Skripta Diskretna Matematika

7/30/2019 Skripta Diskretna Matematika

1/51

D I S K R E T N A M A T E M A T I K A

V J E B E

o o j k d j l o m n k n z d k o j j n n j m z h k n m m k

k d m k o j o d n P H H W F G P H I H F

k m o o m o d n m o l j m n j d D l n m o z m j n j F

d j n j X

w e y t i w e y r x s w e s t e f e s F

x k l d l X

s e x u s t e x D w e u y f y s D s e x q e e x w e s y f i v t e p e F

D P V F I H F P H H W F


2/51

d j

S a d r a j

1 O s n o v n i p r i n c i p i p r e b r o j a v a n j a 3

2 P e r m u t a c i j e s k u p o v a 6

3 K o m b i n a c i j e s k u p o v a 1 0

4 P e r m u t a c i j e i k o m b i n a c i j e s p o n a v l j a n j e m 1 4

5 K o m b i n a t o r n o d o k a z i v a n j e i d e n t i t e t a 1 8

S F I m j F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I V

S F P o l o n j p o n j o j Fn F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F I W

6 R e k u r z i v n e r e l a c i j e 2 1

T F I r o m o n l n n k z j k o n n n m k o ( j n m F F F F F F F F F F F F F F F F F F F F F F P I

T F P x o m o n l n n k z j k o n n n m k o ( j n m F F F F F F F F F F F F F F F F F F F F F P P

T F Q w o d l n j o m o k z j F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F P Q

7 F u n k c i j e i z v o d n i c e 2 5

U F I y n n k j z o d n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F P S

U F P i k o n n j l n n k j z o d n F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Q H

8 F o r m u l a u k l j u i v a n j a i i s k l j u i v a n j a 3 3

9 T e o r i j a g r a f o v a 3 6

W F I o d F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Q T

W F P o z n o o F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Q W

W F Q n k o F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F R Q

W F R l n n o F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F R S

W F S e m d o j l F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F R U

W F T i l o r m l o n o o F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F R V

P


3/51

y n o n n o j n j

1 O s n o v n i p r i n c i p i p r e b r o j a v a n j a

k o n n k AD |A| m o o z n o j l m n @ k d n l A k e F

T e o r e m 1 . 1 . P r a v i l o s u m e .

f o j l m n n j m o n o o m d j n k n k o Ai D i {1, 2, . . . , n}(Ai Aj = , i, j {1, 2, . . . , n} , i = j) j d n k j m o j l m n k o Ai F

ni=1

Ai

=ni=1

|Ai|.

P r i m j e r 1 . 1 . s z d e d f m o m o d o o d o m D o m l o n o m F o o j d m o k D d o n

z n F x k o l k o n n m o m o d o z d e d f c

R j e e n j e . d m o l j d k o D AB n k j k z d e d f D M n k j k m o k D

Zz n D

C o n z d e d f F t n o j d j d

M

Z = Z

C = C

M =

,

o m o m o m j n l o m D o d n o n o

|AB| = |M Z C| = |M| + |Z| + |C| = 2 + 3 + 2 = 7.

D e f i n i c i j a 1 . 2 . x k j n ND i {1, 2, . . . , n} D Ai k o D A1 A2 . . . An o z n m o K a r t e z i j e v p r o d u k t k o

Ai DA1 A2 . . . An = {(a1, a2, . . . , an) : ai Ai} .

T e o r e m 1 . 3 . P r a v i l o p r o d u k t a .

x k j n ND z k o n n k o Ai D i {1, 2, . . . , n} j d

|A1 A2 . . . An| =ni=1

|Ai|.

T e o r e m 1 . 4 . P r i n c i p b i j e k c i j e .

x k S

T

k o n n k o F d j o j l m n k S

j d n k o j l m n k T

k o m o

k o o o j j k j z m k

S k

TF

P r i m j e r 1 . 2 . h z d e d o l d f D m o m o o k o z d o f g F s z d e d f m o m o

d o n P n n D z f g n S D z d g d h n Q n n F x k o l k o n n m o m o d o z d e

d h c

R j e e n j e . x k j AD

k z m d o e h DAB

k z m e f DBC

k

z m f g D CD

n k j k z m d o g h F k o m p AD

m o m o

j k n o d n o j k (p1, p2, p3)D d j j p1 AB D p2 BCD p3 CD F d m j n o m n j k j l o d k d o m o d j

|AD| = |AB BC CD | = |AB| |BC| |CD| = 2 5 3 = 30.

Z a d a t a k 1 . 3 . y d d o j o d n d j l l j o j T H H F[Rj. 24]

Q


4/51

y n o n n o j n j

U p u t a . f o j o d n d j l l j o d n o o j n = p11 p22 . . . pkk k z n o n o k o j

(1 + 1) (2 + 1) . . . (k + 1) F h o k 3

Z a d a t a k 1 . 4 . y d d o j n o n n n j l o j (x, y)

k o j z d o o l j j n j d n d

x2 + y2 5.

[Rj. 8]

U p u t a . k o l k o j x 3 j n o j d j n j n o o j D d k l D o j o m o l j k d j

x {0, 1, 2}F u k o d o l d o k o n n o j n j D j l o n o o j j n j k o m o j d n o m l j o m n o l z o j c

t o j d n n n j d o m o m o k o d ( n n Si =

(x, y) Z2 : x2 + y2 = i , i {0, 1, . . . , 5} Ft l o n o o m n o l z o j m o n o o j l m n k o

Si c

P r i m j e r 1 . 5 . x k d n n k o X = {1, 2, . . . , 100}

S = {(a, b, c) : a, b, c, X, a < b, a < c} .

u o l k o j |S|c

R j e e n j e . f o j a m o l o k o j o j z k X\ {100}D k m o l o j a z o k D o j b o j c d D z o j m o m o z n

100 an n @ l o k o j o j z m

a + 1 I H H D k l j o A F u o n n o D

n o j j

|S| =99a=1

(100 a)2 =99a=1

a2 =99 100 199

6= 328350.

