α = 2 2 α 2 = ( 2) 2 = 2 x = α, y = 2 x, y X 0, X 1.X 2,... x 0 X 0, x 1 X 1, x 2 X 2.. Zorn A, B A B A B A B A B B A A B N 2

1. 2. 3. 4. 5. 6. 7. 8. N Z 9. Z Q 10. Q R 2 1. 2. 3. 4. Zorn 5. 6. 7. 8. 9. x x x y x, y α = 2 2 α x = y = 2 1

α = 2 2 α 2 = ( 2) 2 = 2 x = α, y = 2 x, y X 0, X 1.X 2,... x 0 X 0, x 1 X 1, x 2 X 2.. Zorn A, B A B A B A B A B B A A B N 2

1 R, A, C B,... a, b, x,... 6 1. N Z Q R C. 2. {x N : 0 x 5} = {0, 1, 2, 3, 4, 5}. 3. {0, 1} {x R : x 3 x = 0}. 1 7 a A a A a / A a A A = B A B x(x A x B) x... x... x...... x A B A B x(x A x B) {x A : C(x)} A x x... p q C(x) p q p q A p q p q {a, b} a b {a}, {a 1,..., a n } A B A B A 1 A n A B A B A 1 A n 2 1. 0 {0, 1}. 2. 2 / {0, 1}. 3. {0, 1}. x, y R ( x < y f(x) f(y) ) 3 C = {A i : i I} 4 1. A i C i I A i 2. A i i I A i C 1. {a, b} {c, d} = {a, b, c, d}. 2. {0, 1, 2} {1, 2, 3} = {1, 2}. 3. C = {{0, 1}, {1, 2}, {3}} C = {0, 1} {1, 2} {3} = {0, 1, 2, 3}. 8 0, Z, Q, 5 N 1. f : R R x, y R ( x < y f(x) < f(y) ) 2. f : R R 3. 0 = 1 2 = 3 0 = 1 9 C = {A n : n N} C = {x : n N(x A n )}, C = {x : n N(x A n )}. 3

10 1.,, a f : A B b f : A B c f : A B d f : R R a A B = A B c e f : R R b A B = A B = A f a R f : R R f(a) g a R f : R R f(a) h f : R R a i f : R R 2. C n {A B : A, B C n } {A B : A, B 3. C n } (n N) C = a P Q n N C n P Q a F N k F A k C b x = 1 x + 1 = 2 b F 0,..., F n x 2 2 = 1 3 = 2 A k A k C c P Q P k F 1 k F n Q Q P 4. A 5. A = A 6. { } = 7. = 8. C = {{0, 1}, {1, 2}, {1, 3}} C C 9. {a} = {b} a = b 10. A (B C) = (A B) (A C) 11. A (B C) = (A B) (A C) 12. A B A B = B 13. A C B C A B C 14. (A B) C = A (B C) C A 15. X A X A c A c = {x X : x / A} a A B =, A c B = B = b (A B) c = A c B c c (A B) c = A c B c 16. A B = {a A : a / B} 17. A + B = (A B) (B A) a A + B = B + A, A + = A, A + A = b A + (B + C) = (A + B) + C 18. C = {A i : i N} C 0 = C, C n+1 = c C (b) 4

