Математика и информатика. Решение логико-познавательных задач. Учебное пособие для студентов вузов

E

A B

C

ǸȐș. 2.13

ȼɬɨɪɨɟ ɞɟɣɫɬɜɢɟ — ɷɬɨ ɞɨɩɨɥɧɟɧɢɟ (Ⱥ ȼ) (ɪɢɫ. 2.14).

E

A B

C

ǸȐș. 2.14

Ɍɪɟɬɶɟ ɞɟɣɫɬɜɢɟ — ɷɬɨ ɩɟɪɟɫɟɱɟɧɢɟ (Ⱥ ȼ) ɋ (ɪɢɫ. 2.15).

E

A B

C

ǸȐș. 2.15

41

ɉɪɢɦɟɪ 28. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɚ Ⱥ, ȼ ɢ ɋ ɹɜɥɹɸɬɫɹ ɩɨɞɦɧɨɠɟɫɬɜɚɦɢ ɭɧɢɜɟɪɫɚɥɶɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ȿ. ɋ ɩɨɦɨɳɶɸ ɤɪɭɝɨɜ ɗɣɥɟɪɚ ɬɪɟɛɭɟɬɫɹ

ɢɡɨɛɪɚɡɢɬɶ ɪɟɡɭɥɶɬɚɬ ɜɵɩɨɥɧɟɧɢɹ ɫɥɟɞɭɸɳɢɯ ɨɩɟɪɚɰɢɣ: Ⱥ\ ȼ ȼ\ ɋ . ɉɟɪɜɨɟ ɞɟɣɫɬɜɢɟ — ɷɬɨ Ⱥ\ ȼ (ɪɢɫ. 2.16). ȼɬɨɪɨɟ ɞɟɣɫɬɜɢɟ — ɷɬɨ

Ⱥ\ ȼ (ɪɢɫ. 2.17). Ɍɪɟɬɶɟ ɞɟɣɫɬɜɢɟ — ɷɬɨ ȼ\ ɋ (ɪɢɫ. 2.18). ɑɟɬɜɟɪɬɨɟ

ɞɟɣɫɬɜɢɟ — ɷɬɨ ȼ\ ɋ (ɪɢɫ. 2.19). ɉɹɬɨɟ ɞɟɣɫɬɜɢɟ — ɷɬɨ ɪɟɡɭɥɶɬɚɬ ɜɵɩɨɥɧɟɧɢɹ ɨɩɟɪɚɰɢɢ ɩɟɪɟɫɟɱɟɧɢɹ Ⱥ\ ȼ ȼ\ ɋ (ɪɢɫ. 2.20).

E E

A B A B

C C

ǸȐș. 2.20

42

ɉɪɢɦɟɪ 29. ɋ ɩɨɦɨɳɶɸ ɤɪɭɝɨɜ ɗɣɥɟɪɚ ɬɪɟɛɭɟɬɫɹ ɢɡɨɛɪɚɡɢɬɶ ɪɟ-

ɡɭɥɶɬɚɬ ɜɵɩɨɥɧɟɧɢɹ ɫɥɟɞɭɸɳɢɯ ɨɩɟɪɚɰɢɣ: (Ⱥ ȼ) \ ɋ .

E E

A B A B

C C

E

A B

C

ǸȐș. 2.23 ( (Ⱥ ȼ) \ ɋ )

ɉɪɢɦɟɪ 30. Ⱦɚɧɨ: ȿ={1; 2; 3; 4; 5; 6; 7; 8; 9; 10}, Ⱥ={1; 3; 4; 8}, ȼ={2; 3; 5; 8; 10}. ɇɚɣɬɢ: Ⱥ ȼ , Ⱥ , ȼ , Ⱥ ȼ , Ⱥ ȼ , Ⱥ ȼ .

Ɋɟɲɟɧɢɟ.

Ⱥȼ ={1; 2; 3; 4; 5; 8; 10} ={6; 7; 9}.

Ⱥ={2; 5; 6; 7; 9; 10}.

ȼ={1; 4; 6; 7; 9}

Ⱥȼ ={3; 8} ={1; 2; 4; 5; 6; 7; 9; 10}.

Ⱥȼ ={1; 2; 4; 5; 6; 7; 9; 10}.

Ⱥȼ ={6; 7; 9}.

ɉɪɢɦɟɪ 31. Ⱦɚɧɨ: ȿ=(– ;+ ), Ⱥ=(–7; 13], ȼ=[0; 18). ɇɚɣɬɢ: Ⱥ ȼ ,

Ⱥ , ȼ , Ⱥ ȼ , Ⱥ ȼ , Ⱥ ȼ .

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Ɋɟɲɟɧɢɟ.

Ⱥȼ = ( 7;18) =(– ; –7] [18;+ ).

