44 3,,, {}. A = {1, 2, 3, 4, 5}.,,,,,,. B = {1, 2, 3,, 99}. N = {0, 1, 2, 3, }.. x, P (x) x, {x P (x)} P (x) x., x 2 1 = 0 C = {x x R x 2 1 = 0}., C C

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1 3,. 16,,,, , (Georg Cantor),,, ,,.,., ,,.,,,. ( ).,. x 2 1 = 0. C... A, B, C,, a, b, c,. a A, a A, a A. a A, a A a A, a A. N ( 0). Z, Z +, Z. Q, Q +, Q. R, R +, R. C,. 2.,.

2 44 3,,, {}. A = {1, 2, 3, 4, 5}.,,,,,,. B = {1, 2, 3,, 99}. N = {0, 1, 2, 3, }.. x, P (x) x, {x P (x)} P (x) x., x 2 1 = 0 C = {x x R x 2 1 = 0}., C C = {1, 1}., R.,, E U., ;,.,.,,..,.,,.,, ;, Z.,.,, Z, Q R, Z Q R. 3., : (1),,. : {1, 1, 2, 4, 2}={1, 2, 4} (2). : {1, 2, 3}={2, 3, 1} (3) ( ),., a A, a A a A. (4),,,. : {a, {1, 2}, p, {q}}

3 (5),,,,,. S = {a, {1, 2}, p, {q}} , 3, A B, A B, A B,. A B, B A. A B B A. A B x ( x A x B) A B, A B. A B x ( x A x / B), A = {a, b}, B = {a, b, c}, C = {b, c, d}, A B, A C.,,. 3.2 A, B, A B B A, A B, A = B. A = B A B B A A B, A B. :,. 3.3 A B, A B, B A, A B, A B. A B x ( x A x B) x ( x B x / A)

4 46 3 A B A B A B A B, A B., {a, b} {a, b, c}, {a, b, c} {b, c, d} {a, b, c}., A B A B 3 3.4,. = {x x x}, A = {x x R x = 0} x = 0,, A =. : { }. 3.1 A, (1) A, ; (2)A A. :(1) A, x, x x A,,. 1 2,, , 1 = 2,,. (2), A, A A. 3.5 n n, m (m n) m. n,?. 3.1 A = {1, 2, 3}, A : 0,, :. 1, : {1} {2} {3}. 2 : {1, 2} {1, 3} {2, 3}. 3 : {1, 2, 3}., n, m Cn m, C 0 n + C 1 n + C 2 n + + C n n = 2 n

5 ,, A B, A B A B, A B. A B = {x x A x B}, A = {1, 2, 4}, B = {2, 4, 5}, A B = {1, 2, 4, 5}., A B n. A 1, A 2,, A n n, n n i=1 A i, n A i = A 1 A2 An = {x x A 1 x A 2 x A n } i=1 : 3.7 A i = A 1 A2 An i=1 A B, A B. A B, A B A B = {x x A x B} A B, A B., A = {1, 2, 4}, B = {2, 4, 5}, C = {1, 3}, A B = {2, 4}, B C =, B C. A B 3.4. n.

6 48 3 A 1, A 2,, A n n, n n A i, n A i = A 1 A2 An = {x x A 1 x A 2 x A n } i=1 i=1 3.4 : A i = A 1 A2 An i=1 3.8 A B, A B B A ( A B ), A B. A B = {x x A x / B}, A = {1, 2, 4}, B = {2, 4, 5}, A B = {1} B A = {5}. B A 3.5(a). 3.5 A. 3.9 E, A E, A E A, A = E A = {x x E x / A}, E = {1, 2, 3, 4, 5}, A = {1, 2, 4}, B = {1, 2, 3, 4, 5}, C =, A = {3, 5}, B =, C = E. A 3.5(b) A B, A B B A A B, A B. A B = {x (x A x / B) (x / A x B)}

7 A B = (A B) (B A) A B A B, A = {a, b, c}, B = {b, d}, A B = {a, c, d}. A B = (A B) (A B) A B. A B C 3, : (1) A B = B A. (2) A = A. (3) A A =. (4) A B = (A B) ( A B). (5) (A B) C = A (B C). (6) A (B C) = (A B) (A C) ,.,, E, A B C E 3. (1) A A = A, A A = A. (2) A B = B A, A B = B A. (3) (A B) C = A (B C), (A B) C = A (B C). (4) A (B C) = (A B) (A C), A (B C) = (A B) (A C). (5) A (A B) = A, A (A B) = A. (6) A = A, A E = A. (7) A E = E, A =. (8) A A = E. (9) A A =. (10) = E, E =.