2 x = u y = x = u = v x y {{x}, {x, y}} = {{u}, {u, v}} 1 {x} {u} x = u { } = {x, y} = {u, v} y u v y = u { } y = u = x y x C = {{0, 1}, {1, 2}} C = {0, 1} y = v {1, 2} = {0, 1, 2} 18 1. a A, b B (a, b) 1. {x A : x = x} = A, {x A : x P(P(A B). x} =. 2. 16 2. B = {x A : x / x} B / A. 17 11 A A P(A) A 12 X P(A) X A 19 A, B 13 A = {0, 1} A A B = {(a, b) : a A, b B}, {0}, {1}, {0, 1} P(A) = {, {0}, {1}, {0, 1}}. A B A A A 2 A n+1 = A n A 14 A n P(A) 2 n A = {a 1,..., a n } 20 (A B) C A (B C) X P(A) a k (k = 1,..., n) X k a k k = 1,...n 2 n P(A) 2 n ((a, b), c) (a, (b, c)) (a, b, c) (A B) 15 x y (x, y) C A (B C) A B C (x, y) = (u, v) x = u y = v 16 (x, y) = {{x}, {x, y}} 17 (x, y) = (u, v) x = u y = v x = y x y x = y (x, y) = {{x}, {x, x}} = {{x}} (u, v) = {{u}, {u, v}} u = v (u, v) = {{u}} ((a, b), c) (a, (b, c)). 5

21 1. {{ }} = { } 2. A B A B A, B 3. A 0 = A n (n = 0, 1, 2,...) A n+1 = A n {A n } a A 1, A 2, A 3 b n m A n A m 4. P( ) = { } 5. P({ }) 6. P({a}) 7. P({a, b}) 8. P({a, b, c}) 9. A B P(A) P(B) 10. P(A) P(B) = P(A B) 11. P(A) P(B) = P(A B) 12. A B = P(A) P(B) 13. X P(A) X = 14. P(A) = A 15. {1, 2, 3} {2, 3} 16. A n A 2 17. A = 18. a, b (A {a}) (B {b}) = 19. B C = (A B) (A C) = 20. (A B) X = (A X) (B X) 21. (A B) (X Y ) = (A X) (B Y ) 22. A = B = A B = 23. A X, B Y A B X Y 24. A B 25. A 0 P(A), B 0 P(B) A 0 B 0 P(A B) 26. P(A B) A 0 B 0 A, B 27. A 0 =, A n+1 = P(A n ) (n = 0, 1, 2,...) A n 6

3 A, B P(A) = {X : X A}. 26 1. X = {(x, y) X 2 : x = y} 2. Z E men m n 3 E A B = {(a, b) : a A, b B}. 27 E X x (a, b) a, b X {y X : xey} E x [x] E x/e {[x] E : x X} X/E 22 R < O = {(a, b) R 2 : a < b} R 2 2. 1/ O R 2 3. a b a b a < b a < b (a, b) O 29 C x y x = y 1/ 1 23 X X R X 2 (x, y) R 30 E X xry [x] E = [y] E xey. 24 1. X 2 R = R 31 C X 3 2. R = X 2 X A C A, X R A, B C, A B A B =, C = X 32 f : A B R f b R A b = {a A : f(a) = b} C = {A b : b R} A 25 X E X 2 X 33 X 1. E X X/E X 1. xex ( x X) 2. xey yex ( x, y X). 3. xey, yez xez ( x, y, z X). 28 1. Z a b a b 2. C X E xey ( A C)[x, y A] E X 7

34 1. A = {0, 1, 2, 3} 2 E E = {(0, 0), (1, 1), (2, 2), (3, 3), (0, 1), (1, 0)} a E A b C = A/E C 11. E X E 2. f : N N N a b f(a) = f(b) C X N F E = F 3. a b f(a) f(b) 12. X n 4. A Z Z E xey x y A E Z S = {(x, y) R 2 : (x 1, y + 2) R} a A = Z b A = c A = {0} d A = {1} e A = mz (m ) f A = g A = {x Z : 10 x 10}. h A = N. 5. E, F A 2 A E F A 6. E F 7. A N 2 (x, x) A (x, y) A (y, x) A x, y N x 0 = x, x n = y, (x i, x i+1 ) A (i = 0,..., n 1) x 0,..., x n N x y N 8. E, F A 2 A E F 9. X n E X X/E n [x] E 10. X = R 2 {(0, 0)} 0 X x y ( λ 0)( x = λ y) C = X/E 13. X Y 14. R R 2 S R x 1 y 2 8

