离散数学教程(耿素云屈婉玲北京大学出版社)的全部习题解答.pdf

1 2 beta 4 3 WORKED OUT ANDTEXIFIED BY 4 E mail xiaoxinpan HONORED REVIEWER chouxiaoya September 1 2004 8 BA Thanks a trillion Bow 1CEDGF5H IKJ L M NPO5Q5R SPT U V SXW Y Z NX I a b N 2002 c 6 d e 1 fN 2003 c 1 d e 2gihEj 2kmlonqp rtsoumv wyx zi f N fI r q o N i umvf i umv N q u v J GSE S f o aNE email G K 3 a n beta Ho f vm G H NEHo o f N G m m fN H m g HG i m G m NE i f t r s v H s H 10 4 K iI 02CS a Iq a m f m f m BBS chouxiaoya S tedy S akaru S yitianxing S xuening a t q G l S i q zG Contents 1 m 3 2 f X 22 3 52 7 66 2 Chapter 1 1 1 2 2 1 4 9 16 25 36 49 64 81 100 121 144 169 196 3 1 8 27 64 4 0 1 2 5 2 3 6 a b c d e f g h i j k l m n o p q r s t u v w x y z 2 1 x y x y R x2 y2 1 2 4 k k Z 3 x x N x 8 4 x y z x y z N x2 y2 z2 5 x x R x2 5x 6 0 3 1 4 5 6 8 9 f G 4 1 Proof A B B C A B x x B x C A B A B A C x A A C Q E D 2 A a B a C a b A B B C A C 3 A a B a b C a b b c A B B 3 C A C 4 A a B a b C a b b c A B B C A C 5 A a B a C a A B B C A C 6 1 0 1 a b c 2 a b a c b c 3 a b c a b c a b a c b c a b c 2 0 1 1 2 3 2 1 2 3 1 2 3 1 2 3 3 0 1 2 4 0 1 1 2 1 2 5 0 1 1 1 2 1 1 1 1 1 1 7 8 1 4 2 1 3 5 3 2 3 4 5 4 2 3 4 5 5 4 6 1 1 4 4 9 1 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 12 15 16 18 21 24 27 30 32 64 2 3 7 6 5 4 3 2 1 4 5 4 7 6 5 4 3 2 1 0 3 4 5 6 9 12 15 18 21 24 27 30 10 P A a PP A a a 1 A B B 1 A B A B 8 1 A B A B A B 9 1 A B 021 A B C D E F A B A A B Q E D 12 G HJI K L Lemma 1 1M2N OP m A P B Q A B A A B Proof R S 11TU Q E D Lemma 1 2M2N OP m A P B Q A B A B Proof A B x x A B 5 x x A B V W X Y Z x x A x B M x x A x B x x A x B T 2a bdcfe2g 1 x x A x B h i Y Z A B Q E D 1 j A B A C Ak l mnkA B C Proof A B A C A A B C A bdcfe2g 1 A B C Lemma 1 1 A B C 3 1 Q E D 2 j A B A C k l mnkA B C Proof A B A C A B C bdcfe2g 1 A B C Lemma 1 2 Q E D 3 j A B A C k l mnkA B C Proof A B A C A B C bdcfe2g 1 A B C Lemma 1 2 Q E D 4 j A B A C Ak l mnkA B C Proof A B A C A A B C A bdcfe2g 1 A B C Lemma 1 1 Q E D 13 1 G HJI K L Lemma 1 3M2N OP m A P B Q A B A P A B B Proof 6 M o N Ox x A B x A x B m U x A T 2a prq 1 Y z Proof G H 7 8 kA C y A B C A B C 021 A C B 3 1J 021 A C B 021 A B Lemma 1 1 A B 9 1 A B A C A C A B C 8 1 A B C bdcfe2g 1 A B C 021 w H J 2 x x A x C y x JX oB 2 x A B C P