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Elements of Discrete Mathematics Liu Mohapatra 3rd

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Page 1: Elements of Discrete Mathematics Liu Mohapatra 3rd

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Page 2: Elements of Discrete Mathematics Liu Mohapatra 3rd

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Page 3: 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

m in a is a l L rb a ru c h a m w gn ( 1 9 1998 ) w h e r e h e w a s��

w a s c da Um,ers, T o k o Jap a n L i K T H n n . r a r) C h a ir P m fe s s o r a t N a 【io n a l C e n tra l

L l u . m , t H . .w h .. T aj. m a n d M c i Y i c h i H o n o ta ry C h a ir P ro fe s s o r a t N a tio n a l T s in g H u a

H e is th e a u M a r a r d m a m bo r o f e igh t bo o k s a n d m o n o g r a ph s A n d o v e r 200 ce c h n ic a l

T is a tm m b c t o f A c a d e m ia s in in a F e H o w o f th e ! n 5 1 I t u r e o T E I e c tri c a l a n d E le c tro n ic s

E t g m m a n d a F e u o w o f the A s so c ia to n fo r C o m p u tin g M a c h in e ry

D a g a P r a s a d M o h a p a tr a

D e r g r p r w d M e b a w a re c e iv ed his p h D fro m In d ta n In s titu te o f T e c h n o io g K h a ru gp u r

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

D e p a n m - ! o f C o m p u rc r Sc ie n c e am d E n g in e e rin g a t th e N . tiu n a l In stitu te o f T . c h n u lo gy

R m r lz in 19 96 w h e r e hc is n o w * x s is m n l P ro fe s s o r H is re M a r c h .. t, r e s ts , n . In rlr d isc re lc

m u h r m a t, r s w ft. u e e n g in e e rin g re a l tim e sy u c m \ a n d d i, lr ib . Te d c o m p u tin g H e h a s

p u b h M c d m o te m thin ! pa pe rs in th e se fie ld s D r M n h u p a tr a h a s b e e n te a c h in g d is c re te

m a fh r m u u b au N rr R n u rk c b forepasnen y e a r s H c h a s re c e i. . d y o u . g S c ie n lis l A w a rd fo r

th c y e ar 2 006 b y O rissa B igya n A c ad e m y C u rre n tly h e lb a m e m be r ljj IE E E

V \

Ele m e n ts o f D is c re teM a the m a tic s

A C o m p u te r O rie n te d A pp ro a c h

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C h u n g m u n g L iu

L " " 1 13 , ¢ q f lli" " is

L ll" " C b a '"p«" W '

D u r g a P r a s a d M o h a p a tr a

T a ta M c G r a w H ill P u b lis h in g C o m p a n y L im ilM

N E W D E L H I

M L b r o w H ill u gi, r

N e w U ¢lh i N c w Y o r k S l im .. S a n F ra r' ' " , \ . \ lm d h w C « = =

K u . I · L u n , fn J r 1 im m lm m M U n d M c u c o C tt M ila n

S m Ju a n S a n tiw s iw a p o n d » r T o q rr

Page 4: Elements of Discrete Mathematics Liu Mohapatra 3rd

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TB U M c G ra w H ill

S p e el/ I In dM E d id o n 2008

Adrpd in In dia by anangemcm w ilh th e M c o m w H ill ( , n n p tm ic ) In r N c w Y , · rk

Sd l s T m to rle e In d ia P l k ls la n , N e p d B u w ll d e s h S ri l a n k » n u ll l\lm m n

C o pyrigh t 0 200 8 by T h c M c G ra w H ill C o m p n n ic s In v A ll R l# lu . RC ' " v c , I N , : " n l l . . I llll\

p u blic a tio n m ay be rc pr. d u c e d o r d is trih u lc d in m y l. '"1l , 1r l, \ a lly '"cl,ll . ' I, , l. . . Ill,

mechanical ph m o c o py in g N c o M in g o r t)lh c m is c n r . l, \r c d II\ . .ln lh , " . .r rr l��

n o l lim ited 1o In a n y n c rw o rk o r o th e r c k tm n ic 5 1''r a c 1'" m n . M n , I. .II '" Im , 1, I, n . I I. . r

di nra n c e learning

iS B N 13 : 97 8 0 0 7 0 66 9 13 0

IS B N 10 0-0 7 0 66 9 13 9

M a n a gin g D ire c to r : A ttiv S h u kla

G e n c n l M a n a ge r P u b lis hin g s E M & Tec h E d : \{bhu M , iji,lh J.

Spo n s o rin g E d ito r Sh a lin i Jhu

Ju n io r S po n s o rin g E d ito r N ilu n j- C ha k r a r a wS e n io r C o py E d ito r : 1]ip iM D � v

Ju n io r M a n a g e r Pto d u c tio n A n ja » R a N n

G e n e ra l M a n ag e r M a rk c lin g H igb e r E d u c a tio n & S c ho o l: M ic /lu u tJ C ru

Pto d u c t M a n a ge r : S E M & T e c h E d B /u G n n �s a n

C o n lm lle r Pm d u c lio n R o je n d e r P G ha m e taA ssI G e n e ra l M a n a g e r Pro d u c tio n B L D o g n a

In fo rm a tio n c o n M in e d in lh is w o rk h a s be e n o b ta in e d by T a ta M c G r a w H ill Fro m s o u rc e sbe lie v e d to b e re lia ble H o . E v e r n e ith e r T itta M c G ra w H ill n o r its a u th o m g u a r a n te e ch ea c c u ra c y o r c o m p le te n e s s o f a n y in fo rm a tio n p u blis h e d h e re in a n d n e ith e r T a ta M c G ra wH ill n o r iB a u b o rs sha ll be re spo n s ib le Fo r a n y e rro rs o m is s io n s o r d a m a g c x a ris in g o u l o fu se o f lh is in fo rm a tio n T h is w o rk is p u blish e d w ith th e u n d e rs ta n d in g tha t T a ta M c G ra wH ill a n d i« a u th o rs a re s u pplyin g in fo rm a tio n b u t u e n o t a tte m p tin g [o re n d e r e n g in e e rin g o ro M e r p ro h s , in n a l se rv ic e s Ir s u c h s e rv ic e s u e re q u ire d th e a s sis ta n c e o f o 1 a pp r o p r ia tep m ff io n a 】 sho u rd be so u gh t

T ype s e i a t 7t j C o m po se rs w Z 39 1 M a d ip u r N e w D e lh i 11006 3 a n d p rin te d a t R a s h u jy aC o v e r in in te r a t R a sh triy a prin te rsR Y L L C R A X R O JA L

M C G ro w H in t , p o n , e s

T u m y p {i re r\ls b e lo v e d w ife M a m i

u rn t d u u k h te r H a r a P r iy a

Vu u r lo v e e n c o u r a g e n te n t

tltttl x llp p o r t a re m y g re a te s t in s p ira tio n

D r D u r g a P r a s a d M o h a p a tr a

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h>?4' ) 1 4 " <

t! :

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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