T e o r e m 1 . 5 . P r i n c i p k o m p l e m e n t a .

x k e k o n n k o k d j

A SF d j |S\ A| = |S| |A| .

P r i m j e r 1 . 6 . u o l k o m o d n o j m n j o d 10n

k o j d z n m n k R c

R j e e n j e . t n o j d k o j m o d 1, 2, . . . , n

z n m n k R F t d n o n j n m j E

o j k o l k o m o d n o j m n j o d 10n k o j n d z n m n k R F o d n o j m n j o d 10n

m 10n 1F o d n o j m n j o d 10n m o m o o m k o n! z n m n k @ n o k m o

n k o l k o n l D j d n k o j n m n l j j o n j k o j d m o n l A F h k l D o d n o j m n j o d

10n

k o j n d z n m n k R m

9n

1 D k z n m n k m o l o k o j o m R D j o m o m o o d l j k d z n m n k j d n k H F u o n n o D o d o o n n j z d k j

(10n 1) (9n 1) = 10n 9n.

D e f i n i c i j a 1 . 6 . x k j S

k F P a r t i t i v n i s k u p P(S)

j k k o j d o d k o k S

F

T e o r e m 1 . 7 . x k j S

k o n n k |S| = n N0 F d j |P(S)| = 2n F

D o k a z . k o l k o j S

z n k D j n o j d j j d n o d k k S

m o z n k D d n j j d D

j j 20 = 1 F o m o d d j n NF x k j S = {a1, a2, . . . , an} F o m o m o n n n z o d l j n

nD k o d n j o d j d n o d k k

SF k o l k o j n

i! o m m j n z o j I D o n d

ai n l z D k o l k o j H o n d n n l z o d k o d n o m m n z o m F t n o j d m o n o j n n o d d l m o o d k o k

S d k o m o d k o d o o n o j d n n n n z d l j n

nF x

R


5/51

y n o n n o j n j

k o m m j n n o m n z m o n l z o j I l o j H D d k l D k n z o m 2n F m n j k j D z k l j j m o d j

|P(S)| = 2nF y m j d o k z o o F

P r i m j e r 1 . 7 . x k o o l j l o n n n j o j k o j j j d n k o j o k n k k o j d o o l j Fy d d m n o j F

R j e e n j e . d m o k o o d n n k o n l F

nF

F

F

j bF

F

F

P

I

I P F F F i F F F n

k o m o l j o j m b

o d o o n o b

o k n k k o j d o o l j F o j m o j b

j d n k

m o n o k n k n l o F k o k n k j j d n o z n n o o d n o j m d m n z j m

o z j o m d o n j l j o o l j F x k o o d n d o n j l j o o l j o k n k d m n z j i j m o m o o d n n i + 1 n n @ l o k o j o j z m I n i + 1 D k l j o A D n l o n o D y k o o d n m o m o o d n

n j + 1 n n F h k l D o k n k d m n z j i j n d n o j l o m (n i + 1) (n j + 1) F u o n n o D n o j n j j

ni=1

nj=1

[(n i + 1) (n j + 1) i j] =

ni=1

[(n i + 1) i]

nj=1

[(n j + 1) j]

=

ni=1

(n + 1) i i2

nj=1

(n + 1) i i2

= (n + 1) ni=1

i ni=1

i22

=

(n + 1) n (n + 1)

2 n (n + 1) (2n + 1)

6

2

=

n (n + 1) (n + 2)

6

2.

S


6/51

m j k o

2 P e r m u t a c i j e s k u p o v a

D e f i n i c i j a 2 . 1 . x k j A = {a1, a2, . . . , an} k k o j m n r a z l i i t i h l m n D n k j r {1, 2, . . . , n} F

n r ! o k (b1, b2, . . . , br) m o n o z l l m n z A n z m o r p e r m u t a c i j a k AF

N a p o m e n a 2 . 2 . e k o j r = nD o n d m o d d o p e r m u t a c i j i k AF

P r i m j e r 2 . 1 . x k j A = {a, b, c, d}

F x P ! m j k A

F

R j e e n j e .

(a, b)D (a, c) D (a, d)D(b, a)

D(b, c)

D(b, d)

D

(c, a)D

(c, b)D

(c, d)D

(d, a) D (d, b)D (d, c) F

N a p o m e n a 2 . 3 . (a, a)

D(b, b)

D(c, c)

D(d, d)

n i s u P ! m j F k o m j d j v a a n p o r e d a k F

D e f i n i c i j a 2 . 4 . o d n o jn

d n m o n! = n (n 1) . . . 2 1 @ n f a k t o r i j e l A D d o d n o 0! = 1 F

N a p o m e n a 2 . 5 . m j d k o j l z o @ 7! = 5040

A D o k o k o l n o

o m l k o j @ z l k n

A d o o o k m X

n!

2nn

e

n, lim

n

n!2n

ne

n = 1.

T e o r e m 2 . 6 . x k j A n z n k n N l m n D n k j r {1, 2, . . . , n} Dr

! m j k

A m

Pnr =n!

(n r)! .