Ⱥ=(– ; -7] (13;+ ).

ȼ=(– ; 0) (18;+ ).

Ⱥȼ =[0; 13] =(– ; 0) (13;+ ).

Ⱥȼ =(– ; 0) (13;+ ).

Ⱥȼ =(– ; -7) (18;+ ).

Ⱦɥɹ ɤɨɧɟɱɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɟɝɨ ɦɨɳɧɨɫɬɶ ɟɫɬɶ ɱɢɫɥɨ ɷɥɟɦɟɧɬɨɜ ɷɬɨɝɨ ɦɧɨɠɟɫɬɜɚ. Ɇɨɳɧɨɫɬɶ ɥɸɛɨɝɨ ɛɟɫɤɨɧɟɱɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɛɨɥɶɲɟ ɦɨɳɧɨɫɬɢ ɥɸɛɨɝɨ ɤɨɧɟɱɧɨɝɨ. ɋɪɟɞɢ ɛɟɫɤɨɧɟɱɧɵɯ ɦɧɨɠɟɫɬɜ ɧɚɢɦɟɧɶɲɟɣ ɦɨɳɧɨɫɬɶɸ ɨɛɥɚɞɚɟɬ ɦɧɨɠɟɫɬɜɨ N ɜɫɟɯ ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɟɥ ɢ ɜɫɟ ɦɧɨɠɟɫɬɜɚ, ɟɦɭ ɪɚɜɧɨɦɨɳɧɵɟ (ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ ɫɱɟɬɧɵɟ ɦɧɨɠɟɫɬɜɚ). Ɇɨɳɧɨɫɬɶ ɦɧɨɠɟɫɬɜɚ R ɜɫɟɯ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ (ɬɚɤ ɧɚɡɵɜɚɟɦɚɹ ɦɨɳɧɨɫɬɶ ɤɨɧɬɢɧɭɭɦɚ) ɛɨɥɶɲɟ, ɱɟɦ ɦɨɳɧɨɫɬɶ ɫɱɟɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ.

Ɇɨɳɧɨɫɬɶ ɤɨɧɟɱɧɨɝɨ ɦɧɨɠɟɫɬɜɚ Ⱥ (ɱɢɫɥɨ ɷɥɟɦɟɧɬɨɜ ɜ ɦɧɨɠɟɫɬɜɟ Ⱥ) ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ m(Ⱥ). Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɦɨɳɧɨɫɬɶ ɩɭɫɬɨɝɨ ɦɧɨɠɟɫɬɜɚ m(Ø)=0.

Ⱦɥɹ ɥɸɛɵɯ ɞɜɭɯ ɤɨɧɟɱɧɵɯ ɦɧɨɠɟɫɬɜ ɫɩɪɚɜɟɞɥɢɜɨ ɫɥɟɞɭɸɳɟɟ ɭɬɜɟɪɠɞɟɧɢɟ: m(Ⱥ ȼ) m(A) m(B) m(A B) .

ɉɪɢɦɟɪ 32. ɗɤɡɚɦɟɧ ɩɨ ɦɚɬɟɦɚɬɢɤɟ ɫɞɚɜɚɥɢ 250 ɚɛɢɬɭɪɢɟɧɬɨɜ. Ɉɰɟɧɤɭ ɧɢɠɟ ɩɹɬɢ ɩɨɥɭɱɢɥɢ 180 ɱɟɥɨɜɟɤ. ȼɵɞɟɪɠɚɥɢ ɷɤɡɚɦɟɧ 210 ɱɟɥɨɜɟɤ. ɋɤɨɥɶɤɨ ɱɟɥɨɜɟɤ ɩɨɥɭɱɢɥɢ ɨɰɟɧɤɭ ɬɪɢ ɢɥɢ ɱɟɬɵɪɟ?

ɉɭɫɬɶ Ⱥ — ɦɧɨɠɟɫɬɜɨ ɱɟɥɨɜɟɤ, ɜɵɞɟɪɠɚɜɲɢɯ ɷɤɡɚɦɟɧ. Ɍɨɝɞɚ m(Ⱥ) 210 . ɉɭɫɬɶ ȼ — ɦɧɨɠɟɫɬɜɨ ɱɟɥɨɜɟɤ, ɩɨɥɭɱɢɜɲɢɯ ɨɰɟɧɤɭ ɧɢɠɟ

ɩɹɬɢ. Ɍɨɝɞɚ m(ȼ) 180 . Ɉɱɟɜɢɞɧɨ, ɱɬɨ Ⱥ ȼ — ɦɧɨɠɟɫɬɜɨ ɱɟɥɨɜɟɤ, ɫɞɚɜɚɜɲɢɯ ɷɤɡɚɦɟɧ. Ɍɨɝɞɚ m(Ⱥ ȼ) 250 . ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɫɩɪɚɜɟɞɥɢɜɨ ɫɥɟɞɭɸɳɟɟ ɭɬɜɟɪɠɞɟɧɢɟ:

m(Ⱥ ȼ) m(A) m(B) m(A B) ,

ɝɞɟ A B — ɦɧɨɠɟɫɬɜɨ ɱɟɥɨɜɟɤ, ɫɞɚɜɲɢɯ ɷɤɡɚɦɟɧ ɧɚ ɬɪɢ ɢɥɢ ɱɟɬɵɪɟ.

