Математика и информатика. Решение логико-познавательных задач. Учебное пособие для студентов вузов
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ǫȓȈȊȈ 2
ǴǵǶǮǭǹǺǪǨ
2.1. ǷȖȕȧȚȐȍ «ȔȕȖȎȍșȚȊȖ»
ɋɨɡɞɚɬɟɥɟɦ ɬɟɨɪɢɢ ɦɧɨɠɟɫɬɜ ɹɜɥɹɟɬɫɹ ɧɟɦɟɰɤɢɣ ɭɱɟɧɵɣ Ƚɟɨɪɝ Ʉɚɧɬɨɪ (1845—1918). Ɉɧ ɨɩɪɟɞɟɥɢɥ, ɱɬɨ «ɦɧɨɠɟɫɬɜɨ ɟɫɬɶ ɦɧɨɝɨɟ, ɦɵɫɥɢɦɨɟ ɧɚɦɢ ɤɚɤ ɟɞɢɧɨɟ».
ȼ ɨɛɵɱɧɨɣ ɠɢɡɧɢ ɦɵ ɱɚɫɬɨ ɭɩɨɬɪɟɛɥɹɟɦ ɫɥɨɜɨ «ɦɧɨɠɟɫɬɜɨ»: ɦɧɨɠɟɫɬɜɨ ɥɸɞɟɣ, ɤɧɢɝ, ɡɚɤɨɧɨɜ ɢ ɬ.ɞ. Ɍɟɦ ɫɚɦɵɦ ɤɚɠɞɵɣ ɪɚɡ ɦɵ (ɩɨɪɨɣ ɧɟɨɫɨɡɧɚɧɧɨ) ɝɪɭɩɩɢɪɭɟɦ ɨɛɴɟɤɬɵ ɩɨ ɩɪɢɡɧɚɤɚɦ, ɫɜɨɣɫɬɜɚɦ, ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦ.
Ɇɧɨɠɟɫɬɜɨ — ɝɪɭɩɩɚ ɢɥɢ ɧɚɛɨɪ ɨɛɴɟɤɬɨɜ (ɩɪɟɞɦɟɬɨɜ), ɨɛɥɚɞɚɸɳɢɯ ɤɚɤɢɦ-ɥɢɛɨ ɨɛɳɢɦ ɞɥɹ ɧɢɯ ɜɫɟɯ ɫɜɨɣɫɬɜɨɦ ɢɥɢ ɩɪɢɡɧɚɤɨɦ.
ȼɵɫɤɚɡɚɧɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɹɜɥɹɟɬɫɹ ɧɟ ɨɩɪɟɞɟɥɟɧɢɟɦ, ɚ ɥɢɲɶ ɪɚɡɴɹɫɧɟɧɢɟɦ. Ɉɩɪɟɞɟɥɟɧɢɹ ɩɨɩɪɨɫɬɭ ɧɟ ɫɭɳɟɫɬɜɭɟɬ, ɩɨɫɤɨɥɶɤɭ ɩɨɧɹɬɢɟ «ɦɧɨɠɟɫɬɜɨ» ɹɜɥɹɟɬɫɹ ɩɟɪɜɢɱɧɵɦ (ɤɚɤ ɬɨɱɤɚ, ɱɢɫɥɨ ɢ ɞɪ.), ɧɚ ɨɫɧɨɜɚɧɢɢ ɤɨɬɨɪɨɝɨ ɫɬɪɨɹɬɫɹ ɨɫɬɚɥɶɧɵɟ ɩɨɧɹɬɢɹ ɦɚɬɟɦɚɬɢɤɢ.
Ɇɨɠɧɨ ɝɨɜɨɪɢɬɶ ɧɟ ɬɨɥɶɤɨ ɨ ɦɧɨɠɟɫɬɜɚɯ, ɷɥɟɦɟɧɬɚɦɢ ɤɨɬɨɪɵɯ ɹɜɥɹɸɬɫɹ ɦɚɬɟɪɢɚɥɶɧɵɟ ɨɛɴɟɤɬɵ, ɧɨ ɢ ɨ ɦɧɨɠɟɫɬɜɚɯ, ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɵɯ — ɧɟɤɨɬɨɪɵɟ ɚɛɫɬɪɚɤɬɧɵɟ ɩɨɧɹɬɢɹ (ɱɢɫɥɚ, ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ ɮɢɝɭɪɵ, ɫɢɦɜɨɥɵ ɢ ɬ.ɩ.).
Ɇɧɨɠɟɫɬɜɚ ɨɛɨɡɧɚɱɚɸɬɫɹ ɩɪɨɩɢɫɧɵɦɢ ɛɭɤɜɚɦɢ ɥɚɬɢɧɫɤɨɝɨ ɚɥɮɚɜɢɬɚ (A, B, C, …), ɚ ɢɯ ɷɥɟɦɟɧɬɵ — ɫɬɪɨɱɧɵɦɢ (a, b, c, …).
ɉɪɢɧɚɞɥɟɠɧɨɫɬɶ ɷɥɟɦɟɧɬɨɜ (ɬɨɱɟɤ) ɤ ɦɧɨɠɟɫɬɜɭ ɨɛɨɡɧɚɱɚɟɬɫɹ ɡɧɚɤɨɦ « ». Ɍɚɤ, ɡɚɩɢɫɶ a A ɨɡɧɚɱɚɟɬ, ɷɥɟɦɟɧɬ a ɩɪɢɧɚɞɥɟɠɢɬ ɦɧɨɠɟɫɬɜɭ A. ȿɫɥɢ ɠɟ a ɧɟ ɩɪɢɧɚɞɥɟɠɢɬ ɦɧɨɠɟɫɬɜɭ A, ɬɨ ɩɢɲɭɬ a A.
ɉɪɢɦɟɪ 1. ɉɭɫɬɶ A — ɦɧɨɠɟɫɬɜɨ ɱɟɬɧɵɯ ɱɢɫɟɥ. Ɍɨɝɞɚ 2 A , ɚ 5 A .
ȿɫɥɢ ɱɢɫɥɨ ɷɥɟɦɟɧɬɨɜ ɦɧɨɠɟɫɬɜɚ ɤɨɧɟɱɧɨ, ɬɨ ɦɧɨɠɟɫɬɜɨ ɧɚɡɵɜɚ-
ɟɬɫɹ ɤɨɧɟɱɧɵɦ, ɢɧɚɱɟ — ɛɟɫɤɨɧɟɱɧɵɦ.