x A B C 2 U A B C A B C 4 5U 7 Q E D 14 Proof B E B 9 1 A A B 1 A B A B 8 1 A C A C A A C 8 1 E C 1 C 9 1 Q E D 15 A B D G C F H 16 1 3 4 3 4 2 3 17 1 2 3 18 1 1 2 3 2 3 4 19 1 A B 2 A 3 B 20 G HJI K L Lemma 1 5M2N OP m A B C D Q A B C D A C B D Proof 8 M o N Ox x A C x A x C m v2 x A x C x A x B x C x D 1 A C C 1 A C A C 8 1 B C B C T 1 Q E D 21 1 j A B Ak l mnkA B Proof A B A x x A B x A Y z2 x x A x B x A m U x x A x B x A x A x A x B Y 2 3 W x x A x B x A x A x A x B h i Y Z x x A x B x A x A x A x B T 2a bdcfe2g 1 x x A x A x B x A x A x B T 2a 021 9 x x A x A x B x A x A x A x B T 2a 8 1 x 1 x B 1 x A x B T 2a 1 x 1 1 x A x B T 2a 6 1 x x A x B T 2aJ9 1 x x A x B h i Y Z A B Q E D 2 j A B Ak l mnkB A Proof A B A x x A B x A Y z2 x x A x B x A m v2 x x A x B x A x A x A x B Y 2 3 W x x A x B x A x A x A x B h i Y Z x x A x B x A x A x A x B T 2a bdcfe2g 1 x x A x A x B x A x A x A x B T 2a 8 1 x 1 x B x A 1 x B T 2a 1 x 1 x B x A 1 T 2a 6 1 x x B x A T 2aJ9 1 x x B x A h i Y Z B A Q E D 3 j A B Ak l mnkB Proof 7 8 B A B A B A 1 7 4 A B A B B 1 7 4 A A B 1 7 5 A A B 1 7 2 A A A B A 1 7 5 10 Q E D 4 j A B A Bk l mnkA B Proof 7 8 A B A B A A A B A Y 1 A A Y 1 A B A B A B A B A A A B 1 A A B A B A B A A B 3 1 A B Y 1 A B B Y 1 A B B 3 1 A B B A B A B B 1 Q E D 22 1 Lemma 1 5 P Lemma 1 6 2 j A a C b B D a b A B C D A B 6 C D A C a b B a b c D a b d A B C D A B 6 C D 23 Proof 2 U x B x C x B x C x B x C y x A 2 x A B P x A C A B A C4 5 x A x A B P x A C J9 U A B A C4 5U M x B x C s r A B C D Q A B A C B C Q E D 24 8 HU A B C A C B A B C A B C P A B C A C B C G HU A B C A C B 11 Proof A B C A B C 021 A B C 3 1 A C B 021 A C B 3 1 A C B 021 Q E D w HU A B C A B C Proof A B C A B C 021 A B C 3 1 A B C bdcfe2g 1 A B C 021 Q E D HU A B C A C B C Proof A B C A B C 021 A B C 9 1 A B C A 6 1 A B C A C C 4 521 A C B A C C 021 A C B C 8 1 A C B C bdcfe2g 1 A C B C 021 Q E D 25 1 A 2 A B 3 B A 26 1 Proof G H F A C B C X QM o N Ox x A B x A x B x A x B 12 x A x C x B x C 1 z z A x z z B x z T 2a t u21 z z A x z z B x z h i Y Z z z A z B x z x z T 2a 3 1J 021 z z A z B x z T 2a Y 1 z z A z B x z T 2a bdcfe2g 1 z z A z B x z h i Y Z z z A B x z m U x A B Q E D 30 1 Proof M o N OP m x x P A P B x P A x P B m U x A x B x A B S 26 T S 