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ゆずさこうい
5 years ago
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3 i JR

4 ii A B B A

5 iii

6 iv

7 White Orange Red Blue 4 X Y X Y X Y X Y (1) a b c d 1 Blue 2 Orange 3 White Orange 4 Red White

8 2 a-blue b-red c-white d-orange 1. 2 Orange a d 2. Orange a 3. 3 White b 4. 4 Red c 5. Blue d Orange a 8. Orange d 9. 3 White c Red b 11. Blue a Orange a d 7 Orange a Orange d A B A B R1 A A R2 A B A B R3 A B B A R4 A B B A R5 A B A B R6 1 A B

9 3 A B R7 2 A B A B R8 A B B A R9 A A R10 A A A A R11 A B A B C C 1.1. (1) a b c d 1 Red Orange 2 Blue Orange Red 3 Blue 4 Red Blue (2)

10 4 a b c d 1 Orange Red 2 White Red 3 White Blue 4 Blue Red White 5 Orange Blue (3) a b c d 1 Red Blue 2 Orange Blue White 3 Red Orange 4 White Red 5 Orange Blue 1.2.

11 5 a b c d a b c d 1. a b c d a b c d 6. a b c d 7. a 8. 2 a 9. a b c d a 10. a b c d a 11. a b c d a a 14. b b 16. a b c d b 17. a b c d b 18. a b c d b

12 b 21. c c 23. a b c d c 24. a b c d c 25. a b c d c c 28. d R12 C C R13 x C D C D D x x C x R14 C C R15 C C R16 1 C C R17 2 C C R18 M P S M S P

13 7 R19 P M S M S P 2 R15 3 R R = (2) 1.3. Violet White Orange Blue Pink Yellow Green Scarlet Red 9 a b c d e f g h i 1 Pink 2 Blue Blue 3 Violet Pink 4 Red Blue 5 Green White 6 Yellow Red 7 Scarlet White

14 8 a-violet b-red c-scarlet d-pink e-orange f-blue g-green h-yellow i-white ,880 Blue 6 Red a b c 4 Blue d e f 2 Blue f Red b 6 h Yellow Pink 1 Pink b d f h b f h Red Blue Yellow Pink d 3 a Violet 5 7 c Scarlet g Green i White 1.2. (1) a b c d e f g h i 1 Red 2 Pink White 3 Scarlet Blue 4 Yellow 5 Yellow Violet 6 White Scarlet Red 7 Orange Yellow

15 9 (2) a b c d e f g h i 1 White Scarlet 2 Violet 3 Pink 4 Pink Green 5 Scarlet 6 Pink Violet 7 Yellow Blue Violet 8 Blue Red 9 Blue 10 Orange (3) a b c d e f g h i 1 Orange 2 Red Blue

16 10 3 Scarlet Green 4 White 5 Violet 6 Orange Violet 7 White Green Red 8 Pink Red 9 Violet 3 10 Blue Scarlet 11 Yellow Red

17 A B A B B A 1.4. (1) (2) (3) (4)

18 12 (5) (6) (7) (8) (a) (b) (A)-(D) (A): (a) (b) (B): (a) (b) (C): (a) (b) (D): (a) (b) (1) (a) (b) (2) (a) (b) (3) (a) 20 (b) (4) x, y (a) x < y + 1 (b) x y (5) x, y (a) x < y + 1 (b) x y

19 deduction

20 14 (1) 1.6. (1) (2)

21 15 (3) (4) (5) (6) (7) (8) (9) (10) I don t like either tea or coffee. I don t like tea and I don t like coffee either. (11) (12) (13)

22 16 (14) n n n 400 n 1. n n 2. n n 4 n n n (1) (p1) (p2) (p3) (p4) (p5) (c1) (c2) (c3) (c4) (c5) (2) (p1) (p2) (p3) (p4)

23 17 (p5) (p6) (c1) (c2) (c3) (c4) René Descartes *1 *1

18 384- *2 *3 3 4 3 trivia Gottlob Frege 1848-1925 *4 Bertrand Russell,

com/aristotles-moral-philosophy/ *3 41 *4 http://philosophyofscienceportal.

