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1 tbasic.org * 1 [ ] *1 1
2 proposition statememt truth value True False p, q p not p p p q p and q p q p q p or q p q p q if p, then q q p p p is equivalent to q p q x x 0 p(x 0 ) p(x) 1 q(x, y,...) p(x) x p(x) for all x, p(x) xp(x) x(p(x)) x p(x) p(x) for some, p(x) xp(x) x(p(x)) x s.t. p(x) *2 1.2 A a A a A a A A, B A B A B A B B A *2 s.t. such that there exisit x such that p(x)
3 3 a, b, c, a, b, c, {a, b, c, } A p(x) A x A a p(a) A {a A p(a)} U U U U {x U p(x)} {x p(x)} x(x ) A B A B A B A B A B (x A B) ((x A) (x B)) A B = {x (x A) (x B)} A B A B A B A B (x A B) ((x A) (x B)) A B = {x (x A) (x B)} A B = A B B A A B A B A B (x A B) ((x A) (x B)) A B = {x (x A) (x B)} U U A A A c A c x A c x U A (x U) (x A) A c = {x U x A} = {x x A} *3 1.3 N N = {1, 2, 3, } *4 Z Z = {0, ±1, ±2, ±3, } R C *3 *4 0 1
4 4 N Z R C a { a ; a 0 a = a ; a < 0 a a A A a 0 A a a 0 a a 0 A min(a) A a 1 A a a 1 a a 1 A max(a) A M R A A x x M m R A A x m x
5 P(n) n (1) (2) (1) P(1) (2) n P(n) P(n + 1) n P(n)
6 n 3 = ( n) 2 ( ) P(n) ( ) n (1) n = = 1 2 (2) n * n 3 = ( n) 2 (1) n n + 1 ( n + (n + 1)) 2 = ( n) 2 + 2(( n)(n + 1) + (n + 1) 2 (2) n = n(n + 1) 2 (3) *6 (3) (2) 2 (4) ( n + (n + 1)) 2 = ( n) 2 + n(n + 1) 2 + (n + 1) 2 = ( n) 2 + (n + 1)(n + 1) 2 = ( n) 2 + (n + 1) 3 (4) 1 (1) ( n + (n + 1)) 2 = n 3 + (n + 1) 3 n + 1 n + 1 (1),(2) n ( ) *5 *6
7 7 2 2 P(n) n (1) (2) (1) P(1) (2) n P(1),..., P(n) P(n + 1) n P(n) 2 (2) *7 [= ] P(n) n (1) (2) (1) P(1) (2) n P(1),..., P(n) P(n + 1) n P(1),..., P(n) *8 (1) n = 1 (1) P(1) (2) n P(1),..., P(n) (2) P(n + 1) P(1),..., P(n) P(1),..., P(n), P(n + 1) n *7 *8 2 2 n n
8 8 [ =] (2) P(n) 2 (2) P(1),..., P(n) 1 (2) P(n) P(1),..., P(n) (2) * A N A 1, 2, 3,... A A A A A n N n A P(n) n A (1) 1 A 1 A 1 A (2) n 0 2 (2) P(1),..., P(n 0 ) P(n 0 + 1) 1 A,..., n 0 A n A n A n 0 2 (2) n P(n) n A A A A N N B N B *9 (2)
9 9 B B B B = {n N x B s.t. n x} B B B B b 0 = max B B b 0 B b 0 b 1 b 1 B b 1 B b 0 = max B b 1 b 0 b 0 = b 1 B b 0 = max B B n N n B ( ) P(n) n B (1) B B b 1 b 1 B (2) 1,..., n B n + 1 B n B n < m B m m x B x n + 1 m n + 1 m x n + 1 B n B B n + 1 B n + 1 B (1) (2) 2 B = N B B B
10 n, m N m = qn + r, 0 r < n q, r 1 q m n r n, m Z, n 0 m = qn + r, 0 r < n q, r 1 q m n r r (1) 30 = (2) 30 = (3) 30 = 4 ( 7) + 2 (4) 30 = 5 ( 7) ( ). a, b Z (a 0) b = ca c Z a b a b a b b a a b a, b Z (a 0) b = 0 0 = 0 a a 0 a 0
