91 7,.,, ( ).,,.,.,. 7.1 A B, A B, A = B. 1), 1,.,. 7.1 A, B, 3. (i) A B. (ii) f : A B. (iii) A B. (i) (ii)., 6.9, (ii) (iii).,,,. 1), Ā = B.. A, Ā, Ā,.
92 7 7.2 A, B, C. (1) A = A. (2) A = B B = A. (3) A = B, B = C A = C. A = B. (3)., f : A B g : B C. g f : A C, A = C. 7.1, A, B,. A = B, A, A A., A, A, A.,,,. 7.2,,.,,,.,.,,.,. 7.3 A, B. f : A B A = f(a). f 1 : A f(a) f 1 (x) = f(x) (f f 1 )., f 1, A = f(a). 7.2 N = {1, 2, 3,... } A., f : N A A.
7.2. 93 ( 7.2 (3)), A A = B B.. f : N A, A f(n), A.,., A, g : A N, A.,, A = {a 1, a 2,... },. ( 6.1 )., A,.,, N A. 7.4 E = {2, 4, 6,... }., f : N E f(n) = 2n, f. N : 1 2 3... f E : 2 4 6...,., E N,., E N, E = N. ( 6.3). 7.5 Z., n 1, n, f(n) = 2 n 2, n, f : N Z.
94 7 7.6 N. K N. K k 1 < k 2 < k 3 < < k n < (7.1). n k n N K, K. 7.7 ( ).,. B, A B., f : N B. f A K = f 1 (A). f 1 : K A f 1 (n) = f(n),., K N, 7.6, K., g : N K., f 1 g : N A, A.,,. 7.8 P = {2, 3, 5, 7, 11,... }., P N, P ( 1.21)., 7.6, P., f : N P, f., 7.7 7.6.,, K (7.1).,., K, k 1., K (7.1), ( 13 ). 7.9 A, B, A B =. (1) A, B, A B.
7.2. 95 (2) A, B, A B. (3) A, B, A B. (1) g : N A h : N B., f : N A B f(2n 1) = g(n), f(2n) = h(n), n N,, f, A B., A = {a 1, a 2,... }, B = {b 1, b 2,... }, A B a 1, b 1, a 2, b 2,...,. (2) A g : N A., B, B = n h : [n] B., h(k), k [n], f(k) = g(k n), k {n + 1, n + 2,... }, f : N A B., A B., B, A,. (3) 6.12. 7.10 ( ) A, B, A B., A 1,..., A n, A 1 A n. A, B. B\A B ( 7.7)., A B\A, 7.9, A (B\A) = A B., A 1 A n = (A 1 A n 1 ) A n, n. 7.1 A 1,..., A n, A 1 A n., A 1 A n.
96 7 7.3 7.11 ( ) A, B. f : A B, A., A, f : A B. f A = f(a) ( 7.3)., B, f(a), 7.7, f(a)., A., g : N B., A, A = n h : [n] A. f = g h 1 : A B. A, h : N A. 7.12 ( ) A, B. f : A B, B., B, f : A B. A, g : N A. h = f g : N B, x B, h 1 ({x}) N.,. φ(x) = min h 1 ({x}), x B,, φ : B N. h(φ(x)) = x, h φ = i B ( )., φ : B N., 7.11, B., B, B = n h : [n] B., φ : N [n] k, k [n], φ(x) = 1, k {n + 1, n + 2,... },,., h φ g 1 : A B. B, h : N B., h g 1 : A B.
7.4. 97 X ( ) a 1, a 2, a 3,..., a n,... (7.2). ( ) A., a : n a n ( 4.5 ), A A = {a n n N}., 7.12 A., A,., a 1 1, a 2, a 3,..., a 1 2., 1, 2 3.,,. A,., (7.2), A., A., (7.2), 1, [n] N A., 7.12.,,,.,,,.,,., 11.3. 7.4 7.13 f : N N N f(x, y) = 2 x 1 (2y 1), x, y N,, f.