ɉɪɢɦɟɪ 2. Ɇɧɨɠɟɫɬɜɨ ɩɟɪɫɨɧɚɠɟɣ ɛɚɫɟɧ Ʉɪɵɥɨɜɚ — ɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ, ɚ ɜɫɟ ɧɚɬɭɪɚɥɶɧɵɟ ɱɢɫɥɚ — ɛɟɫɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ.
ɋɪɚɡɭ ɨɝɨɜɨɪɢɦ, ɱɬɨ «ɦɧɨɠɟɫɬɜɨ» ɧɟ ɫɥɟɞɭɟɬ ɩɨɧɢɦɚɬɶ ɛɭɤɜɚɥɶɧɨ ɢ ɬɨɥɤɨɜɚɬɶ ɟɝɨ ɤɚɤ ɫɨɜɨɤɭɩɧɨɫɬɶ, ɫɨɞɟɪɠɚɳɭɸ «ɦɧɨɝɨ» ɷɥɟɦɟɧɬɨɜ.
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ȼɫɬɪɟɱɚɸɬɫɹ ɦɧɨɠɟɫɬɜɚ, ɧɟ ɫɨɞɟɪɠɚɳɢɟ ɧɢ ɨɞɧɨɝɨ ɷɥɟɦɟɧɬɚ. Ɍɚɤɢɟ ɦɧɨɠɟɫɬɜɚ ɧɚɡɵɜɚɸɬ ɩɭɫɬɵɦɢ ɢ ɨɛɨɡɧɚɱɚɸɬ ɫɢɦɜɨɥɨɦ .
ɉɪɢɦɟɪ 3. Ɇɧɨɠɟɫɬɜɨ ɥɸɞɟɣ, ɱɟɣ ɪɨɫɬ ɫɨɫɬɚɜɥɹɟɬ 3 ɦɟɬɪɚ, — ɩɭɫɬɨɟ ɦɧɨɠɟɫɬɜɨ.
ɋɭɳɟɫɬɜɭɸɬ ɪɚɡɧɵɟ ɫɩɨɫɨɛɵ ɡɚɞɚɧɢɹ ɦɧɨɠɟɫɬɜ. Ʉɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɦɨɠɧɨ ɡɚɞɚɜɚɬɶ ɩɟɪɟɱɢɫɥɟɧɢɟɦ ɜɫɟɯ ɟɝɨ ɷɥɟɦɟɧɬɨɜ, ɧɚɩɪɢɦɟɪ, A={a1; a2; a3;...; an}, ɝɞɟ a1, a2, a3,…, an — ɨɛɴɟɤɬɵ ɥɸɛɨɣ ɩɪɢɪɨɞɵ.
ɉɪɢɦɟɪ 4. ɉɭɫɬɶ B — ɦɧɨɠɟɫɬɜɨ ɩɥɚɧɟɬ ɫɨɥɧɟɱɧɨɣ ɫɢɫɬɟɦɵ. Ɍɨɝɞɚ B = {ȼɟɧɟɪɚ, Ɂɟɦɥɹ, Ɇɚɪɫ, Ɇɟɪɤɭɪɢɣ, ɇɟɩɬɭɧ, ɉɥɭɬɨɧ, ɋɚɬɭɪɧ, ɍɪɚɧ, ɘɩɢɬɟɪ}.
Ɂɚɞɚɬɶ ɦɧɨɠɟɫɬɜɨ ɦɨɠɧɨ ɬɚɤɠɟ ɫ ɩɨɦɨɳɶɸ ɩɪɚɜɢɥɚ, ɩɨɡɜɨɥɹɸɳɟɝɨ ɨɩɪɟɞɟɥɢɬɶ, ɹɜɥɹɟɬɫɹɥɢɞɚɧɧɵɣɨɛɴɟɤɬɷɥɟɦɟɧɬɨɦɦɧɨɠɟɫɬɜɚɢɥɢɧɟɬ.
ɉɪɢɦɟɪ 5. ɉɭɫɬɶ ɋ — ɦɧɨɠɟɫɬɜɨ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ x2 – 3x + 2 = 0. Ɍɨɝɞɚ ɦɧɨɠɟɫɬɜɨ ɋ = {x | x2 – 3x+2 = 0}. ȼ ɬɚɤɨɣ ɡɚɩɢɫɢ ɩɪɚɜɢɥɨ, ɡɚɞɚɸɳɟɟ ɦɧɨɠɟɫɬɜɨ, ɨɬɞɟɥɟɧɨ ɜɟɪɬɢɤɚɥɶɧɨɣ ɱɟɪɬɨɣ (ɜɟɪɬɢɤɚɥɶɧɚɹ ɱɟɪɬɚ | ɞɨɫɥɨɜɧɨ ɨɛɨɡɧɚɱɚɟɬ ɬɚɤɢɯ, ɱɬɨ). Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɷɥɟɦɟɧɬɚɦɢ ɦɧɨɠɟɫɬɜɚ ɋ ɹɜɥɹɸɬɫɹ ɱɢɫɥɚ 1 ɢ 2, ɬ.ɟ. ɋ = {1;2}.
Ɂɚɩɢɫɶ A B ɨɡɧɚɱɚɟɬ, ɱɬɨ ɤɚɠɞɵɣ ɷɥɟɦɟɧɬ ɦɧɨɠɟɫɬɜɚ A ɩɪɢɧɚɞɥɟɠɢɬ ɦɧɨɠɟɫɬɜɭ B. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɧɨɠɟɫɬɜɨ A ɧɚɡɵɜɚɸɬ ɩɨɞɦɧɨɠɟɫɬɜɨɦ ɦɧɨɠɟɫɬɜɚ B. Ʉɚɠɞɨɟ ɧɟɩɭɫɬɨɟ ɦɧɨɠɟɫɬɜɨ A ɢɦɟɟɬ, ɤɚɤ ɦɢɧɢɦɭɦ, ɞɜɚ ɩɨɞɦɧɨɠɟɫɬɜɚ: ɩɭɫɬɨɟ ɦɧɨɠɟɫɬɜɨ ɢ ɫɚɦɨ ɦɧɨɠɟɫɬɜɨ A.