Ɍɨɝɞɚ m(Ⱥ ȼ) m(A) m(B) m(A B) . ɉɨɞɫɬɚɜɥɹɹ ɭɠɟ ɢɡɜɟɫɬɧɵɟ ɞɚɧɧɵɟ, ɩɨɥɭɱɚɟɦ m(Ⱥ ȼ) 210 180 250 140.

Ɉɬɜɟɬ: 140 ɱɟɥɨɜɟɤ ɩɨɥɭɱɢɥɢ ɨɰɟɧɤɭ ɬɪɢ ɢɥɢ ɱɟɬɵɪɟ.

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DzȖȕȚȘȖȓȤȕȣȍ ȏȈȌȈȕȐȧ Ȓ ȋȓȈȊȍ 2

(Ⱥ ȼ) (ȼ ɋ) .

3. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ=(–13; 5], ɚ ɦɧɨɠɟɫɬɜɨ ȼ=[–13; 0). ɇɚɣɞɢɬɟ

Ⱥ ȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ.

4. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ=(0; 11], ɚ ɦɧɨɠɟɫɬɜɨ ȼ=[0; 11). ɇɚɣɞɢɬɟ

Ⱥȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ.

5.ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ=(–3; 3), ɚ ɦɧɨɠɟɫɬɜɨ ȼ=[3; + ). ɇɚɣɞɢɬɟ

Ⱥȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ.

6.ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ=(– ; + ), ɚ ɦɧɨɠɟɫɬɜɨ ȼ=[0; 19). ɇɚɣɞɢɬɟ

Ⱥȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ.

7.ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ=(– ; 0), ɚ ɦɧɨɠɟɫɬɜɨ ȼ=[–14; 27). ɇɚɣɞɢɬɟ

Ⱥȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ.

8.ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ=(–3; + ), ɚ ɦɧɨɠɟɫɬɜɨ ȼ=[–12; 12]. ɇɚɣɞɢɬɟ

Ⱥȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ.

9. ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ=(–7; 19), ɚ ɦɧɨɠɟɫɬɜɨ ȼ=(0; 9). ɇɚɣɞɢɬɟ

Ⱥȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ.

10.ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ=(–4; 0), ɚ ɦɧɨɠɟɫɬɜɨ ȼ=[4; 8]. ɇɚɣɞɢɬɟ

Ⱥȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ.

11.ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ={5; 7; 9; 11; 15; 20; 22}, ɚ ɦɧɨɠɟɫɬɜɨ

ȼ=[–5; 5]. ɇɚɣɞɢɬɟ Ⱥ ȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ.

12.ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ={2; 4; 6; 8; 10; 12}, ɚ ɦɧɨɠɟɫɬɜɨ ȼ=(3; 9). ɇɚɣɞɢɬɟ Ⱥ ȼ , Ⱥ ȼ , A \ B , ȼ\ Ⱥ.

13.ɂɡɨɛɪɚɡɢɬɟ ɫ ɩɨɦɨɳɶɸ ɤɪɭɝɨɜ ɗɣɥɟɪɚ ɪɟɡɭɥɶɬɚɬ ɜɵɩɨɥɧɟɧɢɹ ɫɥɟɞɭɸɳɢɯ ɨɩɟɪɚɰɢɣ: (Ⱥ ɋ) ȼ

14.ɂɡɨɛɪɚɡɢɬɟ ɫ ɩɨɦɨɳɶɸ ɤɪɭɝɨɜ ɗɣɥɟɪɚ ɪɟɡɭɥɶɬɚɬ ɜɵɩɨɥɧɟɧɢɹ ɫɥɟɞɭɸɳɢɯ ɨɩɟɪɚɰɢɣ: (Ⱥ\ ɋ) (ȼ\ ɋ) .

15.ɂɡɨɛɪɚɡɢɬɟ ɫ ɩɨɦɨɳɶɸ ɤɪɭɝɨɜ ɗɣɥɟɪɚ ɪɟɡɭɥɶɬɚɬ ɜɵɩɨɥɧɟɧɢɹ ɫɥɟɞɭɸɳɢɯ ɨɩɟɪɚɰɢɣ: (Ⱥ ȼ) (ȼ ɋ) .