2 T 3 x P A B Q E D 2 G HJI K L Lemma 1 7M2N OP m A B C Q C A C A B 15 C B C A B Proof C A C A A A B Lemma 1 4 C A B a 9s E HU C B C A B Q E D Lemma 1 8M2N OP m A B C Q C A C B C A B Proof C A B C A C B C A C A B C B C A B Lemma 1 7 C A B C A B C A B T 2a Y 1 Q E D w H x2TU Proof M o N OP m x x P A P B x P A x P B m v2 x A x B x A B Lemma 1 8 x P A B Q E D 31 193 32 10 O 55v A B A C B C Z A B A C B C 2 A B C Zr T 2 r T 2 s N2 2 2 u 9 2v T m 2 33 lim k Ak lim k Ak 0 1 34 lim k Bk 0 1 lim k Bk 16 35 lim k Ak 0 lim k Ak 0 36 G H K L Lemma 1 9M2N O 2W F P G Q 2 n n N k k N k n F k G k n1 n1 N k k N k n1 F k n2 n2 N k k N k n2 G k Proof 2W2 x A x B x xA x xB x P 02 E U 7 8 n max n1 n2 k k N k n k n1 k n2 w X E Hr Q E D w H x2T 1 G H S KU lim k Ak lim k Bk lim k A k Bk Proof M o N Ox x lim k Ak lim k Bk x lim k Ak x lim k Bk m v2 n0 n0 N k k N k n0 x Ak n0 n0 N k k N k n0 x Bk U n0 n0 N k k N k n0 x Ak n0 N k k N k n0 x Bk V W 8 Y Z n0 n0 N k k N k n0 x Ak k k N k n0 x Bk T 2a 8 1 n0 n0 N k k N k n0 x Ak k N k n0 x Bk 2W2 1 n0 n0 N k k N k n0 x Ak k N k n0 x Bk h i Y Z 2 Q n1 n2 n1 n2 n1 n2 x Ak x Bk x limk Ak Bk 4 5U 2 ha bi hc di a a b c c d 3 ha bi hc di a a b c c d a a b c c d 4 ha bi a a b a a b a 5 ha bi ha bi a a b 6 ha b ci hha bi ci ha bi a a b 7 ha bi ha bi a 8 ha bi 1 hb ai hb ai b b 3 PKQSRUTWV XZYha hb cii a a b b c 6 ha b ci a a b a a b c 4 VKXh i ha a i a a a 3 5 7 Q R W PKQSRUT 5 1 A B 2 A B A B 3 A B C 6 1 Proof x y hx yi A C B D x A y C x B y D aZb 22 x A x B x A y D y C x B y C y D cSdZe f g h i x A x B y C y D cSdZe fkjml8i x A B C D aZb knoY A C B D A B C D T Q E D 2 Proof x y hx yi A B C D x A x B y C y D aZb x A y C x B y D cSdZe fUp q i x A y C x B cSdZe fkjml8i x A y C x B x A y C x D cSdZe fUr s i x A y C x B D cSdZe f g h i x A y C x B y D aZb x A y C x B y D cSdZe fktvuxw ySi hx yi A C hx yi B D z SaZb hx yi A C hx yi B D aZb hx yi A C B D S SaZb knoY A B C D A C B D T Q E D 7 1 Proof x y hx yi A B C x A B y C z SaZb x A x B y C S SaZb x A x B y C aZb x A x B y C 0 cSdZe f 8 i x A x B y C x A 0 cSdZe f Si x A x B y C x A y C y C cSdZe f 4 i x A y C x B y C cSdZe f g h i x A y C x B y C cSdZe fktvuxw ySi hx yi A C hx yi B C z SaZb hx yi A C hx yi B C aZb 23 hx yi A C B C S SaZb knoY A B C A C B C