8 50 3 (11) ( A) = A. (12) A B = A B. (13) (A B) = A B, (A B) = A B; A (B C) = (A B) (A C), A (B C) = (A B) (A C). 13. : (1).. : P Q,, P = Q, P Q Q P. x x P x Q (2).,,. 3.2 : (1) : A (B C) = (A B) (A C). (2) : A A = E. (3) : (A B) = A B. (1) x, 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), A (B C) = (A B) (A C). (2) x, x A A x A x A x A x / A x A x A 1 x E, A A = E. (3) x, x (A B) x / (A B)

9 (x / A) (x / B) (x A) (x B) x ( A B), (A B) = A B A (A B) = (A E) (A B) = A (E B) = A E = A., A (A B) = A.,. (14) A B A, A B B. (15) A A B, B A B. (16) A B A. (17) A B A B = A A B = B A B = A (B C) = (A B) (A C).. x, x A (B C) x A x / (B C) x A x (B C) x A x ( B C) x A (x B x C) x A (x / B x C) (x A x / B) (x A x C) x (A B) (A C), A (B C) = (A B) (A C).. A (B C) = A (B C) = A ( B C) = (A B) (A C) = (A B) (A C) 3.5 A B A B =, A B = E, A = B.

10 52 3 A = A E = A (B B) = (A B) (A B) = (A B) = (B B) (A B) = (B A) B = E B = B, (1) (B (A C)) (A B C). (2) ((A B C) (A B)) ((A (B C) A). (1) (B (A C)) (A B C) = (B (A C)) (B (A C)) = B ( (A C) (A C)) = B E = B (2) A B A B C, A A (B C), ((A B C) (A B)) (A (B C) A) = (A B) A = (A B) A = (A A) (B A) = (B A) = B A A = {a 1, a 2,, a n }, n, A n, CardA = n A = n.. A n, A., 0, = 0. A, A.,,,. 3.2 ( ) A B, A B = A + B A B.

11 (1) A B, A B =, A B = A + B (2) A B, A B = {a 1, a 2,, a k }, A = {a 1, a 2,, a k, x 1, x 2,, x n }, B = {a 1, a 2,, a k, y 1, y 2,, y m }, A + B A B = n + k + m + k k A B = n + m k A B = A + B A B, A B = A + B A B , 21, 4,? E 50, A, B, E = 50, A = 16, B = 21, A B = 4., A B = A + B A B = = 33 E A B = = 17, 17., n., A n P 1, P 2,, P n. A P i A i, i = 1, 2,, n; A i Aj (i j) A P i P j ; A i Aj Ak (i j k) A P i P j P k A 1 A2 An A P 1, P 2,, P n ( )A n A 1 A2 An = n A i 3.8? i=1 1 i<j n A i Aj + 1 i<j<k n + ( 1) n+1 A 1 A2 An A i Aj Ak X 1 250, X 2 3 7

12 54 3 A B C X 2 3 7, A = = 125, B = = 83, C = = 35, A 250 B = = 41, A 250 C = = 17, B 250 C = = 11, A B 250 C = = 5, 2 3 7, A B C = A + B + C A B A C B C + A B C = = 179, X ,..,,,.,. n,., x.,, , 12, 15, 10, 8, 11, A B C. 3 x, y 1, y 2, y 3,, 10 x,, 8 x,, 11 x, , 11 x + 10 x + x + y 1 = x + 8 x + x + y 2 = x + 8 x + x + y 3 = x + 10 x + 8 x + x + y 1 + y 2 + y = 30

