Elements of Discrete Mathematics Liu Mohapatra 3rd

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A B O U T T H E A U T H O R S

C lm n g L a n n g L iu

F rd b « r C h m g L aw (D a . e ) L il= re c e i. N M s B S c d e g re e ( 19 56 ) a l th c N a tio n a l C h e n g

K a a g L n i. Etsi in T a i. A n a n d h is s c ie n c e M a s te r a n d E le c u ic a l E n g in e e rin g d e g , e e s ( 1960 )

a n d M s s c ic n ec D o c u r d e g n e t l9 62 t al u e M a s sa c h u s e n s In s titu te o f T e c h n o lo g H e w a s o n

b e facu] D f th e M a q \ a r h ..q p m ln g im tc o f T e c h n o lo gy (l 96 19 7 1 a n d th e U n i, e rs ity n f

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a n d M E fr o m m o d a l E n gin e e rin g c o H e ge (n o w h IT i R o . Rk e la H e jo in e d th e fa c u lry u t th e

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h>?4' ) 1 4 " <

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GD NTENTS

p r efa c e to the T h ir d E d itio n

p r elin c e to th e F ir s t E d itio n

L is t o f S y m b o ls

1 S e t s a n d P r o p o s itio n s

1 1 In tr o d u c tio n 1

1 2 C o m b in a tio n s o f S e ts 5

1 3 F in ite a n d In fin ite S e ts 8

1 4 U n c o u n ta b ]y In fin ite S e ts 1 0

1 5 M a th e m a tic a l In d u c tio n 1 2

1 6 p r in cip le o f In clu sio n a n d E x c lu sio n 2 0

1 7 M u ltis e ts 2 5

p r o p o s i tion s 2 6 �

M L o gic a l C o n n e c tiv e s 2 6

0 C o n d itio n al a n d B ic o n ditio n als 2 9

\b / I w e lW e ll F o r m e d F o r m u la s 3 2

W T a u to lo gie s 3 3

m T h e o ry o f Infe r e n c e fo r S ta te m e n t C alc u lu s 3 6

1 1 4 1 V a lid ity U s fn g Ty u th T a ble s 36

1 1 4 2 R u le s o f In je r e n c e 3 6

1 1 4 3 c o n s is te n c y o$ P T e m is e s 4 o

1 1 6 T Iie state m e n t F u n c tio n , v a riM e A n d Q u a n tH 'Y *

1 1 6 1 p r e dic a te F o r m u la s 4 3

W §

l

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§v iilc o n te n ts

v a lid F o r m u la s a n d E q u iu a le n e e s 4 6

'

, % . T he o ry o f Irence To r p r e dic a te c a tc u tu s 4 8

1 1 8 3 F o r m u la s In u o tu iTtg M o r e th a n O n e Q ra n tifie r 4 9

1 1 9 M e th o d s o f P r o o f 5 o

1 2 o E u clid e a n A 1g o rith m 5 5

1 2 0 1 E u c lid e a n p r im e 55

p r o ble m s 5 6

2 p e r m u ta tio n s , C o m b in a tio n s ,

a n d

D is c r e te P r o b a b iliw 6 7

2 1 In tr o d u c tio n 6 7

2 2 1h e R u le s o f S u m a n d P ro d u c t 6 7

2 3 P e r m u ta tio n s 6 8

2 4 C o m h in n tin n s 7 4

2 5 G e n e ra tio n o f P e rm u ta tio n s a n d C o m b in a tio n s ? 8

2 6 D is c re te P ro b a b ility 8 J

2 7 C o n d itio n a l P r o b a b ility 8 7

2 8 B a y e s T h e o r e m 8 9

2 9 In fo r m a tio n a n d M u tu a l In fn r m a tio n 9 2

P m b l p m s 9 7

3 R e la tio n s a n d P u n c tio n s 1 0 6

3 1 In tr o d u c tio n 1 0 6

3 2 A R ela tio n al M o d e l fo r D a ta B a s e s 1 1 0

3 3 P ro p e r tie s o f B in a r y R e la tio n s 1 12

3 4 C lo s u r e o f R e la tio n s J r4

3 5 W a r sh a 1T s A 1g o rith m 1 1 7

3 6 E q u iv a le n c e R e la tio n s a n d P a r titio n s 1 2 o

3 7 P a r tia l O r d e rin g R ela tio n s a n d L a ttic e s 1 2 53 8 C h a in s a n d A n tic h a in s 13 1

