2. 2.1 2.2 kuri@ice.uec.ac.jp ( 2007/10/30/16:46) 1
. 1. 1 + 2 = 5. 2. 180. 3. 3 3. 4.. 5.. 2
2.1 1.,,,,. 2., ( ) ( ).,,,, 3.,. 4.,,,. 3
1.,. 1. 1 + 2 = 5. (, ) 2. 180. (, ) 3. 3, 3. (, ) 4.. (, ) 5.. (, ) 4
2. ( ) 1. p, q 2. (True) T, (False) F 3. T, F 4.,,,,. p q p q p q p q p q p q p T T T T T T T F T F T F F T F F F T T F T F F T F F F F T T T T 5
1. p q : p q. p q. 2. p q : p q. p q. p q p q p q p q p q p q p T T T T T T T F T F T F F T F F F T T F T F F T F F F F T T T T 6
3. p q : p q ( ). p q. 1) (p q) = F, (p = T ) (q = F ). 2) p = F (p q) = T. p q p q p q p q p q p q p T T T T T T T F T F T F F T F F F T T F T F F T F F F F T T T T 3) p q, p q. 7
4. (p q) = T, p q, q p. 5. p q (p q) (p q). 6. p q,, (p q) = T, p q., p q. 7. i) p q, ii) p q., p q p = q. p q p q p q p q p q p q p T T T T T T T F T F T F F T F F F T T F T F F T F F F F T T T T 8
8. p: p. p p T F F T 9
1.,. i. : p, q ii. : p q, p q, p q 2., (, ). 3.,,. i. p, q. ii. p q (p q), ( p) q. 10
3. p,. 1. p T = T, p T = p. 2. p F = p, p F = F. 3. p p = T, p p = F. 4. ( p) = p. ( ) i. p = q, p q ii. p q, (p q) = T (p q) = T, (p q) = T iii. p q, p q. 11
. 1. p T = T, p T = p. 2. p F = p, p F = F. 3. p p = T, p p = F. 4. ( p) = p. 1. p T p T p T T T T T F T T F 2. p F p F p F T F T F F F F F 3. p p p p p p T F T F F T T F 4. p p ( p) T F T F T F 12
4. p, q, r,. 1. p q = q p, p q = q p. ( ) 2. (p q) r = p (q r), (p q) r = p (q r). ( ) 3. (p q) r = (p r) (q r), (p q) r = (p r) (q r). ( ) 4. p p = p, p p = p. ( ) 5. (p q) p = p, (p q) p = p. ( ) 13
( = ( ) ) i. ( ) ii. ( )= T ( ) = T, ( )= T = T = T = T = F,, p q p q T T T T F F F T T F F T 14
(p q) r = (p r) (q r) ( ) ( ) p q r p q p r q r T T T T T T T T T T F T F F F F T F T T T T F T T F F T F F F F F T T T T F T T F T F T F F F F F F T F F F F F F F F F F F F F,, =. 15
(p q) r = (p r) (q r) ( ) ( ) ( )= T ( ) = T i. ) ((p q) r) = T. (p q) = T r = T p = T q = T, r = T 1) p = T r = T (p r) = T. 2) q = T r = T (q r) = T. ((p r) (q r)) = T. ( )= T. ii. ) ((p r) (q r)) = T. (p r) = T (q r) = T p = T r = T q = T r = T. p = T q = T r = T ((p q) r) = T. ( ) = T. i, ii, = 16
5. ( ) p, q,. 1. (p q) = p q. 2. (p q) = p q. 17
1. (p q) = p q. 2. (p q) = p q. ( ) 1. 1. 2. 2. p q p q (p q) p q p q p q (p q) p q T T T F F F F T F F T F T F F T F F T T F T T F T F F F T T F F F T T T T F T T, 1, 2,.,. 18
1. (p q) = p q ( ) (, (p q) = T p q = T ) i. ) (p q) = T p q = F. p q = F p = F q = F., p = T q = T, p q = T., (p q) p q. ( (p q) p q) = T ii. ) p q = T p = T q = T, p = F q = F., p q = F, (p q) = T., (p q) p q. ( (p q) p q) = T 1, 2, (p q)= p q. ( : = ) 19