ɉɪɢɦɟɪ 6. ɉɭɫɬɶ A — ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɤɜɚɞɪɚɬɨɜ, ɚ B — ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɱɟɬɵɪɟɯɭɝɨɥɶɧɢɤɨɜ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ A B.
ȿɫɥɢ ɞɥɹ ɞɜɭɯ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɨɞɧɨɜɪɟɦɟɧɧɨ ɫɩɪɚɜɟɞɥɢɜɵ ɭɬɜɟɪɠɞɟɧɢɹ A B ɢ ȼ Ⱥ, ɬɨ ɦɧɨɠɟɫɬɜɚ Ⱥ ɢ ȼ ɫɨɫɬɨɹɬ ɢɡ ɨɞɧɢɯ ɢ ɬɟɯ ɠɟ ɷɥɟɦɟɧɬɨɜ. Ɍɚɤɢɟ ɦɧɨɠɟɫɬɜɚ ɧɚɡɵɜɚɸɬɫɹ ɪɚɜɧɵɦɢ ɢ ɩɢɲɭɬ Ⱥ = ȼ.
ɉɪɢɦɟɪ 7. ɉɭɫɬɶ Ⱥ — ɦɧɨɠɟɫɬɜɨ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ x+2=7, ȼ — ɦɧɨɠɟɫɬɜɨ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ x+4=9. Ɍɨɝɞɚ ɦɧɨɠɟɫɬɜɨ Ⱥ = = {x | x+2=7}, ɚ ɦɧɨɠɟɫɬɜɨ ȼ = {x | x+4=9}. Ɉɱɟɜɢɞɧɨ, ɱɬɨ Ⱥ = {5}, ȼ = {5}, ɬ.ɟ. Ⱥ = ȼ.
Ɇɧɨɠɟɫɬɜɚ, ɷɥɟɦɟɧɬɚɦɢ ɤɨɬɨɪɨɝɨ ɹɜɥɹɸɬɫɹ ɱɢɫɥɚ, ɧɚɡɵɜɚɸɬɫɹ ɱɢ-
ɫɥɨɜɵɦɢ ɦɧɨɠɟɫɬɜɚɦɢ.
ɇɚɩɨɦɧɢɦ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ ɪɚɡɛɢɜɚɟɬɫɹ ɧɚ ɞɜɚ ɩɨɞɦɧɨɠɟɫɬɜɚ — ɪɚɰɢɨɧɚɥɶɧɵɯ ɢ ɢɪɪɚɰɢɨɧɚɥɶɧɵɯ ɱɢɫɟɥ.
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Ɋɚɰɢɨɧɚɥɶɧɵɦ ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨ, ɤɨɬɨɪɨɟ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ qp , ɝɞɟ ɪ ɢ q — ɰɟɥɵɟ ɱɢɫɥɚ, ɩɪɢɱɟɦ q 0. ɂɪɪɚɰɢɨɧɚɥɶɧɵɦ ɧɚ-
ɡɵɜɚɟɬɫɹ ɜɫɹɤɨɟ ɜɟɳɟɫɬɜɟɧɧɨɟ ɱɢɫɥɨ, ɤɨɬɨɪɨɟ ɧɟ ɹɜɥɹɟɬɫɹ ɪɚɰɢɨɧɚɥɶ-
ɧɵɦ. ȼɫɹɤɨɟ ɪɚɰɢɨɧɚɥɶɧɨɟ ɱɢɫɥɨ qp ɹɜɥɹɟɬɫɹ ɥɢɛɨ ɰɟɥɵɦ, ɥɢɛɨ ɟɝɨ
ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɤɨɧɟɱɧɨɣ ɢɥɢ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɛɟɫɤɨɧɟɱɧɨɣ ɞɟɫɹɬɢɱɧɨɣ ɞɪɨɛɢ. ɂɪɪɚɰɢɨɧɚɥɶɧɨɟ ɠɟ ɱɢɫɥɨ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɧɟɩɟɪɢɨɞɢɱɟɫɤɨɣ ɛɟɫɤɨɧɟɱɧɨɣ ɞɟɫɹɬɢɱɧɨɣ ɞɪɨɛɶɸ. ɇɚɩɪɢɦɟɪ, ɪɚɰɢɨɧɚɥɶ-
ɧɵɟ ɱɢɫɥɚ 34 ɢ 13 ɩɪɟɞɫɬɚɜɥɹɸɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɫɥɟɞɭɸɳɢɦɢ ɞɟɫɹ-
ɬɢɱɧɵɦɢ ɞɪɨɛɹɦɢ: 0,75 ɢ 0,333...; ɢɪɪɚɰɢɨɧɚɥɶɧɵɟ ɱɢɫɥɚ 2 ɢ ʌ
ɩɪɟɞɫɬɚɜɥɹɸɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɟɩɟɪɢɨɞɢɱɟɫɤɢɦɢ ɛɟɫɤɨɧɟɱɧɵɦɢ ɞɟɫɹɬɢɱɧɵɦɢ ɞɪɨɛɹɦɢ: 1,41421356... ɢ 3,14159... .
ɉɪɢɦɟɧɹɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ ɱɢɫɥɨɜɵɯ ɦɧɨɠɟɫɬɜ:
•R — ɦɧɨɠɟɫɬɜɨ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ;
•I — ɦɧɨɠɟɫɬɜɨ ɢɪɪɚɰɢɨɧɚɥɶɧɵɯ ɱɢɫɟɥ ( I R );
•Q — ɦɧɨɠɟɫɬɜɨ ɪɚɰɢɨɧɚɥɶɧɵɯ ɱɢɫɟɥ ( Q R );
•N — ɦɧɨɠɟɫɬɜɨ ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɟɥ. Ɇɧɨɠɟɫɬɜɨ N ɫɨɞɟɪɠɢɬ ɱɢɫɥɚ, ɢɫɩɨɥɶɡɭɟɦɵɟ ɞɥɹ ɫɱɟɬɚ (ɰɟɥɵɟ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɱɢɫɥɚ) ɢ ɹɜɥɹɟɬɫɹ ɩɨɞɦɧɨɠɟɫɬɜɨɦ ɦɧɨɠɟɫɬɜɚ Q ( N Q );
•Z — ɦɧɨɠɟɫɬɜɨ ɰɟɥɵɯ ɱɢɫɟɥ. Ɇɧɨɠɟɫɬɜɨ Z ɫɨɞɟɪɠɢɬ ɰɟɥɵɟ ɨɬɪɢɰɚɬɟɥɶɧɵɟ ɱɢɫɥɚ, 0, ɰɟɥɵɟ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɱɢɫɥɚ ɢ ɹɜɥɹɟɬɫɹ ɩɨɞɦɧɨɠɟɫɬɜɨɦ ɦɧɨɠɟɫɬɜɚ Q ( Z Q );
Ɇɧɨɠɟɫɬɜɨ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ R ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨɜɨɣ ɩɪɹɦɨɣ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ ( φ;φ ).