11 11 * 10 a b ±a ± b 2.1. a, b, c, x, y Z (1) a b, a c a (bx + cy) (2) a b, b c a c (3) a b (b 0) a b (4) a b, b a a = b (1) a b, a c k, l Z b = ka, c = la bx + cy = kax + lay = (kx + ly)a a (bx + cy) (2) a b, b c k, l Z b = ka, c = lb c = lb = lka a c (3) a b k Z b = ka b = k a k 1 a b (4) (3) a b a b b a b a (1) x, y 2.3. a 15 a 6 a = 3 (1) a 3 *10 0 0
12 a b a b a, b Z a, b 0 a b a b gcd(a, b) * , 2, 3, , 2, 5, , a, b Z a, b 0 a b gcd(a, b) a 0 b = 0 a 0 gcd(a, 0) = a a, b 0 a, b C x C x x a x b x min( a, b ) C N C a b *11 a b a, b
13 (> 1) (> 1) * 12 p a N(a > 1) a p a = p , 3, 5, 7 4, 6, p a, b p ab p a p b n n p p ab p a p b ( ) (1) n = 1 n ( ) (2) n ( ) n + 1 ( ) n + 1 n + 1 n ( ) n + 1 = q q ab q a q b ( ) a, b a, b ab a, b a, b a, b > 1 a b q r 1, r 2 a = q 1 q + r 1, 0 < r 1 < q b = q 2 q + r 2, 0 < r 2 < q *12
14 14 q a, q b r 1, r 2 0 r 1 r 2 = (a q 1 q)(b q 2 q) = ab + (q 1 q 2 q aq 2 bq 1 )q q ab q r 1 r 2 q 1 0 q 2 0 r 1 r 2 < ab ab q 1 = q 2 = 0 r 1 = a, r 2 = b qc = r 1 r 2 (c > 1) c p c = ps p r 1 r 2 0 < r 1, r 2 < q r 1 r 2 = ab = qc c < q p < q = n + 1 p r 1 p r 2 p r 1 pd = r 1 qps = qc = r 1 r 2 = pdr 2 p qs = dr 2 0 < d < r 1 < q q d q dr 2 q d q r 2 d, r 2 dr 2 < r 1 r 2 = ab ab p r 2 ( ) a, b n + 1 ( ) n ( ) 2.1. p, p 1,..., p r p p 1 p r i (1 i r) p = p i r (1) r = 1 p p 1 p = p 1 (2) r = n p p 1 p n+1 = p 1 (p 2 p n+1 ) p p 1 p p 2 p n+1 p p 1 p = p 1 p p 2 p n+1 * 13 i (2 i n + 1) p = p i (1), (2) r *13 n p 2 p n+1
15 n N, n > 1 n p 1 < p 2 < < p r n = p e 1 1 pe 2 2 pe r r (e i N, e i > 0) ( ) [ ] s p 1,... p s n = p 1 p 2 p s ( 2) (1) n = 1 ( 2) * 14 (2) 1,..., n ( 2) * 15 n + 1 ( 2) n + 1 n + 1 = n + 1 ( 2) n + 1 x, y > 1 n + 1 = xy x, y < n + 1 x, y n x, y ( ) x = p 1 p t, y = p t+1 p u n + 1 = xy = p 1 p t, p t+1 p u n + 1 ( 2) (1), (2) [ ] n > 1 ( ) 2 (1) n = 1 *14 n > 1 n = p 1 p 2 p s ( 2 ) n > 1 n = 1 ( 2 ) n = 2 n = 2 (2) n = 2 2 *15 n = 1 2 ( 2)
16 16 (2) 1,..., n ( ) n + 1 p 1 < p 2 < < p r q 1 < q 2 < < q r n + 1 = p e 1 1 pe 2 2 pe r r = q f 1 1 q f 2 2 q f s s (e i, f j N, e i, f j > 0) ( 3) p 1 q 1 * 16 p 1 q f 1 1 q f 2 2 q f s s 2.1 i p 1 = q i p 1 q 1 < q 2 <... p 1 = q 1 p 1 = q 1 ( 3) n + 1 p 1 = p e p e 2 2 pe r r = q f q f 2 2 q f s s ( 4) n+1 p 1 < n + 1 ( 4) r = s, p i = q i, e i = f i (1 i s) ( 3) 2 3 ( ) 2.4. n > 1 n p p 2 n n n = ml, (1 < m l < n) m p p 2 = p p m l = n n n n < , 3, 5, * 17 *16 p 1 > q 1 p i q j *17 103, 107, 109, 113
17 ( ). * 18 M p 1, p 2,..., p M L = p 1 p 2 p M + 1 L q q p 1, p 2,..., p M i p i = q L p 1 p 2 p M = 1 p 1 = q 1 L p 1, p 2,..., p M 2.7. L = = , 3, 5, 7, 11, = , , *
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