51(075.8) 22.173 76

-

: , . . ; - , . .

, . . 76 : : / . . . : , 2006. 175 . : . ISBN 985-485-608-9. , . , , . , , . 51(075.8) 22.173 . ., 2006 , 2006

ISBN 985-485-608-9

, , , . , .

. . . . ? : . . , , . , . ? , , . , . 3

, , , . , , . 1960 - , , , . . , . . , . . . , , , , . , , , . . , , . -, . , , . : , , , , , , . , , , , . , , . , , , . , , : 4

( ) , . , , . .1. . . , , , . . , , . , , , , , , . , , . , . ? , , ? , XX , : , ; , ; , ; , . , . , , , , , . , , , , . , . , , , 1

. . . .: , 2000. . 316.

5

, . , . . . : . . , , , , , . , , . , , , , . , - , , . - , , , . . . - , . , , . , . . , . . : , , . ? , , , , , 6

?2. , . , , , , . , , . , , . , , , . , . , : . , , . . , . , , , . , , . , . , , , . . ? , 2

. // . .: , 1968. . 385.

7

, . , , , , , . , , , . , , , . , , ? . , - . , . , . XX : , . , , . , , , 3. . . - , . , , - , 3

., . ? .: , 2004. . 20.

8

. , , , , . - , . . . , , , , . , - . , . , , , , , , . , : , . , - , . ? , , ? , , , , . , , . , , . . , 9

(. 1836. . 3) , . . , , . , , . , , . , . , , , . , , ? , ? . , , , . , . -, . -, -, , . , . , , , - . , 4. 4

. . // . . 1963. 3. . 91.

10

. , . . , , . , - . , , . , . , , , , , , . , , . . . , . . , , . , , . . , . , . , , . , . , - , , . , , . , 11

, , . , . , , . . , . , . , . , , , . ., , , . , . : , , . , . , , , , 5. , , - . XX , , , ., - , - , , , . , 5

. . .: --, 1999. . 2. . 302.

12

. , . , , , . : , , . , . , . - . , . , . , , . - . . , . . , . , . , . , , , , . . . , , , 13

. . , , , , . ? . . , , , , , , , . ? ? , ? , , , . - . . . , , , , . , , , . , , . , , . . , , . , . : . , , , , , .

1 , , . , , , .

. , . , , . , , , , , . , - . , , . . . . , , : . () . . , , . , , , , . . . , 15

, . , . . . ( . ., 1958), . , -. , , , . , , , . ? . . : , , , , , , 6. , , , . , , , , . , , . . XIX , , . 6

. . // . 1961. . 6. . 50.

16

- . , . . , . , , , . , . . , . : , , . , , .

1.1. , . . , , , . , . . : , , , , . - , . , , , . , , , . . . . . . , . ? : . ? , . -, , -, . , , - 17

( ) ? , . , , . , , , . , , . , . , . . , . , . , , , . . , , . , . , , . . . , . , , , , . . . , . . . A, B, C, , X, Y, Z - . , . . . 18

, . , , , . , , a, b, c, , x, y, z. . , x M x M. : x M x M. . esti . x M, x M, , x M, x M. , , . , , . , , . . , , , . . , , . 1. - . 2. . 3. . , , . , , -, , , , . -, . , . , 19

, , , . . , . , , , , , . , , . . , . , , , , , . . , . , . , ? , , . , , . . ? , . . , . . , . . . . . -? , , . . . , . , , , . , , , , . . , , , , , , . , 20- 50 % , . 20

, , . ? , ? , . , , , . , , , , . , , . , , . , . , , , . , . . , , . , , , , , , , . , , , , . . , . , , . , , .

, , . . . : 100 , , , : , . 5 , 8, 10, 20, 23, 30, 50 . , , . ? 21

1. , ? 2. , ? 3. ?

1.2. - , . , . , , . . , , . , , , , . . , . , . , , . , , , {, , , , , , , , , }, n {1, 2, 3, , n}, . . , , {x, y} , {y, x}, , , . . . . - . 22

, , . . , : {, , , , , , , }. , , , , . ? - , , , . , . . . , , . , - , , . , . , , , 1 2. , , {, , 1, , , , , , 2}. , , . , , , , , . . . , n , . . = {1, 2, 3, , n}, n . n. , n , : , , , , , , , , , , , , , , , , , . , n = 18 = {1, 2, 3, , 18}. , n , : 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2 1, 1, 2, 1, 1, 1. , n = 25 = {1, 2, 3, , 25}. , 23

, , ., , , . , .. , . , {1, 2, 3, 4, 5, }. , , . . , , . , , n, {1, 4, 9, , n2}, {1, 8, 27, , k3, }. , , . , , , , . , . , . , . . , . .

, , , . , {x : x P} , , . , . , {n : n 1 n 1 000 000} 1 1 000 000, {x : x } . , . , = {11, 22, 33, 44, 55, 66, 77, 88, 99} , , 11. , . , 24

, , . . , . . . , : A = {x : x }; B = {x : x }; = {x : x }. B , . . , B , , , , . . , {x} x. . {x}, {z : z = x}, x, x , , . . . , , , , . , {{1, 2}, {2, 3}} , {1, 2} {2, 3}, {{1, 2}, {2, 3}} {1, 2, 3}. , , , , {x : x x}, {} . , , , , . . {}. , {} {{}}, . . {} {{}}. , Silentium! (!) , , : ? , ? , , , , . , 25

. , , . , .

1. , {9, 9, 10} , ? 2. = {{5, 6, 7}}, 5? 3. , , {} {, {}} ?

1.3. . , , , , , , . , - , , , . , , A B . , , , - , , A B , . , . , . . , . , . , , , . , . . 26

, , , , , , . , , . . A B, A B ( B A), A B. A B (" x ) ( x A x B ). : " ( , ), (. . X Y : X, Y, X Y), , ( , , ), . . , . , , , . , , , , , . -. , , , . A B, . . A B, : A B, B A, . , M = { } N = { }, . . M N. . , . . A A , .. A A A. 27

, . . A, , , A, , . , .. A, A. , , A. , , , . . . 1. {0, 1} : , {0}, {1}, {0, 1}. 2. {0, 1, 2} : , {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}. 3. , , n , 2n , . . : , . . C B B A, C A. , C B, B A, C A. , . . , , - . , , . , , - , , . U. , , , . . 28

, , . , , U. , . , .

, . . 1.1 A B , , A B, . . A B, A B, . . 1.2 A B, A B . , , . , . , , . . , , , , -, , , , -, . , A B, . . A B, B {. . B} A {. . A}, . . {. . A} {. . B}. {. . A} {. . B} 29

: ({. . A} {. . B}) ({. . A} {. . B} {. . B} {. . A}). , . . A B , A = B, A B, B A: A = B A B B A. A = B, , , A B . , A, , , , B, , , A B B A, . . , A = B. , . . , . . . , . . , , . , , , . , , . . . A B, A B, , , , , . , A B , , A B B A . , 30

, . , . . . . (.: . ., . ., . . . ., 1977). . A B , : A = {1, 2, 3} B = {1, 2, 3}. , A B. , . , . , , . , A , B . , , A = B, . . . A B 1, 2, 3 , : A, B. , A = B, . . . . , . , x y , {x, y} : {z : z = x z = y}. , : . , , , , M = {a, b}, . . , M = {b, a}, a, b M, b, a M, , {a, b} = {b, a}. 31

, , . : 1. . 2. . . A B, A B, , , A B: A B = {x : x A x B}. def , .. , , , , def definito . , A 3- , B , , A B - 3- , . A , B , A B . , , . A B , , A B = , , A B . . 1.3 1.4 A B , A B A B. A B .def

, . 32

. A B, . . A B, A B = A (. . 1.4), A B A . A, B C: A B A A B B. , A B A C B C. , A A = . A A = A ( ), , , , . . . A , , , B , , . A B. A 18 : A = {, , , , , , , , , , , , , , , , , }, B 13 : B = {, , , , , , , , , , , , }. 8 : A B = {, , , , , , , }, A, B. . , , . , : , , . . A , , B , C , 33

. : A (B C). , . . A B, A B, , , A B: A B = {x : x A x B}. , A , , B , , A B . A , B , A B . , . . . - A B, . . A B , A B . . 1.51.7 A B , A B , A B A B = . A B .def

, . . A B, .. A B, A B = B (. . 1.6), A B . , A, B C: 34

A A B B A B. , A B A C B C. , A : A A = A ( ), A = A. . A , , , B , , . A B. A 18 : A = {, , , , , , , , , , , , , , , , , }, B 13 : B = {, , , , , , , , , , , , }. 23 : A B = {, , , , , , , , , , , , , , , , , , , , , , }. 8 , , , , , , , , A B, , 23 , 18 + 13 = 31 , . . (18 + 13) 8 = 23. . : . . : , . , . , : , , , , - . , , , . : : ? ? . . A , , B , . A B A B A B. 35

, , x A x B. . A B {x : x }, . , , : C , . D , . , C = A B, D A B, , , , A B, . . D = A B. . E , , . E A B, E = A B. : , . . A B , A B. A B. . , , . . . A B, A \ B ( A B), , , A, B: A / B = {x : x A x B}. , B A. , A B 36def

, , A \ B , , . A = {x : |x| 1} B = {x : x < 0}, A \ B = {x : 0 x 1}. A B, . . A B, A \ B = . . 1.81.10 A B , A B , B A A B = . A \ B .