ɉɪɢɦɟɪ 8. Ⱦɚɧɵ ɫɥɟɞɭɸɳɢɟ ɱɢɫɥɨɜɵɟ ɦɧɨɠɟɫɬɜɚ: R — ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ, Q — ɪɚɰɢɨɧɚɥɶɧɵɯ ɱɢɫɟɥ, N — ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɟɥ, Z — ɰɟɥɵɯ ɱɢɫɟɥ. Ɍɪɟɛɭɟɬɫɹ ɪɚɫɩɨɥɨɠɢɬɶ ɛɭɤɜɵ, ɨɛɨɡɧɚɱɚɸɳɢɟ ɷɬɢ ɦɧɨɠɟɫɬɜɚ, ɬɚɤ, ɱɬɨɛɵ ɤɚɠɞɚɹ ɩɪɟɞɵɞɭɳɚɹ ɛɭɤɜɚ ɹɜɥɹɥɚɫɶ ɩɨɞɦɧɨɠɟɫɬɜɨɦ ɫɥɟɞɭɸɳɟɣ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɫɩɪɚɜɟɞɥɢɜɨ ɛɭɞɟɬ ɫɥɟɞɭɸɳɟɟ ɭɬɜɟɪɠɞɟ-
ɧɢɟ: N Z Q R .
ɉɭɫɬɶ ɚ ɢ b — ɞɜɚ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɥɚ, ɩɪɢɱɟɦ ɚ < b. Ɇɧɨɠɟɫɬɜɨ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ x, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɯ ɨɩɪɟɞɟɥɟɧɧɵɦ ɧɟɪɚɜɟɧɫɬɜɚɦ, ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨɜɵɦ ɩɪɨɦɟɠɭɬɤɨɦ. Ȼɭɞɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɫɥɟɞɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ:
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ɂɧɬɟɪɜɚɥ (ɚ, b) ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɨɬɪɟɡɤɚ [ɚ, b] ɥɢɲɶ ɬɟɦ, ɱɬɨ ɟɦɭ ɧɟ ɩɪɢɧɚɞɥɟɠɚɬ ɝɪɚɧɢɰɵ ɚ ɢ b. ɗɬɨ ɨɬɥɢɱɢɟ ɢɝɪɚɟɬ ɫɭɳɟɫɬɜɟɧɧɭɸ ɪɨɥɶ ɜɨ ɦɧɨɝɢɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɜɨɩɪɨɫɚɯ. Ʉɪɨɦɟ ɬɨɝɨ, ɢɧɬɟɪɜɚɥ (ɚ, b) ɧɟ ɫɨɞɟɪɠɢɬ ɧɢ ɧɚɢɛɨɥɶɲɟɝɨ, ɧɢ ɧɚɢɦɟɧɶɲɟɝɨ ɱɢɫɥɚ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɜ ɨɬɪɟɡɤɟ [ɚ, b] ɬɚɤɢɦɢ ɱɢɫɥɚɦɢ ɹɜɥɹɸɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ b ɢ ɚ.
ɂɧɬɟɪɜɚɥɵ ɢ ɩɨɥɭɢɧɬɟɪɜɚɥɵ ɦɨɝɭɬ ɛɵɬɶ ɤɚɤ ɤɨɧɟɱɧɵɦɢ, ɬɚɤ ɢ ɛɟɫɤɨɧɟɱɧɵɦɢ, ɨɬɪɟɡɤɢ — ɬɨɥɶɤɨ ɤɨɧɟɱɧɵɦɢ.
ɉɪɢɦɟɪ 9. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = [–11; 3], ɚ ɦɧɨɠɟɫɬɜɨ ȼ = (0; 10]. ɂɡɨɛɪɚɡɢɦ ɞɚɧɧɵɟ ɦɧɨɠɟɫɬɜɚ, ɡɚɞɚɧɧɵɟ ɱɢɫɥɨɜɵɦɢ ɩɪɨɦɟɠɭɬɤɚɦɢ, ɧɚ ɱɢɫɥɨɜɨɣ ɩɪɹɦɨɣ ɢ ɜɵɹɫɧɢɦ ɹɜɥɹɟɬɫɹ ɥɢ ɦɧɨɠɟɫɬɜɨ Ⱥ ɩɨɞɦɧɨɠɟɫɬɜɨɦ ɦɧɨɠɟɫɬɜɚ ȼ.
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ǸȐș. 2.1
ɇɚɩɨɦɢɧɚɟɦ, ɱɬɨ ɩɪɢ ɢɡɨɛɪɚɠɟɧɢɢ ɱɢɫɥɨɜɵɯ ɩɪɨɦɟɠɭɬɤɨɜ ɡɚɤɪɚɲɟɧɧɵɦɢ ɤɪɭɠɤɚɦɢ ɨɛɨɡɧɚɱɚɸɬɫɹ ɤɪɚɣɧɢɟ ɬɨɱɤɢ, ɩɪɢɧɚɞɥɟɠɚɳɢɟ ɩɪɨɦɟɠɭɬɤɭ, ɚ ɫɜɟɬɥɵɦɢ ɤɪɭɠɤɚɦɢ — ɧɟ ɩɪɢɧɚɞɥɟɠɚɳɢɟ ɩɪɨɦɟɠɭɬɤɭ (ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ ɜɵɤɨɥɨɬɵɟ ɬɨɱɤɢ). ɂɡ ɪɢɫ. 2.1 ɜɢɞɧɨ, ɱɬɨ ɧɟ ɜɫɟ ɷɥɟɦɟɧɬɵ ɦɧɨɠɟɫɬɜɚ Ⱥ ɜɯɨɞɹɬ ɜ ɦɧɨɠɟɫɬɜɨ ȼ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ A B .