T Q E D 2 Proof A B C A B B A C Z S A B C B A C z A C B C B C A C d 30 A C B C Z S Y C U S k m Z S aWb Z T Z Z Z W d cSdZe f m T lemma 0 nA B ATU A S kQSRUT 9 oYU A B Z o S U WTK C 8 A B UPo 8 Z o A B WPU K K W S UT X2 A B 2mn T A B U noY R1 R2 ha 1i R3 hb 1i R4 hc 1i R5 ha 1i hb 1i R6 ha 1i hc 1i R7 hb 1i hc 1i R8 ha 1i hb 1i hc 1i B A U noY R1 R2 h1 ai R3 h1 bi R4 h1 ci R5 h1 ai h1 bi R6 h1 ai h1 ci R7 h1 bi h1 ci R8 h1 ai h1 bi h1 ci 10 1 Proof a R S S R a S Sbk aZb 1 K starfi sh 8 24 S0 S S0 R S S0 a S Sbk aZb x x y S x x y R S x x y a S R S U x x y S x x y R S x S x y a S x x y x x y R S x a S S x y a S cSdZe fktvuxw ySi x x y x x y R a x a x a y x x y x x y R a x a y cSdZe f 30 Si i a domR a ranR aZb ZaZb a fl dR ZaZb kn fl dR R Q E D 11 1 R1 R2 ha bi ha ci hb di hc ci hc di hd bi hd di R1 R2 hb di R1 R2 R1 R2 R1 R2 ha bi ha ci hc ci hc di hd bi hd di 2 domR1 a b c domR2 a b d dom R1 R2 domR1 domR2 a b c d 3 ranR1 b c d ranR2 b c d ranR1 ranR2 b c d 4 R1 A ha bi hc ci hc di R1 c hc ci hc di R1 R2 A ha bi ha ci hc ci hc di R2 A ha ci 5 R1 A b c d R2 A c R1 R2 A 6 R1 R2 ha ci ha di hd di R2 R1 ha di hb bi hb di hc bi hc di R1 R1 ha di hc ci hc di 12 1 R 1 h i h i h i 25 2 R R h i h i h i h i 3 R R h i h i R h i R R h i h i h i 4 R R R R ranR 5 domR ranR fl dR domR ranR 13 1 Proof R S U k S R R 1 S U KT Lemma 1 4 R R R 1 R R 1 R T W hx yi noY hx yi R R 1 hx yi R hx yi R 1 aZb hy xi R 1 hy xi R U U 8aZb hy xi R 1 R aZb hy xi R R 1 p q i k R R 1 Z U ST W R Z U R0 noY hx yi hx yi R R 1 hx yi R hx yi R 1 aZb hx yi R hy xi R U U 8aZb hx yi R0 hy xi R0 R R0 hx yi R0 hx yi R0 hy xi R0 hy xi R0 cSdZe fS i hx yi R0 hx yi R0 1 hy xi R0 hy xi R0 1 cSdZe f 8 i hx yi R0 hx yi R0 hx yi R0 hy xi R0 hy xi R0 hy xi R0 hy xi R0 hx yi R0 R0 Z U hx yi R0 hy xi R0 hx yi R0 hy xi R0 0 Z hx yi R0 hy xi R0 cSdZe fS i 26 hx yi R0 cSdZe fkjml8i nR R 1 R0 T S R R 1 R Z U KT Q E D 14 1 R h1 2i h2 3i 2 Proof R2 R x z hx zi R2 hx zi R aZb x z hx zi R2 hx zi R aQZaZb x z y hx yi R hy zi R hx zi R a o x z y hx yi R hy zi R hx zi R S S U o x y z hx yi R hy zi R hx zi R S S op q i 2 x y z hx yi R hy zi R hx zi R Z o R ST S SaZb Q E D 15 A Z Y Rn Y 2 Y W x nx 6 x 8 hx xi R T 2 Y W x nhx xi R T 0 Z UY Y Z W mT Sn Y 2 YW K A Z knA P A A 6 K A A A 6 