24 *2 * trivia Gottlob Frege *4 Bertrand Russell, *5 *2 *3 41 *4 *5

25 19 R R R R R R R R R R R

27 * = 10/2 0 5 > *1

28 22 a, b, c,... F, G, H, F (a), G(b), H(c),... F (a, b), G(c, d, e),...

29 23 F 2 G 3 1 A A A B A B A B A B A B A B x x x A xa A x xa x x x x x(f (x) G(x)) x x x x(f (x) G(x))

30 24 x(f (x) G(x)) F (a) G(a) *2 2.1 ( ). n n ( ) x, y, z a, b, c F, G, H t, s, u 2.2 ( ). 1. n F t 1,..., t n F (t 1, t 2,..., t n ) 2. A, B x A, (A B), (A B), (A B), xa, xa 3. L ( F (a) G(y)), (F (y) (G(x, y) H(a))) F, F (x), (G(a) ), (H(a) F (a, z), af (a) A, B, C *2

31 25 A B C (A (B C)) ((A B) C) x + y z x + (y z),, A B C A (A ((B C) A)) A B C (A (B C)) A B C B 2. B C A B C A 3. A B C B A C B xc (((A B) C) ((A C) (B C))) 2. (( A B) (( A B) A)) 3. (((A B) A) A) 4. ( (A B) ( A B)) 5. ((A B) ((C A) (C B))) 6. ((A (B C)) ((A B) (A C))) 1 A A A B A B A B B A A B B A xf (x) F (a) a F (x) xa A A A x a

32 26 x(f (x) xg(x)) F (x) xg(x) x a F (a) ag(a) F (a) xg(a) xg(a) x x G(a) F (a) xg(a) x x x 2.3 ( ). xa xa A x x xa xa A x A A F (x, y) G(a, z) x, y, z z( yf (x, y) wg(a, z)) x xf (x) G(x) x F (x) G(x) 2.4 ( ). A x t A x t( )

33 27 A[x/t] x( yf (x, y) zg(y, z))[x/a] x( yf (x, y) zg(y, z)) x x( yf (x, y) zg(y, z))[y/a] x( yf (x, y) zg(a, z)) F (x, y) y xa A[x/a] a A[x/a] xa a Male(x), F emale(x) x x 2 Eq(x, y), P arent(x, y), Couple(x, y), Elder(x, y) x y x y x y x y a, b, c L L P arent(a, b) Male(a) x(p arent(a, x) P arent(x, c)) xcouple(c, x) x(p arent(x, a) P arent(x, c) F emale(c) Elder(c, a)) 1 Nat(x) x 2 Leq(x, y) x y 3 Sum(x, y, z), P rod(x, y, z) x y z x y z L a n n *3 *3 0

34 28 L 4 xp rod(a 4, a 2, x) xp rod(a 9, a 3, x) xp rod(a 12, a 3, x) 7 x( yp rod(a 7, x, y) (Eq(x, a 1 ) Eq(x, a 7 )) 0 x(nat(x) Leq(a 0, x)) 1 0 x(nat(x) Sum(a 0, x, a 1 )) x(nat(x) y(nat(y) Leq(y, x))) x y y x y x(p arent(a, x) y(p arent(a, y) Eq(x, y))) x y y x y (1) Everybody loves somebody (2) Somebody is loved by everybody 2 (2) (1) (1) (2) Love(x, y) x y 2 (1 ) x ylove(x, y) (2 ) y xlove(x, y) 2.4. L

35 x P osreal 11. x, y y 0 x y

36 ( ). U U n U n n U n {, } U n R a 1,..., a n R(a 1,..., a n ) = R(a 1,..., a n ) R a 1,..., a n R(a 1,..., a n ) = R(a 1,..., a n ) 2 2 a b a, b