2.2. ǶȗȍȘȈȞȐȐ ȕȈȌ ȔȕȖȎȍșȚȊȈȔȐ
ɂɡ ɦɧɨɠɟɫɬɜ ɫ ɩɨɦɨɳɶɸ ɨɩɪɟɞɟɥɟɧɧɵɯ ɨɩɟɪɚɰɢɣ ɦɨɠɧɨ ɨɛɪɚɡɨɜɵɜɚɬɶ ɧɨɜɵɟ ɦɧɨɠɟɫɬɜɚ. Ⱦɥɹ ɧɚɝɥɹɞɧɨɝɨ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɨɩɟɪɚɰɢɣ ɧɚɞ ɦɧɨɠɟɫɬɜɚɦɢ ɢ ɪɚɡɥɢɱɧɵɯ ɫɨɨɬɧɨɲɟɧɢɣ ɦɟɠɞɭ ɦɧɨɠɟɫɬɜɚɦɢ ɭɞɨɛɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɢɚɝɪɚɦɦɵ ɗɣɥɟɪɚ—ȼɟɧɧɚ, ɧɚ ɤɨɬɨɪɵɯ ɦɧɨɠɟɫɬɜɚ ɢɡɨɛɪɚɠɚɸɬɫɹ ɜ ɜɢɞɟ ɮɢɝɭɪ, ɨɝɪɚɧɢɱɢɜɚɸɳɢɯ ɫɨɜɨɤɭɩɧɨɫɬɶ ɬɨɱɟɤ ɧɚ ɩɥɨɫɤɨɫɬɢ. Ɂɧɚɦɟɧɢɬɵɣ ɲɜɟɣɰɚɪɫɤɢɣ ɦɚɬɟɦɚɬɢɤ Ʌɟɨɧɚɪɞ ɗɣɥɟɪ ɩɟɪɜɵɣ ɢɫɩɨɥɶɡɨɜɚɥ ɢɞɟɸ ɢɡɨɛɪɚɠɚɬɶ ɦɧɨɠɟɫɬɜɚ ɫ ɩɨɦɨɳɶɸ ɤɪɭɝɨɜ, ɚ ɩɨɡɞɧɟɟ ɚɧɚɥɨɝɢɱɧɵɣ ɩɪɢɟɦ ɩɪɢɦɟɧɢɥ ɢɡɜɟɫɬɧɵɣ ɚɧɝɥɢɣɫɤɢɣ ɥɨɝɢɤ Ⱦɠɨɧ ȼɟɧɧ, ɨɬɫɸɞɚ ɢ ɧɚɡɜɚɧɢɟ ɞɢɚɝɪɚɦɦɵ ɗɣɥɟɪɚ—ȼɟɧɧɚ.
Ɋɚɫɫɦɨɬɪɢɦ ɤɚɠɞɭɸ ɨɩɟɪɚɰɢɸ ɩɨɞɪɨɛɧɨ.
34
Ɉɛɴɟɞɢɧɟɧɢɟɦ (ɢɥɢ ɥɨɝɢɱɟɫɤɨɣ ɫɭɦɦɨɣ) ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ, ɤɨɬɨɪɨɟ ɫɨɫɬɨɢɬ ɢɡ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ, ɩɪɢɧɚɞɥɟɠɚɳɢɯ ɯɨɬɹ ɛɵ ɨɞɧɨɦɭ ɢɡ ɷɬɢɯ ɦɧɨɠɟɫɬɜ. Ɉɛɴɟɞɢɧɟɧɢɟ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɨɛɨɡɧɚɱɚɟɬɫɹ Ⱥ ȼ (ɪɢɫ. 2.2).
A B
ǸȐș. 2.2
ɉɪɢɦɟɪ 10. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = {Ɍɚɬɶɹɧɚ, Ɉɥɶɝɚ, Ⱥɥɢɧɚ, ɉɟɬɪ, ɂɪɢɧɚ, ȼɢɬɚɥɢɣ}, ɚ ɦɧɨɠɟɫɬɜɨ ȼ = {ɂɜɚɧ, Ɉɥɶɝɚ, Ɇɚɪɢɹ, ɇɢɤɨɥɚɣ, ɉɟɬɪ, ɂɪɢɧɚ}. Ɍɨɝɞɚ Ⱥ ȼ = {Ɍɚɬɶɹɧɚ, Ɉɥɶɝɚ, Ⱥɥɢɧɚ, ɉɟɬɪ, ɂɪɢɧɚ, ȼɢɬɚɥɢɣ, ɂɜɚɧ, Ɇɚɪɢɹ, ɇɢɤɨɥɚɣ}. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ Ⱥ ȼ ɫɨɫɬɨɢɬ ɢɡ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ, ɤɨɬɨɪɵɟ ɯɨɬɶ ɪɚɡ ɩɪɢɫɭɬɫɬɜɭɸɬ ɜ ɤɚɤɨɦ-ɥɢɛɨ ɢɡ ɦɧɨɠɟɫɬɜ Ⱥ ɢɥɢ ȼ.
ɉɪɢɦɟɪ 11. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = {1; 2; 3; 4; 5; 6}, ɚ ɦɧɨɠɟɫɬɜɨ
ȼ = {3; 6; 9; 12}. Ɍɨɝɞɚ Ⱥ ȼ = {1; 2; 3; 4; 5; 6; 9; 12}.
ɉɪɢɦɟɪ 12. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = [–11; 3], ɚ ɦɧɨɠɟɫɬɜɨ ȼ = (0; 10].
Ɍɨɝɞɚ Ⱥ ȼ = [–11; 10] (ɪɢɫ. 2.3).
