untitled

1.1 1.1.1 1.1.2 A, B, C, X, Y, Z 1 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 φ} {x x Z x 0} {x x 2 3x+2 = 0} {,, } {, } {, } {, } {1, 2} 2

{1, 2,, } {,,, } {1, {1, 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) 3 {, } = {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) (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} 4

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 A A A 5 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 φ} A φ 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 6

n N m n N n m<n m N n m N m x (x N m x N n ) (x N n x N m ) N n N m 1.2.5 a, b R (a, b) = {x x R a<x<b} (a, b] = {x x R a<x b} R (1) b a a<x b x (a, b] = (2) a c d b (c, d] (a, b] x x (c, d] c<x d a c c<x a<x d b x d x b a<x b (a, b] x (a, b] x x (c, d] x (a, b] (c, d] (a, b] 1.2.6 Z + = {x x Z x>0} Q + = {x x Q x>0} R + = {x x R x>0} 7 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) 8 A B A A = B 1.2.3 A A B C C 1.2.4

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 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}} 1.3 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 9 10 1.3.1 A B 1.3.2 A B =

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 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 (a, b] (c, d] = (a, d] x x (a, b] (c, d] (a<x b c<x d) c<x b x (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 φ} C = {x x A x φ} B C = B C = A ( 1.3.4) 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] 11 12 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

(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 C B x B x A B C A B ( 1.3.6) A A B C C 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 B 13 (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 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 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 14

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) 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 15 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) (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) 16 A A A B C B C B C B C A (B C) = (A B) C A B 1.3.7

1.3 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 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 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} B = {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 17 18

1.4 1.4.1 A B A B A B A B = {x x A x B} ( 1.4.1) x A B (x A x B) A B x A B x A x B x A B (x A x B) A 1.4.1 A B 1.4.2 =, = 1.4.3 {1, 2, 3} {2, 3, 4} = {1} {2, 3, 4} {1, 2, 3} = {4} N N n = {x x N x n} 1.4.4 A, B, C (1) A B A B 19 (2) B A B C A C (3) C B A B A C (4) (A B) B = (5) (A B) B = A B (1) (2) x x B C x B x C B A x A x A x C x A C B C A C ( 1.4.2) (3) x x A B x A x B C B x C x A x C x A C A B A C ( 1.4.3) (4) x x A B x A x B x B (A B) B = (5) x x A B x A x B x B x (A B) B x B x A 20

x A B x (A B) B A B (A B) B A B A B B (A B) B A B A B B C A C 1.4.2 1.4.3 1.4.5 De-Morgan A, B, C (1) A (B C) = (A B) (A C) (2) A (B C) = (A B) (A C) (1) x x (A B) (A C) x A B x A C x A x B x C x B x C x B C x A x B C x A (B C) (A B) (A C) A (B C) B B C A (B C) A B C B C A (B C) A C A (B C) A B A (B C) A C A (B C) (A B) (A C) ( 1.4.4) (2) x x A (B C) x A x B C x B x C x B x A x B x A B x C x A x C x A C x (A B) (A C) A (B C) (A B) (A C) B C B A B A (B C) B C C A C A (B C) A B A (B C) A C A (B C) (A B) (A C) A (B C)( 1.4.5) 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) 1.4.4 21 22

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) 1.4 1.4.5 1.4.1 A B B A (1) A = B = (2) A = {1, 2, 3} B = {2, 3, 4, 5} (3) A = {x x Z x<0} B = Z (4) A = {x x R x>0} B = {x x R x 1} (5) A = ( 2, 0] (1, 3] B = ( 1, 2] 1.4.2 a c b d(a, b] (c, d] = (a, c] (c, d] (a, b] = (b, d] 1.4.3 (1) C B A C B A B (2) (A B) (A B) = A (3) A B = A B = A (4) C B(A B) C = (5) A (A B) = A B 1.4.4 (1) B A B A = (2) A B A B B A 23 24 1.5 1.5.1 Σ, Γ, Φ, Ψ 1.5.2 { } {} 1.5.3 Γ 1 = {( 1 1, ] n N} Γ (n+ 1 ) (n+ 1) 2 = {(x, x+1] x Z} 1.5.4 Γ(N) = {N n n N} Γ(Q) = {Q a a R} N n Q a 1.2.4 1.2.2 1.5.5 A n = n Γ = {A n n N n 3} A (B C) (A B) C A (B C) (A B) C