6. ( ) 1. p q p, q,,. 2. p q p, q,,. 3. p q p, q,,. 4. p q p, q,,. 5. p q p, q,,. 6. p q p, q,,. 20
( ) 1. p q p, q,,, p q = ( p q). 2. p q p, q,, (p q) = ( p q), p q = ( p) q = ( p q) 3. p q p, q,,, p q = ( p q). 4. p q p, q,, 2, (p q) = ( p q), p q = ( p ( q)) = (p ( q)). 5. p q p, q,, (p q) = ( p q),. 21
6. p q p, q,, (p q) = ( p q), (p q) = ( p q) = ( ( p) ( q)) = (p ( q)). 22
( ) ( ) c ( ) U ( ) φ ( ) ( ) ( ) ( ) T ( ) F ( ) 23
7. p q T, q p T,. 8.. 1. p p 2. (p p) (p p) 3. (p q) p 4. p ( p q) 5. ( q p) (p q) 24
9.. 1. (p p) ( p q) 2. (p q) (q p) 3. [p ( q p)] [(p q) (q p)] 4. [q (r p)] [( q p) r] 25
10..,. p =, q =, r =. 1. (p q) r 2. r (p q) 3. p q r 4. r (p q) ( ) 26
p =, q =, r =. 1. (p q) r,. 2. r (p q),. 3. p q r. 4. r (p q),,. 27
11. 1. p q F, ( p q) q.,. 2. p q T, p (p q).,. 28
1. p q F, ( p q) q.,. ( ) (p q) = F, p = T q = F., ( p q) = (F T ) = T., ( p q) q T F., ( p q) q, F. 29
2. p q T, p (p q). ( ) (p q) = T, p q 1 : 1. p q p q T T T T F F F T T F F T 2. p q p q T T T T F F F T F F F T i) p = T q = T, p (p q) F (T T ). (T T ) = T, (F (T T )) = T. ii) p = F q = T, p (p q) T (F T ). (F T ) = F, (T (F T )) = T. iii) p = F q = F, p (p q) T (F F ). (F F ) = T, (T (F F )) = T. i, ii, iii, p (p q), T. 30
p 1, p 2, p 3, (p 1 p 2 ) p 3, p 1 (p 2 p 3 ), p 2 (p 1 p 3 ). p 1 p 2 p 3.., p 1,..., p n, p 1 p n, p 1 p n. 31
p 1,..., p n, n i=1 p i := { n i=1 p i, n i=1 p i. T, i {1,..., n}, p i = T ; F, i {1,..., n}, p i = F. n i=1 p i := { T, i {1,..., n}, p i = T ; F, i {1,..., n}, p i = F.,. 12.. 1. 2. n n, n p 1,..., p n, p i = p 1 p n. i=1 n p i := p 1 p n. i=1 32
: p 1,..., p n P (p 1,..., p n ), {T, F } n, {T, F } P : {T, F } n {T, F }. P n, p 1,..., p n., p, q, r. P (p, q, r) = p q r P : {T, F } 3 {T, F } P : (p, q, r) v(= P (p, q, r) = p q r) 33
p, t {T, F }, p t := { p, t = T ; p, t = F, t p p t T T T T F F F T F F F T 34
p 1,..., p n n. n (t 1,..., t n ) {T, F } n, p t 1 1 p t n n (t 1,..., t n ) p 1,..., p n. 13. p t 1 1 p t n n = T (p 1 = t 1 ) (p n = t n ) ( ), p t = T (p = t) : t p p t T T T T F F F T F F F T 35
P (p 1,..., p n ) P P := {(s 1,..., s n ) (s 1,..., s n ) {T, F } n, P (s 1,..., s n ) = T }. (P (p 1,..., p n ) = T (s 1,..., s n ) ) 14. P (p 1,..., p n )., P P P = φ, F. P (p 1,..., p n ) = (t 1,...,t n ) e P p t 1 1 p t n n. P (p 1,..., p n ).. 36
( 16 ) P (p, q, r). p q r P (p, q, r) T T T F T T F T T F T F T F F T F T T F F T F F F F T T F F F T P (p 1,..., p n ) = (t 1,...,t n ) e P p t 1 1 p t n n 37