37 31 L a, b, c,... n m m n D (n, m) D (2, 4), (3, 9), (5, 5) (4, 2), (3, 8) n m (n, m) D D n m k 3 M n m k (n, m, k) M {(1923, 191), (8371, 12918), (38418, 9687), (1293, 6563)} 2.6 ( ). U v L U L n U n U v (U, v) L a v a U v(a) n F v F U n v(f )

38 ( ). (U, v) L (U, v) L U ρ x u ρ(x) ρ (U, v) ρ (x u) x u ρ { u y x ρ (x u) (y) = ρ(y) v ρ t v ρ { v(t) t v ρ (t) = ρ(t) t 2.8 ( ). (U, v) L ρ (U, v) (U, v) ρ 1. F (t 1, t 2,..., t n ) (U, v) ρ v(f )(v ρ (t 1 ), v ρ (t 1 ),..., v ρ (t 1 )) = 2. A (U, v) ρ A (U, v) ρ 3. A B (U, v) ρ A B (U, v) ρ 4. A B (U, v) ρ A B (U, v) ρ 5. A B (U, v) ρ (U, v) ρ A B 6. xa (U, v) ρ u U A (U, v) ρ (x u) 7. xa (U, v) ρ u U A (U, v) ρ (x u) A (U, v) ρ (U, v) ρ A (U, v) ρ A

39 A (U, v) (U, v) ρ, ρ (U, v) ρ A (U, v) ρ A (U, v) A (U, v) A U = {0, 1} U 0 1 v(a) = 0, v(b) = 1 v(f ) m, n U v(f )(m, n) m n U 2 F (a, a) F (a, b) F (b, b) F (b, a) F (b, a) F (a, a) F (a, b) F (b, a) F (a, a) ρ u U (U, v) ρ(x u) F (x, x) (U, v) ρ xf (x, x) U u u u (U, v) ρ(x 1) F (a, x) (U, v) ρ xf (a, x) 0 u u U L 2.10 ( ). A 1,..., A n B A 1,..., A n B A

40 34 A 1,..., A n B A 1,..., A n = B A = A = A = B B = A A B A B D x 1. A 1, A 2,..., A n, A = B A 1, A 2,..., A n = A B 2. A A 3. A B, A = B 4. A B, B C = A C 5. = A (B A) 6. ((A B) A) A 7. A B, B = A 8. (A B) ( B A) 9. A, B = A B 10. A 1 A 2 = A i (i = 1, 2) 11. A B = B A 12. A (A A) 13. A i = A 1 A 2 (i = 1, 2) 14. A B, A C, B C = C 15. A B, A = B 16. A B = B A 17. A (A A) 18. (A B) ( A B) 19. (A B) ( A B) 20. (A (B C)) ((A B) (A C)) 21. (A (B C)) ((A B) (A C)) 22. A (A (A B)) 23. A (A (A B)) 24. ((A B) C) ((A C) (B C)) 25. (A (B C)) ((A B) C)

41 = A = xa 27. A[x/a] = xa 28. xa = A[x/a] 29. x(a B) xa xb 30. x(a B), xa = xb 31. x(a B), xa = xb 32. x(a B) = xa xb 33. x(a D) xa D 34. x(a D) xa D 35. x(a B) xa xb 36. xa xb = x(a B) 37. x(a B), xa = xb 38. x(a D) xa D 39. x(a D) xa D 40. ( xa D) x(a D) 41. ( xa D) x(a D) 2.3 A 1,..., A n B A 1,..., A n B x(f (x) G(x)) xf (x) xg(x) (U, v) ρ (1) (U, v) ρ x(f (x) G(x)) (2) (U, v) ρ xf (x) xg(x) (2) (3) (U, v) ρ xf (x) (4) (U, v) ρ xg(x) (3) U u (5) (U, v) ρ(x u) F (x) (1) (6) (U, v) ρ(x u) F (x) G(x) (7) (U, v) ρ(x u) F (x) (5) (4) (U, v) u (8) (U, v) ρ(x u ) G(x) (1) (9) (U, v) ρ(x u ) F (x) G(x) (10) (U, v) ρ(x u ) G(x) (8) x(f (x) G(x))

