1、集合的基本性质.pdf

1、 1 1.1 1.1.1 1.1.2 A, B, C, X, Y, Z 2 a, b, c, x, y, z N, Z, Q R 1.1.3 a A a A a A a A a A a A a A b A a, b A a 1 A, , a n A a 1 , , a n A 1.1.4 1.1.5 3 N 3 2 Q 2 R 3 2 N 2 Q a 1 , , a n , , , , , , , , , , , 1, 2 0, 1, , n 1 x | x x | x Z x 0 x | x 2 3x+2 = 0 , , , , , 1, 2 3 1, 2, , , , , 1, 1, 2,

2、 3 , , x x A x A 1.1.6 A B A B A = B A = B x A x B x A x B A B A = B A = B x x A x B A = B x x A x B x B x A A B A B A B B A A B B A A B A B A B x (x B x A) (x A x B) 4 , = x | x , 1, 2 = x | x 2 3x+2 = 0 N = x | x Z x 0 0, 1, 2 = 2, 0, 1 , = , , , = , , , 1, 2, 1, 3, 2 = 1, 2, 3 A A 1.1 1.1.1 (1) (

3、2) 1 2 1, 2 1, 2, 3 1, 1, 2, 3 (3) 1.1.2 (1) 1, 1, 2 1, 1, 2 1, 2 (2) 1, 1, 2, 1, 2 1, 2, 1, 2 (3) x | x N x N Z (4) x | x Z x 3 0, 1, 2, 3 5 1.2 1.2.1 B A B A B A B A ( 1.2.1) B A x x B x A B A A B A B B A B A x x A x B B A B /A B /A B A B A B /A ( 1.2.2) B /A x x B x A A A B x B 1.2.1 B A 1.2.2 B

4、/A A A 6 A A = A A A A A B A B A B A B 1 A, , B n A B 1 , , B n A 1.2.2 1.2.3 N Z Z Q Q R R /Q Q /Z A x | x A x A A 1.2.4 n N n x | x N x n N N n (1) N 0= (2) n m N n N m x x N n x n n m x n x m N m x N m x x N n x N m N n N m (3) n m N n N m n m 7 n N m n N n m 0 Q += x | x Q x 0 R += x | x R x 0 8

5、 1.2.7 A, B, C (1) A A A (2) B A A B A = B (3) C B B A C A (1) (2) x x B B A x A x A A B x B A = B ( 1.2.3) x x C C B x B x B B A x A C A ( 1.2.4) A B A A = B 1.2.3 A A B C C 1.2.4 9 1.2.7 (2) A = B B A A B 1.2.7 (3) C B B A C B A 1.2.8 A A P(A) P(A) = X | X A X P(A) X A A A A , A P(A) P( ) = 1.2.9

6、P( , ) = , , , , P(0, 1, 2) = , 0, 1, 2, 0,1, 0,2, 1,2, 0,1,2 1.2 1.2.1 (1) (2) 1 2 1, 2 1, 3 1, 1, 2, 3, 3 (3) x | x 2 +3x+2 = 0 x | 2x 2 +3x+1 = 0 x | x N x 5 N, Z Q 1.2.2 a R Q a= x | x Q x a (1) a b Q a Q b (2) a b Q a Q b (3) a R Q a Q a Q 1.2.3 a, b R a c d b (c, d) (a, b) 1.2.4 1, 1, 2 10 1.3

7、 1.3.1 A B A B A B A B A B A B = x | x A x B( 1.3.1) x A B (x A x B) x A B x x x A B x A B (x A x B) A B = A B ( 1.3.2) A B = x x A B x A B A B = x x A x B A B = A B A B A B = A B = x x A x B A A B B 1.3.1 A B 1.3.2 A B = 11 1.3.2 A B A B A B A B A B A B = x | x A x B( 1.3.3) x A B (x A x B) x A B x

8、 A x B x A x B x A B x A B ( x A x B) x A B x A B x A B x A B x A B (x A x B) A B 1.3.3 A B 1.3.3 = = 1.3.4 1, 2, 3 2, 3, 4 = 2, 3 1, 2, 3 2, 3, 4 = 1, 2, 3, 4 1.3.5 ( 1, 1 (0, 2 = (0, 1 ( 1, 1 (0, 2 = ( 1, 2 a c b d (a, b (c, d = (c, b 12 (a, b (c, d = (a, d x x (a, b (c, d ( a x b c x d) c x b x (