3 9 A J o b S ch e d u lin g P r o b le m 13 43 1 0 C o m p a tib le R e la tio n 13 83 1 1 F u n c tio n s J3 9

3 1 2 C o m p o s itio n o f F u n c tio n s 1 4 2

3 1 3 In v e r tib le F u n c tio n 1 4 53 13 1 G r a p h o f In u e r s e F u n c tio n s 1 453 1 3 2 S tep s To r F in d in g th e ln u e r s e d a F u n c tio n 14 73 1 3 3 i r o p e r tie s pf ln u e r s e F u n c tio n 1 5 o

3 1 5 H a sh i n g J 8 h3 1 5 1 H a s h F u n c tio n 1 5 9

c o n tm ts l x l

3 1 5 2 S o m e P o p u ta r H a sh F u n ctio tts 15 93 1 5 3 C o llis io n R e s o lu tio n J 6 o

3 1 5 4 T e c h n iq u e s U s e d to M in im iz e C lu s te r in g 1 633 1 6 P ig e o n h o le P r in c ip le 1 6 4

3 1 6 1 E x te n d e d ffig e o n h o le P T in c ip le 16 5P r o b le m s 1 7 o

E x e r c is e s o n R e c u r s iu e F u n c tio n 1 8 1

E x e r c is e o n H a s h Fh n c tio n 1 8 1

E x e r c is e o n P ig e o n h o le P r in c ip le 1 8 1

P r o g r a m m in g E x e r c is e s 1 8 1

4 G r a p h s a n d P la n a r G r a p h s 1 8 a

4 1 In tr o d u c tio n 1 8 2

J - B a s ic T e rm in o lo g y 1 8 4

J 3 M u ltigr a p h s a n d W e igh te d G r a p h s 1 8 7

4 4 D ig r a p h s a n d R e la tio n s 1 9 0

4 5 R e p r e s e n ta tio n o f G r a p h s 1 9 1

4 5 1 in c id e n c e M a trix 19 1

4 5 2 A dja c e n c y M a tr ix 1 9 2

4 6 0 p e r a tio n s o n G r a ph s 1 9 3

7 P a th s a n d C ir c u its 1 9 6

4 8 G r a p h T r a v e r s a is 1 9 8

4 8 1 Tauersing a G r a p h 1 9 8

§ 2 D e p th F ir s t S e a r c h 19 8

l 8 3 B r e a d th F ir s t S e a r c h 2 0 1

4 8 4 p r o g r a m Im p le m e n ta tio n s 2 0 2

4 8 5 p r o g r a m th a t Im p le m e n ts D e p th F IT st

S e a r c h A lg o r ith m 2 0 5

4 8 6 p r o g r a m th a t Im p le m e n ts B r e a d th

F ir s t S e a r c h A lg o r ith m 2 0 7

S h o r te st p a th s in W e igh te d G ra p h s 2 0 9

E u le ria n P a th s a n d C ir c u its 2 1 2

T h , T r a v e li n g S a le sp e rs o n P r o b le m 2 2 1

4 3 F a c to r s o f a G r a p h 2 2 7

4 P la n a r G r a p h s 2 2 9

4 1 5 1 A ! p lic a tio n s Pf G r a p h C o lo r in g 2 3 8

4 15 2 C h r o m a tic P a r titio n in g 2 3 9

4 1 5 3 C h r o m a tic P o ly n o m ia l 2 3 9

P r o b le m s 2 4 0

p T o g r o m m in g E x e r cis e s

5 T e e s a n d C u t S e ts

T r e e s 2 5 5

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