16.ɂɡɨɛɪɚɡɢɬɟ ɫ ɩɨɦɨɳɶɸ ɤɪɭɝɨɜ ɗɣɥɟɪɚ ɪɟɡɭɥɶɬɚɬ ɜɵɩɨɥɧɟɧɢɹ ɫɥɟɞɭɸɳɢɯ ɨɩɟɪɚɰɢɣ: (Ⱥ ɋ) (ȼ\ ɋ)

17.ɂɡɨɛɪɚɡɢɬɟ ɫ ɩɨɦɨɳɶɸ ɤɪɭɝɨɜ ɗɣɥɟɪɚ ɪɟɡɭɥɶɬɚɬ ɜɵɩɨɥɧɟɧɢɹ ɫɥɟɞɭɸɳɢɯ ɨɩɟɪɚɰɢɣ: (Ⱥ ɋ) \ (ȼ\ ɋ) .

45

18.Ⱦɚɧɨ: ȿ={1; 2; 3; 4; 5; 6; 7; 8; 9; 10} (ɭɧɢɜɟɪɫɚɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ), Ⱥ={2; 4; 6; 8}, ȼ={1; 5; 9}. ɇɚɣɞɢɬɟ: Ⱥ ȼ , Ⱥ , ȼ , Ⱥ ȼ , Ⱥ ȼ ,

Ⱥȼ .

19.Ⱦɚɧɨ: ȿ=(– ;+ ) (ɭɧɢɜɟɪɫɚɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ), Ⱥ=(0; 11], ȼ=[–4; 16).

ɇɚɣɞɢɬɟ: Ⱥ ȼ , Ⱥ , ȼ , Ⱥ ȼ , Ⱥ ȼ , Ⱥ ȼ .

20.ɉɭɫɬɶ ɦɧɨɠɟɫɬɜɨ Ⱥ=[2;7], ɦɧɨɠɟɫɬɜɨ ȼ=[–2; 4], ɚ ɦɧɨɠɟɫɬɜɨ

ɋ=[3;6]. ɇɚɣɞɢɬɟ (Ⱥ\ ȼ) ɋ , (Ⱥ\ ɋ) ȼ, A \ (ɋ B) , (A ȼ) \ ɋ , A (ȼ\ ɋ) .

21.ɇɚ ɩɟɪɜɨɦ ɤɭɪɫɟ ɢɧɫɬɢɬɭɬɚ ɨɛɭɱɚɸɬɫɹ 100 ɱɟɥɨɜɟɤ. ɂɡ 100 ɨɛɭɱɚɸɳɢɯɫɹ 70 ɱɟɥɨɜɟɤ ɡɧɚɸɬ ɚɧɝɥɢɣɫɤɢɣ ɹɡɵɤ, 45 ɡɧɚɸɬ ɮɪɚɧɰɭɡɫɤɢɣ ɹɡɵɤ ɢ 23 ɱɟɥɨɜɟɤɚ ɡɧɚɸɬ ɨɛɚ ɹɡɵɤɚ. ɋɤɨɥɶɤɨ ɨɛɭɱɚɸɳɢɯɫɹ ɧɚ ɩɟɪɜɨɦ ɤɭɪɫɟ ɧɟ ɡɧɚɸɬ ɧɢ ɚɧɝɥɢɣɫɤɨɝɨ, ɧɢ ɮɪɚɧɰɭɡɫɤɨɝɨ ɹɡɵɤɚ?

22.Ʉɨɦɩɚɧɢɹ ɫɬɭɞɟɧɱɟɫɤɨɣ ɦɨɥɨɞɟɠɢ ɢɡ 20 ɱɟɥɨɜɟɤ ɩɨɟɯɚɥɚ ɧɚ ɩɢɤɧɢɤ. Ɍɚɦ ɩɹɬɟɪɨ ɢɡ ɧɢɯ ɨɛɝɨɪɟɥɢ, ɜɨɫɟɦɶ ɛɵɥɢ ɫɢɥɶɧɨ ɩɨɤɭɫɚɧɵ ɤɨɦɚɪɚɦɢ, ɚ 10 ɨɫɬɚɥɢɫɶ ɜɫɟɦ ɞɨɜɨɥɶɧɵ. ɋɤɨɥɶɤɨ ɱɟɥɨɜɟɤ ɨɛɝɨɪɟɥɢ, ɧɨ ɧɟ ɛɵɥɢ ɩɨɤɭɫɚɧɵ ɤɨɦɚɪɚɦɢ? ɋɤɨɥɶɤɨ ɱɟɥɨɜɟɤ ɛɵɥɢ ɩɨɤɭɫɚɧɵ ɤɨɦɚɪɚɦɢ ɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɨɛɝɨɪɟɥɢ?

46

ɉɪɢɦɟɪ 2. ȼɫɟ ɞɟɥɶɮɢɧɵ — ɢɡ ɨɬɪɹɞɚ ɤɢɬɨɨɛɪɚɡɧɵɯ. ɇɢ ɨɞɧɚ ɪɵɛɚ ɧɟ ɩɪɢɧɚɞɥɟɠɢɬ ɤ ɤɢɬɨɨɛɪɚɡɧɵɦ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɢ ɨɞɧɚ ɪɵɛɚ ɧɟ ɹɜɥɹɟɬɫɹ ɞɟɥɶɮɢɧɨɦ.

ɗɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɢɫɬɢɧɧɨ. ɉɭɫɬɶ Ⱥ — ɦɧɨɠɟɫɬɜɨ ɞɟɥɶɮɢɧɨɜ, ȼ — ɦɧɨɠɟɫɬɜɨ ɤɢɬɨɨɛɪɚɡɧɵɯ, ɚ ɋ — ɦɧɨɠɟɫɬɜɨ ɪɵɛ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ Ⱥ ȼ

ɞɟɥɶɮɢɧɨɦ). Ƚɪɚɮɢɱɟɫɤɚɹɢɧɬɟɪɩɪɟɬɚɰɢɹ ɩɪɟɞɫɬɚɜɥɟɧɚɧɚɪɢɫ. 3.2.

B

A C

ǸȐș. 3.2

ɉɪɢɦɟɪ 3. ȼɫɟ ɤɜɚɞɪɚɬɵ — ɪɨɦɛɵ. ɇɟɤɨɬɨɪɵɟ ɪɨɦɛɵ ɢɦɟɸɬ ɨɫɬɪɵɣ ɭɝɨɥ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɟɤɨɬɨɪɵɟ ɤɜɚɞɪɚɬɵ ɢɦɟɸɬ ɨɫɬɪɵɣ ɭɝɨɥ.

ɗɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɥɨɠɧɨ. ɏɨɬɹ ɨɛɚ ɭɬɜɟɪɠɞɟɧɢɹ, ɢɡ ɤɨɬɨɪɵɯ ɫɞɟɥɚɧ ɜɵɜɨɞ, ɩɪɚɜɢɥɶɧɵɟ, ɫɚɦ ɜɵɜɨɞ ɨ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɤɜɚɞɪɚɬɨɜ ɫ ɨɫɬɪɵɦ ɭɝɥɨɦ — ɧɟɜɟɪɧɵɣ. ɉɭɫɬɶ Ⱥ — ɦɧɨɠɟɫɬɜɨ ɤɜɚɞɪɚɬɨɜ, ȼ — ɦɧɨɠɟɫɬɜɨ ɪɨɦɛɨɜ, ɚ ɋ — ɦɧɨɠɟɫɬɜɨ ɪɨɦɛɨɜ, ɢɦɟɸɳɢɯ ɨɫɬɪɵɣ ɭɝɨɥ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ Ⱥ ȼ (ɜɫɟ ɤɜɚɞɪɚɬɵ — ɪɨɦɛɵ), ȼ ɋ ζ Ø (ɧɟɤɨɬɨɪɵɟ ɪɨɦɛɵ ɢɦɟɸɬ ɨɫɬɪɵɣ ɭɝɨɥ). ɇɨ Ⱥ ɋ ɦɨɠɟɬ ɛɵɬɶ ɢ ɩɭɫɬɵɦ ɦɧɨɠɟɫɬɜɨɦ. ɗɬɨɬ ɮɚɤɬ ɯɨɪɨɲɨ ɩɨɹɫɧɟɧ ɧɚ ɪɢɫ. 3.3.

C

B

A

ǸȐș. 3.3

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ɉɪɢɦɟɪ 4. ɋ ɩɨɦɨɳɶɸ ɞɢɚɝɪɚɦɦ ɗɣɥɟɪɚ—ȼɟɧɧɚ ɩɪɨɜɟɪɢɦ ɢɫɬɢɧɧɨɫɬɶ ɫɥɟɞɭɸɳɟɝɨ ɭɬɜɟɪɠɞɟɧɢɹ: «ȿɫɥɢ ɧɢ ɨɞɢɧ ɥɢɰɟɦɟɪ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɧɚɤɚɡɚɧ, ɬɨɡɧɚɱɢɬɧɢɨɞɢɧ, ɩɨɧɟɫɲɢɣɧɚɤɚɡɚɧɢɟ, ɧɟɹɜɥɹɟɬɫɹɥɢɰɟɦɟɪɨɦ».

