ɅɂɌȿɊȺɌɍɊɕ..................................................................................... |
97 |
3
ʑʑʔʓʔʜʗʔ
Ɉɫɧɨɜɭ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɚɬɟɦɚɬɢɤɢ ɧɚɩɪɢɦɟɪ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɢ ɢɧɬɟɝɪɚɥɶɧɨɝɨ ɢɫɱɢɫɥɟɧɢɹ ɫɨɫɬɚɜɥɹɸɬ ɩɨɧɹɬɢɹ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɢ ɛɟɫɤɨɧɟɱɧɨɫɬɢȾɟɤɚɪɬ ɇɶɸɬɨɧ Ʌɟɣɛɧɢɰ ɗɬɢ ɩɨɧɹɬɢɹ ɨɩɪɟɞɟɥɹɸɬ ɨɫɧɨɜɧɨɟ ɫɜɨɣɫɬɜɨ ɱɢɫ ɥɨɜɨɣ ɩɪɹɦɨɣ ɦɧɨɠɟɫɬɜɨ ɬɨɱɟɤ ɥɸɛɨɝɨ ɨɬɪɟɡɤɚ ɩɪɹɦɨɣ ɜɫɸɞɭ ɩɥɨɬɧɨ ɬɨ ɟɫɬɶ ɥɸɛɚɹ ɨɤɪɟɫɬɧɨɫɬɶ ɥɸɛɨɣ ɬɨɱɤɢ ɷɬɨɣ ɩɪɹɦɨɣ ɫɨɞɟɪɠɢɬ ɛɟɫɤɨɧɟɱɧɨɟ ɧɟɫɱɟɬɧɨɟ ɦɧɨɠɟɫɬɜɨ ɬɨɱɟɤ ɷɬɨɣ ɩɪɹɦɨɣ Ⱦɚɥɟɟ ɫɥɟɞɭɸɬ ɨɩɪɟɞɟɥɟɧɢɹ ɩɪɟɞɟɥɚ ɱɢɫɥɨɜɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɢ ɮɭɧɤɰɢɢ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɢ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɫɬɢ – ɨɫɧɨɜɧɵɯ ɚɬɪɢɛɭɬɨɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɚɬɟɦɚɬɢɤɢ Ⱥɥɶɬɟɪɧɚɬɢɜɨɣ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɢ ɛɟɫ ɤɨɧɟɱɧɨɫɬɢ ɹɜɥɹɸɬɫɹ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɨ ɞɢɫɤɪɟɬɧɨɦ ɪɚɡɴɟɞɢɧɟɧɧɨɦ ɢ ɤɨɧɟɱ ɧɨɦ ɂɧɬɭɢɬɢɜɧɨ ɞɢɫɤɪɟɬɧɨɟ ɦɧɨɠɟɫɬɜɨ ɷɥɟɦɟɧɬɨɜ ɞɨɥɠɧɨ ɛɵɬɶ ɫɱɟɬɧɵɦ ɬɨ ɟɫɬɶ ɦɟɠɞɭ ɷɥɟɦɟɧɬɚɦɢ ɞɢɫɤɪɟɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɢ ɦɧɨɠɟɫɬɜɚ ɧɚɬɭɪɚɥɶɧɵɯ ɱɢ ɫɟɥ ɞɨɥɠɧɨ ɫɭɳɟɫɬɜɨɜɚɬɶ ɜɡɚɢɦɧɨɨɞɧɨɡɧɚɱɧɨɟ ɫɨɨɬɜɟɬɫɬɜɢɟ ɬɚɤɨɟ ɱɬɨ ɷɥɟɦɟɧɬɵ ɞɢɫɤɪɟɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɦɨɝɭɬ ɛɵɬɶ ɩɪɨɧɭɦɟɪɨɜɚɧɵ ȿɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɤɨ ɧɟɱɧɵɣ ɧɨɦɟɪ ɩɨɫɥɟɞɧɟɝɨ ɷɥɟɦɟɧɬɚ ɧɟɤɨɬɨɪɨɝɨ ɞɢɫɤɪɟɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɬɨ ɨɧɨ ɤɨɧɟɱɧɨ ɫɨɞɟɪɠɢɬ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɷɥɟɦɟɧɬɨɜ ɟɫɥɢ ɬɚɤɨɣ ɧɨɦɟɪ ɧɟ ɫɭɳɟɫɬɜɭ ɟɬ ɬɨ ɦɧɨɠɟɫɬɜɨ ɛɟɫɤɨɧɟɱɧɨɟ ɧɨ ɫɱɟɬɧɨɟ
Ⱦɢɫɤɪɟɬɧɚɹ ɦɚɬɟɦɚɬɢɤɚ ȾɆ – ɦɚɬɟɦɚɬɢɤɚ ɨɫɧɨɜɚɧɧɚɹ ɧɚ ɩɨɧɹɬɢɹɯ ɞɢɫ ɤɪɟɬɧɨɫɬɢ ɢ ɤɨɧɟɱɧɨɫɬɢ ȾɆ ɢɡɭɱɚɟɬ ɫɜɨɣɫɬɜɚ ɮɭɧɤɰɢɨɧɚɥɶɧɵɯ ɨɬɧɨɲɟɧɢɣ ɨɩɪɟɞɟɥɟɧɧɵɯ ɧɚ ɞɢɫɤɪɟɬɧɵɯ ɢ ɤɨɧɟɱɧɵɯ ɦɧɨɠɟɫɬɜɚɯ ɏɨɬɹ ɜ ȾɆ ɞɨɩɭɫɬɢɦ ɩɪɟɞɟɥɶɧɵɣ ɩɟɪɟɯɨɞ ɤ ɫɱɟɬɧɵɦ ɛɟɫɤɨɧɟɱɧɵɦ ɦɧɨɠɟɫɬɜɚɦ ɦɧɨɝɢɟ ɩɪɢɜɵɱɧɵɟ ɪɟɡɭɥɶɬɚɬɵ ɩɨɥɭɱɟɧɧɵɟ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɚɬɟɦɚɬɢɤɟ ɚɪɢɮɦɟɬɢɤɟ ɚɥɝɟɛɪɟ ɝɟɨ ɦɟɬɪɢɢ ɩɪɢɨɛɪɟɬɚɸɬ ɜ ȾɆ ɫɨɜɟɪɲɟɧɧɨ ɢɧɨɣ ɜɢɞ ɢ ɫɦɵɫɥ ɨɩɟɪɚɰɢɢ ɧɚ ɦɧɨɠɟ ɫɬɜɚɯ ɱɢɫɟɥ ɜɟɤɬɨɪɨɜ ɦɚɬɪɢɰ .
ɋɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɤɥɚɫɫɢɱɟɫɤɨɣ ɢ ɞɢɫɤɪɟɬɧɨɣ ɦɚɬɟɦɚɬɢɤɨɣ ɬɚɤɨɟ ɠɟ ɤɚɤ ɦɟɠɞɭ ɤɥɚɫɫɢɱɟɫɤɨɣ ɢ ɤɜɚɧɬɨɜɨɣ ɦɟɯɚɧɢɤɨɣ ɜ ɮɢɡɢɤɟ ɩɟɪɜɚɹ ɨɩɢɫɵɜɚɟɬ ɫɜɨɣɫɬɜɚ ɩɥɨɬɧɵɯ ɦɚɤɪɨɫɬɪɭɤɬɭɪ ɜɬɨɪɚɹ – ɦɢɤɪɨɫɬɪɭɤɬɭɪ Ⱦɢɫɤɪɟɬɧɨɫɬɶ ɢ ɤɨ ɧɟɱɧɨɫɬɶ ɹɜɥɹɸɬɫɹ ɩɟɪɜɢɱɧɵɦɢ ɩɨɧɹɬɢɹɦɢ ɜ ɩɨɡɧɚɜɚɬɟɥɶɧɨɣ ɞɟɹɬɟɥɶɧɨɫɬɢ ɱɟ ɥɨɜɟɤɚ ɢ ɯɚɪɚɤɬɟɪɢɡɭɸɬ ɦɢɤɪɨɫɬɪɭɤɬɭɪɭ ɟɝɨ ɦɵɲɥɟɧɢɹ ɇɟɩɪɟɪɵɜɧɨɫɬɶ ɢ ɛɟɫ ɤɨɧɟɱɧɨɫɬɶ ɩɪɨɹɜɥɹɸɬɫɹ ɤɚɤ ɪɟɡɭɥɶɬɚɬ ɢɝɧɨɪɢɪɨɜɚɧɢɹ ɩɪɟɪɵɜɢɫɬɨɫɬɢ ɪɟɚɥɶɧɨɝɨ ɦɢɪɚ ɇɚɩɪɢɦɟɪ ɢɧɬɟɝɪɚɥ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɚɬɟɦɚɬɢɤɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɩɪɟ ɞɟɥ ɤɨɧɟɱɧɨɣ ɧɨ ɧɟɩɪɟɪɵɜɧɨɣ ɢɧɬɟɝɪɚɥɶɧɨɣ ɫɭɦɦɵ ɂɫɬɨɪɢɱɟɫɤɢ ɩɪɨɰɟɫɫ ɩɨ ɡɧɚɧɢɹ ɨɤɪɭɠɚɸɳɟɝɨ ɦɢɪɚ ɧɚɱɢɧɚɥɫɹ ɜ ɩɪɟɞɫɬɚɜɥɟɧɢɹɯ ɨ ɟɝɨ ɞɢɫɤɪɟɬɧɨɫɬɢ ɢ ɤɨɧɟɱɧɨɫɬɢ Ɂɟɧɨɧ Ⱦɟɦɨɤɪɢɬ ɉɢɮɚɝɨɪ Ⱥɪɢɫɬɨɬɟɥɶ ɇɚɱɢɧɚɹ ɫ ɷɩɨɯɢ ȼɨɡɪɨɠ ɞɟɧɢɹ Ⱦɟɤɚɪɬ ɇɶɸɬɨɧ Ʌɟɣɛɧɢɰ ɩɨɧɹɬɢɟ ɞɢɫɤɪɟɬɧɨɫɬɢ ɨɬɫɬɭɩɢɥɨ ɧɚ ɜɬɨɪɨɣ ɩɥɚɧ ɚ ɧɚ ɩɟɪɜɨɟ ɦɟɫɬɨ ɩɪɟɬɟɧɞɨɜɚɥɢ ɧɟɩɪɟɪɵɜɧɵɟ ɞɟɤɚɪɬɨɜɵ ɩɟɪɟɦɟɧɧɵɟ ɮɭɧɤɰɢɢ ɢ ɩɪɨɰɟɫɫɵ
4