. , , , . , (. . 1.81.10): (A \ B) B = A B (A \ B) (A B) = A. , B A, A B = B , (A \ B) B = A (. . 1.9). A B , . . A B = , A \ B = A, A B (. . 1.10). , A, B C: A \ B A. , A B (A \ C) (B \ C) (C \ B) (C \ A). , A : A \ A = , A \ = A \ A = . . A , , , B , , . A \ B. 37

A 18 : A = {, , , , , , , , , , , , , , , , , }, B 13 : B = {, , , , , , , , , , , , }. A \ B 10 : A \ B = {, , , , , , , , , }, A, B. 8 , , , , , , , A B, . . A B, A \ B 18 8 = 10 , 18 13 = 5 . . A B, , A \ B B \ A, (A \ B) (B \ A) = . , A \ B = {, , , , , , , , , }, B \ A = {, , , , }. . , , , . , . . A -, , B , . A \ B B \ A. A \ B -, , , B \ A , , -, . . A B A A B, . . A B: A \ B = A \ (A B). . . 38

, : : , , , . , . , A , , B , , , A B, . . . U , A, U \ A A A: A = {x : x U x A}. A A U, A. , U R, . , , , (. 1.11). . 1.12 A, . . A, .def

: . . 39

, : ( ) . , , , , , , , ( !). . 1/4 , 1/4 , 1/2 . (1/2 + 1/4 + 1/4 = 1) , ! . , , , . , , , U. , A, U: A A = U A A = . , , . . A , , , U . A, . . A = U \ A. A 18 : A = {, , , , , , , , , , , , , , , , , }. 33 , , 15 : A = {, , , , , , , , , , , , , , }. . A B : A \ B = A B, . . A \ B A B. . , , . 40

U . A , B , , C , . : A B \ C. . , , . . A B B A. . . 1.13 U , A B , A B.

A , ( ), B , ( ). , B A. . A B, A D B, , , A B: A D B = (A \ B) (B \ A). , A B , , A D B , , , , 41def

. A = {x : 0 x 2} B = {x : 1 x 3}, A D B = {x : 0 x < 1 2 < x 3}. A B, . . A B, A \ B A D B . . 1.141.16 A B , A B , A B A B = . A D B .

, , A D B = (A B) \ (A B). A B . A B , . . A B = , A D B = A B (. . 1.16). A B: A D B = A = B. , A : A D A = , A D = A D A = A. . A , , , B , , . A D B. A 18 : A = {, , , , , , , , , , , , , , , , , }, B 13 : B = {, , , , , , , , , , , , }. 42

15 : A D B = {, , , , , , , , , , , , , , }, A \ B B \ A , , A B, A B, A D B 23 8 = 15. . A B : A \ B = A (A D B), A \ B = A D (A B). A B , . , , . , , , . , , , . A , , B , , A D B , . , A D B . , , A \ B A D B. : , , . . A N, B P. : A B, A B, A \ B, B \ A, A D B. , , , , , 43

, . : . A B N P; A B N P; A \ B , N P, B \ A , P N; , A D B N P. : , , , , , . - , . . , , , . . , : A B = (A D B) (A B) = (A \ B) (A B) (B \ A). n(S) S. A B n(A B) A B, , . . A B . , A B, , A B, . . A B n(A B) n(A B) = n(A) + n(B) n(A B). . A \ B, B \ A A B , A B A B = (A \ B) (A B) (B \ A) , n(A B) = = n(A \ B) + n(A B) + n(B \ A). A B A = (A \ B) (A B) 44

B = (B \ A) (A B) , n(A) = n(A \ B) + n(A B) n(B) = = n(B \ A) + n(A B). n(A) + n(B) n(A B) = n(A \ B) + n(A B) + n(B \ A) + n(A B) n(A B) = n(A \ B) + n(A B) + + n(B \ A) = n(A B), . . 100 75 , 60 , 45 . ? A , , B , . , , n(A B) = n(A) + n(B) n(A B) = 75 + 60 45 = 90.

1. , ? 2. , A B = = A \ (A \ B) ? 3. , ? 4. , A B, . . A B = A B, A = B ? 5. , A B : A B = B, A B = A, A \ B = ?

1.4. , , . , , , . , . 45

, . . . , , , : 1) a + b = b + a; 2) ab = ba; 3) a + (b + c) = (a + b) + c; 4) a (bc) = (ab)c; 5) a(b + c) = ab + ac, , a, b, c. () , , , . () , , , , . () , . . , a b , , . , , , . . , . 1. . A B : A B = B A, A B = B A. 46

, . , A B B A , A B, , A B B A , A B. , , , , , , . . . . + + . , . . + + , + + . . : A D B = B D A, , . . A \ B B \ A. , 1.3. , . 1.17, 1.18 A B, A \ B B \ A .

2. . A, B C : A (B C) = (A B) C, 47 A (B C) = (A B) C.

, . , A (B C) (A B) C , A, B C ( . 1.19) , A (B C) (A B) C A, B C ( . 1.20).

, , . ! abc : (ab)c a(bc). abcd 5 : ((ab)c)d, (a(bc)d), a((bc)d), a(b(cd)) (ab)(cd). abcde 14 . , . , 5 a(bcde), 2 (ab)(cde), 2 (abc)(de) 5 (abcd)e. , n . . , , . {, , }. () : , , ( ), , , ( ) . , . . 1887 . : 1883 , 1885 , . : 48

, , , , . . . : A D (B D C) = (A D B) D C, , . . A \ (B \ C) (A \ B) \ C. . , , . . . , A, B C , A, B C, (. . 1.23), . . A D B D C = (A B C) [(A \ (B C)) (B \ (C A)) (C \ (A B))]. . . 1.21 1.22 A, B C, A \ (B \ C) (A \ B) \ C .

, . 1.22, (A \ B) \ C = A \ (B C). , , A D A = , A B : A D (A D B) = B. 49

, , , . . , : A D B D C = (A B C) \ [(A B) (A C) (B C)]. (*) . , A B C = .

. 1.23 1.24 (*), . . , A, B C . A, B C, , , , . . A B C = , . 1.25, (*) . : (A D B D C) \ (A B C) = (A B C) \ [(A B) (A C) (B C)]. : , . , A, B C A B C. A, B C, (. . 1.19), . A B, A C B C, . . n(A B), n(A C) n(B C), n(A B ), (. . 1.20) A B . n(A B ), A B . , : n(A B ) = n(A) + n(B) + n() n(A B) n(A ) n(B ) + n(A B ). 50

, , , 1.1. . A , , B , , C , U . , n(U) = 100, n(A) = 50, n(B) = 23, n(C) = 30, n(A B) = 20, n(A C) = 8, n(B C) = 10, n(A B C) = 5. n(A B ) : n(A B ) = 50 + 23 + 30 20 8 10 + 5 = 70. , , 100 . 100 70. , ? n(B) n(A B) n(B C) + n(A B C) = 23 20 10 + 5 = 2. ! . , , . , . 3. . A, B C : A (B C) = (A B) (A C), A (B C) = (A B) (A C). , . . . 1.26 A, B C, A (B C) . 1.27 A (B C).

51

: ? , , , , , , , . , , , . - , . , , , , . : , . , , , . , , : . , . , , , , . , : , , . . . , , , . , , . , , : ) ; ) ; ) ; ) ; ) . , , . , .

, . . : A (B C) = (A B) (A C). 52

: A (B C) (A B) (A C) (A B) (A C) A (B C). , x , . . x A (B C). , x A x B C. x A, x A B x A C. , , x (A B) (A C). x B C, x B x C, x A B x A C. , , x (A B) (A C), . . . . x (A B) (A C). , x A B x A C. x, x A x A. x A, x A (B C). x A x A B x A C, , x B x C. , x B C , x A (B C). . , . , , . . A (B C) = (A B) (A C), a + (b c) = (a + b) (a + c), , . . . , . . a(b + c) = ab + ac, : A (B C) = (A B) (A C). . , . . : A (B C) = (A B) (A C). 53

: A (B C) (A B) (A C) (A B) (A C) A (B C). . x A (B C). , x A x B C. , x A, , : x B x C. x B, , A, x A B, x (A B) (A C). x C x A, , x A C, x (A B) (A C). . . x (A B) (A C). , x A B x A C. x A B, x A x B. x A C, x A x C. , x A x B x C, , x B C. , x A (B C). . , . . : A ; B ; C . , . , . , A B , . , B C = , A (B C) = A = A. , A B , A C , , (A B) (A C) , . . A. A, B C : A (B C) = A = (A B) (A C). 54

, B C , A (B C) , A C = , (A B) (A C) = A B . A, B C : A (B C) = A B = (A B) (A C). . , . , , , , . . p, q, r. p q p q, . p q p q, . , , : p (q r) (p q) (p r), p (q r) (p q) (p r). , . , , , , , , . . , , -: A B , A , B , ? . , , , . . A, B C : A (B C) = (A B) (A C). . , , , . , , . x A (B C) (x A) (x B C) (x A) ((x B) (x C))

55

((x A) (x B)) ((x A) (x C)) (x A B) (x A C) x (A B) (A C)

. A, B C : A (B C) = (A B) (A C). . , , , , . x A (B C) (x A) (x B C) (x A) ((x B) (x C)) ((x A) (x B)) ((x A) (x C)) (x A B) (x A C) x (A B) (A C).

, , . , . ( ) p ~p . , p x , ~ p x . , : ~ (p q) ~ p ~ q, ~ (p q) ~ p ~ q. , , , . , , - , - . , , , : p q , -p -q, p q , -p -q. , , . . A B , . . :

56

U \ (A B) = (U \ A) (U \ B), U \ (A B) = (U \ A) (U \ B), U , A B. . . 1.281.30 U . U \ (A B) . 1.28 U \ (A B) . 1.30 . . 1.29 U \ A, U \ B.

. , , , . . x U \ (A B) x A B ~ ( x A B) ~ ((x A) (x B)) ~ (x A) ~ (x B) (x A) (x B) (x U \ A) (x U \ B) x (U \ A) (U \ B)

. x U \ (A B) x A B ~ ( x A B) ~ ((x A) (x B)) ~ (x A) ~ (x B) (x A) (x B) (x U \ A) (x U \ B) x (U \ A) (U \ B)

. 100 75 , 60 , 45 . , ? A , , B , , U 100 . , -

57

, (U \ A) (U \ B). (U \ A) (U \ B) = U \ (A B). , , , n(A B)=90, n((U \ A) (U \ B)) = n(U \ (A B)) = 100 90 = 10, . . 10 . . , : x A B x A x B , x A B x A x B . . , . U \ (A B) = (U \ A) (U \ B). x U \ (A B), x A B, x A x B, . . x U \ A x U \ B. , x (U \ A) (U \ B). x (U \ A) (U \ B), x U \ A x U \ B, x A x B. , x A B, . . x U \ (A B), . U \ (A B) = (U \ A) (U \ B). x U \ (A B), x A B, x A x B, . . x U \ A x U \ B. , x (U \ A) (U \ B). x (U \ A) (U \ B), x U \ A x U \ B, x A x B. , x A B, . . x U \ (A B), , , . . A, B C : A (B D C) = (A B) D (A C). . : A (B D C) (A B) D (A C) (A B) D (A C) A (B D C). . x A (B D C). , x A x B D C. , x A, , , : x B \ C x C \ B, . . x B, x C, x C, x B. , x A, x A B, x A C, A C C, x A C, x A B, A B B. , x (A B) \ (A C) x (A C) \ (A B) x (A B) D (A C). . . x (A B) D (A C). : x (A B) \ (A C) x (A C) \ (A B), x A B,

58

x A C, x A C, x A B. x A B, x A C, x A x B, x A x C. , x A x B, x C, . . x A (B \ C). x , x A (B D C). x A C, x A B, x A x C, x B, . . x A (C \ B), x A (B D C). , . , , , D A D (B C) (A D B) (A D C). . A, B C : A (B \ C) = (A B) \ (A C). , , , , .