ɉɭɫɬɶ Ⱥ — ɦɧɨɠɟɫɬɜɨ ɥɢɰɟɦɟɪɨɜ, ɚ ȼ — ɦɧɨɠɟɫɬɜɨ ɥɸɞɟɣ, ɩɨɧɟɫɲɢɯ ɧɚɤɚɡɚɧɢɟ. Ɍɨɝɞɚ ɨɱɟɜɢɞɧɨ, ɱɬɨ Ⱥ ȼ Ø ɢ ɞɚɧɧɨɟ ɭɬɜɟɪɠɞɟ-

ɧɢɟ ɢɫɬɢɧɧɨ (ɪɢɫ. 3.4).

ǸȐș. 3.4

Ɇɧɨɝɢɟ ɧɚɭɤɢ ɢ ɢɫɤɭɫɫɬɜɚ Ⱦɪɟɜɧɟɝɨ ɦɢɪɚ ɧɚɜɫɟɝɞɚ ɭɲɥɢ ɜ ɩɪɨɲɥɨɟ ɢ ɩɪɟɞɫɬɚɜɥɹɸɬ ɢɧɬɟɪɟɫ ɢɫɤɥɸɱɢɬɟɥɶɧɨ ɤɚɤ ɩɚɦɹɬɧɢɤɢ ɫɬɚɪɢɧɵ. Ɉɞɧɚɤɨ, ɧɟɤɨɬɨɪɵɟ ɢɡ ɧɢɯ ɩɟɪɟɠɢɥɢ ɜɟɤɚ, ɢ ɜ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɧɟ ɩɨɬɟɪɹɥɢ ɫɜɨɸ ɚɤɬɭɚɥɶɧɨɫɬɶ. Ʉ ɢɯ ɱɢɫɥɭ ɨɬɧɨɫɢɬɫɹ ɢ ɥɨɝɢɤɚ Ⱥɪɢɫɬɨɬɟɥɹ.

3.2. ǶșȕȖȊȕȣȍ ȗȖȕȧȚȐȧ ȔȈȚȍȔȈȚȐȟȍșȒȖȑ ȓȖȋȐȒȐ

Ɉɫɧɨɜɨɩɨɥɨɠɧɢɤɨɦ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɥɨɝɢɤɢ ɹɜɥɹɟɬɫɹ ɜɟɥɢɤɢɣ ɧɟɦɟɰɤɢɣ ɦɚɬɟɦɚɬɢɤ Ƚɨɬɮɪɢɞ ȼɢɥɶɝɟɥɶɦ Ʌɟɣɛɧɢɰ (1646—1716). Ʌɟɣɛɧɢɰ ɛɵɥ ɧɚɢɛɨɥɟɟ ɛɥɢɡɨɤ ɤ ɪɟɚɥɶɧɨ ɨɫɭɳɟɫɬɜɥɟɧɧɨɦɭ ɜɩɨɫɥɟɞɫɬɜɢɢ ɡɚɦɵɫɥɭ ɩɨɫɬɪɨɟɧɢɹ ɭɧɢɜɟɪɫɚɥɶɧɨɝɨ ɥɨɝɢɱɟɫɤɨɝɨ ɢɫɱɢɫɥɟɧɢɹ, ɫ ɩɨɦɨɳɶɸ ɤɨɬɨɪɨɝɨ ɫɩɨɪɵ ɦɟɠɞɭ ɥɸɞɶɦɢ ɦɨɠɧɨ ɛɵɥɨ ɛɵ ɪɚɡɪɟɲɚɬɶ ɩɨɫɪɟɞɫɬɜɨɦ ɜɵɱɢɫɥɟɧɢɣ. Ɋɚɡɜɢɜɚɹ ɢɞɟɢ Ʌɟɣɛɧɢɰɚ, ɢɪɥɚɧɞɫɤɢɣ ɦɚɬɟɦɚɬɢɤ Ⱦɠɨɪɞɠ Ȼɭɥɶ (1815—1864) ɩɨɥɨɠɢɥ ɧɚɱɚɥɨ ɫɨɡɞɚɧɢɸ ɚɩɩɚɪɚɬɚ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɥɨɝɢɤɢ, ɤɨɬɨɪɚɹ ɜ ɨɬɥɢɱɢɟ ɨɬ ɨɛɵɱɧɨɣ ɚɥɝɟɛɪɵ ɨɩɟɪɢɪɭɟɬ ɧɟ ɱɢɫɥɚɦɢ, ɚ ɜɵɫɤɚɡɵɜɚɧɢɹɦɢ.

ɋɭɠɞɟɧɢɹ ɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɥɨɝɢɤɟ ɧɚɡɵɜɚɸɬɫɹ ɜɵɫɤɚɡɵɜɚɧɢɹɦɢ ɢɥɢ ɥɨɝɢɱɟɫɤɢɦɢ ɜɵɪɚɠɟɧɢɹɦɢ.