9、c, b x x (a, b (c, d ( a x b c x d) a x d x (a, d 1.3.6 A B = x | x A x C = x | x A x B C = B C = A ( 1.3.4) A B C = C B B C = A 1.3.4 1.3.7 A, B, C (1) A A B B A B (2) A B A A B B 13 (3) B A C A B C A (4) C A C B C A B (1)(2) (3) x x B C x B B A x A x C C A x A x A B C A ( 1.3.5) (4) x x C C A x A

10、C B x B x A B C A B ( 1.3.6) A A B C C B 1 .3 .5 1.3.6 1.3.7 (3) (4) 1.3.8 A 1 , A 2 , B 1 , B 2 (1) B 1 A 1 B 2 A 2 B 1 B 2 A 1 A 2 14 (2) B 1 A 1 B 2 A 2 B 1 B 2 A 1 A 2 (1) B 1 A 1 A 1 A 1 A 2 B 1 A 1 A 2 B 2 A 2 A 2 A 1 A 2 B 2 A 1 A 2 B 1 A 1 A 2 B 2 A 1 A 2 B 1 B 2 A 1 A 2 (2) B 1 B 2 B 1 B 1

11、A 1 B 1 B 2 A 1 B 1 B 2 B 2 B 2 A 2 B 1 B 2 A 2 B 1 B 2 A 1 B 1 B 2 A 2 B 1 B 2 A 1 A 2 1.3.9 A, B, C (1) A = A A = (2) A A = A A A = A (3) A B = B A A B = B A (4) A (B C) = (A B) C A (B C) = (A B) C (5) A (A B) = A A (A B) = A (6) A (B C) = (A B) (A C) A (B C) = (A B) (A C) (1) (2) x A x A x A x x

12、A A x A A A = A x A x A x A A A = A (3) x A x B x B x A x x A B x B A 15 A B = B A x A x B x B x A A B = B A (4) A A B B C A B A (B C) C B C B C A (B C) C A (B C) A B A (B C) C A (B C) (A B) C A (B C) A (B C) (A B) C A (B C) = (A B) C ( 1.3.7) A A B C B A (B C) A B A (B C) B C B C C A (B C) C A (B C

13、) A B A (B C) C A (B C) (A B) C (A B) C A (B C) A (B C) = (A B) C ( 1.3.8) (5) A A A B A A (A B) A A A (A B) A (A B) = A A (A B) = A (6) x x A (B C) x A x B C x B x A x B x A B 16 x C x A x C x A C x (A B) (A C) A (B C) (A B) (A C) A A B B C A B A (B C) A A C B C A C A (B C) A B A (B C) A C A (B C)

14、(A B) (A B) A (B C) A (B C) = (A B) (A C) ( 1.3.9) A (B C) = (A B) (A C) ( 1.3.10) (A B) (A C) = (A B) A) (A B) C) = (A (A B) (C (A B) = A (C (A B) = A (C A) (C B) = (A (A C) (B C) = A (B C) A A A B C B C B C B C A (B C) = (A B) C A B 1.3.7 17 A A A B C B C B C B C A (B C) = (A B) C A B 1.3.8 A A A

15、A B C B C B C B C B C A (B C) = A B A C ( A B) (A C) 1.3.9 A A A A B C B C B C B C B C A (B C) = A B A C ( A B) (A C) 1.3.10 18 1.3 1.3.1 A B A B (1) A = B = (2) A = 1, 2, 3 B = 3, 4, 5 (3) A = x | x Z x 0 (4) A = x | x R x 0 B = x | x R x 1 (5) A = ( 2, 0 (1, 4 B = ( 1, 2 (3, 5 1.3.2 a, c, b, d R a c b d (a, b) (c, d) = (c, b) (a, b) (c, d) = (a, d) 1.3.3 1.3.9 (1) A (B C) (A B) C (2) (A B) C A (B C) (3) A (A B) = A 1.3.4 A (B C) = (A B) (A C) A (B C) = (A B) (A C) A (B C) = (A B) (A C) 1.3.5 (1) B A A B = B (2) B A A B = A 1.3.6 A B = A C A B = A C B = C 1.3.7 A B = A B