, , . . , .

1. , A, B C : A (B C) = (A B) (A C) A (B C) = = (A B) (A C) ? 2. , A, B A (A B) = A A (A B) = A ? 3. , A, B C A (B C) = (A B) C C A ?

1.5. , , . ., . , 59

, , , , , . , . , , , . ., . : , , , , . . : (. . ), (. . ), , , (. . ). , , . , , , , , , (, , ) , . , , . . , , () (). . , , , . . . : . , . , , : - . , . - , , , , , . , . , ( ), . -, . , , ( ) . -

60

, . , , , , , . , ( ), , , .

. X Y . (, ), x X - y Y, , f ( f) X Y, f : X Y. , f function (), g, h . . . X , Y = N . f : X Y , x X y = f(x) Y . (, ) f, y Y, - x X, y = f(x). f : X Y : f X Y. , , : x X , y Y, . . y Y, y f(x); x X , Y, . . , , y1, y2 Y , y1 = f(x), y2 = f(x). , X , , Y . x X y Y, , f : XY. , , . . f : X Y : X, , f D( f ); Y, , f 61

R( f ); (), . , f : X Y x D( f ) y R( f ), . . y = f(x), x. , , , ( ). , R( f ) D( f ). . X = {k, m, n, p, q} , Y = {, , , } , , f : X Y : f(k) = , f(m) = , f(n) = , f(p) = , f(q) = . , f . , X, . . f, Y, , . , , . . , . y = f(x) , , , , y = x2, y = x3 + 1, y = x . . R, R+. R+, R. , , . , , . . , , , : , . , . . , , . : , , .

62

. ( ), , .

: ; ; . : 1. X = Y = {a, b}. , {a, b}, . , fi, i = 1, 2, 3, 4. :x a b f1(x) a a f2(x) a b f3(x) b a f4(x) b b

f1 f4 () , R( f1 ) = {a} R( f2 ) = {b}. X , f2 f3 X , f2(x) = x x X, . . . , x X () f(x) Y. , x X. 2. , , . x R [x] , , x. , [0] = 0, [1] = 1, [2] = 2, [3,4] = 3, [3,4] = 4, [ 2] = 1. x R, {x} , x [x], . . {x} = x [x]. , {0} = 0, {3} = 0, {5} = 0, {0,3} = 0,3, {0,3} = 0,7, { 2} = 2 1. , , : 1 f(x) = [x] + {x} (1 + (1)[x]). 2 63

! , , f(x) = x, x, 2n x 2n + 1, n f(x) = [x], x, 2n + 1 x 2n + 2, n . , f(x) Y x X, , . , , , , . 3. , . . , , , : 1, x , D( x) = 0, x . , , , , . , , . , . , , , -. , . ? , , . , -, , . , . , . , . , .

64

, . . , , : , , , , , , , , . , , . . , 16 : . . , : . , , . . , , , , .

. , , . , . . (), . , , , . , , , , . , . . f X Y. A X f : X Y, f(A), f(x) Y x A: f(A) = {y : y = f(x), x A}. 65def

, . . . , , , , , : , , , , . , . f, , , . , . . X = {k, m, n, p, q} , Y = {, , , } , , f : X Y : f(k) = , f(m) = , f(n) = , f(p) = , f(q) = . A = {k, m, n, p}, B = {m, p, q} C = {k, q}. , : f(A) = {, , , }, f(B) = {, , } f(C) = {}. . f : X Y A X. y f(A), x A, $ x A, y = f(x): y f(A) $ x A, y = f(x). . : A B f(A) f(B). , f(A) = A D( f ) = , D( f ) f. . . f : X Y A, B X : f(A B) = f(A) f(B) f(A B) f(A) f(B). . . , f(A B) f(A) f(B). y f(A B), $ x A B , y = f(x). x A x B, y = f(x). 66

y f(A) y f(B). , y f(A) f(B). , f(A) f(B) f(A B). y f(A) f(B), y f(A) y f(B). , , , A B, , , $ x1 A , y = f(x1) $ x2 B , y = f(x2). x1 A B, y = f(x1), x2 A B, y = f(x2). , y f(A B). , . f(A B) f(A) (B) , , , . , f(A B) f(A) f(B). X = {a, b, c}, Y = {d, e}. A = {a, b}, B = {b, c} f : XY : f(a) = d, f(b) = e, f(c) = d. , A B = {b}, f(A B) = {e}. , f(A) = {d, e} f(B) = {e, d}, , f(A) f(B) = {d, e}, . . {e} = f(A B) f(A) f(B) = {d, e}, f(A B) f(A) f(B). , . , , A , , B , . f , , . . f(A B), , , . . f(A) f(B). , . . f : X Y A, B X : f(A D B) f(A) D f(B) f(A \ B) f(A) \ f(B). , , , . 67

. f X Y. B Y f : X Y, f 1(B), x X, f(x) B: f 1(B) = {x : f(x) B}. , , f, , , : , , , , . , , . . X = {k, m, n, p, q} , Y = {, , , } , , f : X Y : f(k) = , f(m) = , f(n) = , f(p) = , f(q) = . B = {}, C = {, , } D = {, , , }. , : f 1(B) = {k, q}, f 1(C) = {m, n, p}, f 1(D) = {k, m, n, p, q}. . f : XY B Y. x f 1(B) , f(x) B: x f 1(B) f(x) B. . : C D f 1(C) f 1(D). , f 1(B) = B R( f ) = , R( f ) f. . . f : X Y B, D Y : f 1(B D) = f 1(B) f 1(D), f 1(B D) = f 1(B) f 1(D). . . f 1(B D) f 1(B) f 1(D). x f 1(B D), 68def

f(x) B D. f(x) B f(x) D. x f 1(B) x f 1(D). , x f 1(B) f 1(D). , f 1(B) f 1 (D) f 1(B D). x f 1(B) f 1(D), x f 1(B) x f 1(D). f(x) B f(x) D, f(x) B D. , x f 1(B D). , . 1 f (B D) = f 1(B) f 1(D) , , , . . f : X Y B, D Y : f 1(B \ D) = f 1(B) \ f 1(D) f 1(B D D) = f 1(B) D f 1(D). , f : X Y A f 1(f(A)), A X f(f 1(B)) B B Y. , B f, . . B R( f ), f ( f 1(B)) = B.

1. , X , Y , X Y, X Y ? 2. , (), . . , , , , , , ? 3. , N, n n- , mn M, m m- ? 69

? , . : , , , . , . : . , , , : abcde f ghi k l m n o p q r s t u v w x y z 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 Lempereur Napolon ( ), , 666 , . , . . , 1886 , . . . - ? -, . , , , 20 + 5 + 30 + 60 + 5 + 80 + 5 + 110 + 80 + 40 + + 1 + 60 + 50 + 20 + 5 + 50 + 40 = 661 666. ? : 5 , , article lempereur. -, , . . , j. , - , , , . 70

, . . . , , : . , , , , . , , , . . , , , . , , . : : . , , . , , , . , - . , , . . . , , , , , , .

2 , , . .

(), , . , , , . . , . . . , , , , . . . . ( . ., 1913) , . . , , , . . . 72

, , , - - -. , . . , , , , , , , , . : , . , , , . , , , . : 7. , . , , : . , , , , . . , , .7

. // . 1975. . 30, . 1. . 199.

73

. , , . , . - , , , . , . . : , , . . , , . , , , .

2.1. . . . , , . , , , . : , ? , . , , . , , XIII . XX , , , 74

. . , , . , , , , , , . . ( combinatio ). , , , . . : , , , , , . , , . , , , . . , . , . . 7 5 . ? , ? . 7 , 5 . , 7 + 5 = 12 . , , 7 5 , 5 , 5 . . , , , 5 + 5 + 5 + 5 + 5 + 5 + 5 = 7 5 = 35 . . , 75

. , , , , , . . . A n , B m A B = , A B n + m . . A B, . . A B. , A, 1 n, A n . , A B, A B = . , , B, n + 1, n + 2, n + 3 . . n + m, B m . A B , 1 n + m, , , A B n + m . . A n , B, A, m , A B n + m . . . , , 15 , 10 5 . , 15 , 10 , 5 . 15 + 10 + 5 = 30 . k . k , n1 1- , n2 2- , , nk k- , n1 + n2 + + nk. 76

, . . A B n , B C m . , A C B? n A B, m , A C nm. . A n , . . A = {ai : i =1, 2, , n}, B m , . . B = {bj : j =1, 2, , m}, C, , . . C = {(ai, bj) : i = = 1, 2, , n; j =1, 2, , m}, nm . . , C : C1 = {(a1, bj): j = 1, 2, , m}, C2 = {(a2, bj): j = 1, 2, , m}, . Ci = {(ai, bj): j = 1, 2, , m}, . Cn = {(an, bj): j = 1, 2, , m}. , C1 C2 = , C1 , a1, C2 , a2. , Ci Ck = i k. , C Ci, i = 1, 2, , n, . . C = C1 C2 Cn. , (ai, bj) , C, . . (ai, bj) , (ai, bj) i, (ai, bj) C1 C2 Cn. Ci , i =1, 2, , n, m , nm, . . A n , B A m , A B nm . . . , . 77

( ), (, , , ). , 25 = 10 . . , k n1 , n2 k- , nk , k n1n2 nk . , , . . , . : 5 , 20 , 12 ?. 52012 = 1200 . , , : 5 , 20 , 12 5 + 20 + 12 = 37 . , . . , , 15 , 10 5 . , 15 10 , 1510 = 150 . , 15 5 , 155 = 75 . , 10 5 , 105 = 50 . , , 150 + 75 + 50 = 275 . 78

, , , , .