Ȼɭɪɧɵɣ ɢɧɬɟɪɟɫ ɤ ȾɆ ɨɛɨɡɧɚɱɢɥɫɹ ɜ ɤɨɧɰɟ XX ɜɟɤɚ ɢ ɫɬɢɦɭɥɢɪɨɜɚɥɫɹ ɫɨ ɡɞɚɧɢɟɦ ɗɐȼɆ ɧɨɜɵɦɢ ɬɨɱɤɚɦɢ ɡɪɟɧɢɹ ɢ ɧɨɜɵɦɢ ɨɬɤɪɵɬɢɹɦɢ ɫɪɟɞɢ ɤɨɬɨɪɵɯ ɧɚ ɩɟɪɜɨɟ ɦɟɫɬɨ ɫɥɟɞɭɟɬ ɩɨɫɬɚɜɢɬɶ ɜɵɹɫɧɟɧɢɟ ɱɢɫɬɨ ɞɢɫɤɪɟɬɧɨɣ ɫɬɪɭɤɬɭɪɵ ɧɚɫɥɟɞɫɬɜɟɧɧɨɝɨ ɜɟɳɟɫɬɜɚ Ʉɪɢɤ ɍɨɬɫɨɧ ɪɚɫɲɢɮɪɨɜɤɭ ɝɟɧɟɬɢɱɟɫɤɨɝɨ ɤɨɞɚɢɦɟɸɳɟɝɨ ɞɢɫɤɪɟɬɧɭɸ ɫɬɪɭɤɬɭɪɭ ɞɢɫɤɪɟɬɧɵɣ ɯɚɪɚɤɬɟɪ ɨɛɦɟɧɚ ɢɧɮɨɪɦɚɰɢɟɣ ɦɟɠɞɭ ɧɟɣɪɨɧɚɦɢ ɝɨɥɨɜɧɨɝɨ ɦɨɡɝɚ Ɇɨɞɟɥɢɪɨɜɚɧɢɟ ɷɬɢɯ ɩɪɨɰɟɫɫɨɜ ɜ ɰɢɮɪɨɜɵɯ ɫɢɫɬɟɦɚɯ ɜɨɫɩɪɢɹɬɢɹ ɯɪɚɧɟɧɢɹ ɩɟɪɟɪɚɛɨɬɤɢ ɢ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɢɧɮɨɪɦɚɰɢɢ ɜ ɦɚɲɢɧɚɯ ɠɢɜɵɯ ɨɪɝɚɧɢɡɦɚɯ ɢ ɢɯ ɨɛɴɟɞɢɧɟɧɢɹɯ ɩɨɪɨɞɢɥɨ ɰɟɥɵɣ ɪɹɞ ɧɨɜɵɯ ɧɚɭɱɧɵɯ ɞɢɫɰɢɩɥɢɧ ɬɟɨɪɢɹ ɢɧɮɨɪɦɚɰɢɢ ɤɨɞɢɪɨɜɚɧɢɹ ɤɢɛɟɪɧɟɬɢɤɢ ɢ ɬ ɩ ɨɫɧɨɜɭ ɢɡɭɱɟɧɢɹ ɤɨɬɨɪɵɯ ɫɨɫɬɚɜɥɹɟɬ ȾɆ ȼ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ȾɆ ɜɤɥɸɱɚɟɬ ɜ ɫɟɛɹ ɬɚɤɢɟ ɦɨɳɧɵɟ ɢɧɫɬɪɭɦɟɧɬɵ ɤɚɤ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɥɨɝɢɤɚ ɬɟɨɪɢɹ ɦɚɬɟɦɚɬɢɱɟ ɫɤɢɯ ɫɬɪɭɤɬɭɪ ɝɪɚɮɨɜ ɢ ɫɟɬɟɣ ɚɥɝɨɪɢɬɦɨɜ ɤɨɦɛɢɧɚɬɨɪɢɤɚ Ɋɨɥɶ ɢ ɡɧɚɱɟɧɢɟ ȾɆ ɜ ɪɚɡɜɢɬɢɢ ɫɨɜɪɟɦɟɧɧɨɝɨ ɟɫɬɟɫɬɜɨɡɧɚɧɢɹ ɢ ɝɭɦɚɧɢɬɚɪɧɵɯ ɧɚɭɤ ɬɪɭɞɧɨ ɩɟɪɟɨɰɟ ɧɢɬɶ ɛɨɥɶɲɢɧɫɬɜɨ ɤɚɱɟɫɬɜɟɧɧɨ ɧɨɜɵɯ ɨɬɤɪɵɬɢɣ XXI ɜɟɤɚ ɫɜɹɡɚɧɨ ɫ ɚɞɟɤɜɚɬ ɧɵɦ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɨɛɴɟɤɬɨɜ ɨɤɪɭɠɚɸɳɟɝɨ ɦɢɪɚ – ɨɛɴɟɤɬɨɜ ȾɆ.
