2 (2) okamotoy@uec.ac.jp 2015 4 17 2015 4 15 14:10 ( ) (2) 2015 4 17 1 / 40
( ) 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) 2015 4 17 2 / 40
( ) 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) 2015 4 17 3 / 40
3 3 ( ) (2) 2015 4 17 4 / 40
1 2 3 4 5 ( ) (2) 2015 4 17 5 / 40
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) 2015 4 17 6 / 40
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) 2015 4 17 7 / 40
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) 2015 4 17 8 / 40
Constructive Solid Geometry Constructive Solid Geometry POV-Ray http://en.wikipedia.org/wiki/file:csg tree.png ( ) (2) 2015 4 17 9 / 40
P, Q P Q A, B A B ( ) (2) 2015 4 17 10 / 40
1 2 3 4 5 ( ) (2) 2015 4 17 11 / 40
( ) 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) 2015 4 17 12 / 40
P Q P Q P Q P Q Q P P Q P Q P Q P Q Q P ( ) (2) 2015 4 17 13 / 40
( ) 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) 2015 4 17 14 / 40
( ) ( ) P, Q P Q P Q 100 P = Q = 100 P Q = 100 P Q = 100 ( ) (2) 2015 4 17 15 / 40
( ) 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) 2015 4 17 16 / 40
P P P P P P P T F T F T T ( ) (2) 2015 4 17 17 / 40
( ) P P P P = P P = ( ) (2) 2015 4 17 18 / 40
( ) 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) 2015 4 17 19 / 40
( ) ( ) 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) 2015 4 17 20 / 40
Augustus De Morgan (1806 1871) http://commons.wikimedia.org/wiki/file:de Morgan Augustus.jpg ( ) (2) 2015 4 17 21 / 40
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) 2015 4 17 22 / 40
P, Q P Q Q P 100 P Q = 100 Q P = 100 ( ) (2) 2015 4 17 23 / 40
(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) 2015 4 17 24 / 40
(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) 2015 4 17 25 / 40
( ) ( ) (2) 2015 4 17 26 / 40
1 2 3 4 5 ( ) (2) 2015 4 17 27 / 40
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) 2015 4 17 28 / 40
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) 2015 4 17 28 / 40
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) 2015 4 17 28 / 40
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) 2015 4 17 28 / 40
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) 2015 4 17 28 / 40
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) 2015 4 17 28 / 40
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) 2015 4 17 28 / 40
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) 2015 4 17 28 / 40
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) 2015 4 17 28 / 40
( ) ( ) (2) 2015 4 17 29 / 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) ( ) ( ) (2) 2015 4 17 30 / 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) ( ) ( ) (2) 2015 4 17 30 / 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) 2015 4 17 30 / 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) 2015 4 17 30 / 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) R P (Q R) ( ) (2) 2015 4 17 30 / 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) 2015 4 17 30 / 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) 2015 4 17 30 / 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) ( ) ( ) (2) 2015 4 17 30 / 40
1 2 3 4 5 ( ) (2) 2015 4 17 31 / 40
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) 2015 4 17 32 / 40
A, B A = B x A x B ( ) (2) 2015 4 17 33 / 40
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) 2015 4 17 34 / 40
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) 2015 4 17 35 / 40
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) 2015 4 17 35 / 40
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) 2015 4 17 35 / 40
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) 2015 4 17 35 / 40
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) 2015 4 17 35 / 40
( ) 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) 2015 4 17 36 / 40
( ) 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) 2015 4 17 36 / 40
( ) 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) 2015 4 17 36 / 40
( ) 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) 2015 4 17 36 / 40
1 2 3 4 5 ( ) (2) 2015 4 17 37 / 40
3 3 ( ) (2) 2015 4 17 38 / 40
( ) OK OK ( ) (2) 2015 4 17 39 / 40
1 2 3 4 5 ( ) (2) 2015 4 17 40 / 40