離散数学 第 2回 集合と論理 (2):集合と論理の対応

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1 2 (2) :10 ( ) (2) / 40

2 ( ) 1 (1) (4 10 ) 2 (2) (4 17 ) 3 (3) (4 24 ) 4 (1) (5 1 ) 5 (2) (5 8 ) 6 (4) (5 15 ) 7 (3) (5 22 ) 8 (1) (5 29 ) 9 (2) (6 5 ) (6 12 ) ( ) (2) / 40

3 ( ) 10 (1) (6 19 ) 11 (2) (6 26 ) 12 (3) (7 3 ) 13 (4) (7 10 ) 14 (4) (7 17 ) 15 (5) (7 24 ) ( ) (7 31 ) (8 7 ) ( ) (2) / 40

4 3 3 ( ) (2) / 40

5 ( ) (2) / 40

6 A, B A B A B = {x x A x B} A = {a, b, c, d, e, f } B = {a, b, c, g, h} A B = {a, b, c} A d e f a b c g h B ( ) (2) / 40

7 A, B A B A B = {x x A x B} A = {a, b, c, d, e, f } B = {a, b, c, g, h} A B = {a, b, c, d, e, f, g, h} A d e f a b c g h B ( ) (2) / 40

8 A, B A B A B = {x x A x B} A = {a, b, c, d, e, f } B = {a, b, c, g, h} A B = {d, e, f } A d e f a b c g h B A B A \ B ( ) (2) / 40

9 Constructive Solid Geometry Constructive Solid Geometry POV-Ray tree.png ( ) (2) / 40

10 P, Q P Q A, B A B ( ) (2) / 40

11 ( ) (2) / 40

12 ( ) P (Q P) P Q Q P P (Q P) T T T T T F T T F T F T F F T T ( ) (2) / 40

13 P Q P Q P Q P Q Q P P Q P Q P Q P Q Q P ( ) (2) / 40

14 ( ) P, Q P Q P Q P Q P P Q P Q (P Q) ( P Q) T T F T T T T F F F F T F T T T T T F F T T T T ( ) (2) / 40

15 ( ) ( ) P, Q P Q P Q 100 P = Q = 100 P Q = 100 P Q = 100 ( ) (2) / 40

16 ( ) P, Q P Q (P Q) (Q P) P Q P Q P Q Q P (P Q) (Q P) (P Q) ((P Q) (Q P)) T T T T T T T T F F F T F T F T F T F F T F F T T T T T ( ) (2) / 40

17 P P P P P P P T F T F T T ( ) (2) / 40

18 ( ) P P P P = P P = ( ) (2) / 40

19 ( ) P, Q (P Q) P Q (P Q) P Q P Q P Q P Q P Q (P Q) ( (P Q)) ( P Q) T T F F F T F T T F F T F T F T F T T F F T F T F F T T T F T T P Q P Q P Q P Q (P Q) ( (P Q)) ( P Q) T T F F F T F T T F F T T F T T F T T F T F T T F F T T T F T T ( ) (2) / 40

20 ( ) ( ) P, Q (P Q) P Q OR NOT NOT AND (P Q) P Q AND NOT NOT OR P = Q = (P Q) = P Q = ( ) (2) / 40

21 Augustus De Morgan ( ) Morgan Augustus.jpg ( ) (2) / 40

22 P, Q P Q Q P P Q P Q Q P Q P (P Q) ( Q P) T T T F F T T T F F T F F T F T T F T T T F F T T T T T ( ) (2) / 40

23 P, Q P Q Q P 100 P Q = 100 Q P = 100 ( ) (2) / 40

24 (1) P P P P P P P Q Q P P Q Q P (P Q) P P (P Q) P P P ( P) P T P P F P P F F P T T (P Q) R P (Q R) (P Q) R P (Q R) ( ) (2) / 40

25 (2) (P Q) R (P R) (Q R) (P Q) R (P R) (Q R) P (Q R) (P Q) (P R) P (Q R) (P Q) (P R) (P Q) (P R) P (Q R) (P Q) (Q R) P R ( ) (2) / 40

26 ( ) ( ) (2) / 40

27 ( ) (2) / 40

28 P (Q R) (P Q) (P R) P (Q R) (P P) (Q R) P (P (Q R)) P ((P Q) R) (P (P Q)) R ((P Q) P) R (P Q) (P R) ( ) (2) / 40

29 P (Q R) (P Q) (P R) P (Q R) (P P) (Q R) P (P (Q R)) P ((P Q) R) (P (P Q)) R ((P Q) P) R (P Q) (P R) ( ) (2) / 40

30 P (Q R) (P Q) (P R) P (Q R) (P P) (Q R) P (P (Q R)) P ((P Q) R) (P (P Q)) R ((P Q) P) R (P Q) (P R) P P P ( ) (2) / 40

31 P (Q R) (P Q) (P R) P (Q R) (P P) (Q R) P (P (Q R)) P ((P Q) R) (P (P Q)) R ((P Q) P) R (P Q) (P R) (P Q) R P (Q R) ( ) (2) / 40

32 P (Q R) (P Q) (P R) P (Q R) (P P) (Q R) P (P (Q R)) P ((P Q) R) (P (P Q)) R ((P Q) P) R (P Q) (P R) (P Q) R P (Q R) ( ) (2) / 40