hA Ai S T 2 Y P A h i S T Z UY Y nhA i S h Ai S hA Ai S T 2 27 8 27 Tn Y 2 Y nh i T T 2 Y nA P A A A A hA Ai T T Z UY Y nh Ai T hA i T h i T T AX Z Y Rn Y 2 Y 6 T 2 Y P x hx xi R T Z UY Y V X n Z W 8 RX k U KT P x y z hx yi R hy zi RTK Z SYhx yi R hy zi R hx zi R oT Sn Y 2 Yh i S T 2 Yh i S T Z UY Y x y z hx yi S hy zi S x y z hx zi S T Tn Y 2 Y A h i T T 2 Yh i T T Z UY Y x y z hx yi T hy zi T x y z hx zi T T 16 1 R h0 10i h1 9i h2 8i h3 7i h4 6i h5 5i h6 4i h7 3i h8 2i h9 1i h10 0i S h0 4i h3 3i h6 2i h9 1i h12 0i 2 Rn Y 2 Yh0 0i R T 2 Y h5 5i R T Z UY s mT S Z UYh0 10i R h10 0i 0 6 10 T 28 0 Yh0 10i R h10 0i h0 0i R T Sn Y 2 Yh0 0i S T 2 Yh3 3i S T 0 Z UYh0 4i S h4 0i S T S Z UY Yh12 0i S h0 4i S h12 4i S T 17 M R 1111 1110 1110 1001 1 23 4 G R Rn Y 2 Y nh3 3i R T 2 Y nh1 1i R T Z UY aZb T S Z UY nh1 2i R h2 1i R 1 6 2 T 0 Y nh2 0i R h0 3i R h2 3i R T 18 R1 ha ai ha bi hb bi hc bi hc ci R2 ha bi hb ci R3 ha bi ha ci hc ai hc bi R4 ha ai hb bi hc ci M R1 110 010 011 M R2 010 001 000 M R3 011 000 110 M R4 100 010 001 R1 K Y 2 Y 0 mT 2 Yha ai R1 T 0 Z UYha bi R1 hb ai R1 T S Z UY 0 Y 0 mT R2 K Y 2 Yha ai R2 2 Y 0 mT 0 Z UYha bi R2 hb ai R2 T S Z UY 0 Yha bi R2 hb ci R2 ha ci R2 T R3 K Y 2 Yha ai R3 2 Y 0 Yha ci R3 hc ai R3 ha ai R3 T R4 K Y 2 Y 0 mT 2 Yha ai R4 T Z UY 0 mT S Z UY 0 Y 0 mT 19 R1 ha ai ha bi hb ai hb bi hb ci hc ai hc ci R2 ha ai ha bi hb bi hc ai hc bi R3 ha bi ha ci hb ai hb ci hc ai hc bi R4 ha ai ha bi ha ci hb ci hc ai a b c G R1 a b c G R2 a b c G R3 a b c G R4 30 R1 K Y 2 Y 0 Yhc ai R1 ha bi R1 hc bi R1 T R2 K Y 2 Yhc ci R2 2 Yha ai R2 0 Z UYha bi R2 hb ai R2 T S Z UY 0 Y 0 mT R3 K Y 2 Yha ai R3 2 Y 0 mT Z UY 0 Yha bi R3 hb ai R3 ha ai R3 T R4 K Y 2 Yhb bi R4 T 2 Yha ai R4 T 0 Z UYha bi R4 hb ai R4 T S Z UYha ci R4 hc ai R4 a 6 c 0 Yhc ai R4 ha bi R4 hc bi R4 T 20 M R 100 010 011 000 a b c d 1 2 3 G R1 31 21 M R1 1 10 001 M R2 01 01 00 M R2 R1 M R1 M R2 1 10 001 01 01 00 0 1 00 oYR2 R1 h1 i T 22 Proof Zak 2 19 3 aP p q i CYc di C a bi C ca 0 UT W ha1 b1i a2 b2ii ha2 b2i a3 b3ii R a1 b1i C a3 b3i C a1a2 0 a2a3 0T ba Z ab 0 sgn a sgn b K noY 3 a1a2 0 a2a3 sgn a1 sgn a2 sgn a2 sgn a3 a sgn a1 sgn a3 S a1a3 0 a knha1 b1i a3 b3ii R T S C R U KT Q E D C R a bi a bi C a 0 a bi a bi C a b yi w T 34 1 P T