1.5.6 Γ Γ Γ Γ Γ x Γ( X Γ x X) Γ = {x X Γ x X} x Γ x Γ x Γ x Γ x Γ( X Γ x X) 1.5.7 Γ Γ Γ Γ Γ x Γ( X Γ x X) Γ = {x X Γ x X } x Γ x Γ x Γ x Γ x Γ( X Γ x X) Γ = {A, B} Γ = A B Γ = A B 1.5.8 Γ = {{0, 1, 2}, {1, 2, 3}, {2, 3, 4}} Γ = {2} 25 Γ = {0, 1, 2, 3, 4} 1.5.9 1.5.5 Γ Γ = 1.5.3 Γ 1 Γ 2 Γ 1 = {0} Γ 2 = R X Γ Γ X X ΓΓ Γ Γ Γ 1.3.7(1)(2) 1.3.8 1.5.10 Γ Σ (1) X Γ Y Σ X Y Γ Σ (2) X Γ Y Σ Y X Σ Γ (1) x x Γ X Γ x X X Y Σ X Y x X X Y x Y Y Σ x Y x Σ Γ Σ (2) x x Σ Y Σ x Y X Γ Y Σ Y X x Y Y X x X X Γ x X x Γ Σ Γ 1.5.10 Σ = {A} Σ = Σ = A 26

(1) X Γ X A Γ A (2) X Γ A X A Γ 1.3.7(3)(4) I i I A i {A i i I} I Γ I = Γ i I A i = iγ = {A i i I} I {A i i I} 1.5.3 Γ 1 Γ 2 N Z 1.5.4 Γ(N) Γ(Q) N R Γ = {A i i I} i IA i i IA i Γ Γ x i IA i ( i I x A i ) x i IA i ( i I x A i ) I = N N n i IA i i IA i i na i i na i I = N = N N 0 i 0A i i 0A i I = N n+1 i IA i i IA i i na i i na i i<n+1a i i<n+1a i A 0 A n A 0 A n De-Morgan 1.5.11 (1) ( i IA i ) ( j JB j ) = {A i B j i I j J} (2) ( i IA i ) ( j JB j ) = {A i B j i I j J} (3) ( i IA i ) ( j JB j ) = {A i B j i I j J} (4) ( i IA i ) ( j JB j ) = {A i B j i I j J} (5) A i IA i = {A A i i I} (6) A i IA i = {A A i i I} (1) x x {A i B j i I j J} i I j J x A i B j x A i x i IA i x B j x j JB j x ( i IA i ) ( j JB j ) {A i B j i I j J} ( i IA i ) ( j JB j ) A i {A i i I} A i B j {A i B j i I j J} A i A i B j 1.5.10(3) {A i i I} {A i B j i I j J} i IA i {A i B j i I j J} j JB j {A i B j i I j J} ( i IA i ) ( j JB j ) = {A i B j i I j J} (2) x x ( i IA i ) ( j JB j ) x i IA i x j JB j i I j J x A i x B j A i B j {A i B j i I j J} x A i B j x {A i B j i I j J} ( i IA i ) ( j JB j ) {A i B j i I j J} A i {A i i I} A i B j {A i B j i I j J} A i B j A i 1.5.10(4) {A i B j i I j J} {A i i I} 27 28

{A i B j i I j J} i IA i {A i B j i I j J} j JB j {A i B j i I j J} ( i IA i ) ( j JB j ) (3) x x ( i IA i ) ( j JB j ) x i IA i x j JB j i I j J x A i x B j A i B j {A i B j i I j J} x A i B j x {A i B j i I j J} ( i IA i ) ( j JB j ) {A i B j i I j J} A i B j {A i B j i I j J} A i {A i i I} A i B j A i 1.5.10(3) {A i B j i I j J} {A i i I} {A i B j i I j J} i IA i {A i B j i I j J} j JB j {A i B j i I j J} ( i IA i ) ( j JB j ) (4) x x ( i IA i ) ( j JB j ) x i IA i x j JB j i I j J x A i x B j A i B j {A i B j i I j J} x A i B j x {A i B j i I j J} 29 {A i B j i I j J} ( i IA i ) ( j JB j ) A i B j {A i B j i I j J} A i {A i i I} A i A i B j 1.5.10(4) {A i i I} {A i B j i I j J} i IA i {A i B j i I j J} j JB j {A i B j i I j J} ( i IA i ) ( j JB j ) {A i B j i I j J} (5) x x A i IA i x A x i IA i x i IA i i I x A i i I x A A i x {A A i i I} A i IA i {A A i i I} i I i IA i A i i I A A i A i IA i {A A i i I} A i IA i (6) x x {A A i i I} i I x A A i x A i I x A i x A x i IA 30