D o k a z . l m n r ! m j m o m o o d n n n n D d n n 1 D d F r ! n n r + 1 n n F o j

Pnr = n (n 1) . . . (n r + 1) = n (n 1) . . . (n r + 1) (n r)!(n r)! =

n!

(n r)! .

P r i m j e r 2 . 2 . u o l k o n z o l o d l j n S m o z k d D k o d n o m o m

m j z l m o l n D n o l m j m o n o z l l n F @ r k d m Q H

l o D o d k o j S m o l n D o l o l n F A

R j e e n j e . x o m j o m o m o o l o k o j o d S m o l n k D n m z o m j o o j R z o D

o n j D o o j P52 n n z o n o o m j o F h o D o m j o m o m o o n n

k n o P253 n n F d D z l o k o j n n o n j n j o o m j D o o j P253 n n o n j n j

o l m j F h k l D k o n n o j n j j

P52 P253 = 5 4 25 24 23 = 276000.

P r i m j e r 2 . 3 . x z j U m l d Q d j o j k F x k o l k o n n l j d m o m o o l o d k o d

@ A d j o j k n j d n l o k D

@ A m l d n l z n o j o l j d n j o j o z j n m j d n d j o j k F

T


7/51

m j k o

R j e e n j e .

@ A f l o k o d d j o j k m o m o o n V z l m j @ k o d j d j o j k l o k n j d n o m

o d I F d o V F m j d D k l j o A F h j o j k n o l o k m o m o o d n 3!

n n F

w l d @ k o j m U A m o m o o d n o l U m j D o m o m o n n 7! n n F h k l D j n j j

8

3!

7! = 3!

8!F

@ A o d m o n j j m l d d D o m o m o n n 7!

n n F d d m o m o m j j o Q

d j o j k F d j o j k m o m o m j z m o d o D d o D d F o d m o

m l d F h k l D z d j o j k m m o T n n D l n o D n k o n o m o m j l d j o j k D z d

m m o S n n D k o n n o z R n n F j n j j 7! 6 5 4 F

P r i m j e r 2 . 4 . x o l n l z m + n

z l k n j o d k o j j m

m m k D o l z l j

k n j n o F

@ A x k o l k o n n m o m o k n j o d n o l k o l m o d n m m j d k n j z m m k c

@ A u o l k o j o d k n j n o l k o l m o d m m k k n j d j d n d o d c

R j e e n j e .

@ A u n j z m m k m o m o o d n m!

n n D o l k n j n n!

j k o n n o j n j

m! n!F@ A f l o k o d m m m k k n j m o m o n o l m j n n + 1 n n @ m m k k n j

o m l o k m o n l o k o j m o d o d o (n + 1)

! o m j D k l j o A F n o l o k

m m k k n j m o m o m j n m! n n D j o o j z m j o l k n j D o m o m o n

n!n n F u o n n o D o o j k n o

(n + 1) m! n! = m! (n + 1)! n n F

P r i m j e r 2 . 5 . u o l k o m n o j z m P H H H H U H H H H k d z n m n k k o o j m o n o

z l F

R j e e n j e . x o m m j m o n l z l o k o j z n m n k z k {2, 3, . . . , 6} F w o m o o n o o m o l j k d j z n m n k n D k d j n n F

I

k o l k o j z n m n k n D n j m o m o z n Q n n F d n j z n m n k d m o m o

z n R n n @ l o k o j n z n m n k o m o n k o j j o d n k o A D o l

z n m n k n 8 7 6

n n F h k l D o o m l j m m o 3 4 6 7 8

n n F

P

k o l k o j z n m n k n n D n j m o m o z n P n n D z d n j n S D o l Q o n

8 7 6 D o n m d j j o 2 5 6 7 8h k l D k o n n o j n j j

6 7 8 (3 4 + 2 5) = 6 7 8 22 = 7392 F

Z a d a t a k 2 . 6 . x k j S

k o d n o j j z n m n k z k {1, 3, 5, 7}

k d n j d n

z n m n k n o n l j F

@ A y d d |S|F [Rj. 64]@ A y d d

nS

nF [Rj. 117856]

U p u t a . o k j n k k o m n o o j k

SF

Z a d a t a k 2 . 7 . x k o l k o n n m o z m j V k l n o k l o k o d n k o j d j k l n

n d j c

U


8/51

m j k o

@ A f z d o d n j F [Rj. 8!]

@ A k o d k l z l k j F

Rj. (8!)2

@ u l n d o l j o m k k o j m n l z F A

P r i m j e r 2 . 8 . x k o l k o n n

nl j d m o j o k o o k l o o l c h o d m m o j d n k m k o

j d n z d o m o d o o j o m F

@ A f z d o d n j F

@ A k o d e n j w k o j d j d n o d o d o F (n 2)

R j e e n j e .

@ A w o m o z m l j n d n n F x j j o l m o l j d n o m l n o d D o m o m o n n n!n n F d D k j o d o j m o k D d m o d k o d o n l j o n o

n D

d k l D o d o o j (n

1)!F

h n n j d z m o j d n o o n j m m o o k o m D d j j n o d o l l j d

m o m o z m j n (n 1)!

n n F

@ A e n j w k m o m o m j d n m l o k o m j d n o n j m o k o m D e n j w k n

o l o k m o m o o d n P n n D o l n 2 l j d n (n 2)! n n F h k l D o d o o j

2 (n 2)!F

Z a d a t a k 2 . 9 . y d d o j n n n k o j m o m o o j n

n o o k o o k l o o l k o d

@ A m k n l n j D [Rj. n! (n 1)!]@ A k n j d d o o m F [Rj. 2n (n 1)!]