42 36 xf (x) xg(x) x(f (x) G(x)) xf (x) xg(x) 2. x(f (x) G(x)) xf (x) xg(x) (U, v) ρ (1) (U, v) ρ x(f (x) G(x)) (2) (U, v) ρ xf (x) xg(x) (2) (3) (U, v) ρ xf (x) (4) (U, v) ρ xg(x) (3) (U, v) u (5) (U, v) ρ(x u) F (x) (1) (U, v) ρ(x u) F (x) Gx) (6) (U, v) ρ(x u) F (x) (7)(U, v) ρ(x u) G(x) (6) (5) (7) (4) (U, v) u (8) (U, v) ρ(x u ) G(x) (1) (9) (U, v) ρ(x u ) F (x) G(x) (10) (U, v) ρ(x u ) F (x) (11) (U, v) ρ(x u ) G(x) (11) (8) (10) (1)(2)(3)(4)(5)(7)(8)(9)(10) (6) (11) (7) (10) U = {u, u } v(f ) = {u }, v(g) = {u} (U, v)

43 ( 5 ) A = B A A = A B A, A = B

44 L E J L. E. J. Brouwer C I C. I. Lewis Graham Priest 1948-

45 x(f x Gx) the + The present king of France is bald F (x) B(x) I(x, y) x(f (x) ( y(f (y) I(x, y)) B(x)))

46 40 x(f (x) ( y(f (y) I(x, y)) B(x))) F (x) x

47 A B A B 3.1

48 42 A B A B B A A B B A A E I O A I I X X X Y M A B A b A B A B

49 43 A B A B A B A b 1. X m 2. Y M 2 Y m X m XYm Xym

50 44 XYm Xym M m M m M m X y 1. X M 2. M Y X m 2 M y

51 45 M m X Y X Y F t, t F t F t

52 ( ). p, q, r, A, B A A B A B A B ( ). v p v p v(p) 3.3 ( ). v v 1. p v v(p) = p v 2. A v A v A v 3. A B v A B v A B v 4. A B v A B v A B v 5. A B v A v B v A B v 3.4 ( ). A 1,..., A n B A 1,..., A n B A A

53 47 A 1,..., A n B A 1,..., A n = B A = A x, y, z x y = ((x y) + x) x y = x y 3 x y = ((x y) + x) + y 4 x = x x + 0 = x 6 x + x = 0 7 x 0 = 0 8 x 1 = x 9 x x = x 10 x (y + z) = (x y) + (x z) 11 x + (y + z) (x + y) + z 12 x + y = y + x 13 x (y z) (x y) z 14 x y = y x 1 A A A A = 4 A (A + 1) = 3 ((A (A + 1)) + A) + (A + 1) = 10 (((A A) + (A 1)) + A) + (A + 1) = 9 ((A + (A 1)) + A) + (A + 1) = 8 ((A + A) + A) + (A + 1) = 11 (((A + A) + A) + A) + 1 = 6 ((0 + A) + A) + 1 = 12 ((A + 0) + A) + 1 = 5 (A + A) + 1 = = = 5 1

54 48 A A x(f x Gx) 3.2 F a Gb 3.2 x(f (x) G(x), ( xf (x) xg(x)), xf (x), F (a), F (a) G(a), F (a), G(a) x(f (x) G(x), ( xf (x) xg(x)), xg(x), G(b), F (b) G(b), F (b), G(b) 3.2