98 7, f. z N, z = 2 k w 2 2 w., x = k + 1 x N,, w w = 2y 1 y N, z = f(x, y)., f., z N z = 2 k w, k, w, f. 7.14 N N. 7.13. N N. N N (x, y) 7.1,., (x, y) f(x, y) = (x + y 2)(x + y 1) 2. (7.3). + y, (x, y) N N, (7.3) y 5 4 3 2 1 1 2 3 4 5 x 7.1: N N (x, y) 7.2 (7.3) f : N N N,. 7.15 A, B, A 1, B 1. A = A 1, B = B 1 A B = A 1 B 1.
7.4. 99, g : A A 1 h : B B 1. f : A B A 1 B 1 f(x, y) = (g(x), h(y)), f. 7.16 ( ) A, B, A B., A 1,..., A n, A 1 A n. A = B = N, 7.15 7.14, A B = N N = N., A B., A 1 A n = (A 1 A n 1 ) A n. 7.17 A, B, A B. B, B = n B = {b 1, b 2,..., b n }. k = 1, 2,..., n, A k = {(x, b k ) x X} X Y., x (x, b k ) A A k, A k., n A B =, 7.10. A B. 7.3 A 1,..., A n, A 1 A n., A 1 A n.,.,,,, n, m N, n Z, (7.4) m.,.,,. A k
100 7 7.18 Q., N, Z, 7.16, N Z., f(m, n) = n, m N, n Z, m, f : N Z Q., 7.12, Q. Q (, N ),. 7.18,,,., ( ). 7.4. 7.5. 4.5, A 1, A 2, A 3,..., A n,,...,, n A n, (A n n N). A n = n N n=1 A n. 7.19 A 1, A 2,..., B 1 = A 1, \ n 1 B 2 = A 2 \A 1,..., B n = A n A k,..., (7.5), B 1, B 2,...., B 1, B 2,...,. A n = B n. (7.6) n=1 n=1
7.4. 101 m n x B m B n. m < n. B n, x A n, x n 1 A k (7.7)., x B m, B m A m x A m., (7.7) 2, B m B n =., (7.6),, n n A k = B k, n = 1, 2,..., (7.8). n = 1 (7.5). n 1 n (7.8)., n+1 ( n ) ( n A k = A k A n+1 = ( n ) = B k B n+1 = n+1 B k ) ( \ n A k A n+1 A k ), (7.8) n + 1.,, (7.8). 2), (7.6). x n=1 A n, n N x A n., (7.8) n A n B k, x n=1 B n., (7.6).,, (7.6). 7.20 A 1, A 2,... R,.,. A = n=1 2), n,,,. A n B k
102 7 A,., A n.,. A 1 A 2 {}}{{}}{{}}{ x (7.9), A,.,. x A, x A n 1, x n g : A N., g(x) = A n nχ n (x), x A, n=1., χ n (x) A n., f(x) = g(x) 1 n=1 A n + {y A g(x) y x}, x A,., x A 1, 1 0., f(x), (7.9) x, f : A N. 7.21 A 1, A 2,... R.,. A = n=1 A n 7.19, B 1, B 2,.... B n A n B n,., M = {n N B n } N. 7.19, A = B n = B n (7.10) n=1 n M., M, (7.10), A., M
7.4. 103, M = {n 1, n 2,... } C k = B nk., A =, C 1, C 2,... R, 7.20 A. n. a 0, a 1,..., a n Z, a 0 0, x n a 0 x n + a 1 x n 1 + + a n 1 x + a n = 0 (7.11)., (7.11) n..,.., 2 p, q Z, p 0, q/p, 1 px q = 0 C k. 2 3, 3 2 + 1, x 2 + 6x + 7 = 0, (x 1) 3 2 = 0,.,. 7.22.,. (7.11), h = n + a 0 + a 1 + + a n., m, h h m E m., E m m, m., E m.
104 7, E m A m. 1 n n, E m., A m., A, A = A m (7.12) m=1., α, m, α A m., (7.12)., (7.12)., 7.21, A,. A. 7.22 7.20, A n,., A 1, A 2,...,,.,. 11.