P r i m j e r 2 . 1 0 .

@ A k o l k o n l z o j100!

c

@ A k o j m k o n n o m o o jp

l z o j n!

n o k o c

R j e e n j e .

@ A m j m o d o j100!

z o n o l k o n l k o l k o o j S o j l j j n j o o m

n o k o F @ k o j P S o j n! n o k o d o j m o j d n n l n n j o o m k j D l o j P n o o j l j j o d o j S F A u o n n o D o j

100!z

100

5

+

100

25

= 24

n l F @ @ A d j l z d k j o j n j n o z o j d k o n o m l F A

@ A o o jp

o j n!

n o k o o j l j j o n o l k o k o l k o l z k o

o d o j z k {1, 2, . . . , n} n o k o F f o j p m o j m k o j d j l j pD n d j l j

p2 o j l j j j d n o m D m o n m k o j d j l j

p2D l n

p3j o j d n o m D d F h k l D

k n o j o j l j n j o o o j p

o j n!

n o k o j

k=1

n

pk

F

N a p o m e n a 2 . 7 . m

k=1

n

pk

j k o n n D j z o d n o j k k d j pk > n j d

n

pk

= 0 F

V


9/51

m j k o

Z a d a t a k 2 . 1 1 . n

k n d d z n k o o d l j d o l n o m k o m j o m F k o d l n o k o m j

n k n d d m o m k j F l o j d n k k n d d n k o m d o E

j l n o k o m j l n o m j o F s z n j k o l k o o o l j z n k k n d d FRj.

3n 2n2

100%

U p u t a . o o l j n o d D j F o n k o j m n k o z n o j k o d o o j o d k o j m

n k o k n d d z l o n o d l n k o m j D z m o n k o j m z l Q l n k o m j F

W


10/51

u o m n j k o

3 K o m b i n a c i j e s k u p o v a

D e f i n i c i j a 3 . 1 . x k n r o d n o j k d j n r F x k j A k k o j d n z l o j E k F

r! l n o d k k

Ak m o d j

r k o m b i n a c i j a k

AF

P r i m j e r 3 . 1 . y d d Q ! k o m n j k {a, b, c, d}

F

R j e e n j e .

{a, b, c} , {a, b, d} , {a, c, d} , {b, c, d} .

N a p o m e n a 3 . 2 . x j n o d k 3

D e f i n i c i j a 3 . 3 . x k j n o d n r n n n j l o j D n r F B i n o m n i k o e f i c i j e n t j o jn

r

=

n!

r! (n r)!k o j m o

n o

r F

N a p o m e n a 3 . 4 . r > n d ( n m o

n

r

= 0 F

T e o r e m 3 . 5 . x k n

r

o d n o j k d j n r F f o j r ! k o m n j n! l n o k j

Cnr =

n

r

.

D o k a z . n m o d r

! m j n

! l n o k m Pnr =

n!

(n r)! F u o d k o m n j o d k n j n D o z j d n

r! k o m n j d o j o n o

r! z l

r! m j F k l j j m o X

Cnr =Pnrr!

=n!

(n r)! r! =

n

r

.

P r i m j e r 3 . 2 . x k o l k o n n m o m o o d o d S o o z o d R o o U d n

@ A k o n m k j D

@ A k o d d o n o P o o D

@ A k o d d m Q o o D

@ d A k o d o d n o o d n n d c

R j e e n j e .

@ A s z o d 4 + 7 = 11 l j d m o m o z S D o d o o j

11

5

n n F

@ A P o o m o m o z n

4

2

n n D o l o j l j d n

7

3

D j n j j

4

2

7

3

F

@ A o n o o m m o d @ d j n k n 3 A l j F

I

y d n o n o Q o o D m m o

43

72

= 4 7

2

= 25

n n F

P

y d n o n o R o o D m m o

4

4

7

1

= 7

n n F

I H


11/51

u o m n j k o

h k l D m m o k n o Q P n n F

N a p o m e n a 3 . 6 . x o n o l o n j j o d Q o o D z m o d o l V l j d j o d j

o o F o j l j k o j o j o o z n F o c

@ d A o o n m j m o z j d n o o o j d n o d n D o j n m W l j d o d k o j m o m o

z o d S l j d D o d n o n o D m m o

9

5

n n F

P r i m j e r 3 . 3 . x k o l k o n n m o m o o d j l P R H d n Q j d n k o o j n k n k o d

@ A k n d n j z j o j n o D d n j z h k n m m k D n i n l k D

@ A k n d n j z h k n m m k c

R j e e n j e .

@ A x j j o d m o d n z k n D z m o d o k d n z d k n D o l

d n m j m o k n F o n m d j

240

80

160

80

n n F

@ A z m l j m o o k o @ A d j l z d k D m o o n m o o m l j n j n o k o j o o j

k o j @ m l j j z h k n m m k A D j o d o o

24080

16080

3!

n n F

T e o r e m 3 . 7 . x k

n

r o d n o j k d j

n r F n o m n k o j n j d F@ I A S i m e t r i j a .

n

r

=

n

n r

.

@ P A P a s c a l o v a f o r m u l a . n

r

=

n 1r 1

+

n 1

r

.

D o k a z . o m o n d n n F

@ A e l k n n F

@ I A n

r

=

n!

r! (n r)! =n!

(n r)! [n (n r)]! =

n

n r

.

@ P A n 1r 1

+

n 1

r

=

(n 1)!(r 1)! (n r)! +

(n 1)!r! (n r 1)!

=(n 1)!

(r 1)! (n r 1)!