Proof 3 C R1 R2 6Zn x x A hx xi R1 R2 o xW y0 amb C x x A hx xi R1 R2 T A k Z x A hx xi R1 R2 R1 R2 P 2 U k SV P o U KT Q E D 2 P T Proof 3 R1 R2 6Kn x x A hx xi R1 hx xi R2 S Z W 70 a b k x x A hx xi R1 R2 T A Z b x A hx xi R1 R2 K R1 R2P 2 U k SV P o U KT S YR2 R1P o U KT Q E D 3 P Sa T Proof 2UY 3 A a b c R1 E R2 ha ci hc ai IA Z R1 R29 U o WV P U kTU W Yr R2 R1 zZT 2 Y 3 R1 R2 IA R1 R2 r R1 R2 r R2 R1 IA9k o U KT R1 R29k o U K r R1 R2 r R2 R1 P Sa o U KT Q E D 4 P Sa T Proof 2UY 3 A a b c R1 ha bi hb ai IA R2 ha ci hc ai IA Z R1 R2 9 C WT R1 R2 ha bi ha ci hb ai hc ai hc bi IAPZ n hc bi R1 R2 hb ci R1 R2 SV P o U KT Z S YR2 R1 zZT 2 Y 3 R1 R2 IA R1 R2 R1 R2 R2 R1 IA9k o U KT R1 R29k o U K R1 R2 R2 R1P Sa o U KT Q E D 3 sgn signum sgn x 1x 0 40 35 Proof 5 m R Z U ST x y A hx yi R hx yi R 1 cSdZe f 8 i hx yi R hx xi R R 2 hy xi R d H 2 m R ST x y z A hx yi R hy zi R hy xi R hy zi R R Z U hx zi R d H 2 S Z U N k W W 8 R A 0 o KT Q E D 36 Proof 1 Bik 7 Z 0 2 T 2 W j k 1 2 m j 6 k noY Bij Bik Aij B Aik B aZb Aij Aik B B 0 Si p q i B B 1 g Si 3 A B 1 B 1 A g n i 1 Ai B Sbk aZb n i 1 Ai B g h i m k 1 Aik B Ai B nm Z k m k 1 Bik aZb 2 Sbk aZb S 2 N ZaZb2 17 K T 2 A B K S g T 41 Q E D 37 Proof 0kRU mT Q E D A R 1 6 11 16 2 7 12 17 3 8 13 18 4 9 14 19 5 10 15 20 38 Proof 1 A aZbK A T 2 Ai1 Bj1 Ai2 Bj2 A Ai1 Bj1 Ai2 Bj26 x x Ai1 Bj1 Ai2 Bj2 aZb x x Ai1 x Bj1 x Ai2 x Bj2 p aZb x x Ai1 x Ai2 x Bj1 x Bj2 cSdZe fUp q i x x Ai1 x Ai2 x x Bj1 x Bj2 Z Z Za i x x Ai1 Ai2 x x Bj1 Bj2 p aZb Ai1 Ai26 Bj1 Bj26 aZb Ai1 Ai2 Bj1 Bj2 1 2 g Ai1 Bj1 Ai2 Bj2 p W Ai1 Bj1 Ai2 Bj2 R Ai1 Bj16 Ai2 Bj2 Ai1 Bj1 Ai2 Bj2 T 3 A A A i 1 2 1 2 g m i 1 Ai n j 1 Bj 1 2aZb 1 i m 1 j n Ai Bj g h i A AaZb S Ai Bj A Lemma 1 3 nAi Bj Ai Ai Bj BjT U A k N Z g 9 1 2 b C A 1 s Q 2 s ST 42 Q E D 39 1 R h1 2i h1 3i h2 1i h2 3i h3 1i h3 2i IA A R 1 2 3 4 2 1 A R 1 1 2 3 4 R 1 IA 2 A R 2 1 2 3 4 R 2 h1 2i h2 1i IA 3 A R 3 1 3 2 4 R 3 h1 3i h3 1i IA 4 A R 4 1 2 3 4 R 4 h2 3i h3 2i IA 5 A R 5 R 5 R h1 2i h1 3i h2 1i h2 3i h3 1i h3 2i IA 40 Proof 6 Y sP aZb0nC A A A R1 B B A R2 A B 8 knoY x y A hx yi R1 A A A R1 x A y A aZb A A R1 x A y A D E A A R1 x A y A B B A R2 A B LIM B A A R1 x A y A B A R2 A B S S U o A A R1 x A y A B A R2 A B D E B A R2 x A y A A B cSdZe fkjml8i