Ɉɫɧɨɜɧɨɣ ɭɩɨɪ ɩɪɢ ɷɬɨɦ ɞɟɥɚɟɬɫɹ ɧɚ ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɢ ɚɫ ɫɨɰɢɚɰɢɢ ɨɫɧɨɜɧɵɯ ɩɨɧɹɬɢɣ ɢ ɪɟɡɭɥɶɬɚɬɨɜ ɬɟɨɪɟɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ ɋɨɨɬɜɟɬ ɫɬɜɟɧɧɨ ɤɚɠɞɨɟ ɜɧɨɜɶ ɜɜɨɞɢɦɨɟ ɩɨɧɹɬɢɟ ɢ ɩɨɥɭɱɚɟɦɵɣ ɬɟɨɪɟɬɢɱɟɫɤɢɣ ɪɟɡɭɥɶ ɬɚɬ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɩɪɢɦɟɪɨɦ ɪɟɲɟɧɢɹ ɤɨɧɤɪɟɬɧɨɣ ɡɚɞɚɱɢ ɉɨ ɜɨɡɦɨɠɧɨɫɬɢ ɫ ɰɟɥɶɸ ɛɨɥɟɟ ɭɫɩɟɲɧɨɝɨ ɭɫɜɨɟɧɢɹ ɢ ɡɚɩɨɦɢɧɚɧɢɹ ɦɚɬɟɪɢɚɥɚ ɨɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɨɩɪɟɞɟɥɟɧɢɹ ɬɟɨɪɟɦɵ ɢ ɜɵɜɨɞɵ ɩɨ ɤɚɠɞɨɦɭ ɪɚɡɞɟɥɭ ɧɭɦɟɪɭɸɬɫɹ ɢ ɜɵɞɟ ɥɹɸɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɧɨɜɧɨɝɨ ɬɟɤɫɬɚ
ɉɟɪɜɚɹ ɝɥɚɜɚ ɩɨɫɨɛɢɹ ɩɨɫɜɹɳɟɧɚ ɢɡɥɨɠɟɧɢɸ ɤɥɚɫɫɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɦɧɨ ɠɟɫɬɜ ɨɫɧɨɜɚɧɧɨɣ ɧɚ ɞɜɭɯ ɬɚɤ ɧɚɡɵɜɚɟɦɵɯ ɢɧɬɭɢɬɢɜɧɵɯ ɩɪɢɧɰɢɩɚɯ Ʉɚɧɬɨɪɚ ɨɛɴɟɦɧɨɫɬɢ ɢ ɚɛɫɬɪɚɤɰɢɢ Ɂɞɟɫɶ ɨɫɨɛɨɟ ɜɧɢɦɚɧɢɟ ɭɞɟɥɹɟɬɫɹ ɪɚɡɥɢɱɢɸ ɜ ɩɨɧɹ ɬɢɹɯ ɨɬɧɨɲɟɧɢɹ ɩɪɢɧɚɞɥɟɠɧɨɫɬɢ ɢ ɜɤɥɸɱɟɧɢɹ ɱɚɫɬɨ ɨɬɨɠɞɟɫɬɜɥɹɟɦɵɯ ɫɬɭ ɞɟɧɬɚɦɢ ȼ ɷɬɨɣ ɠɟ ɝɥɚɜɟ ɩɨɞɪɨɛɧɨ ɪɚɫɫɦɨɬɪɟɧɵ ɫɩɟɰɢɚɥɶɧɵɟ ɜɢɞɵ ɨɬɧɨɲɟ ɧɢɣ ɧɚ ɞɢɫɤɪɟɬɧɵɯ ɦɧɨɠɟɫɬɜɚɯ ɹɜɥɹɸɳɢɯɫɹ ɨɫɧɨɜɧɵɦ ɢɧɫɬɪɭɦɟɧɬɨɦ ɚɧɚɥɢɡɚ ɞɢɫɤɪɟɬɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɫɬɪɭɤɬɭɪ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɢ ɩɨɪɹɞɤɚ
ȼɨ ɜɬɨɪɨɣ ɝɥɚɜɟ ɩɨɫɨɛɢɹ ɪɚɫɫɦɨɬɪɟɧ ɪɹɞ ɩɪɢɦɟɪɨɜ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɫɬɪɭɤ ɬɭɪ ɩɨɫɬɪɨɟɧɧɵɯ ɧɚ ɤɨɧɟɱɧɵɯ ɦɧɨɠɟɫɬɜɚɯ ɢ ɩɨɤɚɡɚɧɨ ɢɯ ɨɬɥɢɱɢɟ ɨɬ ɚɧɚɥɨɝɢɱ ɧɵɯ ɫɬɪɭɤɬɭɪ ɧɚ ɦɧɨɠɟɫɬɜɟ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ ɢ ɯɨɪɨɲɨ ɡɧɚɤɨɦɵɦ ɫɬɭɞɟɧ ɬɚɦ ɢɡ ɤɭɪɫɚ ɜɵɫɲɟɣ ɦɚɬɟɦɚɬɢɤɢ
ɇɚɤɨɧɟɰ ɜ ɡɚɤɥɸɱɢɬɟɥɶɧɨɣ ɬɪɟɬɶɟɣ ɝɥɚɜɟ ɩɟɪɜɨɣ ɱɚɫɬɢ ɩɨɫɨɛɢɹ ɪɚɫɫɦɨɬ ɪɟɧ ɤɥɚɫɫɢɱɟɫɤɢɣ ɨɛɪɚɡɟɰ ɞɢɫɤɪɟɬɧɨɣ ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɫɬɪɭɤɬɭɪɵ ɛɭɥɟɜɨɣ ɚɥ ɝɟɛɪɵ ɢ ɨɩɪɟɞɟɥɟɧɧɵɯ ɜ ɧɟɣ ɛɭɥɟɜɵɯ ɮɭɧɤɰɢɣ ɗɬɚ ɚɥɝɟɛɪɚ ɢ ɮɭɧɤɰɢɢ ɩɨɥɭɱɢ ɥɢ ɲɢɪɨɤɨɟ ɩɪɢɦɟɧɟɧɢɟ ɜ ɫɢɦɨɞɢɱɟɫɤɨɣ ɥɨɝɢɤɟ ɢ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɬɟɯɧɢɤɟ ɨɫɨ ɛɟɧɧɨ ɩɪɢ ɫɢɧɬɟɡɟ ɫɨɜɪɟɦɟɧɧɵɯ ɰɢɮɪɨɜɵɯ ɭɫɬɪɨɣɫɬɜ ɢ ɫɢɫɬɟɦ
5
ʟʏʖʓʔʚ
ʛʜʝʕʔʠʡʑʏǡ ʣʢʜʙʥʗʗǡ ʝʡʜʝʧʔʜʗʮ
ʚˈˍ˙ˋˢ ͒ Ǥ ʛːˑˉˈ˔˕˅˃ ˋ ˑ˒ˈ˓˃˙ˋˋ ː˃ˇ ːˋˏˋ
ɈɋɇɈȼɇɕȿ ɉɈɇəɌɂə ɌȿɈɊɂɂ ɆɇɈɀȿɋɌȼ
Ɉɩɪɟɞɟɥɟɧɢɟ Ɇɧɨɠɟɫɬɜɨɦ Ɇ ɧɚɡɵɜɚɟɬɫɹ ɨɛɴɟɞɢɧɟɧɢɟ ɜ ɟɞɢɧɨɟ ɰɟɥɨɟ ɨɩɪɟɞɟɥɟɧɧɵɯ ɪɚɡɥɢɱɢɦɵɯ ɨɞɧɨɬɢɩɧɵɯ ɨɛɴɟɤɬɨɜ ɚ ɤɨɬɨɪɵɟ ɧɚɡɵɜɚɸɬɫɹ ɷɥɟ ɦɟɧɬɚɦɢ ɦɧɨɠɟɫɬɜɚ
ɚ Ɇ.