ȼɵɫɤɚɡɵɜɚɧɢɟ — ɷɬɨ ɥɸɛɨɟ ɭɬɜɟɪɠɞɟɧɢɟ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɝɨ ɢɦɟɟɬ ɫɦɵɫɥ ɝɨɜɨɪɢɬɶ, ɱɬɨ ɨɧɨ ɢɫɬɢɧɧɨ ɢɥɢ ɥɨɠɧɨ.

ȼɵɫɤɚɡɵɜɚɧɢɹ ɩɪɢɧɹɬɨ ɨɛɨɡɧɚɱɚɬɶ ɫɬɪɨɱɧɵɦɢ ɢɥɢ ɩɪɨɩɢɫɧɵɦɢ ɛɭɤɜɚɦɢɥɚɬɢɧɫɤɨɝɨɚɥɮɚɜɢɬɚ: A, B, C, D, a, b, c, d… .

ɉɪɢɦɟɪ 5. ɉɪɢɦɟɪɚɦɢ ɜɵɫɤɚɡɵɜɚɧɢɣ ɦɨɝɭɬ ɫɥɭɠɢɬɶ ɫɥɟɞɭɸɳɢɟ ɭɬɜɟɪɠɞɟɧɢɹ.

1.Ɇɚɪɫ ɞɚɥɶɲɟ ɨɬ ɫɨɥɧɰɚ, ɱɟɦ ȼɟɧɟɪɚ (ɢɫɬɢɧɧɨɟ ɜɵɫɤɚɡɵɜɚɧɢɟ).

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2.Ɇɨɫɤɜɚ — ɫɬɨɥɢɰɚ Ɋɨɫɫɢɢ (ɢɫɬɢɧɧɨɟ ɜɵɫɤɚɡɵɜɚɧɢɟ).

3.ɑɢɫɥɨ 26 — ɩɪɨɫɬɨɟ (ɥɨɠɧɨɟ ɜɵɫɤɚɡɵɜɚɧɢɟ).

4.11 < 17 (ɢɫɬɢɧɧɨɟ ɜɵɫɤɚɡɵɜɚɧɢɟ).

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɤɚɠɞɨɟ ɜɵɫɤɚɡɵɜɚɧɢɟ ɢɥɢ ɢɫɬɢɧɧɨ, ɢɥɢ ɥɨɠɧɨ. ȼɨɨɛɳɟ, ɫɭɳɟɫɬɜɭɸɬ ɪɚɡɧɵɟ ɜɚɪɢɚɧɬɵ ɨɛɨɡɧɚɱɟɧɢɹ ɢɫɬɢɧɧɨɫɬɢ ɢ ɥɨɠɧɨɫɬɢ ɜɵɫɤɚɡɵɜɚɧɢɣ (ɬɚɛɥ. 3.1).

ǺȈȉȓȐȞȈ 3.1

ȼɵɫɤɚɡɵɜɚɧɢɹ ɦɨɝɭɬ ɛɵɬɶ ɨɛɪɚɡɨɜɚɧɵ ɫ ɩɨɦɨɳɶɸ ɫɥɨɜ ɢɥɢ ɫɢɦɜɨɥɨɜ, ɨɞɧɚɤɨ ɧɟ ɤɚɠɞɵɣ ɧɚɛɨɪ ɫɥɨɜ ɢɥɢ ɫɢɦɜɨɥɨɜ (ɞɚɠɟ ɨɫɦɵɫɥɟɧɧɵɣ) ɹɜɥɹɟɬɫɹ ɜɵɫɤɚɡɵɜɚɧɢɟɦ.

ɇɚɩɪɢɦɟɪ, ɜɨɫɤɥɢɰɚɬɟɥɶɧɵɟ ɢ ɜɨɩɪɨɫɢɬɟɥɶɧɵɟ ɩɪɟɞɥɨɠɟɧɢɹ ɜɵɫɤɚɡɵɜɚɧɢɹɦɢɧɟɹɜɥɹɸɬɫɹ(«Ʉɚɤɨɝɨɰɜɟɬɚɷɬɨɬɞɨɦ?», «Ɂɞɪɚɜɫɬɜɭɣɬɟ!»).