4 A 3. a b, b a a = b. A A R A 2 A (A, ) (a, b) R arb 39 X = P({0, 1}) A E A C = {X i : i I} A (1) {0, 1} X i (2)i j X i X j =, (3) A = i I X i {0} {1} E A a A [a] E = {x A : xea} a E A/E A 4.1 (order) (preorder) 35 A O 1. a O a ( a A) 2. a O b, b O c a O c ( a, b, c A) A/ = {[a] : a A} O 36 A [a] [b] a b a b a b A/ 37 A a b, b a (well- a = b definedness) 38 A A 4.2 40 A A a b (a b b a) 41 A a b (a b b a) [a] [b], [b] [b] [a] = [b]. 9

42 1. X a b a b X 2. X = {0, 1, 2} O = {(0, 0), (1, 1), (2, 2), ((0, 1), (1, 2)} O X U 1 V U V, 3. X = {0, 1, 2} O = {(0, 0), (1, 1), (2, 2), ((0, 1), (1, 3)} O X (A, 1 ) (A, 2 ) 4. R[X] R f(x) g(x) f(x) g(x) 5. N + = N {0} m n n m 6. A = P(X) A X = {0, 1, 2} 7. C α β α β 8. p N + m n p m n 9. A = P(N) B C B \ C 10. N 2 (a, b) (c, d) (3a + 1)3 d (3c + 1)3 b a N 2 b N 2 {(a i, b i ) : i N}, i N (a i+1, b i+1 ) (a i, b i ) u v u v u v 11. (A, A ) (B, B ) f : A B x A y f(x) B f(y) ( x, y A). A = P(X) U 2 V V U. 10

5 F : a b F 1 : b a 43 F A B 49 A B F : X Y, A X, B Y (*) a A (a, b) F 44 F = {(a, b) R 2 : 2a + b = 0} R R a R 2a + b = 0 (a, b) F b R F : R R 45 F : A B a A b B i.e., F (a) = b, F : a b 46 F = {(x, y) R 2 : y = x 2 } 1. F R R F : R R 2. F 0 S = {r R : r 0} F : R S 48 F : A B 1. F ranf = B. 2. F x x F (x) F (x ) ( x, x A). 3. F F 4. F : A B F 1 = {(b, a) B A : (a, b) A B} F b B 1. F (A) = {F (a) : a A} A F F : A B a A 2. F 1 (B) = {a X : F (a) B} B F (a, b) F b B F (a) 50 1. F : X Y ranf = F (X). 2. A 1 A 2 X F (A 1 ) F (A 2 ) Y. 3. B 1 B 2 Y F 1 (B 1 ) F 1 (B 2 ) X. 51 F : R R F : x x 3 x domf = R, ranf = R, F ({1, 2}) = {0, 7}, F 1 ({0}) = { 1, 0, 1} 52 1. A = {, { }} F : A A F ( ) = { }, F ({ }) = F ({ }) { } F (A) F [A] F A 2. F 1 (B) F 1 B F F 1 3. F R R F : 53 F : A B, G : B C F G R R G F G F (a) = G(F (a)) 4. F : 2 4 47 F : A B G F = {(a, c) A C : (a, b) F, (b, c) G( b B)} 1. A F domf 54 2. {b B : b = F (a) ( a A)} F ranf ranf = {F (a) : a A} 11