Ɇɧɨɠɟɫɬɜɨ ɦɨɠɧɨ ɨɩɢɫɚɬɶ ɭɤɚɡɚɜ ɤɚɤɨɟ-ɬɨ ɫɜɨɣɫɬɜɨ ɩɪɢɫɭɳɟɟ ɜɫɟɦ ɷɥɟɦɟɧɬɚɦ ɷɬɨɝɨ ɦɧɨɠɟɫɬɜɚ
Ɂɚɦɟɱɚɧɢɟ ȼɨɨɛɳɟ ɝɨɜɨɪɹ ɩɨɧɹɬɢɟ ɦɧɨɠɟɫɬɜɚ ɫɱɢɬɚɟɬɫɹ ɩɟɪɜɢɱɧɵɦ ɢɫɯɨɞɧɵɦ ɩɨɧɹɬɢɟɦ ɢ ɤɚɤ ɬɚɤɨɜɨɟ ɧɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɉɪɢɜɟɞɟɧɧɨɟ ɜɵɲɟ ɨɩɪɟɞɟ ɥɟɧɢɟ ɫɥɟɞɭɟɬ ɫɤɨɪɟɟ ɫɱɢɬɚɬɶ ɭɬɨɱɧɟɧɢɟɦ ɩɨɧɹɬɢɹ ɦɧɨɠɟɫɬɜɚ
Ɇɧɨɠɟɫɬɜɨ ɜɫɟ ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɨɝɨ ɹɜɥɹɸɬɫɹ ɱɢɫɥɚɦɢ ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨ ɜɵɦ ȼ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɛɭɞɟɦ ɩɪɟɠɞɟ ɜɫɟɝɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɢɦɟɧɧɨ ɬɚɤɢɟ ɦɧɨ ɠɟɫɬɜɚ Ɇɧɨɠɟɫɬɜɨ ɷɥɟɦɟɧɬɚɦɢ ɤɨɬɨɪɨɝɨ ɹɜɥɹɸɬɫɹ ɞɪɭɝɢɟ ɦɧɨɠɟɫɬɜɚ ɧɚɡɵɜɚ ɟɬɫɹ ɤɥɚɫɫɨɦ ɢɥɢ ɫɟɦɟɣɫɬɜɨɦ.
Ɇɧɨɠɟɫɬɜɨ ɫɨɞɟɪɠɚɳɟɟ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɷɥɟɦɟɧɬɨɜ ɧɚɡɵɜɚɟɬɫɹ ɤɨɧɟɱɧɵɦ ɉɪɢ ɩɨɞɫɱɟɬɟ ɤɨɥɢɱɟɫɬɜɚ ɷɥɟɦɟɧɬɨɜ ɭɱɢɬɵɜɚɸɬɫɹ ɬɨɥɶɤɨ ɪɚɡɥɢɱɧɵɟ ɧɟɩɨɜɬɨ ɪɹɸɳɢɟɫɹ ɷɥɟɦɟɧɬɵ
Ɇɧɨɠɟɫɬɜɨ ɧɟ ɫɨɞɟɪɠɚɳɟɟ ɷɥɟɦɟɧɬɨɜ ɧɚɡɵɜɚɟɬɫɹ ɩɭɫɬɵɦ ɢ ɨɛɨɡɧɚɱɚɟɬ ɫɹ ɫɢɦɜɨɥɨɦ .