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ǸȐș. 2.3 |
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ɉɟɪɟɫɟɱɟɧɢɟɦ (ɢɥɢ ɥɨɝɢɱɟɫɤɢɦ ɩɪɨɢɡɜɟɞɟɧɢɟɦ) ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɧɚ-
ɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ, ɤɨɬɨɪɨɟ ɫɨɫɬɨɢɬ ɢɡ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ, ɩɪɢɧɚɞɥɟɠɚɳɢɯ ɨɞɧɨɜɪɟɦɟɧɧɨ ɤɚɠɞɨɦɭ ɢɡ ɷɬɢɯ ɦɧɨɠɟɫɬɜ. ɉɟɪɟɫɟɱɟɧɢɟ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɨɛɨɡɧɚɱɚɟɬɫɹ Ⱥ ȼ (ɪɢɫ. 2.4).
35
A B
ǸȐș. 2.4
ɉɪɢɦɟɪ 13. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = {Ɍɚɬɶɹɧɚ, Ɉɥɶɝɚ, Ⱥɥɢɧɚ, ɉɟɬɪ, ɂɪɢɧɚ, ȼɢɬɚɥɢɣ}, ɚ ɦɧɨɠɟɫɬɜɨ ȼ = {ɂɜɚɧ, Ɉɥɶɝɚ, Ɇɚɪɢɹ, ɇɢɤɨɥɚɣ, ɉɟɬɪ, ɂɪɢɧɚ}. Ɍɨɝɞɚ Ⱥ ȼ = {Ɉɥɶɝɚ, ɉɟɬɪ, ɂɪɢɧɚ}. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ Ⱥ ȼ ɫɨɫɬɨɢɬ ɢɡ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ, ɤɨɬɨɪɵɟ ɩɪɢɧɚɞɥɟɠɚɬ ɦɧɨɠɟɫɬɜɭ Ⱥ ɢ ɦɧɨɠɟɫɬɜɭ ȼ ɨɞɧɨɜɪɟɦɟɧɧɨ.
ɉɪɢɦɟɪ 14. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = {1; 2; 3; 4; 5; 6}, ɚ ɦɧɨɠɟɫɬɜɨ
ȼ = {3; 6; 9; 12}. Ɍɨɝɞɚ Ⱥ ȼ = {3;6}.
ɉɪɢɦɟɪ 15. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = [–11; 3], ɚ ɦɧɨɠɟɫɬɜɨ ȼ = (0; 10].
Ɍɨɝɞɚ Ⱥ ȼ = (0; 3] (ɪɢɫ. 2.5).
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ǸȐș. 2.5
Ɋɚɡɧɨɫɬɶɸ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ, ɤɨɬɨɪɨɟ ɫɨɫɬɨɢɬ ɢɡ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ ɦɧɨɠɟɫɬɜɚ Ⱥ, ɤɨɬɨɪɵɟ ɧɟ ɩɪɢɧɚɞɥɟɠɚɬ ɦɧɨɠɟɫɬɜɭ ȼ. Ɋɚɡɧɨɫɬɶ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɨɛɨɡɧɚɱɚɟɬɫɹ A \ B (ɪɢɫ. 2.6).
A B
ǸȐș. 2.6
ɉɪɢɦɟɪ 16. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = {Ɍɚɬɶɹɧɚ, Ɉɥɶɝɚ, Ⱥɥɢɧɚ, ɉɟɬɪ, ɂɪɢɧɚ, ȼɢɬɚɥɢɣ}, ɚ ɦɧɨɠɟɫɬɜɨ ȼ = {ɂɜɚɧ, Ɉɥɶɝɚ, Ɇɚɪɢɹ, ɇɢɤɨɥɚɣ,
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ɉɟɬɪ, ɂɪɢɧɚ}. Ɍɨɝɞɚ A \ B = {Ɍɚɬɶɹɧɚ; Ⱥɥɢɧɚ; ȼɢɬɚɥɢɣ}, ɚ ȼ\ Ⱥ= {ɂɜɚɧ; Ɇɚɪɢɹ; ɇɢɤɨɥɚɣ}. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ A \ B ɫɨɫɬɨɢɬ ɢɡ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ, ɤɨɬɨɪɵɟ ɩɪɢɧɚɞɥɟɠɚɬ ɦɧɨɠɟɫɬɜɭ Ⱥ ɢ ɧɟ ɩɪɢɧɚɞɥɟɠɚɬ ɦɧɨɠɟɫɬɜɭ ȼ, ɚ ɦɧɨɠɟɫɬɜɨ ȼ\ Ⱥ ɫɨɫɬɨɢɬ ɢɡ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ, ɤɨɬɨɪɵɟ ɩɪɢɧɚɞɥɟɠɚɬ ɦɧɨɠɟɫɬɜɭ ȼ ɢ ɧɟ ɩɪɢɧɚɞɥɟɠɚɬ ɦɧɨɠɟɫɬɜɭ Ⱥ.
ɉɪɢɦɟɪ 17. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = {1; 2; 3; 4; 5; 6}, ɚ ɦɧɨɠɟɫɬɜɨ
ȼ = {3; 6; 9; 12}. Ɍɨɝɞɚ A \ B ={1; 2; 4; 5}, ɚ ȼ\ Ⱥ={9; 12}.
ɉɪɢɦɟɪ 18. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = [–11; 3], ɚ ɦɧɨɠɟɫɬɜɨ ȼ = (0; 10]. Ɍɨɝɞɚ A \ B = [–11; 0] (ɪɢɫ. 2.7). Ɍɚɤ ɤɚɤ ɜ ɦɧɨɠɟɫɬɜɨ ȼ ɬɨɱɤɚ 0 ɧɟ ɜɯɨ-
ɞɢɥɚ, ɬɨ ɩɪɢ ɜɵɱɢɬɚɧɢɢ ɷɬɚ ɬɨɱɤɚ ɨɫɬɚɧɟɬɫɹ ɜ ɦɧɨɠɟɫɬɜɟ Ⱥ (ɤɪɭɝɥɚɹ ɫɤɨɛɤɚ ɦɟɧɹɟɬɫɹ ɧɚ ɤɜɚɞɪɚɬɧɭɸ).
