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1 P P N p() N : p() N : p() N 3,4,5, L N : N : N p() N : p() N : p() N p() N p() p( ) N : p() k N : p(k) p( k ) k p(k) k k p( k ) k k k 5 k 5 N : p() p() p( )
2 p q p q p q p q p q p q p q p q p q x y p q t u r s p q p p q p q p q p p p q q p p p q P Q [] p, q P Q [] P Q P Q [ p q] P Q Q P [ q p] p q imply / mpl / (); (). +(tht). ; implictio
3 ( ) A A p q q p A( ) ( A) ( A) (A) A A p ( q r) ( p q) ( p r) ( A ) ( A) ( ) ( A ) A A p q p q ( A ) ( A) ( ) ( ) ( A ) ( ) ( A) ( A ) A A A A p, q P,Q p q p, q, r p q p, q ~ p ot, p p' p p c P p q or p q c p c c p q d p q ( P Q) ( PQ ) q p q p q p q p q q p p if d oly if q iff,,,, [] p q p q p q p q p q p q p q ( p q) q p ( q p) p q ( p q) q p ( q p) p q q p T:true F:fulse T F F T T T T T F T F T T F F F T T T T T F F T F F F F p, q p q ~ ~ ~~ T T T F F T T F F F T F F T T T F T F F T T T T [ p q] [ q p] p q q ~p T T T T T F F F F T T T F F T T p p p p q q p ( p q) r p ( q r) p p p ( q r) ( p q) ( p r) pq p q p p q p q p q p q p q p q p q
4 P f ( p, q, L, r), Q g( p, q, L, r) ( P Q) ( P Q) P( p, q) Q( p, q) P Q P Q P Q P Q [] IfThe Implies P Q P Q P Q Q P P Q Q P P Q P Q x U P Q P Q p( q( U p( q( p q P P Q Q P P Q Q P x x x x x x x x, p() [Def] p(, q(, r( x x : x x : p( x p( p,q x; p( p(,q( x : p( p( x [Def] X x x R ; x x N; p( ) p( q( x : p( q( x p( q( x U : p( q ( P Q U P Q P Q x P x Q x U : p( q( { x p( } { x q( } x : p( x : p( x : p( x : p( x[ p( q( ] x[ p( q( ] xp( X : p( ) xp( X : p( ) xp( X : p( ) xp( X : p( ) P Q, Q R P R
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6 p, q, r, p', q', r' R x px qx r p' x q' x r' x pp' qq' rr' p, q, r, s, p', q', r', s' R rx s, r' x s', x px q p' x q' pp' qq' rr' ss' rx s r' x s' px q p' x q' p q r s rx s r' x s' p' q' r' s' p, q, p', q' Q p q p' q' p p' q q' Ax x A E p, q, r, p', q', r' Q p q r p' q' r' Ax x ( x ) A E p p' q q' r r' Ax x ( A E) x s, t, u, s', t', u' R, s t s' t' s s' t t', s t s' t' s s' t t',, c s t uc s' t' u' c s s' t t' u u' p, q, p', q' R A pa qe p' A q' E p p' q q' A pa qe p' A q' E p p' q q' A ke A E e A A pa qe A p' A q' E p p' q q' Ax x Ay y A( x, ( x, ( x, A E A A E k x x k k x x ( k ) x k x [] s ke te s' t' e s s' t t'
7 x y x y x y x y x kx 3 x kx 3 x x 3k ( k )( x 3) k x 3 x 3 k 4 x 3 k x k x x 3 k 4 x 3 Y log X Y log X log X log X ( X ) log X log X X Y X Y logy log X X Y X Y X Y logy log X X X Y X X Y X X Y Y X log Y log X X Y X Y X. Y X kx 6 ( x3) log3 ( kx6) log3 ( x 3) log9 ( kx 6) log3 ( x 3) kx 6 ( x 3) x3 kx6 x3 A A 3A E O c d A A ( d) A ( d c) E O c d A 3A E O ( d) A ( d c) E 3A E ( d 3) A ( d c ) E O d c d 3 A E d 3 A A ke A 3A E O p, q, p', q' R k 3k k, A E, E A ke pa qe O A pa qe p' A q' E d 3 d c q p ; A E p p' q q' p d 3 A A ke p q A d c c d p, q d 3 A E, E A d c c d
8 p q A p q A' ' p q p q p q p q A A p q p q " p q" " q p" x : p( x : p( x : p( x : p( p q p p', q q' p q p q q p P Q x y x y, z N z ( " p q" " q p" q p p q q q q p p q p q q p ( ) P() N ; P( ) ( ) k N ;[ P( k ) P( k )] P( k) P( k) P(k) P( k) ( ) P(), P () N; P( ( ) kn;[ P( k) P( k ) P( k )] ) ( ) P() ( ) k N;[ P() L P( k) P( k )] N; P( ) P( ), P( k) P( k ) P() P( k ) P(k) N; P( ) P() p q P() P( k ) P( k), P( k ) P(k) p(k) k : p( k)
9 , ( 4 ) N ; ( ) ( ) k kn [ k, ( k ) ( k 4k ) k k k k ( k ) k k { k k ( k ( k ) } ( k ) ] 4k ) x N x ( ) x k ( x kn [( k kx ( k ( k ( k { ( k ) x} k ( ( { ( k ) x} x ( ( k { ( k) x} kx N : x y, N L 3 4 xy x y x y x y x y ( x xy : kn x k k x y y k k N ( ) q( ) p ( ) q () kn ( k ) q ( k ) ( k ), q ( k ) ) x] x k y k x k y k x k y k k k x y xy x y p p p ( k ) q ( k ) p p ( k ) q ( k ) [] k k k 6, 39 ( 4)
10 ( ) R ( ) ( ) ( ) def ( ) ( ) P( ( ) (, ) A Q R P ( ) P( x P( x P( ) A A AE EA E ( )( ) ( )( ) ( ) 3 3 ( ) A c 3c ( c)( c c c) G x ( x )( x x L x ) L x, y I xy x 3y ( x 3)( y ) 6 f ( ) ( ) ( ) f '( c) 5 x 3y 5 y x ( x, ( 3k, k), k I 3 3 x xyy 9 x D y N x, r r ( ) L L ( r,,,, ) r ( x x LL x x p p q c ( k) k ki ( k) ( x ) ( x ) ( A, x ) x )!!) x L x L( ) ( ) ( )( ) ( 3 ( x x x LL x L ( 3! ( ) h ( h) h h LL h h h N x R L ( ) ( )( ) L! 3! 3 e x x x x L! 3! ( c) ( c)( c ) ( c) []! p q r c c ( p q r ) p! q! r! p q r p q
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More information1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
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