13 x = 8, y 1 = 5, y 2 = 1, y 3 = 5, ,,,., A, A A,, P (A)( 2 A ). P (A) = {x x A} A, A, A =, B = {1, 3, 5}, C = {1, {2, 3}}. A B C. A =, 0,, P (A) = { }. B 0 :. 1 : {1} {3} {5}. 2 : {1, 3} {1, 5} {3, 5}. 3 : {1, 3, 5}., P (B) = {, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, {1, 3, 5}}., P (C) = {, {1}, {{2, 3}}, {1, {2, 3}}} A = n. A P (A). 0 A 0, Cn 0. A A 1, Cn 1. A n A A n, Cn n. P (A) = Cn 0 + Cn Cn n = 2 n A B, (1) A B P (A) P (B). (2) P (A) = P (B) A = B. (1), x, x P (A) x A x B x P (B)

14 56 3 P (A) P (B)., y, y A {y} P (A) {y} P (B) y B A B., A B P (A) P (B) (2) P (A) = P (B) (P (A) P (B)) (P (B) P (A)) (A B) (B A) A = B 3.5.2,,. n x y( x = y) ( ), < x, y >. x < x, y >, y < x, y >., (1) {x, y} ( ), {x, y} {y, x}. < x, y > < y, x > x y. (2) {x, y} x y, < x, y > x y., < x, y > : (1) x y, < x, y > < y, x >. (2) < x, y >=< u, v > x = u y = v., 3 4 n n (n 3), n 1 < x 1, x 2,, x n 1 >, x n, n < x 1, x 2,, x n >. < x 1, x 2,, x n >=<< x 1, x 2,, x n 1 >, x n >, n M n < x 1, x 2,, x n >. n < a 1, a 2,, a n >=< b 1, b 2,, b n > a i = b i, i = 1, 2,, n A, B, A, B A B, A B. A B = {< x, y > x A y B}

15 3.5 57, < x, y > A B, x A y B; < x, y >/ A B, x / A y / B. A = {1, 2, 3}, B = {a, b}, A B = {< 1, a >, < 1, b >, < 2, a >, < 2, b >, < 3, a >, < 3, b >} B A = {< a, 1 >, < a, 2 >, < a, 3 >, < b, 1 >, < b, 2 >, < b, 3 >} A A = {< 1, 1 >, < 1, 2 >, < 1, 3 >, < 2, 1 >, < 2, 2 >, < 2, 3 >, < 3, 1 >, < 3, 2 >, < 3, 3 >} B B = {< a, a >, < a, b >, < b, a >, < b, b >}, A B B A,. n, A 1, A 2,, A n n, A 1 A 2 A n = {< x 1, x 2,, x n > x i A i, 1 i n} A 1, A 2,, A n., A 1 = A 2 = = A n = A, A 1 A 2 A n = A n. A B C 3, : (1), A B B A( A B A B ). (2), (A B) C A (B C)( A B C ). (3) A B, A = B =. (4), A (B C) = (A B) (A C) (B C) A = (B A) (C A) A (B C) = (A B) (A C) (B C) A = (B A) (C A). < x, y >, < x, y > A (B C) x A y (B C) x A (y B y C) (x A y B) (x A y C) < x, y > (A B) < x, y > (A C) < x, y > (A B) (A C), A (B C) = (A B) (A C). (5) A C B D A B C D,. < x, y >, < x, y > A B x A y B A C B D x C y D

16 58 3 < x, y > C D, A B C D. (5),. A = B =, A C B D. A B, A C B D. : x A, B, y B, x A y B < x, y > A B < x, y > C D x C y D x C A C. B D. A = B, A C, B D., A =, B = {1}, C = {2}, D = {4}, A C = B D, B D. A B =, B D, A C A B C,. (1) A B = A C B = C. (2) A (B C) = (A B) (B C). (3) A, A A A. : (1). A =, B = {1}, C = {2}, A B = A C, B C. (2). A = B = {1}, C = {2}, A (B C) = {1} {< 1, 2 >} = {1} (A B) (B C) = {1} = (3). A =, A A A , A, A A, A. A = {x z( z A x z)} 3.14 A 1 = {{a, b, d}, {c, d}, {d, e, f}}, A 2 = {b, {c, d}}, A 3 = {{a, b}}, A 4 =,. A 1 = {a, b, c, d, e, f} A2 = b {c, d} A3 = {a, b}