p q i B A R2 x B y B W aZb B B A R2 x B y B hx yi R2 aZb kgS Y m A x y A A R1 x A y A B B A R2 x B y B T A x y A A R1 x A y A hx yi R1 aZb hx yi R1 R1 R2 LIM hx yi R2 W aZb 43 B B A R2 x B y B aZb Q E D 41 Proof 5 YR3 2 ST x y hx yi A B x A y B z SaZb hx xi R1 hy yi R2 R1 R2 2 hhx yi hx yii R3 R3aZb C YR3 Z U ST x1 x2 y1 y2 hhx1 y1i hx2 y2ii R3 hx1 x2i R1 hy1 y2i R2 R3aZb hx2 x1i R1 hy2 y1i R2 R1 R2 Z U hhx2 y2i hx1 y1ii R3 R3aZb ST x1 x2 x3 y1 y2 y3 hhx1 y1i hx2 y2ii R3 hhx2 y2i hx3 y3ii R3 hx1 x2i R1 hy1 y2i R2 hx2 x3i R1 hy2 y3i R2 R3aZb hx1 x2i R1 hx2 x3i R1 hy1 y2i R2 hy2 y3i R2 cSdZe fUp q i hx1 x3i R1 hy1 y3i R2 R1 R2 hhx1 y1i hx3 y3ii R3 R3aZb k C R3 o U KT Q E D 42 W X W m Ip gSnyu oTa W S Z T 2 hA 42i KP Z oTa d e a b c e m oT 46 B K XZY k lcm 1 2 10 k N UXlcm 1 2 10 2520 T 1 B N C T 47 45 1 23 69 1827 54 4 A n7 N N Y B1 1 2 6 18 54 B2 1 3 9 27 54 B3 1 3 6 18 54 B4 1 3 9 18 54 T Zak 2 31 2 C A g Q5 ZP pP op mT 48 1 Proof 5 YR B 2 ST x x B x A B A hx xi R R ST x y z hx yi R B hy zi R B hx yi R B B hy zi R B B R BaZb hx yi R hx yi B B hy zi R hy zi B B p aZb hx yi R x B y B hy zi R y B z B z SaZb 46 hx yi R hy zi R x B y B z B cSdZe fUp q i i hx zi R x B y B z B R hx zi R x B z B cSdZe fkjml8i hx zi R hx zi B B z SaZb hx zi R B R BaZb S k R B I U KT Q E D 2 Proof 5 YR B 2 ST x x B x B x B cSdZe fS i x A x B B A hx xi R R 2 hx xi R hx xi B B z SaZb hx xi R B B p aZb hx xi R B R BaZb C YR B S Z U ST x y hx yi R B hy xi R B hx yi R B B hy xi R B B R BaZb hx yi R hx yi B B hy xi R hy xi B B p aZb hx yi R x B y B hy xi R y B x B z SaZb hx yi R hy xi R cSdZe fkjml8i x y R S Z U ST m W 1 8 S k C AT R A U y y C x x C hy xi R TU k C B S L k x y n x B y BTS n hy xi R BT k C R B B U KT Q E D 49 Proof 5 YR 2 ST x y hx yi A B x A y B z SaZb hy yi R2 hx xi R1 R1 R2 ST x1 x2 x3 y1 y2 y3 hhx1 y1i hx2 y2ii R hhx2 y2i hx3 y3ii R hy1 y2i R2 hx1 x2i R1 y1 y2 hy2 y3i R2 hx2 x3i R1 y2 y3 RaZb hy1 y2i R2 hy2 y3i R2 hy1 y2i R2 hx2 x3i R1 y2 y3 hx1 x2i R1 y1 y2 hy2 y3i R2 hx1 x2i R1 y1 y2 hx2 x3i R1 y2 y3 cSdZe f g h i g Y W 4W zZT 1W z R2 hy1 y3i R2 T 48 2W z hy1 y2i R2 y2 y3 hy1 y3i R2 T 3W z hy2 y3i R2 y1 y2 hy1 y3i R2 T 4W z R1 hx1 x3i R1 y1 y3 T 4W z 9S hhx1 y1i hx3 y3ii RT R QSRUT S nR I U KT Q E D 50 Proof 5 YR 2 ST x y hx yi