55 1. F = {(a, b) R 2 : a + b = 1} R R 2. F = {(a, b, c) R 3 60 f : X X X 0 = X, : a + b + c = 0} X F : R 2 n+1 = f(x n ) (n N) R {X 3. F = {(a, b) R 2 : a 2 n } n N = b} R R 4. F = {(a, b) R 2 : a = b 2 } R R 61 1. f : X Y g : Y 56 F : R R 1. F (x) = x 2, 2. F (x) = x 3 + 1, 3. F (x) = sin x + cos x. 57 1. F : R R, x x + 1 2. G : R R, x x 2 3. H : R S, x x 2 S = {r R : r 0} 4. K : S S, x x 2. 58 F : R 2 R 2 F (x, y) = (x + y, xy) ranf R 2 59 f : A B 1. X Y A f(x) f(y ) B 2. C B f(f 1 (C)) C 3. f C B f(f 1 (C)) = C 4. X A A f 1 (f(a)) 5. f A = f 1 (f(a)) 6. X 1, X 2 A f(x 1 X 2 ) = f(x 1 ) f(x 2 ) 7. C 1, C 2 B f 1 (C 1 ) f 1 (C 2 ) = f 1 (C 1 C 2 ) 8. f 1 (C 1 ) f 1 (C 2 ) = f 1 (C 1 C 2 ) P(X) g(y) = f 1 ({y}) g X g 2. f : X Y a f b f(x A) Y f(a) A X 12

6 F : X Y 1. A X F (A) = {F (a) : a A}. 2. B X F 1 (B) = {a A : F (a) B}. 3. domf = X, ranf = F (x). 62 h (g f) = (h g) f 66 X Y Y X 67 1. Y = { } Y = 2. X X = 3. X m, Y n Y X 4. P(X) {0, 1} X A χ A 1. X Y F : X Y, x x X Y (inclusion map) 2. X X X (identity map) id X 68 {X i : i I} 3. F : X Y, X 0 X F I X i X 0 {(x, y) F : i I x X 0 } F X 0 F X 0 : X 0 Y X i (*) f : I i I X i 4. F : X Y X, (x, y) x (*) i I f(i) X i. G : X Y Y, 69 X 0 = {1, 2}, X 1 = {3, 4} i=0,1 (x, y) y X i 5. A X χ A : X {0, 1} A characteristic function 2. g(0) = 1, g(1) = 4, { 3. h(0) = 2, h(1) = 3, 1 x A χ A (x) = 0 x / A. 4. k(0) = 2, k(1) = 4. X 0 X 1 X 0 X 1 1. f(0) = 1, f(1) = 3, 63 1. ran(f X 0 ) = F (X 0 ) 70 1. X i = i I 2. i I X i = 3. π : X Y X, (x, y) x 2. {X 0, X 1 } Y X 0 X 1 i {0,1} X i 4. A X χ A 3. X i X i.e., X i = X ( i I) i I X i = X I 5. χ A B (x) = χ A (x)χ B (x), χ A c(x) = 1 1. X k = i I X i χ A (x). f i I X i f(k) X k 64 f : X Y, g : Y X g f X k = X (i) f (ii) 2. i {0,1} X i f (f(0), f(1)) X 0 X 1 g 3. 65 f : X Y, g : Y Z, h : Z W 13

71 1. F : X Y, X 0 X 1 X (F X 1 ) X 0 = F X 0 2. A = {0, 1, 2} N χ A : X {0, 1} a χ A (0), χ A (1), χ A (2), χ A (3) b {(x, y) N 2 : χ A (x)χ A (y) = 1} c {(x, x+1) N 2 : χ A (x)χ A (x+1) = 1} 3. F (x) = χ A (x)χ A c(x) 0 4. F (x) = χ A (x) + χ A c(x) 1 5. A, B X F (x) = max{χ A (x), χ B (x)} F : X {0, 1} F A B 6. X 0 = {0, 1}, X 1 = {2, 3} i=0,1 X i 7. i = 0, 1, 2 X i n i i {0,1,2} X i 8. X Y Z (X Y ) Z 9. A B = a X A X B = b X A X B X A B 10. {a i } i=0 RN 14