ɇɟ ɹɜɥɹɸɬɫɹ ɜɵɫɤɚɡɵɜɚɧɢɹɦɢ ɢ ɩɪɟɞɥɨɠɟɧɢɹ «Ɉɧ ɫɟɪɨɝɥɚɡ» ɢɥɢ «Ɍɪɟɭɝɨɥɶɧɢɤ — ɪɚɜɧɨɛɟɞɪɟɧɧɵɣ» — ɜ ɧɢɯ ɧɟ ɭɤɚɡɚɧɨ ɨ ɤɚɤɨɦ ɱɟɥɨɜɟɤɟ ɢɞɟɬ ɪɟɱɶ ɢɥɢ ɤɚɤɨɣ ɬɪɟɭɝɨɥɶɧɢɤ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ. Ɍɚɤɢɟ ɩɪɟɞɥɨɠɟɧɢɹ ɫ ɧɟɢɡɜɟɫɬɧɵɦ ɱɥɟɧɨɦ (ɩɟɪɟɦɟɧɧɨɣ) ɧɚɡɵɜɚɸɬ ɧɟɨɩɪɟɞɟɥɟɧɧɵɦɢ ɜɵɫɤɚɡɵɜɚɧɢɹɦɢ (ɨɧɢ ɛɭɞɭɬ ɪɚɫɫɦɨɬɪɟɧɵ ɞɚɥɟɟ).

Ɉɬɦɟɬɢɦ, ɱɬɨ ɩɪɟɞɥɨɠɟɧɢɟ «ɇɟɤɨɬɨɪɵɟ ɥɸɞɢ ɫɟɪɨɝɥɚɡɵ» ɭɠɟ ɹɜɥɹɟɬɫɹ ɢɫɬɢɧɧɵɦ ɜɵɫɤɚɡɵɜɚɧɢɟɦ.

Ɂɚɦɟɱɚɧɢɟ 1. ɉɨɜɟɫɬɜɨɜɚɬɟɥɶɧɨɟ ɩɪɟɞɥɨɠɟɧɢɟ «ə ɥɝɭ» ɧɟ ɹɜɥɹɟɬɫɹ ɜɵɫɤɚɡɵɜɚɧɢɟɦ, ɩɨɫɤɨɥɶɤɭ, ɟɫɥɢ ɨɧɨ ɢɫɬɢɧɧɨ (ɬ.ɟ. ɹ ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɥɝɭ), ɡɧɚɱɢɬ ɹ ɧɟ ɥɝɭ, ɚ ɝɨɜɨɪɸ ɩɪɚɜɞɭ! ɂ ɧɚɨɛɨɪɨɬ. ɗɬɨ ɩɪɢɦɟɪ ɥɨɝɢɱɟɫɤɨɝɨ ɩɚɪɚɞɨɤɫɚ. Ʌɨɝɢɱɟɫɤɢɟ ɩɚɪɚɞɨɤɫɵ ɧɟ ɨɬɧɨɫɹɬɫɹ ɤ ɜɵɫɤɚɡɵɜɚɧɢɹɦ.

Ɇɚɬɟɦɚɬɢɱɟɫɤɢɟ ɭɬɜɟɪɠɞɟɧɢɹ ɞɨɥɠɧɵ ɛɵɬɶ ɱɟɬɤɢɦɢ ɢ ɨɞɧɨɡɧɚɱɧɨ ɩɨɧɢɦɚɟɦɵɦɢ. ɗɬɚɰɟɥɶɞɨɫɬɢɝɚɟɬɫɹɢɫɩɨɥɶɡɨɜɚɧɢɟɦɥɨɝɢɱɟɫɤɢɯɫɜɹɡɨɤ.

ɋɭɳɟɫɬɜɭɟɬ ɩɹɬɶ ɥɨɝɢɱɟɫɤɢɯ ɫɜɹɡɨɤ: ɱɚɫɬɢɰɚ «ɧɟ», ɫɨɸɡɵ «ɢ», «ɢɥɢ», «ɟɫɥɢ…, ɬɨ…», «ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ…».

ȼɵɫɤɚɡɵɜɚɧɢɹ, ɫɮɨɪɦɢɪɨɜɚɧɧɵɟ ɛɟɡ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɥɨɝɢɱɟɫɤɢɯ ɫɜɹɡɨɤ, ɧɚɡɵɜɚɸɬɫɹ ɩɪɨɫɬɵɦɢ (ɷɥɟɦɟɧɬɚɪɧɵɦɢ) ɜɵɫɤɚɡɵɜɚɧɢɹɦɢ.

ɇɚɩɪɢɦɟɪ: «ȼ Ɇɨɫɤɜɟ ɡɢɦɨɣ ɱɚɫɬɨ ɢɞɟɬ ɫɧɟɝ».

ȼɵɫɤɚɡɵɜɚɧɢɹ, ɫɮɨɪɦɢɪɨɜɚɧɧɵɟ ɢɡ ɩɪɨɫɬɵɯ ɜɵɫɤɚɡɵɜɚɧɢɣ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɥɨɝɢɱɟɫɤɢɯ ɫɜɹɡɨɤ, ɧɚɡɵɜɚɸɬɫɹ ɫɥɨɠɧɵɦɢ (ɫɨɫɬɚɜ-

ɧɵɦɢ).

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