1

n r +1

r

= (n 1)!(r 1)! (n r 1)! nr (n r)

=n!

r! (n r)! =

n

r

.

I I


12/51

u o m n j k o

@ A u o m n o n n n F

@ I A f o j l j n j d n k j o j r ! l n o d k o n! l n o k F f o j d n n j d n k j o j

(n r) ! l n o d k o n ! l n o k F d o j o o j d n k D j z k z o r

! l n o o d k j d n o z n n o o d d m o j d n (n r)

! l n o d k o n o F

@ P A k o l k o j n = r

d n j l n o j d D j j n

n = n 1

n 1= 1

F o m o d d j

r < nF m d n j j z z l j n o j

r! k o m n j

n! l n o k F o k m o d j

o m j d n k o j d n n F o m j m o o z o l j n k S

k o j m n

l m n F x k j

x SF r ! k o m n j m o m o o d j l d j @ d j n k n 3 A m l j r ! l n o d k o k S

X

o n k o j d l m n xX A = {X S : |X| = r, x X}D

o n k o j n d l m n x

XB= {X S : |X| = r, x / X}

F

j d |A| =

n 1r 1

@ k k m l j A o j o d l m n x j o r 1 l m n o d

o l n 1

l m n k S

A D|B| =

n 1

r

F y o j d m l j

A

B n j d j

r! l n o d k o k

SF y m j d o k z o o F

Z a d a t a k 3 . 4 . x k n

k

o d n o j k d j n k F u o l k o m n n n z o d l j n n k o j d

kn l

n kj d n c

Rj.

n

k

P r i m j e r 3 . 5 . M e t o d a k u g l i c a i t a p i a .

x k o l k o n n m o m o n N j d n k k l o d m N z l k j c

R j e e n j e .

. . . . . . o m j m o n z k l d k o n o n j o j l F u l l j o o d d d j o j k j D

k l z m d d d o j k j D d F k l n k o n z d n j d z d n j o j k j F h k l D

m m o n z o d n

k l m 1 d F k n z o d j o n o j d n o d k l o k j m F E

d m o m o z m j n

n + m 1

m 1

n n D o j o d o o n n j z d k F

Z a d a t a k 3 . 6 . x k m n o d n o j F y d d o j n n n j l o o j n j n j j d n d

x1 + x2 + . . . + xm = n.

Rj. n + m 1m 1 P r i m j e r 3 . 7 . u o l k o m n o k

(x1, x2, x3, x4) N4 k d j x1x2x3x4 = 9000F

R j e e n j e . 9000 = 23 32 53

F

k k o o j W H H H j o l k xi = 2i 3i 5i Y i, i {0, 1, 2, 3} D i {0, 1, 2} D i {1, 2, 3, 4} F h

l o x1x2x3x4 = 9000D m o j d

1 + 2 + 3 + 4 = 3,

1 + 2 + 3 + 4 = 2,

1 + 2 + 3 + 4 = 3.

o o j

63

z l j n j j d n d D

53

d D

63

z d n j F y d o o n n j z d k j

6

3

2

5

2

= 4000

F

I P


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u o m n j k o

P r i m j e r 3 . 8 . u o l k o m n j k j l o o j n o j m o d (0, 0)

d o(m, n) N2

c

@ A f z d o d n j F

@ A u o j o l z o k o m (p, q)D d j j p N, p < m q N, q < n F@ A u o j n o l z m n o m [(p, q), (p + 1, q)]D d j j p N, p < m 1 q N, q < nF

R j e e n j e . x j j m j m o d k o j o j m o o d k n j d n o o j d n k o d

d o n j k F k n z o d x N0 l o h 9 @ d n o A y N0 l o q 9 @ o A o d j o n o

j d n o d(x0, y0) N20 d o (x0 + x, y0 + y) F x F q q q h h q h q q h h h n m o d j o d (x0, y0) d o

(x0 + 6, y0 + 6) F

@ A x z o d l j n m + n k o j d m l o h 9 n l o q 9 m

m + n

m

@ z m o m m j n k o j m

l o o h 9 D n o l m d m o l o o q 9 A D o j o d o o o o m l j F

@ A x j j z (0, 0) d o m o (p, q)D z m z n j (m, n)F d o m o m o z n

p + q

p n n D d n

m p + n qm p

n n F h k l D o o j

p + qp

m p + n qm p

F

@ A o o m l j m o o d m o @ n j k A o d z o n k o j o l z m m n o m D

m m n o m o l z k o j o d o d (0, 0)

(p, q)

D z n j (p + 1, q)

D z m (m, n)

F

k n o m

m + n

m

F v o m

p + q

p

m p 1 + n qm p 1

F u o n n o D j n j j

m + n

m

p + q

p

m p 1 + n qm p 1

F

D e f i n i c i j a 3 . 8 . x k j

n o d n o j F x k j

A k

2n l m n F S p a r i v a n j e j j k

An

d o l n o d k o F

P r i m j e r 3 . 9 . x k j n NF y d d k o l k o m z l n j k A k o j d 2n l m n F

R j e e n j e . h m o d n n j n j F

@ A m j k A m (2n)!F m j j d n o n j k A m n!F h k l D z k o n j k

Am o m o d o o n o

2n n! m j k

A@ k l n n j D k o j j d o l n k D

m o m o n m P A F s z d z l n j n k k o n m o m o d o d j

j d n k m j F k o D z m o n j d o m o m j F h k l D

m o n j k A

m

(2n)!

2n

n!F

@ A x j j o d 2n

l m n z m o P D z m o d o l 2n 2 j o P D d F z d n j o l P m o

n! k F x o D n j n m n o d k k o d o n d o l n k o D o j j n j

n1i=0

2n2i

2

n!