Ɇɧɨɠɟɫɬɜɨ ɦɨɠɟɬ ɛɵɬɶ ɡɚɞɚɧɨ ɩɟɪɟɱɢɫɥɟɧɢɟɦ ɫɩɢɫɤɨɦ ɫɜɨɢɯ ɷɥɟɦɟɧ ɬɨɜ ɩɨɪɨɠɞɚɸɳɟɣ ɩɪɨɰɟɞɭɪɨɣ ɢɥɢ ɨɩɢɫɚɧɢɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɩɪɟɞɢɤɚɬɨɦ ɤɨɬɨɪɵɦ ɞɨɥɠɧɵ ɨɛɥɚɞɚɬɶ ɟɝɨ ɷɥɟɦɟɧɬɵ ɉɪɢɱɟɦ ɜ ɩɨɫɥɟɞɧɟɦ ɫɥɭɱɚɟ ɧɟɨɛɯɨɞɢɦɨ ɮɨɪɦɭɥɢɪɨɜɚɬɶ ɨɩɢɫɚɧɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɷɥɟɦɟɧɬɨɜ ɦɧɨɠɟɫɬɜɚ ɞɨɫɬɚɬɨɱɧɨ ɤɨɪɪɟɤɬɧɨ ɞɥɹ ɬɨɝɨ ɱɬɨɛɵ ɦɧɨɠɟɫɬɜɨ ɛɵɥɨ ɨɩɪɟɞɟɥɟɧɨ ɜɩɨɥɧɟ ɨɞɧɨɡɧɚɱɧɨ Ⱦɨɛɚɜɢɦ ɱɬɨ ɦɧɨɝɢɟ ɱɢɫɥɨɜɵɟ ɦɧɨɠɟɫɬɜɚ ɦɨɝɭɬ ɛɵɬɶ ɡɚ ɞɚɧɵ ɜɫɟɦɢ ɬɪɟɦɹ ɭɤɚɡɚɧɧɵɦɢ ɫɩɨɫɨɛɚɦɢ ɧɚɩɪɢɦɟɪ ɦɧɨɠɟɫɬɜɨ ɱɟɬɧɵɯ ɨɞɧɨ ɡɧɚɱɧɵɯ ɱɢɫɟɥ
6
ɉɪɢɦɟɪ 1 ɇɟɤɨɬɨɪɵɟ ɩɪɢɦɟɪɵ ɦɧɨɠɟɫɬɜ ɡɚɞɚɧɧɵɯ ɪɚɡɥɢɱɧɵɦɢ ɫɩɨɫɨ
ɛɚɦɢ
ɚ M1 |
^1;2; 3; 4`; |
||||
ɛ M 2 |
^x |
|
x Z 4 x 9`; |
||
|
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ɜ M3 |
^x |
|
x 2n 1 n N`. |
||
|
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Ɇɨɳɧɨɫɬɶɸ ɤɨɧɟɱɧɨɝɨ ɦɧɨɠɟɫɬɜɚ Ɇ ɧɚɡɵɜɚɟɬɫɹ ɤɨɥɢɱɟɫɬɜɨ ɟɝɨ ɷɥɟɦɟɧ ɬɨɜ Ɉɛɨɡɧɚɱɚɟɬɫɹ M ȿɫɥɢ A
B ɬɨ ɦɧɨɠɟɫɬɜɚ Ⱥ ɢ ȼ ɧɚɡɵɜɚɸɬɫɹ ɪɚɜɧɨɦɨɳ
ɧɵɦɢ Ɉɩɪɟɞɟɥɟɧɢɟ ȿɫɥɢ ɜɫɟ ɷɥɟɦɟɧɬɵ ɦɧɨɠɟɫɬɜɚ Ⱥ ɹɜɥɹɸɬɫɹ ɬɚɤɠɟ ɷɥɟɦɟɧ
ɬɚɦɢ ɦɧɨɠɟɫɬɜɚ ȼ ɬɨ ɝɨɜɨɪɹɬ ɱɬɨ ɦɧɨɠɟɫɬɜɨ Ⱥ ɜɤɥɸɱɚɟɬɫɹ ɫɨɞɟɪɠɢɬɫɹ ɜ ɦɧɨɠɟɫɬɜɟ ȼ
Ⱥ
ȼ
Ⱥ ȼ
Ɉɩɪɟɞɟɥɟɧɢɟ ȿɫɥɢ Ⱥ ȼ ɬɨ ɦɧɨɠɟɫɬɜɨ Ⱥ ɧɚɡɵɜɚɟɬɫɹ ɩɨɞɦɧɨɠɟɫɬɜɨɦ ɦɧɨɠɟɫɬɜɚ ȼ ɬɚɤɠɟ ɝɨɜɨɪɹɬ ɱɬɨ ȼ ɩɨɤɪɵɜɚɟɬ Ⱥ ȿɫɥɢ ɩɪɢ ɷɬɨɦ Ⱥ z ȼ ɬɨ ɦɧɨɠɟɫɬɜɨ Ⱥ ɧɚɡɵɜɚɟɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦ ɩɨɞɦɧɨɠɟɫɬɜɨɦ ɦɧɨɠɟɫɬɜɚ ȼ ɢ ɨɛɨ ɡɧɚɱɚɟɬɫɹ Ⱥ ȼ
Ɂɚɦɟɱɚɧɢɟ ɇɟ ɫɥɟɞɭɟɬ ɫɱɢɬɚɬɶ ɪɚɜɧɨɫɢɥɶɧɵɦɢ ɨɬɧɨɲɟɧɢɹ ɩɪɢɧɚɞɥɟɠ ɧɨɫɬɢ ɢ ɜɯɨɠɞɟɧɢɹ ɨɞɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɜ ɞɪɭɝɨɟ Ɇɨɠɧɨ ɩɪɢɜɟɫɬɢ ɫɥɟ ɞɭɸɳɢɣ ɩɪɢɦɟɪ ɉɭɫɬɶ Ⱥ – ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɫɬɭɞɟɧɬɨɜ ɞɚɧɧɨɣ ɝɪɭɩɩɵ ɚ ȼ – ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɭɱɟɛɧɵɯ ɝɪɭɩɩ ɞɚɧɧɨɝɨ ɢɧɫɬɢɬɭɬɚ Ɂɞɟɫɶ A B ɧɨ A B ɩɨ ɫɤɨɥɶɤɭ ɷɥɟɦɟɧɬɵ ɷɬɢɯ ɦɧɨɠɟɫɬɜ ɪɚɡɧɨɪɨɞɧɵ ɗɬɨɬ ɩɪɢɦɟɪ ɩɨɤɚɡɵɜɚɟɬ ɬɚɤɠɟ ɱɬɨ ɷɥɟɦɟɧɬɚɦɢ ɦɧɨɠɟɫɬɜ ɦɨɝɭɬ ɹɜɥɹɬɶɫɹ ɞɪɭɝɢɟ ɦɧɨɠɟɫɬɜɚ