17 A4 = A, A A, A. A = {x z( z A x z)} 3.14, A 1 = {d}, A 2 = b {c, d}, A 3 = {a, b}.,.,, ;,., :,.. ;, A = {{a, b}, {a, c}}, A A A A. A = {a, b, c} A = {a} A = a b c A = a A = a A A = (a b c) a = (b c) a ,, ;,.,, A B, A B A B. :, A. A, A, A. A, ℵ 0 (ℵ ; ℵ 0 ) A, A = ℵ 0 ; A, ℵ A, A = ℵ., A A ( ), A {a 1, a 2,, a n, }.

18 60 3 A f a 1, a 2,, a n,, a 1 = f(1), a 2 = f(2),, a n = f(n),. :, 0 1. : (1). (2) (1) A = {x x, x 2 = 4}. (2) A = {x x 10 }. (3) A = {(x, y) x, y, (x, y) }. 2.. (1). (2) 3 7. (3) A = { 3, 2, 1, 0, 1, 2, 3}. 3.. (1). (2). (3) { }. (4) { }. (5) {a}. (6) {a}. (7) {a} {a, {a}}. (8) {a} {a, {a}}. 4. A B, A B A B? 5. A B C, A B, B C, A / C? 6. A B C 3,,. (1) A B B C, A C. (2) A B B C, A C. (3) A B B C, A C. (4) A B B C, A C. (5) A B B C, A / C. (6) A B B C, A / C. 7. A = {1, 2, 3, 4}, B = {2, 3, 5}, A B A B A B B A A B. 8. E = {1, 2, 3, 4, 5}, A = {1, 4}, B = {1, 2, 5}, C = {2, 4},.

19 (1) A B. (2) (A B) C. (3) (A B) (A C). (4) (A B). (5) A B. (6) B C. (7) P (A) P (C). (8) P (A) P (C). 9. A B C 3,,. (a) A B = A C, B = C. (b) A B = A C, B = C. (c) A B = A C, B = C. 10. A B, A B = A A B = (1) A (B C) = (A B) C. (2) (A B) C = (A C) (B C). (3) (A B) C = (A C) (B C). (4) A (A B) = A B (1) ((A B) B) (A B). (2) A (B A) B. (3) ((A B C) (B C)) A. (4) (A B) (C (A B)) , C 40, Java 35, Perl 10, 3 5,. 3? ,,,. 20 3, , 1400, : , 2, 4., ( ) ,. (1) {, { }}. (2) {a, b, {a, b}}. (3) {1, {, 1}}.

20 < x + 2, 4 >=< 5, 2x + y >, x y. 19. A = {a, b}, A P (A) P (A) A. 20.? (a) A B =. (b) A B = B A. (c) (A B) C = A (B C). 21. A B C, A B A C, B C? 22. A B C, D,,?,. (1) (A B) (C D) = (A C) (B D). (2) (A B) (C D) = (A C) (B D). (3) (A B) (C D) = (A C) (B D). (4) (A B) C = (A C) (B C). 23. A = {{1, 2}, {1, 3}, {2, 3}},. (1) A. (2) A. (3) A. (4) A (1). (2). (3) 0 1/2. (4) 7.

A股招股说明书.doc

A股招股说明书.doc 150,000,000 1.00 4.20 0.113 4.087 150,000,000 630,000,000 17,000,000 613,000,000 2000 12 22 23 000 511 512 5-1-3 5-1-10 5-1-11 5-1-12 5-1-19 5-1-34 5-1-39 5-1-41 5-1-43 5-1-47 5-1-62 5-1-84 5-1-88 5-1-92