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–11 |
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ǸȐș. 2.7
ɇɚɣɞɟɦ ɬɟɩɟɪɶ ȼ\ Ⱥ. ȼ\ Ⱥ=(3; 10] (ɪɢɫ. 2.8). Ɍɚɤ ɤɚɤ ɜ ɦɧɨɠɟɫɬɜɨ Ⱥ ɬɨɱɤɚ 3 ɜɯɨɞɢɥɚ, ɬɨ ɩɪɢ ɜɵɱɢɬɚɧɢɢ ɷɬɚ ɬɨɱɤɚ ɧɟ ɨɫɬɚɧɟɬɫɹ ɜ ɦɧɨɠɟɫɬɜɟ ȼ (ɤɜɚɞɪɚɬɧɚɹ ɫɤɨɛɤɚ ɦɟɧɹɟɬɫɹ ɧɚ ɤɪɭɝɥɭɸ).
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–11 |
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ǸȐș. 2.8 |
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ɋɢɦɦɟɬɪɢɱɟɫɤɨɣ ɪɚɡɧɨɫɬɶɸ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ, ɫɨɫɬɨɹɳɟɟ ɢɡ ɷɥɟɦɟɧɬɨɜ, ɤɚɠɞɵɣ ɢɡ ɤɨɬɨɪɵɯ ɩɪɢɧɚɞɥɟɠɢɬ ɢɥɢ ɬɨɥɶɤɨ ɦɧɨɠɟɫɬɜɭ Ⱥ, ɢɥɢ ɬɨɥɶɤɨ ɦɧɨɠɟɫɬɜɭ ȼ. ɋɢɦɦɟɬɪɢɱɟɫɤɚɹ ɪɚɡɧɨɫɬɶ ɦɧɨɠɟɫɬɜ Ⱥ ɢ ȼ ɨɛɨɡɧɚɱɚɟɬɫɹ Ⱥ ȼ (ɪɢɫ. 2.9).
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A B
ǸȐș. 2.9
ɉɪɢɦɟɪ 19. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = {Ɍɚɬɶɹɧɚ, Ɉɥɶɝɚ, Ⱥɥɢɧɚ, ɉɟɬɪ, ɂɪɢɧɚ, ȼɢɬɚɥɢɣ}, ɚ ɦɧɨɠɟɫɬɜɨ ȼ = {ɂɜɚɧ, Ɉɥɶɝɚ, Ɇɚɪɢɹ, ɇɢɤɨɥɚɣ, ɉɟɬɪ, ɂɪɢɧɚ}. Ɍɨɝɞɚ Ⱥ ȼ = {Ɍɚɬɶɹɧɚ; Ⱥɥɢɧɚ; ȼɢɬɚɥɢɣ; ɂɜɚɧ; Ɇɚɪɢɹ; ɇɢɤɨɥɚɣ}. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ Ⱥ ȼ ɫɨɫɬɨɢɬ ɢɡ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ, ɤɨɬɨɪɵɟɩɪɢɧɚɞɥɟɠɚɬɢɥɢ ɬɨɥɶɤɨɦɧɨɠɟɫɬɜɭ Ⱥ, ɢɥɢɬɨɥɶɤɨ ɦɧɨɠɟɫɬɜɭ ȼ.
ɉɪɢɦɟɪ 20. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = {1; 2; 3; 4; 5; 6}, ɚ ɦɧɨɠɟɫɬɜɨ
ȼ = {3; 6; 9; 12}. Ɍɨɝɞɚ Ⱥ ȼ = {1; 2; 4; 5; 9; 12}.
ɉɪɢɦɟɪ 21. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = (–5; 5], ɚ ɦɧɨɠɟɫɬɜɨ ȼ = [–5; 5).
ɇɚɣɞɟɦ Ⱥ ȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ. |
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ɉɪɢɦɟɪ 22. |
ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = (– ; + ), ɚ |
ɦɧɨɠɟɫɬɜɨ |
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ȼ = [–7; 29). ɇɚɣɞɟɦ Ⱥ ȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ. |
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Ⱥ ȼ = (– ; + ), Ⱥ ȼ = [–7; 29), |
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A \ B = (– ;–7) [29;+ ), ȼ\ Ⱥ= Ø. |
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ɉɪɢɦɟɪ 23. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = {1; 3; 5; 7; 9; 11; 13; 15}, ɚ ɦɧɨ- |
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ɠɟɫɬɜɨ ȼ = [15; 23). ɇɚɣɞɟɦ Ⱥ ȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ. |
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Ⱥ ȼ = {1; 3; 5; 7; 9; 11; 13} [15; 23), |
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5; 7; 9; 11; 13}, ȼ\ Ⱥ= (15; 23). |
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ɉɪɢɦɟɪ 24. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = {1; 3; 5; 7; 9; 11; 13; 15}, ɚ ɦɧɨ- |
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ɠɟɫɬɜɨ ȼ = [5; 11]. ɇɚɣɞɟɦ Ⱥ ȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ. |
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Ⱥ ȼ = {1; 3} [5; 11] {13; 15} , |
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ɉɪɢɦɟɪ 25. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ = (–3; 8], ɚ ɦɧɨɠɟɫɬɜɨ ȼ = (8; 15). ɇɚɣɞɟɦ Ⱥ ȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ.
Ⱥ ȼ = (–3; 15], Ⱥ ȼ = Ø, A \ B = (–3; 8], ȼ\ Ⱥ= (8; 15).