=1

n!n1i=0

(2n 2i) (2n 2i 1)2

=(2n)!

2n n! .

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m j k o m n j o n l j n j m

4 P e r m u t a c i j e i k o m b i n a c i j e s p o n a v l j a n j e m

D e f i n i c i j a 4 . 1 . n r ! t o r k u (x1, x2, . . . , xr) @ n n n o z l A l m n k S n z m o r p e r m u t a c i j o m s p o n a v l j a n j e m F

P r i m j e r 4 . 1 . A = {a, b, c} F y d d P E m j o n l j n j m k AF

R j e e n j e .

(a, a)D

(a, b)D

(a, c)D

(b, a)D

(b, b)D

(b, c)D

(c, a)D

(c, b)D

(c, c)

x m n j X k o l k o j r

! m j o n l j n j m k o d n

l m n c

T e o r e m 4 . 2 . o o j o n o nr r ! m j o n l j n j m n! l n o k F

D o k a z . l n n r

! o k m o m o o d n n

n n D d k o n n

n n @ j l m n

m j o n l j A D . . .D r ! n n n n F m D n r ! o k m o m o o d n nr n n F

Z a d a t a k 4 . 2 . x k o l k o n n m o m o T o k o j m m o n o n n m k o l n m o d j l z E

m I H d j k o d k o d j d o j o j d n o k c

Rj. 610

N a p o m e n a 4 . 3 . u d o n m o m l n o n n m k o l n m D z d k o n o l o n j F

D e f i n i c i j a 4 . 4 . K o n a n i m u l t i s k u p M

n k S

j n (S, m)

d j j m : S N0 n k j k

d j

xS

m(x)k o n n o j F

x S o j

m(x)z o m o k r a t n o s t o d

xF

D e f i n i c i j a 4 . 5 . x k j M = (S, m)

m l k F n r

! o k (x1, x2, . . . , xr) , xi M, i {1, 2, . . . , n} D

k d j o j o j l j n j l m n xi m n j o d m (xi) z o m o r p e r m u t a c i j a m u l t i s k u p a F e k o j xS

m(x) = r D o o m o o p e r m u t a c i j i m u l t i s k u p a M F

P r i m j e r 4 . 3 . M = {a,a,b,b,c,c,c} = {a2, b2, c3} u o l k o m m j m l k Mc

R j e e n j e . w o m o z m l j n d n n F

1 . n a i n . f o j m j U ! l n o k j 7!

F u k o j M

m l k D m o o j l n k m E

j @ a d a n z l k j m o A F k m j l m n k z l o m m j o m

m l k w F o j n o j

7!

2! 2! 3! F

2 . n a i n . y d m o o d m j @ o d d m A n k o j m o s m j e s t i t i a! o F o m o m o n n

7

2

n n F c ! o z m m o m o o d n

5

3

n n D z b! o j o o m o j o j d n n n

2

2

F

k n o j o j m j m l k j d n k

7

2

5

3

2

2

=

7!

2! 3! 2! F

e n l o n o o d j m o o j m j o n o m l k F

T e o r e m 4 . 6 . f o j m j m l k {xm11 , xm22 , . . . , xmkk }D m j ki=1

mi = ND j d n k j

N

m1

N m1m2

. . .

N m1 m2 . . . mk1

mk

=

N!

m1! m2! . . . mk!

Z a d a t a k 4 . 4 . y d d o j n n n z o @ n z o n l D j d n d o j k A k o j m j P n l D Q j d n S

d o j k F

Rj.

10!

2! 3! 5!

P r i m j e r 4 . 5 . u o l k o m o z n m n k o j k o j m z n m n k l m n m l k 74, 22, 41, 61cR j e e n j e . d k m o j z j n j m n n k o l k o l j o n o o o m k o l k o z n m n k

d n o z n m n k o j F o k o l j n o d m o m l k o z o j k n o S j

l m n z n m n k n o o j F

I R


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I

74, 21

D

74, 41

D

74, 61

f o j m j k o o d j m l k o j d n k j

5!4!

D z o j k n o j o j o m l j

j d n k #1 = 3 5!4! = 15P

73, 21, 41

D

73, 21, 61

D

73, 61, 41

#2 = 3 5!

3!= 60

Q

73, 22#3 =

5!

2! 3! = 10

R

72, 22, 61

#4 = 2 5!2! 2! 1 = 60

S

72, 21, 41, 61

D

71, 22, 41, 61

#5 = 2 5!

2!= 120

h k l D k j o j k n o # =5

i=1 #i = 265@ y k o l m n m o m o l o n o j " m z d A

P r i m j e r 4 . 6 . o k d j o j4n!

d j l j 23n

3n

R j e e n j e . o m o m o m l k M =

a41, a

42, . . . , a

4n

F f o j m j o m l k @ p r i r o d a n b r o j ! A

j

(4n)!4!4!...4!