55 49 x(f (x) G(x)) (1) ( xf (x) xg(x)) (2) xf (x) (3) xg(x) (4) F (a) (5) G(b) (8) F (a) G(a) (6) F (b) G(b) (9) F (a) (7) F (b) G(a) G(b) (10) 3.2 x(f x Gx) = xf x xgx x(f (x) G(x)) (1) ( xf (x) xg(x)) (2) xf (x) xg(x) F (a) G(b) F (a) G(a) F (b) G(b) F (a) G(a) F (b) G(b) 3.3 x(f x Gx) = xf x xgx xf (x) a F (a) F (a) xg(x) G(b) G(b)

56 A A 2. A B A B 3. (A B) A B 4. A B A B 5. (A B) A B 6. A B A B 7. (A B) A B 8. xa A[x/a] a a A[x/a] 9. xa A[x/b] b 10. xa A[x/b] b 11. xa A[x/a] a a A[x/a] 12.

57 51 A. A A B. A B (A B). A B A B. A B (A B). A B A B. A B (A B). A B xa... a.... A[x/a] xa. A[x/b]

58 52 xa. A[x/b] xa... a.... A[x/a] x((f (x) G(x)) H(x)) (H(x) ( F (x) G(x))) (1) 2 (H(a) ( F (a) G(a))) (2) x((f (x) G(x)) H(x)) x((f (x) G(x)) H(x)) x(h(x) ( F (x) G(x))) x(h(x) ( F (x) G(x))) (H(a) ( F (a) G(a))) (1) (2) (H(a) ( F (a) G(a))) H(a) ( F (a) G(a)) (3) ( F (a) G(a)) F (a) G(a) (4) x((f (x) G(x)) H(x)) x(h(x) ( F (x) G(x))) (H(a) ( F (a) G(a))) H(a) ( F (a) G(a)) x((f (x) G(x)) H(x)) x(h(x) ( F (x) G(x))) (H(a) ( F (a) G(a))) H(a) ( F (a) G(a)) F (a) G(a) (3) (4) F (a) G(a) (5)

59 53 x((f (x) G(x)) H(x)) x(h(x) ( F (x) G(x))) (H(a) ( F (a) G(a))) H(a) ( F (a) G(a)) F (a) G(a) F (a) G(a) (5) x((f (x) G(x)) H(x)) a (F (a) G(a)) H(a) (6) x((f (x) G(x)) H(x)) a a x((f (x) G(x)) H(x)) a x(h(x) ( F (x) G(x))) (H(a) ( F (a) G(a))) H(a) ( F (a) G(a)) F (a) G(a) F (a) G(a) (F (a) G(a)) H(a) (F (a) G(a)) H(a) (6) (F (a) G(a)) H(a) (F (a) G(a)) H(a) H(a) H(a)

60 54 x((f (x) G(x)) H(x)) a x(h(x) ( F (x) G(x))) (H(a) ( F (a) G(a))) H(a) ( F (a) G(a)) F (a) G(a) F (a) (F (a) G(a)) H(a) (F (a) G(a)) H(a) (7) G(a) (F (a) G(a)) H(a) (F (a) G(a)) H(a) (F (a) G(a)) F (a) G(a) (8) x((f (x) G(x)) H(x)) a x(h(x) ( F (x) G(x))) (H(a) ( F (a) G(a))) H(a) ( F (a) G(a)) F (a) G(a) F (a) (F (a) G(a)) H(a) (F (a) G(a)) H(a) F (a) G(a) (8) G(a) (F (a) G(a)) H(a) (F (a) G(a)) H(a) F (a) G(a) x((f (x) G(x)) H(x)) (H(x) ( F (x) G(x)))