x A i IA i {A A i i I} A i IA i i I A i i IA i i I A i IA i A A i A i IA i = {A A i i I} 1.5.12 Γ X, Y Γ X Y Y X ΓΓ 1.5.4 Γ(N) Γ(Q) ( 1.2.4 1.2.2) 1.5.3 Γ 1 1.5.13 Γ Γ X, Y Γ X Y X Y = ΓΓ Γ Γ = {A 1,, A n } A 1,, A n 1.5.2 1.5.3 Γ 2 1.5.5 Γ 1.5.14 Γ A Γ A A P(A) Γ A Γ P(A) 1.5.3 Γ 1 Γ 2 R 1.5.4 Γ(N) N Γ(Q) Q Γ B B AΓ A Γ Γ Γ Γ A A a, b, c, x, y, z A, B, C, X, Y, Z Σ, Γ, Φ, Ψ 31 1.5 1.5.1 i I B i A i i I B i i I A i i I B i i I A i 1.5.2 Γ Γ (1) Γ = {{0,1,2,4}, {1,3,4,5}, {0,2,4,5}} (2) Γ = {( x, x] x R x>0} (3) Γ = {A a a R a>0} A a = {<x, y> <x, y> R R a+y x a+y} 1.5.3 (1) ( i I A i ) ( i IB i ) = i I(A i B i ) (2) ( i I A i ) ( i IB i ) = i I(A i B i ) (3) A ( i I A i ) = i I(A A i ) (4) A ( i I A i ) = i I(A A i ) 1.5.4 (1) {} (2) { } (3) {(0, x] x N x>0} (4) {(0,3), (1,4), (2,5)} (5) {{0,3}, {1,4}, {2,5}} 1.5.5 Γ x, y Γ X Γ x, y X 1.5.6 Γ= {A i i N} n N B n = i n A i C n = i n A i (1) {B i i N} {C i i N} (2) i 0A i = i 0B i i 0A i = i 0C i 1.5.7 32

1.6 a, b {a, b}a b 1.6.1 a, b <a, b> <a 1, b 1 > = <a 2, b 2 > a 1 = a 2 b 1 = b 2 a b <a, b> <b, a> <a, b>a b 1.6.2 A, B {<x, y> x A y B} A B A B ( 1.6.1) <x, y> A B x A y B <x, y> A B x A y B A = A = A B = B A A B C (A B) C A (B C) (A B) C = A (B C) A B, A B A B A B A B A B 1.6.3 B 1 A 1 B 2 A 2 B 1 B 2 A 1 A 2 <x, y> <x, y> B 1 B 2 x B 1 y B 2 B 1 A 1 B 2 A 2 x A 1 y A 2 <x, y> A 1 A 2 B 1 B 2 A 1 A 2 ( 1.6.1) A 1 B1 B 1 B 2 B 2 A 2 1.6.1 A 1 A 2 A A B B 1.6.1 A B A B 33 1.6.4 A, B, C (1) A (B C) = (A B) (A C) (B C) A = (B A) (C A) (2) A (B C) = (A B) (A C) (B C) A = (B A) (C A) (3) A (B C) = (A B) (A C) 34

(B C) A = (B A) (C A) (1) <x, y> <x, y> A (B C) x A y B C x A (y B y C) y B x A y B <x, y> A B y C x A y C <x, y> A C <x, y> (A B) (A C) A (B C) (A B) (A C) A A B B C C B C A B A (B C) A C A (B C) (A B) (A C) A (B C) ( 1.6.2) (B C) A = (B A) (C A) (2) <x, y> <x, y> (A B) (A C) <x, y> A B <x, y> A C (x A y B) (x A y C) y B y C y B C x A y B C <x, y> A (B C) (A B) (A C) A (B C) A A B C B B C C A (B C) A B A (B C) A C A (B C) (A B) (A C)( 1.6.3) (B C) A = (B A) (C A) (3) <x, y> <x, y> A (B C) x A y B C x A y B y C x A y B <x, y> A B y C <x, y> A C <x, y> (A B) (A C) A (B C) (A B) (A C) <x, y> <x, y> (A B) (A C) <x, y> A B <x, y> A C <x, y> A B x A y B x A <x, y> A C y C x A y B y C y B y C y B C <x, y> A (B C) (A B) (A C) A (B C) ( 1.6.4) (B C) A = (B A) (C A) A A B A A C A A (B C) B C B C 1.6.2 35 36

A A A B A A C A ( B C) B C B C 1.6.3 A A A B A A C A (B C) B C B C 1.6.4 1.6.4 1.6.5 n n 2 n a 1,, a n n <a 1,, a n > n <a 1,, a n > = <b 1,, b n > 1 i n a i = b i n n 1.6.6 n n 2 A 1,, A n n {<x 1,, x n > 1 i n x i A i } A 1,, A n A 1 A n 1 i n A i i 1.6.7 A 1 A n 37 1 i n A i = A A n A n A n = {<x 1,, x n > 1 i n x i A} A 1 = A n 1.6.8 A 1,, A n (1) 1 i n A i = A 1 A n = (2) 1 i n B i A i B 1 B n A 1 A n (3) 1 i n i 1.6 1.6.1 A B A B A B B A 1.6.2 1.6.4 (1) (B C) A = (B A) (C A) (2) (B C) A = (B A) (C A) (3) (B C) A = (B A) (C A) 1.6.3 (1) A ( i IB i ) = i I(A B i ) (2) A ( i IB i ) = i I(A B i ) 1.6.4 {B i i I} ( i IB i ) ( i IB i ) = i I(B i B i ) 1.6.5 1 i n A i A 1 A n 38