1. , A, n , B, m , , . . A B , A B n + m ? 2. , . . : 5 , 4 , 3 , 60 ? 3. , 5 , 4 3 , 47 ?

2.2. : , . . , , . , , , , , . . ( . ., 1983), , . . a, b c, , , : {a, b, c} = {b, a, c} = {c, b, a}. 79

, . , , , . . M n n , . . M {1, 2, , n}. , 33 : , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . , , . ., . , , , , . ., . . , , , . , , : (a, b, c), (b, a, c), (c, b, a). , : (, ), (, ). , . , , , . , . : , . : , , , : , . : (, , ), (, , ), (, , ), (, , ), (, , ), (, , ). . , , , , 80

, . , . , . , , 4 7 . - , , . , . . , 3 1-, 2- 3- . , . 1- , . . 3 , 2- , . . 2 , 3- , . . . , 3 2 1 = 6 . (. 2.1), () , .

81

, , , , , , 24 , : (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ), (, , , ). , . , , : . . . , , . . , , . . , . 1 n n-, n! ( -), . . n! = 123(n1)n. , . , . n Pn (P permutation ). : . . Pn n : Pn = n(n 1) 321 = n! . n . n . , n 1 . n 1 82def

n1 . Pn1 ( Pn1). , n , , n , n1 , . nPn1, . . Pn = n Pn1. : Pn = n Pn1 = n (n 1) Pn2 = = n (n 1) (n 2) 2 1. , . , n! , n + 1 n (n + 1)! = n! (n + 1). : 1!= 1, 2!=1!2 = 2, 3!=2!3 = 6, 4!=3!4 = 24, 5!=4!5 = 120, 6!=5!6 = 720, 7!=6!7 = 5040, 8!=7!8 = 40 320, 9!=8!9 = 362 880, 10!=9!10 = 3 628 800. , n n n( n + 1) , . . 1 + 2 + 3 + + n = . 2 n , , . . n! , , , n n! ! , (n + 1)! = n!(n + 1) n = 0, 1! = 0!1. , 0! = 1, . . , . 83

, , P3 = 3! = 123 = 6. : , , , , , . . , 24 , 120 . . , , - . , : , , , . , -. P4 = 4! = = 1234 = 24. , , . , , . . , , , . , P3 = 3! = 6. , ? . n , , , , , Pn1 = (n 1)!. , , 7 ? , . , : , , , , . , n 1 , n = 7. (n 1)! = 6! = 123456 = 720. 84

. III , . , , . , . , , P4 = 4! = 24, , , . , , : [] . , 6 120 . , , , . , , 33 , . , , : , , , , , , , , . , . - 33 , 33! . , , . , 10!, 3,6 , , , , , . 10! , , . , . n m (m n) . n m? 85

, , , , . . 12. , , , , 43 = 12 . : (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ).

, . . , . . , m n (m n), , . n = 4 m = 2. , . 2 4 43 = 12 . , , . , , , , 1 2. - (. 2.2) , , . n m : n ; n 1 ; .. m- n (m 1) . , n(n 1) (n (m 1)) . . . , : m 86

, n ? : n m? n m m An (A arrangement ). : . m . An , m n, : n! m An = n ( n - 1) ... ( n - m + 1) = ( n - m)! . n . , m , n . m1 n1 m m1 . An--1 ( 1 m-1 An-1 ). , n m , , n , n1 m1, . 87

m n m n An--1, 1

. . m m An = n An--1 . 1

:m m m An = n An--1 = n (n 1) An--22 = 1 m = n (n 1) (n 2) An--33 = = n (n 1) (n 2) (n m + 1).

m n n m + 1 n n m, . . n! n(n 1)(n 2)(n m + 1) = . ( n - m)! , n m: n! m . An = ( n - m)! , , , . , :1 2 3 An = n, An = n(n 1), An = n(n 1)(n 2), 2 3 4 A3 = 32 = 6, A4 = 432 = 24, A5 = 5432 = 120. m An m = 0, , n! 0 0 = 1, , An = 1. , An = ( n - 0)! , - . , : n .

. n m = n, . . An = Pn = n!. , m = n1 n-1 n An = An = n!. . , n n1 , n n-1 , , . . An = An . 88

. , n- m- , . n! m An = . ( n - m)! n- , n m- m . , , , , , , , . . . , , , 1, 2, , n. - , . , 33 33! 2 A33 = = 33 32 = 1056 ( 33 - 2)! , . . 3 6 . , . 3- , . . , 6- . 6! 3 A6 = = 6 5 4 = 120. ( 6 - 3)! . , . : n m (m n)? , , , , , . ? , , , . , . 2.2, . , 12 , , . , , , , , . , . . , , 12:2 = 6 . 89

. . , , . , , , . 45 = 20, , , 20:2 = 10 . . () . , , . , . , , {, , }. . : ; {}, {}, {} 1 ; {, }, {, }, {, } 2 ; {, , }, 3 . n m m C n ( combinaison ). : .m . C n , m n, n ( n - 1) ... ( n - m + 1) n! m Cn = = m! m!( n - m)!

. , . k- () (), k . m- () n- Pm , , m . , An m- m n- C n m- m m Pm C n = An , , m Pm = m!, An = n ( n - 1) ... ( n - m + 1), m Cn m An n ( n - 1) ... ( n - m + 1) n! = = = Pm m! m!( n - m)!

90

, , . n- m , 1, 2, 3, , m m! . nm m+1, m+2, , n (n m)! . , m n Cnm . , m!(n m)! Cnm n , n!. m!(n m)! Cnm = n! m Cn =

n! , m! (n m)!

. , , , , . . , , , . , :1 Cn = n, Cn2 = n(n 1) , Cn3 = n(n 1)(n 2) , 2 6 2 3 4 C3 = 3 2 =3, C4 = 4 3 2 =4, C5 = 5 4 3 2 =5. 2! 3! 4!

Cnm m = 0, , n! Cn0 = = 1, , Cn0 = 1. 0! (n 0)! , , ( 0 ) n- . , , Cnn = 1. . : m mn, , :Cnm = Cnn m .

, n m . m nm , . 91

, m n, , n m n. , Cnm = Cnn m n! n! = . m! (n m)! (n m)! (n n + m)!1 , C50 = C55 = 1, C5 = C54 = 5, C52 = C53 = 10.

. 0 m < n, :Cnm+1 = Cnm + Cnm 1 .

, n+1 . . m , , , , . , . , m1 n. Cnm 1 . m n , . Cnm . Cnm+1 , , , . , n! n! Cnm + Cnm 1 = Cnm+1 , , + = m! (n m)! (m 1)! (n m + 1)! (n + 1)! 6 = . , C54 + C53 = C64 , C75 + C74 = C85 , C96 + C95 = C10 . m! (n m + 1)! Cnm m m m . , Cn +1 + Cn = Cn++1 1

, n m, 0 m < n, m m Cm + Cm+1 + + Cnm = Cnm++1 . 1

m = 1 n : 1+2++n= 92n(n + 1) . 2

. A1, A2, , Am A1 A2 Am, : n(A1 A2 Am) = n(A1) + n(A2) + + n(Am) n(A1 A2) n(A1 A3) n(Am-1 Am) + n(A1 A2 A3) + n(A1 A2 A4) + + + n(Am-2 Am-1 Am) + + (1)m1n(A1 A2 Am). , n(A1) = n(A2) = = n(Am) = n1, n(A1 A2) = = = n(Am-1 Am) = n2, , n(A1 A2 Am) = nm, , n(A1 A2 Am) =

(1)k =1

m

k 1 k Cm

nk .

. . a b n :1 (a + b)n = Cn0 an + Cn an1b + + Cnm anmbm + + Cnn 1 abn1 + Cnn bn =

=

Ck =0

n

k n

a n-k b k .

. : , . , , , . , , ?

( ) , , a + b. : (a + b)2 = a2 + 2ab + b2. (a + b)n n . , , , . , . , Cnm . 93

, n = 3, 4 5, : (a + b)3 = a3 + 3a2b + 3ab2 + b3, (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4, (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5. , a = b = 1, 1 Cn0 + Cn + Cn2 + + Cnn = 2n.

: 1. a b n. 2. , , m nm , Cn = Cn . 3. , .

. , , , , , , , . , 1 C43 C4 . , , . 4! 1 C43 = C4 = = 4 : 3! 1! , , , . . , - , 6 49 . 49 6, . . 49 48 47 46 45 44 49! 6 C49 = = , 6! 43! 6! 49 . : . 94

, 5 , , 6 5 , 43 1. 5 1 C65 C43 . 4 2 C64 C43 . . , , , , . , .

1. , , , , , 120 ? 2. , , 120 ? 3. , 5 n < 9, Cn5 < Cn4 , n = 9, Cn5 = Cn4 , , n > 9, Cn5 > Cn4 ?

2.3. : , . . . , . : , , 95

, . : ? . , , , 1, 3, 6, 10, 27. . , , . 1 (1,1,1), . . . , 3 (3,0,0), (0,3,0), (0,0,3). , , 6 (2,1,0), (2,0,1), (1,2,0), (1,0,2), (0,2,1), (0,1,2). , 10 . , , 27 . : , . , , . . , , . . . , . . , . , . , . . , . , . , , , . 96

, 3 , 2 1 , 6=3+2+1 1, 1, 1, 2, 2, 3. P6 = 6! 6- , . . 6 . . , 1, 2, 3 3!, , . 3! . , . , , 6! , = 60. 3! 2! 1! = 1, , 6! = 60. 3! 2! 1! , , . . , n , n1 1, n2 2, , nk k , n1 + n2 + + nk = n, . , n n1, n2, , nk , n1 + n2 + + nk = n, P n ,n ,...,n . P . : -, - . . - -, - -.1 2 k

. P n ,n ,...,n , n1 + n2 + + nk = n, 1 2 k

P n ,n ,...,n =1 2 k

(n1 + n2 + ... + n k )! n! = n1! n2! ... nk ! n1! n2! ... nk !