ɉɚɪɚɞɨɤɫ Ɋɚɫɫɟɥɚ Ɂɚɞɚɧɢɟ ɦɧɨɠɟɫɬɜ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɢɦ ɩɪɟɞɢɤɚɬɨɦ ɦɨɠɟɬ ɩɪɢɜɟɫɬɢ ɤ ɩɪɨɬɢɜɨɪɟɱɢɹɦ Ɋɚɫɫɦɨɬɪɢɦ ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɦɧɨɠɟɫɬɜ ɧɟ ɫɨ ɞɟɪɠɚɳɢɯ ɫɟɛɹ ɜ ɤɚɱɟɫɬɜɟ ɷɥɟɦɟɧɬɚ Y ^X X X ` ȿɫɥɢ ɬɚɤɨɟ ɦɧɨɠɟɫɬɜɨ ɫɭɳɟ ɫɬɜɭɟɬ ɬɨ ɦɨɠɧɨ ɨɬɜɟɬɢɬɶ ɧɚ ɫɥɟɞɭɸɳɢɣ ɜɨɩɪɨɫ ɩɪɢɧɚɞɥɟɠɢɬ ɥɢ ɨɧɨ ɫɚɦɨ ɫɟɛɟ ɋ ɨɞɧɨɣ ɫɬɨɪɨɧɵ ɟɫɥɢ Y Y ɬɨ Y Y . ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ ɟɫɥɢ Y Y ɬɨ Y Y ɗɬɨ ɩɪɨɬɢɜɨɪɟɱɢɟ ɦɨɠɧɨ ɪɚɡɪɟɲɢɬɶ ɪɚɡɥɢɱɧɵɦɢ ɫɩɨɫɨɛɚɦɢ ɜ ɰɟɥɨɦ ɫɜɨɞɹɳɢɦɫɹ ɤ ɬɨɦɭ ɱɬɨ Y ɧɟ ɹɜɥɹɟɬɫɹ ɦɧɨɠɟɫɬɜɨɦ
7
Ⱦɥɹ ɬɪɟɯ ɦɧɨɠɟɫɬɜ Ⱥ ȼ ɋ ɫɩɪɚɜɟɞɥɢɜɵ ɫɥɟɞɭɸɳɢɟ ɫɨɨɬɧɨɲɟɧɢɹ
A A; A A;
A B B C o A C;
A B B C o A C;
ɋɜɹɡɶ ɦɟɠɞɭ ɜɤɥɸɱɟɧɢɟɦ ɢ ɪɚɜɟɧɫɬɜɨɦ ɦɧɨɠɟɫɬɜ ɭɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦ
A B l A B B A.
Ɂɞɟɫɶ ɡɧɚɤ ɨɛɨɡɧɚɱɚɟɬ ɤɨɧɴɸɧɤɰɢɸ ɥɨɝɢɱɟɫɤɨɟ «ɢª ȼ ɡɚɤɥɸɱɟɧɢɟ ɞɨɛɚɜɢɦ ɱɬɨ Ƚ Ʉɚɧɬɨɪ ɩɪɟɞɥɨɠɢɥ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɨɧɹɬɢɟ
«ɭɧɢɜɟɪɫɚɥɶɧɨɝɨ ɦɧɨɠɟɫɬɜɚª ɭɧɢɜɟɪɫɭɦ ɤɚɤ ɛɵ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɝɨ ɩɨɧɹɬɢɸ ɩɭɫɬɨɝɨ ɦɧɨɠɟɫɬɜɚ ɉɨ ɦɵɫɥɢ Ʉɚɧɬɨɪɚ ɭɧɢɜɟɪɫɚɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ U ɫɨɞɟɪ ɠɢɬ ɜɫɟ ɦɵɫɥɢɦɵɟ ɦɧɨɠɟɫɬɜɚ ɢ ɩɪɢ ɷɬɨɦ ɨɧɨ ɫɚɦɨ ɫɨɞɟɪɠɢɬɫɹ ɜɨ ɦɧɨɠɟɫɬɜɟ ɫɜɨɢɯ ɩɨɞɦɧɨɠɟɫɬɜ ɜ ɤɚɱɟɫɬɜɟ ɷɥɟɦɟɧɬɚ ȼ ɞɚɥɶɧɟɣɲɟɦ ɫɦɵɫɥ ɢ ɫɨɞɟɪɠɚɧɢɟ ɩɨɧɹɬɢɹ ɭɧɢɜɟɪɫɚɥɶɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɛɭɞɭɬ ɪɚɫɤɪɵɬɵ ɛɨɥɟɟ ɩɨɞɪɨɛɧɨ
ɈɉȿɊȺɐɂɂ ɇȺȾ ɆɇɈɀȿɋɌȼȺɆɂ ɂ ɂɏ ɋȼɈɃɋɌȼȺ
Ɉɩɪɟɞɟɥɟɧɢɟ Ɉɛɴɟɞɢɧɟɧɢɟɦ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɋ ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɨɝɨ ɩɪɢɧɚɞɥɟɠɚɬ ɯɨɬɹ ɛɵ ɨɞɧɨɦɭ ɢɡ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ.