61 55 x(f (x) G(x)), x(h(x) G(x)) x(f (x) (H(x) G(x))) (1) (2) x(f (x) G(x)) x(f (x) G(x)) x(h(x) G(x)) x(h(x) G(x)) x(f (x) (H(x) G(x))) x(f (x) (H(x) G(x))) F (a) G(a) H(b) G(b) (1) (2) F (a) G(a) H(b) G(b) (3) x(f (x) (H(x) G(x))) a (4) x(f (x) G(x)) x(f (x) G(x)) x(h(x) G(x)) x(h(x) G(x)) x(f (x) (H(x) G(x))) x(f (x) (H(x) G(x))) a F (a) G(a) F (a) G(a) H(b) G(b) H(b) G(b) F (a) F (a) G(a) G(a) H(b) H(b) G(b) G(b) (F (a) (H(a) G(a))) (3) (4) (F (a) (H(a) G(a))) F (a) (H(a) G(a)) (5) F (a) (H(a) G(a)) H(a) G(a) (6) G(a)

62 56 x(f (x) G(x)) x(h(x) G(x)) x(f (x) (H(x) G(x))) a F (a) G(a) H(b) G(b) F (a) G(a) H(b) G(b) (F (a) (H(a) G(a))) F (a) (H(a) G(a)) x(f (x) G(x)) x(h(x) G(x)) x(f (x) (H(x) G(x))) a F (a) G(a) H(b) G(b) F (a) G(a) H(b) G(b) (5) (6) (F (a) (H(a) G(a))) F (a) (H(a) G(a)) H(a) G(a) x(f (x) (H(x) G(x))) b F (b) (H(b) G(b)) (8) x(f (x) G(x)) x(h(x) G(x)) x(f (x) (H(x) G(x))) a b F (a) G(a) H(b) G(b) F (a) G(a) H(b) G(b) (F (a) (H(a) G(a))) F (a) (H(a) G(a)) H(a) G(a) (F (b) (H(b) G(b))) (8) (F (b) (H(b) G(b))) (H(b) G(b)) (9)

63 57 x(f (x) G(x)) x(h(x) G(x)) x(f (x) (H(x) G(x))) a b F (a) G(a) H(b) G(b) F (a) G(a) H(b) G(b) (F (a) (H(a) G(a))) F (a) (H(a) G(a)) H(a) G(a) (F (b) (H(b) G(b))) F (b) (H(b) G(b)) H(b) G(b) (9) 1. a 1,..., a n 2. U = {1, 2,..., n} a i (1 i n) v(a i ) = i v 3. P 1,..., P m 4. v P i (1 i m) P i (b 1,..., b k ) v(b 1 ), v(b 2 ),..., v(b k ) a, b G(b), H(b), G(a), F (a) U = {1, 2} v a 1 b 2 F G H (U, v) 1 2 F (a) H(b)

64 58 F (b) H(a) 1 2 G(a), G(b) F (a) G(a) H(b) G(b) x(f (x) G(x)) x(h(x) G(x)) F (b) G(a) ρ (U, v) = ρx 1 F (x) (H(x) G(x)) (U, v) = ρx 2 F (x) (H(x) G(x)) U C U U C U C P U C P C P C P C C C P C P C P C 1. A 1,..., A n = B A 1,..., A n B

65 59 2. A 1,..., A n B A 1,..., A n = B *1 *1

66 60 x yf (x, y) G(a 0 ) yf (a 0, y) F (a 0, a 1 ) yf (a 1, y) F (a 1, a 2 ) yf (a 2, y) F (a 2, a 3 ) yf (a 3, y) x yf (x, y) G(a 0 ) 3.4

69 63 4 *1 *1

70 64 * *3 4.1 *2 *3 4.2

72 A B B A *4 A B A A *4

73 (1) (2) (3) (4)

74 68 (5) (1)

75 69 (2) 4.3. (1) (2) (3) 4.3

77 W1H who what when when why how W1H

78 J F J F J F J F

79 (1) 2009 (2) (3) OS 4.5.