. , , . , 97

n . , n! . , ni i =1, 2, , k . , , . ni! i =1, 2, , k , , ni i , n! ni! i =1, 2, , k. : n! , n = n1 + n2 + + nk , n1! n2! ... nk ! . , , . , , 6! P 3,2,1 = = 60. : 3! 2! 1! , 9 , 1- , 2 , 2 , 2 , 1- 1- , 9! P 1,2,2,2,1,1 = = 45 360. 1! 2! 2! 2! 1! 1! , 0! = 1 P n1 ,n2 ,...,nk , n1 + n2 + + nk = n, ni 0 0. , n1 = 3, n2 = 0, n3 = 2, 5! = 10. n = 3 + 0 + 2 = 5 P 3,0,2 = 3! 0! 2! . , , , - . , , . , . 4 + 3 + 2 + 1 = 10, 4 , 3 , 2 1 . n1 = 4, n2 = 3, n3 = 2, n4 = 1 n = 10, 10! 5 6 7 8 9 10 P 4,3,2,1 = = = 457910 = 12 600. 4! 3! 2! 1! 3! 2! , - ? 98

. ( k=n) ( k=2): P m,n m = C nm . , k = n, n1 = 1, n2 = 1, , nn = 1 n! n1 + n2 + + nn = n, P 1,1,,1 = = n! = Pn , . . 1! 1! ... 1! Pn P n ,n ,...,n n1 = n2 = = nn = 1. k = 2, n1 + n2 = n n2 = n n1, n! P n ,n Cnn , P n ,n = P n ,n n = = Cnn . n1! (nn1 )! , , n1 = m, : n! P m,n m = = C nm . m! (n m)!1 2 n 1 1 1 2 1 2 1 1

. m nm , . . P m,n m , C nm . . . (a + b)n n (a + b): (a + b)n = (a + b) (a + b) K (a + b). 14444244443 4 4n

, , (a + b)2 (a + b)2 = (a + b)(a + b) = aa + ab + ba + bb, (a + b)3 (a + b)3 = (a + b)(a + b)(a + b) = aaa + aab + aba + baa + abb + bab + bba + bbb. , (a + b)n n , . . (a + b)(a + b) (a + b), , n a b. , , a b, . . , a b. m P m,nm = Cn , 99

m anmbm Cn . , :

(a + b)n = P 0,n an + P1,n1 an1b ++ P m,nm anmbm ++ P n1,1 abn1+ P n,0 bn = 0 1 m n n = Cn an + Cn an1b + + Cn anmbm + + Cn 1 abn1 + Cn bn. , , , , , n . Cn1 ,n2 ,...,nk , n1 + n2 + + nk = n,2 k k Cn 1 , Cn n1 , , Cn n 1 ... nk 1 , . .

n

n

n

nk n2 n n Cn1 ,n2 ,...,nk = Cn1 Cn n1 Cn n 1 ...nk 1 .

def

. P n1 ,n2 ,...,nk n P n1 ,n2 ,...,nk = Cn1 ,n2 ,...,nk .

, n , n1 1- , n2 2- ni i- i =1, 2, , k. , n n . Cn1 n1 1- . , n2 nn1 , Cn n1 n2 2- . 1- 2- , nn1n2 , n3 3- Cn n1 n2 . , (n n1 )! (n n1 n 2 )! n! n1 ! (n n1 )! n2 !(n n1 n2 )! n3 !(n n1 n2 n3 )! (n n1 n 2 ... nk 1 )! n! = , ... n1 ! n2 ! ... nk ! nk !(n n1 n2 ... nk )!n3 nk n2 n Cn1 Cn n1 Cn n1 n2 Cn n 1 ...nk 1 =

=

. 100

: , , . , . , : n k , n1 , n2 . ., n1 + n2 + + nk = n ? , . . , , , n k , , , n .

, . . , n- . . , (a1 + a2 + + ak)n =

Pn n

1 , 2 ,..., nk

n n a1n a2 ... ak ,1 2 k

(n1, n2, , nk), n1 + n2 + + nk = n n1 0, n2 0, , nk 0. . a1n a2n ... akn , n- k a1+a2++ak ( n = n1 + n2 + + nk ni 0, i =1, 2, , k), n! . n1 a1, n2 a2, , nk ak , . . P n ,n ,...,n = n1! n2! ... nk !1 2 k 1 2 k

, ni = 0, = ai0 = 1 , . . ai , , , , ni! = 0! = 1 . , n = 3, k = 3 (a + b + c) aini

(a + b + c)3 = (a + b + c)(a + b + c)(a + b + c) = = a3 + b3 + c3 + 3(a2b + a2c + b2a + b2c + c2a + c2b) + 6abc. 1, 3 6 . , , , , . a3 P 3,0,0 = 3!/(3!0!0!) = 1, b3 P 0,3,0 = 3!/(0!3!0!) = 1 c3 P 0,0,3 = 3!/(0!0!3!) = 1. a2b aab, aba, baa, . . 101

a b, P 2,1,0 = 3!/(2!1!0!) = 3. , , b2c P 0,2,1 = 3!/(0!2!1!) = 3. , abc P 1,1,1 = 3!/(1!1!1!) = 6, . . a, b, c. , , n = 3 k = 5, (a + b + c + d + e)3 = (a3 + ) + 3(a2b + ) + 6(abc + ), , , , . . , : (a + b + c + d + e)3 = (a3 ++ e3) + 3(a2b ++ e2d) + 6(abc ++ edc). . 5 : abcde , , , 0. (a3 ++ e3) 5 : 3 0, , 30 000, . . 5!/(1!4!) = 5. (a2b ++ e2d) 5 : 2, 1 0, 21 000, . . 5!/(1!1!3!) = 20. , (abc ++ edc) 5 : 1 0, 11 100, . . 5!/(3!2!) = 10. a = b = c = d = e = 1. 53=125, 5 + 320 + 610 = 125, . *** : ? , ( , ) . , . , . :

102

, , , , . . , , , , . . , . . : ? . . aa, bb. , a b , , a b, a. a, , 1- a 2 b, . . 3! P 1,2 = = 3, : abb, bab bba. : 1! 2! aabb, abab, abba.

, . , , , , , . . . . , , . - . . , . . . . (a, b, c, ), (A, B, C, ). , , . .: ?

103

( alternance ) . , AbAb aBaB, abab ABAB. , aa, AA. 2 4! a 2 A, . . P 2,2 = = 6, : 2! 2! aaAA, AAaa , aAaA, AaAa , aAAa, AaaA .

, , aabb AABB, abab ABAB, abba ABBA. , , , . . , . ? , . , , , . , , , . , . , . , .: ?

aa, bbb. 5! P 2,3 = = 10, : 2! 3!aabbb, ababb, abbab, abbba, baabb, babab, babba, bbaab, bbaba, bbbaa. : ababb babba. , , bbb, , . , . . , , , , -

104

. , 8. , . , , , , : .

, . : , , . . , , : , . , , , , , , 33 = 9 : , , , , , , , , . n , , {a1, a2, , an}, nn = n2 . , (ai, aj) ai, aj {a1, a2, , an}, n2. {a1, 2, , an} n3 , n2 n , . . n2n = n3 . . , k , n , . , k n , Ank .8

. . , // . 1978. 2. . 151.

105

A . : A . , . . Ank :k An = n k

. k , n , nk . , , k n , nk, . , , , {, , }, A32 = 32 = 9. , -, . . , , . - , , . , . . . , . . . , k- , , . k- , 33 , , , , , , k A33 = 33k. , , . . . 1-, 2-, , (k1)- 33 , , . : 106

, k- 33 , , 1 . , . , k : , . . k . . . , , . . k- , 32 , k , A32 = 32k. k- k k A33 A32 = 33k 32k. , , . , 1.1. , , : , , . , 33 , , , , 100 A33 = 33100. , , () , . . : , , . 96 . , . , , , . , , .

107

. , Ank = nk , k > n. , , , , A24 = 24 = 16. , , : , , , , , , . . , 8, . , . , 8 , . , , . . , : 0, 1, 2, 3, 4, 5, 6, 7, 9. , , A96 = 96 = 531441. 000000, , 531440. , , 6 A10 1 = 1 000 000 1 = 999 999. 999 999 531 440 = 468 559, . . , . , , , . , . ? ? . . . , : ) ; ) , . 108

. . , , . *** , . , , , , . , , . : , , , , ? . , + , + , + + , , , + + . . , . . , , . . . , , . , , , . , , . . , , ( ), ( ). , , , . : ?

, , . , , . . , . ,

109

. A2 = 23 = 8, :3

, , , ,

, , , .

, . , : ' ' ' ' , ' ' '

( ) ( )

: , , . . , , . . . , : , , , . , . , . . , 9. , ? . . , XX , . , , . , , . . .: , , ?

9

. . , // . . . . 2002. . 61, 5. . 42.

110

, , , (, , ), (, , ). (, , ). , , , . A3 = 32 = 9, :2

, , ,

, , ,

, , .

, , . , , . ddEffEgg. . , , , . , . , . .: ?

, , . A2 = 23 = 8, :3

, , , , , , , . , . , , . , , , , . , , , , , . , , , , , . , , , , . , , , .

111

. , . , : , , , .

. . , . , . . 4 : , , . , 8 . . : . , , . . I. , . , , , . , . , , . , 8 , , , . . ( ), (. 2.3), .

|

|. 2.3

|

, 2 , 1 , 3 2 . . , 112

, , , . . 3 7 (), . . , 3- 7- . C73 = 7 6 5 = 35. , 3! n = 4, m = 8, , m n, n m , , Cnm1 . 1 II. , . , , . 8 3 , . , , , (. 2.4), 3 , , 5 .

||. 2.4

|

, 8 + 3 = 11 , 8 3 . 3 11 8 11 , . . , 3- 11- . 3 C11 = 11109 = 165. , n = 4, m = 8, 3! n n m m Cm1n 1 = Cm+ n 1 . + , m n . . . , m , n , . 113

, m n , C nm . C . : . C nm n m . . C nm :n m C nm = Cm1n 1 = Cm+ n 1 = +

(m + n 1)! m! (n 1)!

