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Διακριτά Μαθηματικά Ι: Εισαγωγή στη Λογική

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  • 1. . . 2 : mboudour@upatras.gr 2014 . : .

2. . : . 3. ( ) , , . 1 . (.) 2 2 + 2 = 4. (.) 3 2 + 3 = 7. (.) 4 4 3 . (.) 5 n , 2n . (.) 6 n, 2n + n . ( .) 7 2 . ( Goldbach.) . : . 4. 1 x + y = y + x, x,y R. (.) 2 2n = n2, n N. (.) 3 3 7 . (.) 4 , 2 + 2 = 4. (.) 5 x y = y x. (.) 6 A2 = 0 A = 0, A. (.) : : , : , : ( ), : ... , : . . : . 5. A B, , , : A B = {x: (x A) (x B)}, A B = {x: (x A) (x B)}, Ac = {x: (x A)}. . : . 6. 1: 0: p p 0 1 1 0 p q p q p q p q p q 0 0 0 0 1 1 0 1 0 1 1 0 1 0 0 1 0 0 1 1 1 1 1 1 . : . 7. : ( ) XOR, |: NAND Sheer. | p q p q p|q 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 0 . : . 8. (p q) (p q) p q p q p q (p q) (p q) (p q) 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 (p q) [(q r) (p r)] = P Q = (q r) (p r), P = (p q) Q p q r r p q q r (p q) (q r) p r Q P 0 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 0 1 0 1 1 1 1 0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 . : . 9. 1 P . P , P. , 1. P P . , P , , , , 0. . : . 10. [p (p q)] q , : p q p q p (p q) [p (p q)] q 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 1 1 1 1 1 p p , p p p p 0 1 0 1 0 0 . : . 11. 2 P Q . P Q P Q P Q . , P Q , . (p q) p q ( De Morgan), ( ): p q p q (p q) p q q p 0 0 0 1 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 0 1 1 1 0 0 0 0 . : . 12. . : . 13. 3 P Q . P Q P = Q P Q . P = Q , P Q . . : . 14. A = (p q) (r s) B = (p r) (q s). A = B. , B , , q s , , , P = A B . p q r s p q r s A P B q s p r 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 1 . : . 15. . : . 16. 1 : 1 p q,(p q) r,(p p) p, 2 (p q) [(p q) (p q)], 3 [(p q) r] (p q), 4 [(p q) (p r)] (q p), 5 ((p q) r). 2 : 1 p q (p q), 2 p|q (p q), 3 p p|p, 4 p q (p|p)|(qIq), 5 Sheer, p q,p q p q. . : . 17. 3 : 1 [p (q r)] [(p q) (p r)], 2 [p (q r)] [(p q) (p r)], 3 [(p q) r] [p (q r)], 4 [(p q) r] [p (q r)], 5 (p q) r p (q r), 6 (q p) (p q), 7 (p q) = (p q). 4 : 1 A = B B = C, A = C. 2 P Q,Q = R R S, P = S. 3 P = Q,Q = R R = P, P Q. . : . 18. 1 1 2 3 4 ( ) 5 6 7 8 1 , . . : . 19. P = Q : ( ) P , ( ) Q . : . : n N . n = 2k, k N. , n2 = 4k2 = 2(2k2), , n2 . : n i=1 i = 1 2 n(n + 1). : x = n i=1 i, , x = 1 + 2 + 3 + ... + n. , x = n + (n 1) + (n 2) + ... + 1. , 2x = (n + 1) + (n + 1) + (n + 1) + ... +(n + 1) = n(n + 1) , , x = 1 2 n(n + 1). . : . 20. P1 P2 Pn = Q n () : (P1 = Q) (P2 = Q) (Pn = Q). : n N, n3 + n . : (i). n N , , n = 2k, k N, n3 + n = 8k3 + 2k = 2(4k3 + k), . (ii). n N , , n = 2k + 1, k N, n3 + n = (8k3 + 12k2 + 6k + 1) + (2k + 1) = 2(4k3 + 6k2 + 4k + 1), . . : . 21. : x,y R,|x + y| |x| + |y|. : (i). x,y 0, x + y 0 , , |x + y| = x + y = |x| + |y|. (ii). x 0 y