Ɉɛɨɡɧɚɱɚɟɬɫɹ ɋ Ⱥ ȼ
Ⱥȼ
Ƚɟɨɦɟɬɪɢɱɟɫɤɨɟ ɢɡɨɛɪɚɠɟɧɢɟ ɦɧɨɠɟɫɬɜ ɜ ɜɢɞɟ ɨɛɥɚɫɬɢ ɧɚ ɩɥɨɫɤɨɫɬɢ ɧɚ-
ɡɵɜɚɟɬɫɹ ɞɢɚɝɪɚɦɦɨɣ ɗɣɥɟɪɚ – ȼɷɣɧɚ.
Ɉɩɪɟɞɟɥɟɧɢɟ ɉɟɪɟɫɟɱɟɧɢɟɦ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɋ ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɨɝɨ ɩɪɢɧɚɞɥɟɠɚɬ ɤɚɠɞɨɦɭ ɢɡ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ
Ɉɛɨɡɧɚɱɟɧɢɟ ɋ Ⱥ ȼ
Ⱥ ɋ ȼ
8
Ⱦɥɹ ɦɧɨɠɟɫɬɜ Ⱥ ȼ ɢ ɋ ɫɩɪɚɜɟɞɥɢɜɵ ɫɥɟɞɭɸɳɢɟ ɫɜɨɣɫɬɜɚ
Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ; |
A B B A; |
A B B A; |
|
$ % & $ % & |
$ % & $ % & |
||
A % & $ % $ & |
A B & $ % $ & |
||
A $ % $ |
|
|
A $ % $ |
A Ⱥ; |
|
|
A . |
Ɉɩɪɟɞɟɥɟɧɢɟ Ɋɚɡɧɨɫɬɶɸ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɫɨɫɬɨ ɹɳɟɟ ɢɡ ɷɥɟɦɟɧɬɨɜ ɦɧɨɠɟɫɬɜɚ Ⱥ ɧɟ ɩɪɢɧɚɞɥɟɠɚɳɢɯ ɦɧɨɠɟɫɬɜɭ ȼ
Ɉɛɨɡɧɚɱɚɟɬɫɹ ɋ Ⱥ \ ȼ
Ⱥ ȼ
Ɉɩɪɟɞɟɥɟɧɢɟ ɋɢɦɦɟɬɪɢɱɟɫɤɨɣ ɪɚɡɧɨɫɬɶɸ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɋ ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɨɝɨ ɩɪɢɧɚɞɥɟɠɚɬ ɜ ɬɨɱɧɨɫɬɢ ɨɞɧɨɦɭ ɢɡ ɦɧɨɠɟɫɬɜ Ⱥ ɢɥɢ ȼ
Ɉɛɨɡɧɚɱɚɟɬɫɹ Ⱥ ' ȼ
Ⱥ ' ȼ A \ B B \ A .
A B
9
Ɉɩɪɟɞɟɥɟɧɢɟ ɋȿ ɧɚɡɵɜɚɟɬɫɹ ɞɨɩɨɥɧɟɧɢɟɦ ɦɧɨɠɟɫɬɜɚ Ⱥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɧɨɠɟɫɬɜɚ ȿ ɟɫɥɢ Ⱥ ȿ ɢ Cȿ ȿ \ A.
A E
Ⱦɥɹ ɦɧɨɠɟɫɬɜ Ⱥ ȼ ɢ ɋ ɫɩɪɚɜɟɞɥɢɜɵ ɫɥɟɞɭɸɳɢɟ ɫɨɨɬɧɨɲɟɧɢɹ
A \ B A; |
A \ $ ; |
A \ $ \ % A B; |
|
A ' % % ' A; |
|
A ' % $ % \ $ % |
|
A \ % & $ \ % $ \ & |
|
A \ % & $ \ % $ \ & |
|
$ % \ & $ \ & % \ & |
|
$ % \ & $ \ & % \ & |
|
A \ % \ & |
$ \ % $ & |
|
$ \ % \ & $ \ % & |
$ ' % ' & $ ' % ' & |
A % ' & $ % ' $ & |
||
A CE$ |
( A CE$ ; |
CE( |
; CE ( &ECEA $ |
CE $ % &EA CEB; |
|
CE $ % CEA CEB. |
|
ɉɪɢɦɟɪ ɂɫɯɨɞɹ ɢɡ ɨɩɪɟɞɟɥɟɧɢɹ ɪɚɜɟɧɫɬɜɚ ɦɧɨɠɟɫɬɜ ɢ ɨɩɟɪɚɰɢɣ ɧɚɞ ɦɧɨɠɟɫɬɜɚɦɢ ɞɨɤɚɡɚɬɶ ɬɨɠɞɟɫɬɜɨ ɢ ɩɪɨɜɟɪɢɬɶ ɟɝɨ ɫ ɩɨɦɨɳɶɸ ɞɢɚɝɪɚɦɦɵ ɗɣ ɥɟɪɚ – ȼɷɣɧɚ
A\ B A\ A B .
ɂɡ ɡɚɩɢɫɚɧɧɵɯ ɜɵɲɟ ɫɨɨɬɧɨɲɟɧɢɣ ɜɢɞɧɨ ɱɬɨ
A\ A B A\ A A\ B A\ B A \ ȼ.
ɑɬɨ ɢ ɬɪɟɛɨɜɚɥɨɫɶ ɞɨɤɚɡɚɬɶ
10