. 1 n. m1 1, m2 2, , mn n. m , , m1 + m2 + + mn = m. 1 0. , . , , mi = 0, . . i , . , n = 4 m = 8, , 11100111110 , 3 1, 2, 5 3 4. mi m, m , n, n 1. : . , : m+n1 m n1 ? (m + n 1)! m n1 , . . P m ,n1 = , m! (n 1)! n m P m ,n1 = Cm1n 1 = Cm+ n 1 , , +C nm = P m ,n1 =(m + n 1)! , m! (n 1)!

. 114

, , , . . . . . 7 : , , . , . , , . . , , A37 = 37 = 2187 . , , , , , . m = 7 n = 3 : , , , C nm = C37 = C77+31 = C97 = 9! / (7!2!) = (98) / 2 = 36 . . , , , (m + n 1)! C nm = , m! (n 1)! m > n. , . , . . C nm , m < n m = n. , 7 2 0, 1, 2, 3, 4, 5, 6. C72 = C22+ 7 1 = C82 = 8! / (2! 6!) = ( 8 7 ) / 2 = 28,

. . 28 . . ( ) ( ) , , . , . n , : 4 = 1+1+1+1 = 1+2+1 = 1+1+2 = 2+1+1 = 2+2 = 1+3 = 3+1. 115

m n ? . . , m : x1 + x2 + + xn = m, xi 0, i = 1, 2, , n. . : , , , . , m m n1 , n , xi, i = 1, 2, , n. , , xi = 0. (m + n 1)! m . C nm = Cm+ n 1 = m! (n 1)! , , , . x1 + x2 + + xn = m xi 0. xi >0 (, m n), I n , Cm1 . 1 , . . , 4 8 . I , . . , . . 2.3, , , , 2 , 1 , 3 2 , , C8411 = C73 = 35. 116

II , . . , , , . . 2.4, , 3 , , 5 , , 3 C84+411 = C11 = 165, . . C48 . , , II? , , , I. . ? , , - ! : . *** , , , , () . , . . , , . . : , : , , , !

, , , , . , , , , , , . . : . , , , , , - .

117

: ?

: , . 4! 3 3 3 C2 = C3+ 21 = C 4 = = 4. , 3! 1! , , , . . : . . . . . . , , .: , ?

, , , , . , , : + , , , , , . 4! 2 2 = 4, C32 = C2+31 = C4 = 2! 2! : (+, +), (+, ), (+, ), (, ), (, ), (, ). , , 10. , . , . : , .

10

. . : // . 1987. 2. . 139.

118

. , , , , , , , : , , , . , , . , , . , , , , . , .: , , , ?

, . m Cn . n = 4 , , , , , 6! 3 3 3 . . m = 3. , C4 = C3+ 41 = C6 = = 20 3! 3! . , , , , : , , : ! , , . , , , . , , , . : , , , .

, , , . , , , . , , . 119

1. , 210 ? 2. , P 1,2 = 3 ? 3. , , 7 , , A10 = 107 ? 4. , A24 = 16 ? 5. , 16 , 5 , C16 = 15 504 ?

2.4. . , , . , , . . , , , . , , - . , . . , . , , : ) () , . . ; 120

) ; ) . , . , 1812 ., : , , , . , , , , . , . , . ? , , , . , , ( hazard ). , . . , , , 1 6. , , , ( ) . : 1, 2, 3, 4, 5, 6. : () (). . () , . , , . 121

(., 1972) , : . , , . , . . , , , , ( : , ). (1, 2), (3, 5), (6, 4), (3, 3) . . A62 = 62 = 36 . , : , , , , , , , A23 = 23 = 8 . , 36 ( ) 6 , 6 C36 . , . , , , . . . , , , 2. , , , . . , . , , , .

. , , . . , (, ) , . () , : (), (), (122

), (, ). , , . . . : ? , , , , 3, 10 : , , , , , , , , . ( ) 10 , , , . , , . , , , - () . , , . . . - , , , . , . . . , , , 0 1 ( probabilite ), . , , , , . . . : n , 123

n . : n- m , , m/n n, . , , . . , . , . , . . n m , m n n . , (), , , , n n. , , : ? , . , , , , 5 , , 55 , . , , , , . . . n m A n (), , . . : () m . n , ( statistik) . 124

, , , , , . , , , , -, . , : . , , , , . , . , , , . , , . 29 343 . 430 . , 1 = { } 430 0,0147. (1) = m = n 29343

(, -), . . , . . . . , 544 777 , 8771 . 2 = { } 8771 0,0161. (2) = m = n 544 777

125

. , 100 , 0, 0,03. 0,01, . , , . , , , , , . . . , , , , . , . , . . . , , , , :

0,075 0,017 0,046 0,016 0,030 0,087

0,009 0,018 0,075 0,012 0,034 0,042

0,031 0,065 0,110 0,023 0,048 0,055 126

0,065 0,025 0,002 0,011 0,005 0,015

0,007 0,004 0,019 0,003 0,007 0,022

? , , , , , , , . . , . . -. . , . , , , . . . , , , () . : ? , , . , . , , . , , , . . . : ( , ) , . . Basic English, , 1000 , . , , , . -, , , , -, . -, , . , , . , .

127

, , , , . , , , , , . .

( ) , . , , , . . , , , , . . , . , 1/2. , , , . , 1, 2, 3, 4, 5, 6 , . . . , , 1, 2, 3, 4, 5, 6, 1/6. = ? , ( 1, 3, 5) , () 3/6 = 1/2. : , , . , , . ( ) , , . 128

, m , . , , , . , , , , . , . , , . . , . , - . -, , . . , . . . : , ; , ; , , , . . . , ? . , , . , 1/2, . , , = { } = { } . , . , , 2/3.

. () = m , n n

129

, , . , , . . . , , . . .

. - , , , , , . ? (2 2 ), P 1,2 = 4!/(2!2!) = 6, : , , , , , . , () = 1/ P 2,2 = 1/6. . , , , , . ? A43 = 432 = 24. , . . ., , (. . .) = 1/ A43 = 1/24.. , 2 : , , , , . , 1- 2 , ?

3 3 C2 = C2+31 = 4!/(3!1!) = 4. -

130

, . . . , 3 ( ) = 1/ C2 = 1/4.

. , .

, , , , . , , , . , . , . : 30 , 10 , 10/30 = 1/3. , , , , , , , . , . , , , .. n , m nm . k . , k l ?

k n . = { k l }. , k Cn . , l , k l . , 0 l m 0 k l n m 0. l l m Cm . k l k l n m Cnm . l k l Cm Cnm . , l k l C m Cn m () = . k Cn

131

. 17 3 3 . , 2 1 ?

, , , . n = 20, m = 17, n m = 3, k = 3, l = 2, k l = 1. , = { 2 1 } 2 1 17 16 3 C17 C3 20 19 18 () = =( )/( ) 0,35. 3 C20 1! 3! 2!

. , 36 , 6 . , 1 2 ?

. = { 6 1 6 2 } C36 . 1 4, ,1 2 C4 , 2 4 C4 . , 3 36 4 4 = 28 , , , 3 C28 . , 3 1 2 C4 C4 C28 . , 1 2 3 36 35 34 33 32 31 4 4 3 28 27 26 C4 C4 C28 =( )/( ) 0,04. 6 C36 6! 1! 2! 3!

() =

. . , 2 , . 36 , (, ), (), (), (), (), : 6 , , 10 , 11 , , 14 . , = { } , = {(, 14), (, 14), (, 14), (, 14)}, = { } = {(, 6), (, 7), , (, 14)}. . , (), . , , . . . , , , , .

132

, . .. 10 . = { 3 }?

, 10 {, }. 10 A2 = 210. , 3 , 7 . 3 7 C10 = C10 = 10! / (3!7!) = (1098) / 3! = 120. , 3 P 3,7 = 10! / (3!7!) = C10 . , :3 10 () = C10 / A2 = 120 / 210 = (1523) / ( 210) = 15 / 128 0,12.

. 10 . = {, 1, 2, 3, 4, 5, 6, 0, 1, 2, 4, 2, 1 }?

, 10 10 {1, 2, 3, 4, 5, 6} A6 = 610. , 1 , 2, 3, 4, 5, 6 1, 2, 4, 2, 1 . (0 + 1 + 2 + 4 + 2 + 1)! = P 0,1,2,4,2,1 = 0! 1! 2! 4! 2! 1! = 10! / (2!4!2!) = 3457910. :10 () = P 0,1,2,4,2,1 / A6 =

3 4 5 7 9 10 610

=

175 0,0006. 67

. , n , k . S = { , , }?

k An = nk, S k An . : k k (S) = An / An =

n (n 1)( n 2) ... (n k + 1) . nk

133

, , , , 0, 5 5 A10 / A10 =

10 9 8 7 6 0,3. 105

. , , , , .

, , : ? {(, ), (, ), (, ), (, )}. {, }. = { {, }, {, }}, () = 2/4 = 1/2. , , ? , ! , , 1000 , , .

1. , , , , , , , 1/ A53 = 1/(543) = 1/60 ? 2. , , 15 2 5 10 , C52 / C15 = ((54)/2) / ((1514)/2) = 2/21? 3. , , P6 / A66 = 6! / 66 = 10/648 ? -, , , . , . . , , (., 2001). XX XXI , , , , , . 134

. . , .