7 a b, b a a = b. A (a b a b, b a) 76 1. (R, ) A = (0, 1] 0 1 max A = 1, min A sup A = 1, inf A = 0. 1 A 0 A 2. (Q, ) B = {x Q : 2 < x 1} max B = sup B = 1 min A, A/ inf A 2 Q 2 B 7.1 7.2 72 A (*) a, b A a b b a 73 R 77 X X 74 (X, ) A X, a X 1. a A a A b A, b a A max A 2. a A a A b A, a b A min A 3. a A b A, b a A (1) (X, ) sup A a < b a b a b 4. a A b A, a b A (X, <) a b a < b inf A a = b (X, ) < a b, b a a, b X a, b, c A a b c a c. 78 (X, <) (2) (1) (2) 5. A A 79 75 1. 2. 80 (X, <) < a < b a < c < b 3. c X 15

81 1. a = sup A, a b b A 2. R I = [0, 1) max I, min I, sup I, inf I 3. R X = {1 1/n : n = 1, 2,...} 4. X = {a} P(X) 5. X P(X) 6. Q A Q 7. Q A, B Q A B( ) A, B 8. A R 9. (N, <) < 10. (Q, <) 11. (X, <) 12. 0 1 X 01100 5 X X x y y x 01 01 < 010 < 0100 a (X, ) b a X X a = {x X : x a} (X a, ) 82 1. X X a b a b a = b 2. X X a b a b a b 16 3.

8 N = {0, 1, 2,...} 5. N σ(n) Z, Q R 6. Z 7. Z 0. N 2 = {(m, n) : m, n N} 1. N 2 (m, n) (m 1, n 1 ) m + n 1 = n + m 1 [(0, 0)] = σ(0) 2. Z = N 2 / = {[(m, n)] : (m, n) N 2 } 10. Z a 1 = 1 a = a a Z [(m, n)] (m, n) 11. Z a Z a + b = 3. Z b + a = 0 b Z [(m, n)]+[(m 1, n 1 )] = [(m+m 1, n+n 1 )] well-defined [(m, n)] [(m 1, n 1 )] = [(m m 1 +n n 1, m 4. σ σ(m + n) = σ(m) + σ(n) σ(m n) = σ(m) σ(n) 8. Z 9. Z a + 0 = 0 + a = a a Z 0 = 9 n 1 + n m 1 )] welldefined 4. σ : N Z σ(n) = [(n, 0)] 0. A = {(a, b) : a Z, b Z {0}} σ(m + n) = σ(m) + σ(n) 1. A + N + Z (m, n) (m 1, n 1 ) m n 1 = n m 1 σ(m n) = σ(m) σ(n) N Z 2. Q = A/ = {[(m, n)] : (m, n) A} 3. Q 5. N σ(n) Z N [(m, n)]+[(m 1, n 1 )] = [(m n 1 +n m 1, n Z n 1 )] well-defined 83 1. N 2 [(m, n)] [(m 1, n 1 )] = [(m m 1, n n 1 )] well-defined 2. [(m, n)] = [(m 1, n 1 )] m + n 1 = n + m 1 4. τ : Z Q τ(n) = [(n, 1)] 3. Z well-defined τ(m + n) = τ(m) + τ(n) + N + Z Z well-defined τ(m n) = τ(m) τ(n) 17

N Z 2. R well-defined 5. Z τ(z) Q Z 3. R Q 4. 5. = 2 R 84 Q a < b ( c)a < c < b x 2 10 85 Q {a n } n=0 p Q +, n N, m 1, m 2 N m 1, m 2 n a m1 a m2 < p. 86 {a n } n=0 C 1. C {a n } n=0 {b n } n=0 p Q + n N m 1, m 2 N (m 1, m 2 n a m1 b m2 < p). 2. R = C/ = {[{a n } n=0] : {a n } n=0 C}. 3. R [{a n } n=0] + [{b n } n=0] = [{a n + b n } n=0] [{a n } n=0] [{b n } n=0] = [{a n b n } n=0] [{a n } n=0] [{b n } n=0] p Q +, m N, n N, [n m a n < b n + p] 4. σ : Q R a Q a {a} n=0 σ σ(a + b) = σ(a) + σ(b) σ(a b) = σ(a) σ(b) a b σ(a) σ(b) 87 1. 18