p q r P (p, q, r) T T T F T T F T T F T F T F F T F T T F F T F F F F T T F F F T (t 1, t 2, t 3 ) = (T, T, F ) p t 1 q t 2 r t 3 = p q r (t 1, t 2, t 3 ) = (T, F, F ) p t 1 q t 2 r t 3 = p q r (t 1, t 2, t 3 ) = (F, F, T ) p t 1 q t 2 r t 3 = p q r (t 1, t 2, t 3 ) = (F, F, F ) p t 1 q t 2 r t 3 = p q r : P = {(T, T, F ), (T, F, F ), (F, F, T ), (F, F, F )}, : P (p, q, r) = (p q r) (p q r) ( p q r) ( p q r). 38
: P (p, q, r) = (p q r) (p q r) ( p q r) ( p q r) P p q r p q r p q r p q r p q r F T T T F F F F F T T T F T F F F T F T F T F F F F F T T F F F T F F T F F T T F F F F F F F T F F F F F F T F F T F F T F T T F F F F F F T T 39
,. (t 1,..., t n ) p 1,..., p n, p t 1 1 p t n n., (t 1,..., t n ) p 1,..., p n, p t 1 1 p t n n. 40
, 13 : p t 1 1 p t n n = T (p 1 = t 1 ) (p n = t n ),. 15. p t 1 1 p t n n = F (p 1 = t 1 ) (p n = t n ) ( ), p t = F (p = t) : t p p t T T F T F T F T T F F F 41
, P (p 1,..., p n ) P P := {(s 1,..., s n ) (s 1,..., s n ) {T, F } n, P (s 1,..., s n ) = T }., P (p 1,..., p n ) P P : P := {(s 1,..., s n ) (s 1,..., s n ) {T, F } n, P (s 1,..., s n ) = F } (P (p 1,..., p n ) = F (s 1,..., s n ) ) 42
, 14 ( ) : P (p 1,..., p n ) =,. (t 1,...,t n ) e P p t 1 1 p t n n 16. P (p 1,..., p n )., P = φ, T. P (p 1,..., p n ) = (t 1,...,t n ) P p t 1 1 p t n n P (p 1,..., p n ). 43
17. P (p, q, r). p q r P (p, q, r) T T T F T T F T T F T F T F F T F T T F F T F F F F T T F F F T P (p 1,..., p n ) = (t 1,...,t n ) P p t 1 1 p t n n 44
p q r P (p, q, r) T T T F T T F T T F T F T F F T F T T F F T F F F F T T F F F T (t 1, t 2, t 3 ) = (T, T, T ) p t 1 q t 2 r t 3 = p q r (t 1, t 2, t 3 ) = (T, F, T ) p t 1 q t 2 r t 3 = p q r (t 1, t 2, t 3 ) = (F, T, T ) p t 1 q t 2 r t 3 = p q r (t 1, t 2, t 3 ) = (F, T, F ) p t 1 q t 2 r t 3 = p q r : P P = {(T, T, T ), (T, F, T ), (F, T, T ), (F, T, F )}., : P (p, q, r) = ( p q r) ( p q r) (p q r) (p q r). 45
18. ( ),. i) ii), i) : 1. P (p, q, r) P (p, q, r). 2. P (p, q, r). 3. ( P (p, q, r)),,. 46
1. P (p, q, r) P (p, q, r) : p q r P (p, q, r) P (p, q, r) T T T F T T T F T F T F T F T T F F T F F T T F T F T F F T F F T T F F F F T F 2. P (p, q, r) : P = {(T, T, T ), (T, F, T ), (F, T, T ), (F, T, F )} ( P (p, q, r) ) P (p, q, r) = (p q r) (p q r) ( p q r) ( p q r) 3.,, : P (p, q, r) = ( P (p, q, r)), ( P (p, q, r)) = ( p q r) ( p q r) (p q r) (p q r), 47
19. 1. 3 p, q, r 2 T,, T. 2. 3 p, q, r, T, 1 T,, T. 48
1. 3 p, q, r 2 T,, T. 2. 3 p, q, r, T, 1 T,, T. 1. p q r P (p, q, r) T T T F T T F T T F T T T F F F F T T T F T F F F F T F F F F F 2. p q r P (p, q, r) T T T F T T F F T F T F T F F T F T T F F T F T F F T T F F F T 49
References [1], ( ),, 2004. [2],,,, 2004. [3],,, 2003. [4],,, 2003. [5] S. Lipschutz,, ( ),, 2004(H16). [6], ( ),, 2006. [7],,,, 1990. 89