, . , 12 , , . , - . , , , . . . : , , . , , - . ( ) . . , , . , . , , , . 135

. , , . , , . , A B = , A B . , . . . A B (A B) = p(A) + p(B). . . n . A m , B k , (A) = m/n (B) = k/n. A B , . . A B = , , A B. , A B m + k (A B) =m+k m k = + = p(A) + p(B), n n n

. . () . , A = { 5 } B = { 10 } , . . A B = . A B = { , 5} (A B) = p(A) + p(B). A62 = 36. (1, 4), (4, 1), (2, 3), (3, 2), B (4, 6), (6, 4), (5, 5). (A B) = 4/36 + 3/36 = 7/36. , , , , . 136

? , , , , . -, . . , , , , . . . . : , . . , n , . . n A1, A2, , An. - , , , , Ai (Ai), Ai, , : (A1) + (A2) + + (An) = 1. , Ai, : (A) =

i

(Ai),

, . , U = A1 A2 An (U) = (A1) + (A2) + + (An) = 1, 0, () = 0, . , . , . 137

1. . () : () 0. 2. . U 1: (U) = 1. 3. . () A B : (A B) = p(A) + p(B). . . , , , - , , . , . . , A B, p(A) p(B) A 0 p(A) 1. , A B, B = A (B \ A) (. 1.3), A B \ A , A (B \ A) = . p(B) = p(A (B \ A)) = p(A) + p(B \ A), p(B) p(A) = p(B \ A). , p(B \ A) 0 , , p(B) p(A) 0 p(A) p(B). , A U, p(A) p(U). p(U) = 1 p(A) 0, 0 p(A) p(U) = 1, . . 0 p(A) 1. , , () . . . , , . , , 9/10, 6/10, , . , - ? 138

: A = { } B = { } , A B = { } (A B) = p(A) + p(B) = 9/10 + 6/10 = 15/10 = 3/2. , , A B, 1, . . 0 p(A B) 1. ? , : 0,9 0,6 , . . , 4 . , , 5 1/3 4 1/2 ? 5 4 . = { 5} B = { 4} . A B = { } : (A B) = p(A) + p(B) = 1/3 + 1/2 = 5/6. . . A1, A2, , An , . . , : (A1 A2 An) = (A1) + (A2) + + (An). , , n = 3. , , . . (A B) = p(A) + p(B). (. 1.4) A, B C , . . A B C = (A B) C. (. 1.4) (A B) C = (A C) (B C)= = = , A, B, C 139

A C = , B C = . , A B C , , (A B C) = ((A B) C) = (A B) + (C) = p(A) + p(B) + (C), . . (A B C) = p(A) + p(B) + (C). . 10 000 10 100 000 , 40 50 000 1000 500 . ? A1 = { 100 000 }, A2 = { 50 000 } A3 = { 500 }. A1 A2 A3, Ai . p(A1) = 10/10 000, p(A2) = 40/10 000 p(A3) = 1000/10 000, , p(A1 A2 A3) = p(A1) + p(A2) + p(A3) = 0,001 + 0,004 + 0,1 = 0,105. , , . ( ), , . . , , , , , , . , , , , . . . . , , 1 11. , . . , . - , . . . , . ( . ., 2004), 140

- , , . , , A = { } , A = { }. A , , , : , , , , , , . . , . . A = U \ A. , A = { } A = { } . . , A p(A) = 1 p( A ). , A A = U, A A = , . . A A , p(U) = 1, 1 = p(U) = p(A A ) = p(A) + p( A ). , p(A) = 1 p( A ). , U, () = 1 p(U) = 1 1 = 0. 2.4, . . 23 . , ? 23 23 , . . {1, 2, , 365}. = { }. , , , . , . . , 141

S. () = 1 ( ) = S, () = 1 (S). , (S) = Ank / Ank n = 365 k = 23, (S) =365 364 ... 341 364 ... 341 = (1 1 )(1 2 )(1 22 ), = 23 22 365 365 365 365 365

, : () = 1 (1 1 )(1 2 ) (1 22 ) 0,507. 365 365 365 , 22 , , , , 0,475, . . 1/2, 23 1/2. . , . ? A = { }, B = { }. A B = { }. , p(A) = 1/2, p(B) = 1/2, p(A B) 1, A B . p(A B) p(A) + p(B), A B , . . A B . 4 : (, ), (, ), (, ), (, ). A B 3 , , p(A B) = 3/4. . n , , m , A, k , B. , A B , . . , m + k l , A B. p(A) = m/n, p(B) = k/n p(A B) = l/n. , A B m + k l , p(A B) =m+k l m k l = + = p(A) + p(B) p(A B). n n n n

142

, , p(A) = 1/2, p(B) = 1/2, p(A B) = 1/4, p(A B) = 1/2 + 1/2 1/4 = 3/4. , . . . . . A B () : p(A B) = p(A) + p(B) p(A B). . A B , . . A B = , p(A B) = p() = 0 p(A B) = p(A) + p(B). A B, . . A B . A, B A B (. 2.5) : A = (A \ B) (A B), B = (B \ A) (A B), A B = (A \ B) (A B) (B \ A). 1.3 .

, A \ B B \ A (. 1.3). , A B A \ B, A B B \ A, . , 1.4 , : C (D \ E) = (C D) \ (C E). C = A B, D = A E = B, (A B) (A \ B) = (A B A) \ (A B B) = (A B) \ (A B) = . 143

, C = A B, D = B E = A, , (A B) (B \ A ) = . , A \ B, A B B \ A . : p(A) = p((A \ B) (A B)) = p(A \ B) + p(A B), p(B) = p((B \ A) (A B)) = p(B \ A) + p(A B), p(A B) = p((A \ B) (A B) (B \ A)) = p(A \ B) + p(A B) + p(B \ A). , p(A) + p(B) = (p(A \ B) + p(A B)) + (p(B \ A) + p(A B)) = = (p(A \ B) + p(A B) + p(B \ A)) + p(A B) = p(A B) + p(A B). p(A B) = p(A) + p(B) p(A B), . . () . . . ? A = { 6 } B = { 6 }. p(A B) p(A B) = p(A) + + p(B) p(A B). , p(A) = 1/6, p(B) = 1/6, p(A B) = 1/62 = = 1/36, p(A B) = 1/6 + 1/6 1/36 = 11/36. , , , , . , ? A = { }, B = { }. A B , . 0,0099. , p(A B) = p(A) + p(B) p(A B) = = 0,0099 + 0,0099 0,00990,0099 0,02. 144

. . A1, A2, , An () : (A1 A2 An) = p( Ai ) 1in1i< jn

p( Ai A j ) +

+

1i< j 0, p(B) = > 0, n n r . B, l n

, , , A B. p(A B) = , p(B A) =r r n p( A B ) = = . m mn p( A) r r n p( A B ) = = . l ln p( B )

, , (A B) = p()p(A B) = p()p( ). , A B , . . . . A B = , . . , , , A , , 0, p(A B) = p(A B) / p() = 0 / p() = 0. . A, , A , . . A 1, p(A B) = p(A B) / p() = p() / p() = 1.

147

. 1, 2, 3 , 4, 5, 6 . . , ? = { } = { 3 }. p() = 3/6 = 1/2 p(A B) = 2/6 = 1/3. p(A B) p(A B) = = (1/3)/(1/2) = 2/3. , p() = 3/6 = 1/2.

. , () > 0, , , p(A B) =p( A B ) . p( B )

, , . . p() 0, p(A B) p() > 0 , p() 0. , , p( B ) = p(A B ) / p( B ). , = { } = { } , A B p() > 0. p(A B). , . . 2, 4, 6, 2, . . A B 2, p(A B) = 1/3. , B 1, 3, 5, 3 5, . . A B 3 5, p(A B ) = 2/3. . l r , p(A B) r p(A B) = . l 148

, , . , n , p() = l/n p(A B) = r/n. p(A B) = (r/n) / (l/n) = r/l. . , . , ? = { }, = { }. 18 , 2 . , p(A B) = 2/18 = 1/9. , p() p(A B) . , , U, , . . U, U. , A B , . . p(A B), p(). . , , , , , , . p(A B).

. p(A B) , , A B, - . : 1. p(A B) 0 , ; 2. (U B) = 1 U , U; 3. p(( ) D) = p(A D) + p(B D) , . . = , , D. 0 p(A B) 1, , p(A B) = 0, A B = , . . A B , p(A B) = 1, A. 149

. 1, 2, , n , . . i j = i j, () : p((1 2 n) D) = p(1 D) + p(2 D) ++ p(n D). , , n = 2, . . : p(( ) D) = p(A D) + p(B D) , = . , ( D) (B D) = . (. 1.4), (A B) D = (A D) (B D), A D B D , p((A D) (B D)) = p(A D) + p(B D). p(( A B) D) =

p(( A B) D) p(( A D) ( B D)) = = p( D) p( D) p( A D) + p( B D) = p (A D) + p(B D). = p( D)

, , . . A = U \ A. . , , U p() > 0, : p(A B) = 1 p( A B). , A A = U A = . , ((A ) ( A )) = . , U p(U B) = 1. (. 1.4) 1 = (U B) = ((A A ) B) = ((A A ) B) / (B) = = p((A B) ( A B)) / (B) = (p(A B) + ( A B)) / (B) = = p(A B) + p( A B). p(A B) = 1 p( A B). p(A B), (B) > 0 ( ) = p(B)p(A B) ,150

(A) > 0, p(B A) (A B) = p(A)p(B A). , , . . (B) > 0 (A) > 0, () : (A B) = (B)p(A B) (A B) = (A)p(B A). , . . () , . ( ) ( ). , , . , , , . . . . . , ? A = { } B = { }. A B = { }. , A B , . . A B , p(A B) = p(A) + p(B) p(A B), p(A B) = p(A)p(B A). 36 . , p(A) = 9/36 = 1/4, 9 36 . p(B A) , , , 35 8 , p(B A) = 8/35. , , , p(A B) = p(A)p(B A) = (1/4)(8/35) = 2/35. :0 2 p(A B) = ( C 92 C 36 9 ) / C36 =

9 8 36 35 2 / = . 2! 2! 35

p(B). B = (A B) ( A B), p(B) = = p(A B) + p( A B), A = { }. -

151

, p( A ) = 27/36, 36 27 = 36 9 . 27 9 27 , p(B A ) = 9/35, p( A B) = p( A )p(B A ) = . = 36 35 4 35 p( A B) , , , , . . p( A B) =1 1 1 27 9 36 35 27 1 2 . (C 27 C 9 ) / C 36 = / = 2 2 1! 1! 2! 4 35

2 27 1 + = . , 35 4 35 4 , , A. :

, p(B) = p(A B) + p( A B) =

p(A B) = p(A) + p(B) p(A B) =

1 4

+

1 4

2 35

=

31 70

.

, , , , . . 1, 2, , n p(1 2 n-1) > 0. 1 2 n 1 2 n-1 1 2 1, , . . , B (A) (B), p(1 2 n-1) > 0 p(1 2 k) > 0 k = 1, 2, , n1. , (Ak 1 2 k-1) . n . . 1, 2, , n p(1 2 n-1) > 0, () : p(1 2 n) = p(1)p(2 A1)p(3 1 2) p(An 1 2 n-1). , () , , . , n = 3, , A, B D. p(A B)>0. , A B (A B) = (A)p(B A). (. 1.4)

152

A B D = (A B) D. , p(A B D) = p((A B) D) = p(A B)p(D A B) = p(A)p(B A)p(D A B), . . p(A B D) = p(A)p(B A)p(D A B).