= (4n)!23n3n N

N a p o m e n a 4 . 7 . l n o D z o k o m n o n n j z n m o d j z z o l k

n!

r!(n r)! o d n o j F

D e f i n i c i j a 4 . 8 .

r! l n

xSm(x) = r m l k M = (S, m) k m o d j r k o m b i n a c i j a s p o n a v -

l j a n j e m k S

F

P r i m j e r 4 . 7 . s P E k o m n j k S = {a, b, c}F

R j e e n j e . {a, b} , {a, c} , {b, c} , {a, a} , {b, b} , {c, c} F

N a p o m e n a 4 . 9 . y o d o l n p o d m u l t i s k u p o v i m l k M = {a, b, c} @ k o d l m n m o E

m o o d k o l k o o d l m o A F

d l m o o d d o j r ! k o m n j o n l j n j m n ! l n o k F x k j S = {a1, a2, . . . , an} z Ed n k F d

xi o z n m o o j o n l j n j l m n ai r ! k o m n j F o l m d o d n o d n j o j n n n j l o o j n j n j j d n d

x1 + x2 + . . . + xn = r. @ I A

j o l m j m o m e t o d o m t a p i a i k u g l i c a D o n o m P r i m j e r u 3 . 5 . n n I P F

T e o r e m 4 . 1 0 . f o j r ! k o m n j o n l j n j m n! l n o k j d n k j

n + r 1

n 1

=

n + r 1

r

F

N a p o m e n a 4 . 1 1 . o n o n o n l m o d m o o m l l j k d k l n k S m o o j o o l j m n o o @ k o n k j d n d @ I A o z n d l m n

xi n o d o z o o n n F A y n j d l j Dk d o o j o n n j n o j o n l j n j o d n o l m n k j n k n j F

P r i m j e r 4 . 8 . f o l o n o d j n d X n k o m D n o m j n k F x k o l k o n n

d n m o n T n d c

R j e e n j e . m o o l n o d k o m l k {S, T, V} D o d n o n o o j n n n j E l o o j n j n j j d n d

x1 + x2 + x3 = 6 F j j o j j d n k

6 + 3 1

3 1

=

8

2

F

I S


16/51


P r i m j e r 4 . 9 . u o l k o m j l o o j n j n j j d n d

x1 + x2 + x3 + x4 + x5 = 50

z j X x1 D x4 D x5 0 Y 2 x3 7 Y x2 2

R j e e n j e . m o j y1 = x1 D y2 = x2 2D y3 = x3 2 D y4 = x4 D y5 = x5 D n k o n o n j d n d l

y1 + y2 + y3 + y4 + y5 = 46 @ P A

z j yi 0 D i {1, 2, . . . , 5} y3 5t d n o n o m j o m o d m o j x c D c N n x 0 D o z n m o j F j y3 5 k o m o o k n o z n j F o l m o n o m k o m l m n X o j j l o o j n j n j

j d n d @ P A z j y3 5 j d n k j z l o j j n j z k k d o d n j o j j n j z j

y3 > 5 D j F y3 6 F f o j j n j z j y3 6 o d j m o j o m z3 = y3 6 D zi = yi D z i = 1, 2, 4, 5 o o m o o j d n d z1 + z2 + z3 + z4 + z5 = 40 F x k j D o j j l o o j n n n n

j n j o n j d n d j d n k j

46 + 5 1

46

40 + 5 140

=

50

4

44

4

F

P r i m j e r 4 . 1 0 . u o l k o m r ! o d k o k {1, 2, . . . , n} k d n d d z o n o j c

R j e e n j e . o m m o o j b1 D b2 D F F F D br k d j d

1 b1 < b2 < < br n r + 1 d o z n m o

a1 = b1 D a2 = b2 + 1 D a3 = b3 + 2 D F F F D ar = br + r 1w o j m a1 . . . ar n m z o n z n j j d

1 a1 < a2 < .. . < ar n o o j j k j z m E o k (a1, a2, . . . , ar) (b1, b2, . . . , br)Fh k l D o n o j o d o o n k o l k o n n m o m o o d

r l m n o d n j

n r + 1 F y d o o j n r + 1

r

F

P r o b l e m i d i s t r i b u c i j e

I F l m o o d d o j o d m

z l o j k n

z l k j k o d

@ A k k j m o m o n j j d n o j k X

n (n 1) . . . (n m + 1).

@ A k k j m o d o z o l j n o m n o o o j k X

nm.

@ A k k j m o d m o j d n o j k D n o l n j j r1 o j k D . . .D rko j k

k! X

n!

r1! rk! , d j j n =ki=1

ri.

@ d A k k j m o d o z o l j n o j o j k D l o d k n k j j n X

n (n + 1) (n + 2) . . . (n + m 1).

P F l m o o d d o j o d m d n n o j k n z l k j k o d

@ A k k j m o n j j d n o j k X n

m

.

@ A k k j m o o z o l j n o j o j k Xn + m 1

m

.

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@ A n j d n k j n d z n X

m o o j j l o o j n j n j j d n d r1 + r2 + . . . + rn = m, ri 1 F

z j si = ri 1 s1 + . . . + sn = m n + 1 F j n j j D o

m n + n 1n 1

=

m 1n 1

.

u o l k o o o j j

n! l n o k c x o o j j d n o n n n z n n j o F

D e f i n i c i j a 4 . 1 2 . n! B e l l o v b r o j D Bn k o o j j n! l n o k F h o d n o D d n m o B0 = 1F

T e o r e m 4 . 1 3 . j d

Bn =

nk=1

n 1k 1

Bnk.