33 P (Q R) (P Q) (P R) P (Q R) (P P) (Q R) P (P (Q R)) P ((P Q) R) (P (P Q)) R ((P Q) P) R (P Q) (P R) (P Q) R P (Q R) ( ) (2) / 40

34 P (Q R) (P Q) (P R) P (Q R) (P P) (Q R) P (P (Q R)) P ((P Q) R) (P (P Q)) R ((P Q) P) R (P Q) (P R) P Q Q P ( ) (2) / 40

35 P (Q R) (P Q) (P R) P (Q R) (P P) (Q R) P (P (Q R)) P ((P Q) R) (P (P Q)) R ((P Q) P) R (P Q) (P R) (P Q) R P (Q R) ( ) (2) / 40

36 P (Q R) (P Q) (P R) P (Q R) (P P) (Q R) P (P (Q R)) P ((P Q) R) (P (P Q)) R ((P Q) P) R (P Q) (P R) ( ) (2) / 40

37 ( ) ( ) (2) / 40

38 2 (P Q) R P (Q R) (P Q) R (P Q) R ( ) ( P Q) R ( ) P ( Q R) ( ) P (Q R) ( ) P (Q R) ( ) ( ) (2) / 40

39 2 (P Q) R P (Q R) (P Q) R (P Q) R ( ) ( P Q) R ( ) P ( Q R) ( ) P (Q R) ( ) P (Q R) ( ) ( ) (2) / 40

40 2 (P Q) R P (Q R) (P Q) R (P Q) R ( ) ( P Q) R ( ) P ( Q R) ( ) P (Q R) ( ) P (Q R) ( ) P Q P Q ( ) (2) / 40

41 2 (P Q) R P (Q R) (P Q) R (P Q) R ( ) ( P Q) R ( ) P ( Q R) ( ) P (Q R) ( ) P (Q R) ( ) (P Q) P Q ( ) (2) / 40

42 2 (P Q) R P (Q R) (P Q) R (P Q) R ( ) ( P Q) R ( ) P ( Q R) ( ) P (Q R) ( ) P (Q R) ( ) (P Q) R P (Q R) ( ) (2) / 40

43 2 (P Q) R P (Q R) (P Q) R (P Q) R ( ) ( P Q) R ( ) P ( Q R) ( ) P (Q R) ( ) P (Q R) ( ) P Q P Q ( ) (2) / 40

44 2 (P Q) R P (Q R) (P Q) R (P Q) R ( ) ( P Q) R ( ) P ( Q R) ( ) P (Q R) ( ) P (Q R) ( ) P Q P Q ( ) (2) / 40

45 2 (P Q) R P (Q R) (P Q) R (P Q) R ( ) ( P Q) R ( ) P ( Q R) ( ) P (Q R) ( ) P (Q R) ( ) ( ) (2) / 40

46 ( ) (2) / 40

47 A, B x A B (x A) (x B) x A B (x A) (x B) x A B (x A) (x B) (x A) (x B) A, B, A B, A B, A B x A, x B, x A B, x A B, x A B ( ) ( ) (2) / 40

48 A, B A = B x A x B ( ) (2) / 40

49 A, B, C A (B C) = (A B) (A C) A B A B A B A B A B C A C B C C A (B C) C A B C A C = (A B) (A C) x A (B C) x (A B) (A C) ( ) (2) / 40

50 A, B, C A (B C) = (A B) (A C) x A (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)) ( ) ( ) ( ) ( ) ( ) (2) / 40

51 A, B, C A (B C) = (A B) (A C) x A (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 (x A) (x B) ( ) (2) / 40

52 A, B, C A (B C) = (A B) (A C) x A (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 (x A) (x B) ( ) (2) / 40

53 A, B, C A (B C) = (A B) (A C) x A (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)) ( ) ( ) ( ) ( ) (P Q) P Q ( ) (2) / 40

54 A, B, C A (B C) = (A B) (A C) x A (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)) ( ) ( ) ( ) ( ) P (Q R) (P Q) (P R) ( ) (2) / 40

55 ( ) A, B, C A (B C) = (A B) (A C) ( ) x A (B C) ((x A) (x B)) ((x A) (x C)) (x A B) (x A C) x (A B) (A C) ( ) ( ) ( ) (2) / 40

56 ( ) A, B, C A (B C) = (A B) (A C) ( ) x A (B C) ((x A) (x B)) ((x A) (x C)) (x A B) (x A C) x (A B) (A C) ( ) ( ) x A B (x A) (x B) ( ) (2) / 40

57 ( ) A, B, C A (B C) = (A B) (A C) ( ) x A (B C) ((x A) (x B)) ((x A) (x C)) (x A B) (x A C) x (A B) (A C) ( ) ( ) x A B (x A) (x B) ( ) (2) / 40

58 ( ) A, B, C A (B C) = (A B) (A C) ( ) x A (B C) ((x A) (x B)) ((x A) (x C)) (x A B) (x A C) x (A B) (A C) ( ) ( ) ( ) (2) / 40

59 ( ) (2) / 40

60 3 3 ( ) (2) / 40

61 ( ) OK OK ( ) (2) / 40

62 ( ) (2) / 40

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