A B x A y B z SaZb hx xi R1 hy yi R2 R1 R2 2 hhx yi hx yii R RaZb C YR S Z U ST x1 x2 y1 y2 hhx1 y1i hx2 y2ii R hhx2 y2i hx1 y1ii R hx1 x2i R1 hy1 y2i R2 hx2 x1i R1 hy2 y1i R2 RaZb hx1 x2i R1 hx2 x1i R1 hy1 y2i R2 hy2 y1i R2 cSdZe fUp q i x1 x2 y1 y2 R1 R2 S Z U hx1 y1i hx2 y2i Zak 2 1 hhx1 y1i hx2 y2ii hhx2 y2i hx1 y1ii Zak 2 1 ST x1 x2 x3 y1 y2 x3 hhx1 y1i hx2 y2ii R hhx2 y2i hx3 y3ii R hx1 x2i R1 hy1 y2i R2 hx2 x3i R1 hy2 y3i R2 RaZb hx1 x2i R1 hx2 x3i R1 hy1 y2i R2 hy2 y3i R2 cSdZe fUp q i hx1 x3i R1 hy1 y3i R2 R1 R2 hhx1 y1i hx3 y3ii R RaZb k C R U KT Q E D 51 49 1 2 4 8 R1 1 23 6 R2 h1 1i h1 2ih1 3i h1 6i h2 1i h2 2ih2 3i h2 6i h4 1i h4 2ih4 3i h4 6i h8 1i h8 2ih8 3i h8 6i R 52 19 T 53 5 W S 0 mT Lemma 2 5 W A A U KT Proof 2 W Y A A x A x A cSdZe f i x A x A Z o x x A x A A A aZb S Z S Y A B A A B B A x x A x B x x B x A aZb x x A x B x B x A Zg h o x x A x B aZb A B W SaZb Y A B C A A B B C x x A x B x x B x C aZb x x A x B x B x C Zg h o 50 x x A x C RIS A C aZb S A U KT Q E D C kn m o S 0 CT 40d7 K 2 28 o 9 n hx1 y1i 6 hx2 y2i T V m f Z n domf PU K n S T f domf T C Y domf f T UP G n domf f T b hx1 y1i hx2 y2i f x1 x2 hx1 y1i 6 hx2 y2iT Zak 2 1 N60ny16 y2T q A B A B Y A B A B T Q E D C 8 T Proof 53 f g A B oY x y z hx yi f g hx zi f g hx yi f hx yi g hx zi f hx zi g p aZb hx yi f hx zi f cSdZe fkjml8i y z f 8 f g S 8 T Q E D 0 2 Y f g A B s t f g T Proof kgS F GZT 76 UT f g A B Y Zak 1 3 k Y f g f g f g T Lemma 3 1 f g f g A B Y f g f g A T W oY f g f g A T Lemma 3 2 oYf g T Q E D 0 3 Y f g 8 f g A B f g T 5 Yf g 8 f g A B T Proof 5 kgS UT 8 kgS 8 T C 76 UT f g 8 C Y dom f g domf domg Zak 2 3 1 A A f g A B A i 8 WaZb0noYf g A B T knoYf g 8 f g A B T S oYf g 8 f g A B T Q E D C Yf g A B f g T Proof 5 kgS UT 54 f g Y f g f f f g f i A B d H noYf g f g A B T C 76 UT f g A B Y f g dom f g Lemma 3 1 A f g A B kn f g A T Zak 1 3 k Y f g f g f g T L 0 oY f g f g A T Lemma 3 2 oYf g T kgS UT 76 UT WP G KnA B 6 T g Y A B 6 ubW k T A n A BT A B 6 T B 6 W N k a B 3 f hx ai x A TNqo A U ZX A X f 8 T A B 8 nf A B T A B T G T YA B 6 A B 6 T q 10 d T Lemma 3 4 H f g A B f g s domg domf f g T Proof d H f g Yg f T x y hx yi g x domg domaZb x domf domg domf z hx zi f domaZb z hx zi f hx zi f cSdZe fS i z hx zi f hx zi g f g z h