80 74 PROSPECTUS

81 A B

83 analogy induction abduction A B extrapolation A B A B

84 78 *1 (1) (2) (3) (1)(2) (3) (2)(3) (1) (1)(3) (2) (1) (3) (2) H B H P P H P H B P H B H B P *1 2008

85 A P B P A B

86 80 A B *2 A B A B A B P 5.1. (1) (2) (3) 5.2. (1) 10 A *2

87 81 (2)

88 P Q 1. P Q 2. Q P 3. C P Q

89 [ /110] *3 *3

90 84 P Q P Q P 5.4. (1) (2) 10km (3) IQ 5.5. (1) (2) 10 16

91 D D (1) D (2) D 5.4

95 (1) (2) (3) (4) (5) (6)

96 90 (7) A B A B A B A B (8)

97 A B s B s A B A

98 92 A 1 A 2... A n B A k s B s A k *1 e e p q p e Re f f(p, q) f(p, q) p q pq B e 1, e 2,..., e n e i p i q i B e i f(p, q) p q 30 1/ /2 20 P (1 P ) 10 + (1 P ) (1/2) 20 = 20 20P P 30 + P (1/20) 200 = 40P 1/3 1/3 * TBS 2000

100 A, B A B A B A B A B A B B 0 A B A 1 A A B 1 A B A B A A A A 1 B B

101 A

102 (1) (2) 2 k 2 k (3) A B A 100 B (a)(b) (a) A B

103 97 (b) A B (c) (a) (b)

104 98

105 99 7 * In God We Trust *1 P. J pp 60-66

106

107 ( ). set a, b, c,... { } {a, b, c,...} a, b, c,... a, b, c,... element a S a S a S a / S {a, b} {b, a} {a, a} {a} 8.2 ( ). empty set 8.3 ( ). C(x 1,..., x n ) n *1 x 1,..., x n x 1 x 1 3 x 2 3 x x 2 2 = x 2 3 C(x 1,..., x n ) x 1,..., x n x 1,..., x n { x 1,..., x n C(x 1,..., x n )} {x x 5 } 5 {0, 1, 2, 3, 4, 5} *2 8.4 ( ). S T S T subset S T S S, S S 8.5 ( ). S T T S S *1 *2 0

108 102 T T S S T S = T 8.6 ( ). S T S = T S T proper subset S T 8.7 ( ). I i I S i {S i i I} I I {S i i I} (S i ) i I (S i ) i 8.8 ( ). S T S T union S T S T S T intersection S T M M M = {x M M(x M)} M = {x M M(x M)} I (S i ) i I S i = (S i ) i I i I S i = (S i ) i I i I S T S T subtraction S \ T S T 8.9. S = {0, 1, 2}, T = {2, 3, 4} S T = {0, 1, 2, 3, 4} S T = {2} S \ T = {0, 1} S 1 S 2 S 3 S 1 S 2 S 3 S 1 S 2 S 3 (S 1 S 2 ) S 3 S 1 (S 2 S 3 ) S 1 (S 2 S 3 ) = (S 1 S 2 ) S 3 S 1 (S 2 S 3 ) = (S 1 S 2 ) S 3

109 A, B, C A B = B A A (B C) = (A B) C A B = B A A (B C) = (A B) C A (B C) = (A B) (A B) A (B C) = (A B) (A B) (A \ B) B = A (A \ B) B = A \ (B C) = (A \ B) (A \ C) A \ (B C) = (A \ B) (A \ C) A A A B B C A C A B A A A B B A C A B C A A B A C A B C x C(x) C (x) {x C(x)} {x C (x)} 8.11 ( ). S S power set P(S) 2 S S = {0, 1, 2} P(S) = {, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}} 8.12 ( ). S x T y x, y { x, y x S, y T } S T cartesian product S T S = {0, 1}, T = {a, b, c} S T = { 0, a, 0, b, 0, c, 1, a, 1, b, 1, c } S 1 (S 2 S 3 ) S 1 S 2 S 3 S 1 S 2... S n x 1, x 2,... x n 1, x n... x 1,..., x n S n S n S 1 = S,