. , , . . , , . . , -, , , -. , ? : A = { }, B = { }, D = { }. A B D. , p(A) = 1/5, p(B A) = 1/4 p(D A B) = 2/3. p(A B D) = p(A)p(B A)p(D A B) =1 1 2 1 . = 5 4 3 30

, 2.3. 5 1,1,1,2,1. A53 = 543 3- , . . . . A11 = 1, . . ; T A11 = 1 1 ; O A2 = 21 = 2 . 1 A11 A11 A2 = 112 = 2, 1 A11 A11 A2 1 1 2 1 p() = . = = A53 5 4 3 30

, , . 153

, . . . , , , . , ? 4- , 10 1, 2, 3, 1, 1, 1, 1, 2, 1, 2, 4 A10 =10987. A22 A32 , A22 = 21 A32 = 32 . , p() =A22 A32 2 1 3 2 1 . = = 4 A10 10 9 8 7 420

() () . . C, p(C) > 0, 1, 2, , n () : p(1 2 n C) = p(1 C)p(2 C 1) p(n C 1 2 n-1). , 1 2 n C = = C 1 2 n. p(1 2 n C) =

p( A1 A2 K An C )

p (C ) p (C ) = p(1 C)p(2 C 1)p(n C 1 2 n-1).

=

p(C A1 A2 K An )

=

. . , : , , . , , , . , . , . -

154

, . , , , . . A , p(A B) p(A), . . p(A B) = p(A). , . A = { }? 36 , p(A) = 4/36 = 1/9. , B = { }. A , B, p(B A) = 2/18 = 1/9. , p(B A) p(A) , , A B. A , B , , A B. , 10 ( ) 20 . B = { }, A = { }. B , 19 9 p(A) = 9/19. B , . . , 19 , p(A) = 10/19. A B. , , p(A B) = p(A), . . A B p(A) > 0, p(B A) p(B A) = p ( B A) p ( A) =p ( B )p( A B ) p( A) =

p ( B )p ( A) p ( A)

= p(B),

. . B A.. A B, B A, A B. A B , . () . . A B :

p(A B) = p(A)p(B).

155

, . : A = { 1- }, B = { 2- }, A B = { }. A B , p(A B) = p(A)p(B) = 1/61/6 = 1/36. , , . . p(A B) = p(A)p(B), p(B) > 0, A B. . , p(A) = p(A B) / p(B), p(B A), p(B A) = p(A), . . A B. , A B p(B) > 0, . . p(B) 0. . . . A B , p(A B) = p(A)p(B). A B . , . A = { 6 } B = { }. , A B . 62 = 36 (i, j), i, j = 1, 2, 3, 4, 5, 6. 6 A, . . (6, j), j = 1, 2, 3, 4, 5, 6, 18 B, . . (i, 1), (i, 3), (i, 5), i = 1, 2, 3, 4, 5, 6. A B c 3 : (6, 1), (6, 3), (6, 5). , p(A) = 6/36 = 1/6, p(B) = 18/36 = 1/2 p(A B) = 3/36 = 1/12 = 1/61/2 = p(A)p(B), . , , , p(B) = 0, p(A B) = p(A)p(B) . , (. 1.4) B = B U = B (A A ) = (B A) (B A ), (B A) (B A ) = , p(B) = (B A) + p(B A ). p(B) = 0 , p(B A) p(B A ) 0. , 0 . , p(B) = 0, p(B A) = 0 p(A B) = 0, 0 = p(A B) = p(A)p(B) = 0. . . A = { }, B = { }. A B ?

156

, A B, . . p(A) = 4/36 = 1/9, p(B) = 18/36 = 1/2 p(A B) = 2/36 = 1/18, 1/18 = 1/91/2, . . p(A B) = p(A)p(B) . , A B . , A = { } B = = { } . B C = { : , } , p(B) = 18/36 = 1/2, p(C) = 4/36 = 1/9, p(B C) = 2/36 = 1/18 1/18 = p(B C) = p(B)p(C) = 1/91/2 = 1/18. A B? : , , . B C .. , p(A B) = p(A)p(B), , .

. , . , , . . , . . . A, B. , C. ? U = {(, ), (, ), (, ), (, )}. A = {(, ), (, )}, B = {(, ), (, )}, C = {(, ), (, )}, A B = {(, )}, A C = {(, )}, B C = {(, )}, p(A) = p(B) = p(C) = 2/4 = 1/2, p(A B) = p(A C) = p(B C) = 1/4 , , 1/4 = p(A B) = p(A)p(B) = 1/21/2, 1/4 = p(A C) = p(A)p(C) = 1/21/2, 1/4 = p(B C) = p(B)p(C) = 1/21/2. . A B , A C B C . . , 1/2 A B , p(A C) = p(A)p(C) p(B C) = p(B)p(C) .

157

1, 2, , n , , , , i j, j i , . . . , 1, 2, , n , Ai1 , Ai2 , ..., Aik , {i1, i2, , ik} {1, 2, , n},

Ai1 Ai2 Ai3 ... Aik .. 1, 2, , n , Ai1 , Ai2 ,..., Ai k

p( Ai1 Ai2 ... Aik ) = p( Ai1 )p( Ai2 )p( Aik ).. 1, 2, , n, , Ai1 Ai2 Ai3 ... Aik ,

p( Ai1 ( Ai2 Ai3 ... Aik )) = p( Ai1 )p( Ai2 Ai3 ... Aik ).

Ai2 , Ai 3 ,..., Aik , p( Ai2 ( Ai3 ... Aik )) = p( Ai2 )p( Ai3 ... Aik ).

:p( Ai1 Ai2 Ai3 ... Aik ) = p( Ai1 )p( Ai2 )p( Ai3 ... Aik ).

, :p( Ai1 Ai2 ... Aik ) = p( Ai1 )p( Ai2 )p( Aik ),

. . () . n . , n -: 1, 2, , n , : p(i j) = p(i)p(j), 1 i < j n, p(i j k) = p(i)p(j)p(k), 1 i < j < k n, p(1 2 n) = p(1)p(2)p(n). , , . . ji, p(i j) = p(i)p(j), .. n > 2 1, 2, , n, , .

158

, p() = 1/2, p(B) = 1/2, p(C) = 1/2 A, B, C, . . p(A B) = p(A)p(B), p(A C) = p(A)p(C) p(B C) = p(B)p(C) , . . p(A B C) = p(A)p(B)p(C), A B C = p(A B C) = 0, p(A)p(B)p(C) = 1/8 . , , . . A B C .. . : A = { }, B = { }, C = { }. ? C = {(O, O), (P, P)}. A B = A C = B C = {(O, O)}, A B C = {(O, O)}. , p() = p(B) = p(C) = 1/2, p(A B) = p(A C) = p(B C) = 1/4, . . , , . , , 1/4 = p(A B C) p(A)p(B)p(C) = = 1/21/21/2 = 1/8, A, B, C . , , 1, 2, , n, n > 2, ( ). , , . . , .. 1, 2, , n , , , A1 , A2 , , An . , A B , A B , A B . , A = (A B) (A B ) (A B) (A B ) = , p(A) = p(A B) + p(A B ). p( B ) = 1 p(B)

p(A B ) = p(A) p(A B) = p(A) p(A)p(B) = p(A)(1 p(B)) = p(A)p( B ).

, A B . A B , A B . , , A, B C , A , B, C, A , B , A , B , C . .. 1, 2, , n , ()

p(1 2 n) = 1 p( A1 )p( A2 )p( An ).

159

. B = 1 2 n, , B B = U, . . , p( B ) + p(B) = p(U) = 1 p(B) = 1 p( B ). , (. 1.4) B = A1 A2 K An = A1 A2 An . ,

1, 2, , n , A1 , A2 , , An . , n : p(1 2 n) = 1 p( A1 A2 K An ) = = 1 p( A1 A2 An ) = 1 p( A1 )p( A2 )p( An ), . , 1, 2, , n p(A1) = p1, p(A2) = p2, p(An) = pn, p(1 2 n) = 1 q1q2qn, qi = 1 pi i = 1, 2, , n.. n , A p. , A ? k = { A k- }, k=1, 2, , n. , , , n 1, 2, , n . p(Ak) = p p( An ) = 1 p = q k. 1 2 n = { A n }. A1 A2 K An = A1 A2 An ? , A . , , , q = 1 p, p(1 2 n) = 1 p( A1 )p( A2 )p( An ) = 1 qn.

, 0 < q < 1, n 1, . . A ( 1) . . , , , 10 . , ? p = 0,0099, q = 1 0,0099 0,99. , ( n = 10) 1 qn = 1 (0,99)10 0,095. , . n p, n , . . n , pn. n q = 1 p, , , , n

160

, qn. 1 qn.. p n 1 qn, q = 1 p.

, , p = 1/6, q = 1 p = 5/6, n ( ) pn = (1/6)n. n ( ) qn. ( n ) 1 qn = 1 (5/6)n. , 1654 : . , , . , . , . , . . 6 , . . , 2 . , , (6, 6), . . 12 , 1/2. 24 , . , . . : 4 6 24 12 ? , 6 n = 4 ( q = 1 p = 1 1/6 = 5/6) 625 671 5 1q =1 =1 = 0,518, 1296 1296 6 . . , , . , , , . , 12 24 ( q = 1 p = 1 1/62 = 35/36) 44

35 1 q24 = 1 36

24

0,491,

. . . 4 6 , 24 12 .

161

, 25 , . . 12 , 1 (35/36)25 0,505. , 25 , 50,5 % , , 24 , 49,1 % . , . , n . , n k . , k = 0 k = n k, 0 k n. . n , , ( p q = 1 p), k n k C n p k q n k =

n! p k q n k . k! (n k )!

. n k , k , nk . , , k nk pkqn-k. k , nk , pkqn-k. , k n , , k . C n k n , , k k n Cn p k q n k , . , . - , , . , (., 1977) 40 . , , , , . . , , 510 . N, i, i- pi. -

162

, . . . i- k n :k Cn p ik (1 p i ) nk =

n! p ik (1 p i ) nk . k! ( n k )!

, k k n k , Cn p q , (. 2.2)

C