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ɉɪɢɦɟɪ 26. ɉɭɫɬɶ Ⱥ — ɦɧɨɠɟɫɬɜɨ ɤɭɪɫɚɧɬɨɜ ɩɟɪɜɨɝɨ ɤɭɪɫɚ, ɚ ȼ — ɦɧɨɠɟɫɬɜɨ ɤɭɪɫɚɧɬɨɜ-ɫɩɨɪɬɫɦɟɧɨɜ. ɇɚɣɞɟɦ Ⱥ ȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ, Ⱥ ȼ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɨɛɴɟɞɢɧɟɧɢɟ A B ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɤɭɪɫɚɧɬɨɜ ɢɧɫɬɢɬɭɬɚ, ɤɨɬɨɪɵɟ ɥɢɛɨ ɭɱɚɬɫɹ ɧɚ ɩɟɪɜɨɦ ɤɭɪɫɟ, ɥɢɛɨ ɡɚɧɢɦɚɸɬɫɹ ɫɩɨɪɬɨɦ. ɉɟɪɟɫɟɱɟɧɢɟ A B — ɷɬɨ ɦɧɨɠɟɫɬɜɨ ɩɟɪɜɨɤɭɪɫɧɢɤɨɜ-ɫɩɨɪɬɫɦɟɧɨɜ. Ɋɚɡɧɨɫɬɶ A \ B — ɷɬɨ ɜɫɟ ɩɟɪɜɨɤɭɪɫɧɢɤɢ, ɧɟ ɡɚɧɢɦɚɸɳɢɟɫɹ ɫɩɨɪɬɨɦ, ɚ ȼ\ Ⱥ — ɷɬɨ ɜɫɟ ɤɭɪɫɚɧɬɵɫɩɨɪɬɫɦɟɧɵ, ɧɨ ɧɟ ɩɟɪɜɨɤɭɪɫɧɢɤɢ. ɋɢɦɦɟɬɪɢɱɟɫɤɚɹ ɪɚɡɧɨɫɬɶ Ⱥ ȼ ɩɨɥɭɱɚɟɬɫɹ ɢɡ ɨɛɴɟɞɢɧɟɧɢɹ ɩɨɫɥɟ ɜɵɱɟɬɚ ɩɟɪɟɫɟɱɟɧɢɹ, ɬ.ɟ. ɷɬɨ ɜɫɟ ɤɭɪɫɚɧɬɵ ɩɟɪɜɨɝɨ ɤɭɪɫɚ, ɧɨ ɧɟ ɫɩɨɪɬɫɦɟɧɵ ɢ ɜɫɟ ɫɩɨɪɬɫɦɟɧɵ, ɧɨ ɧɟ ɩɟɪɜɨɤɭɪɫɧɢɤɢ.
ɉɪɢ ɪɚɛɨɬɟ ɜ ɤɨɧɤɪɟɬɧɨɣ ɩɪɟɞɦɟɬɧɨɣ ɨɛɥɚɫɬɢ ɨɛɵɱɧɨ ɨɝɪɚɧɢɱɢɜɚɸɬɫɹ ɧɟɤɨɬɨɪɨɣ ɫɨɜɨɤɭɩɧɨɫɬɶɸ ɨɛɴɟɤɬɨɜ. Ɂɚɮɢɤɫɢɪɨɜɚɧɧɨɟ ɤɚɤɢɦɥɢɛɨ ɨɛɪɚɡɨɦ ɦɧɨɠɟɫɬɜɨ ɨɛɴɟɤɬɨɜ, ɞɨɩɭɫɬɢɦɵɯ ɩɪɢ ɞɚɧɧɨɦ ɪɚɫɫɦɨɬ-
ɪɟɧɢɢ, ɧɚɡɵɜɚɸɬ ɭɧɢɜɟɪɫɚɥɶɧɵɦ ɦɧɨɠɟɫɬɜɨɦ.
ɉɪɢɦɟɪɚɦɢ ɭɧɢɜɟɪɫɚɥɶɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɹɜɥɹɸɬɫɹ ɱɢɫɥɚ ɜ ɚɪɢɮɦɟɬɢɤɟ, ɡɚɤɨɧɵ ɜ ɸɪɢɫɩɪɭɞɟɧɰɢɢ, ɫɥɨɜɚ ɜ ɹɡɵɤɨɡɧɚɧɢɢ ɢ ɬ.ɞ.
ɍɧɢɜɟɪɫɚɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ ɨɛɨɡɧɚɱɚɟɬɫɹ ɛɭɤɜɨɣ ȿ ɢ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɜɢɞɟ ɧɟɤɨɬɨɪɨɝɨ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ, ɚ ɜɫɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɦɧɨɠɟɫɬɜɚ Ⱥ, ȼ, ɋ,… ɹɜɥɹɸɬɫɹ ɟɝɨ ɩɨɞɦɧɨɠɟɫɬɜɚɦɢ ɢ ɩɪɟɞɫɬɚɜɥɹɸɬɫɹ ɜ ɜɢɞɟ ɤɪɭɝɨɜ (ɪɢɫ. 2.10).
E
A B
C
ǸȐș. 2.10
Ɋɚɫɫɦɨɬɪɢɦ ɦɧɨɠɟɫɬɜɨ Ⱥ, ɤɨɬɨɪɨɟ ɹɜɥɹɟɬɫɹ ɩɨɞɦɧɨɠɟɫɬɜɨɦ ɭɧɢɜɟɪɫɚɥɶɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ȿ (ɪɢɫ. 2.11).
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E
A
ǸȐș. 2.11
Ɇɧɨɠɟɫɬɜɨ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ ɭɧɢɜɟɪɫɚɥɶɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ȿ, ɧɟ ɩɪɢɧɚɞɥɟɠɚɳɢɯ ɦɧɨɠɟɫɬɜɭ Ⱥ, ɧɚɡɵɜɚɟɬɫɹ ɞɨɩɨɥɧɟɧɢɟɦ ɦɧɨɠɟɫɬɜɚ Ⱥ ɞɨ ȿ
ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ Ⱥ (ɪɢɫ. 2.12).
E
A
A
ǸȐș. 2.12
ɉɪɢɦɟɪ 27. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɚ Ⱥ, ȼ ɢ ɋ ɹɜɥɹɸɬɫɹ ɩɨɞɦɧɨɠɟɫɬɜɚɦɢ ɭɧɢɜɟɪɫɚɥɶɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ȿ. ɋ ɩɨɦɨɳɶɸ ɤɪɭɝɨɜ ɗɣɥɟɪɚ ɬɪɟɛɭɟɬɫɹ
ɢɡɨɛɪɚɡɢɬɶ ɪɟɡɭɥɶɬɚɬ ɜɵɩɨɥɧɟɧɢɹ ɫɥɟɞɭɸɳɢɯ ɨɩɟɪɚɰɢɣ: (Ⱥ ȼ) ɋ .
Ȼɭɞɟɦ ɪɟɲɚɬɶ ɞɚɧɧɵɣ ɩɪɢɦɟɪ ɩɨ ɞɟɣɫɬɜɢɹɦ. ɉɟɪɜɨɟ ɞɟɣɫɬɜɢɟ — ɷɬɨ ɜɫɟɝɞɚ ɞɟɣɫɬɜɢɟ ɜ ɫɤɨɛɤɚɯ ( Ⱥ ȼ ) (ɪɢɫ. 2.13).
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