110 104 S n+1 = S S n 8.13 ( ). S 1,..., S n S 1 S 2... S n R n n-ary relation n 1 R S n R S n N N 2 = N N L { x, y x, y N 2, x < y} x, y L x, y x < y R x, y R xry 8.15 ( ). R S (1) x S xrx R reflexive (2) x, y S xry yrx R symmetrical (3) x, y, z S xry yrz xrz R transitive (4) x, y S xry yrx x = y R antisymmetric 8.16 ( ). S R S semiorder, partial order x, y S x y y x S total order 8.17 ( ). S S (S, ) partially ordered set; poset (S, ) totally ordered set (S, ) S (S, ) M S M M x, y M x M y x y (M, M ) (S, ) *3 M M 8.18 ( ). (M, M ) (S, ) M x S y x y y M *3

111 105 upper bound M x S y y x y M lower bound S M S M supremum M (1) x M x M (2) M x M x S M S M infimum M (1) x M M x (2) M x x M (M, M ) (M, M ) (1) S P(S) M, N P(S) M S N M N S P(S) P(S) M M, M M (2) N x, y N x y x y N 8.21 ( ). (S, ) s S x S s x s = x s S maximal element x S x s s = x s S minimal element x S x s s S maximum element x S s x s S minimum element (S, ) (S, ) (S, ) (S, ) (M, M ) (M, M )

112 R Z + R M = {1 1/n n Z + } M M (M, M ) 1 1 M M 8.24 ( ). (S, ) S (M, M ) S well-ordered set (1) N (N, ) Q (Q, ) ({x Q 0 < x}, ) (Q, ) 8.26 ( ). S S equivalent relation x, y Z x 3 y x 3 y 3 3 Z 8.28 ( ). S x S [x] = {y S x y} x equivalent class S/ = {[x] x S} S/ S quotient set S x, y S [x] = [y] x y 8.30 ( ). A B f A B function mapping, map (1) a A b B a, b f (2) a A, b, b B a, b f a, b f b = b. A a a, b f B b f A B f A B f : A B A f domain B f codomain a A, b B a, b f b f(a)

113 107 A 1 A 1... A n a 1,..., a n A 1 A 1... A n f( a 1,..., a n ) f(a 1,..., a n ) Z Q f : Z (Z \ {0}) Q, f(x, y) = x y f Z (Z \ {0}) Q f : Z Z Q, f(x, y) = x y 1, 0 f(1, 0) 8.32 ( ). A id A (a) = a ( a A) id A : A A A identity function 8.33 ( ). f : A B, g : B C g f : A C g f(a) = g(f(a)) ( a A) g f f g composition 8.34 ( ). f : A B A M {f(x) B x M} M f direct image f[m] B N {x A f(x) N} N f inverse image f 1 [N] f : A B g : B C h : C D id B f = f = f id A h (g f) = (h g) f g f[m] = g[f[m]] ( M A) (g f) 1 [M] = f 1 [g 1 [M]] ( M C) f[m M ] = f[m] f[m ], f[m M ] = f[m] f[m ] ( M, M A) f 1 [M M ] = f 1 [M] f 1 [M ], f 1 [M M ] = f 1 [M] f 1 [M ] ( M, M B) 8.36 ( ). f : A B x, y A f(x) = f(y) x = y f injection one-to-one mapping

114 108 x B y A f(y) = x f surjection onto mapping f f bijection 8.37 ( ). (S i ) i I (S i ) i I i I S i f f(s i ) S i (S i ) i I i I (S i ) i I i I S i (S i ) i I i I S i the axiom of choice f : A B, g : B C f, g g f g f f g f g f h : B A h f = id A f h : B A f h = id B S i

JR A B B A

JR A B B A 2012 1 2 2 2 2.1...................................... 2 2.2............................................. 4 2.3................................. 9 2.4........................................ 9 3 10 3.1.............................................