D e f i n i c i j a 4 . 1 4 . h n m o S t i r l i n g o v e b r o j e v e d r u g e v r s t e S(n, k) k o o j k ! l n j n ! l n o k F

N a p o m e n a 4 . 1 5 . o n k d k o o z n k S(n, k) = nk F

T e o r e m 4 . 1 6 . j d

S(n + 1, k) = S(n, k 1) + kS(n, k) .

v k o d m o d j S(n, 1) = S(n, n) = 1, n NF d m o m o k o n l z k o j l k o o m o l n o o j d F

n\k I P Q R S T I I

P I I

Q I Q I

R I U T I

S I I S P S I H I

T I Q I W H T S I S I

I U


18/51

u o m n o n o d o k z n j d n

5 K o m b i n a t o r n o d o k a z i v a n j e i d e n t i t e t a

o o j j l n m o d o k z d n m o d o m d o k o o j n j F o m o d l m o d o k z

n k d n A = B

m z z A

B

k d m o m o k o m n o n o n @ m o E

d k n o m n k o ( j n D k o j l D l n o l p o n j o j D d F A w o d o j o d

k o k F o d ( n m o k S k o j m m o o d d o j l m n n d z l n n F d o m k o k d o k z j m o d j

A =

|S

|F x j j o j k o k d d k n o z d ( n j k

S@ z z

An m d j d j

k k o d ( n SA F o l j d n j k o k j d o k z n j B = |S| z z k l j j m o d j d A = B F o n k d j o n o k o n j k j k k o d o k z l o

B = |S|F

5 . 1 P e r m u t a c i j e

Z a d a t a k 5 . 1 . h o k d n j k o k o m n o n m n D j F m o d o m d o k o o j n j F

@ A Pnr = nPn1r1

x k j S k r E m j k X o d n l m n F x l j o j n j d n d j o j l m n k

S m o m P F T F x d n o j n j o j z n n d n n F k

rE m j

o d j o d n j l m n o m

x1n o m m j

(r 1)E m j o m o l n 1 l m n k X = X\ {x1} F i l m n n o m m j m o m o o d n n n n D o l n z o d r 1 l m n z

(n 1)E l n o k A m o m o o d n Pn1r1 n n F

@ APn+1r = P

nr + rP

nr1

x k j S

k r

E m j k X

o dn + 1

l m n F x l j o j n j d n d j o j

l m n k S

m o m P F T F p k j m o n k l m n x

z k A

F k r

E m j

o d A m o m o o d j l d d j n k n k o n o o o m o j l j j l l m n x m j l n F f o j m j k o j m

x o j l j j j

rPr1n j m o m o o d j d n o d n o z j z l m n x n o l o z j o d (r 1)E o k z nE l n o k X \ {x} F f o j m j k o j m n o j l j j

xj j d n o n o o j

rE m j

nE l n o k

X

\ {x

}F

@ A

n

m

m

r

=

n

r

n rm r

x k j

X k o d

n l m n n k j

S k n o

(Y, Z) k d j

Z Y X

|Z| = r, |Y| = mF o d k

Y X k d j

|Y| = mm o m o o d n

n

m

n n D j d n o m k d

m o o d l k Y o d k Z Y k d j |Z| = r m o m o o d n

m

r

n n F o j o j

n o (Y, Z) j d n k l j o j n d n F d n D k S j j k j k o m

n o (Y, Z)

m j Z X

D|Z| = r

DY X Z

D|Y| = m r

F f j k j j d n

Y = Y

\Z

F o d k Z

X

k d j

|Z

|= r

m o m o o d n n

r n n F t d n o m k d m o o d l k

Z o d k

Y X Z k d j |Y| = mr m o m o o d n nrmr

F y o o k z j

d j o j l m n k S

j d n k d n o j n d n F

h m o l k o j n j F o m o d o m m o z o z n j l j d l j F d

j n l j o j n o j n n n k o j o dn

d l m o z m o z m

n j k o j d o l ( n l D z E

m o d d l o l ( n l o d r n j k o j d ( n l F x d n o j n o o d n n j l j o d m o

rn j z ( n l D o n d o d o l

n r m j o d m o n j m r k o j o l ( n l Fy d j m o o j l n o k o d l k o j d o l ( n l ( n l F

@ d A

nk=0

m

k

n

r k

=

m + n

r

o n o n o m o o n d l k o o j n j z d k F o j m o k

S m o o d

rE l n

k z m m k o l m j d k n d d m o d k o j j m m l d n d j o j k F d n n j o j n n n k o j m o z m o o d

m + no o o d

r l n o k D j F o j l m n k

SF

m j m o d k S m o m o o d j l n d j n k n k o o n o o o j k k o j d l j o jm l d k m o

0, 1, 2, . . . , nF

km l d m o m o o d n

mk

n n D j d n o m k d

I V


19/51

S F P o l o n j p o n j o j Fn

m o o d l m l d k m o m o d o n r k d j o j k n nrk

n n F o l j n k o

d j o j l m n k S

F

z n j D n k X Y d j n k n k o k d j |X| = n |Y| = mF e k o j S k o d k o Z X Y k d j |Z| = r D o n d d n n o z n o j l m n k SF m j d

j k S j k j n j o m d j n k n k o Sk D k {0, 1, . . . , n}D m l m n k Sk n o

(P, R) k d j

P X, R Y |R| = k, |P| = r k F f j k j j d n Z = P R P = Z

X, R = Z

Y

F o n d l j o o j n j F

@ A

ni=0

i

n

i

= n2n1

m l m o o k j X z k n o d n

o z d o j n j j d n k o j n z m o k n

o n m o o d o o l j m n o o o l l j d k o j n j m d m j F f d d k n m o m o

o d n n

n n o n z k o o d o l n 1 l j d o d l j d l l n d m j D d n

n j o j n n d o m k o n k F k k k m o o m k o d

o d m o i

o dn

o k o j d m j m n j m @ d k l D n i

n n A o d m o k n F

f d d j l n k o z o l j n z j m o o j ini

z j d n o i o d 0 d o n F

z n j D z n

E l n k X

d n n o j l m n k S

n o (x, Y)

m

j x

X

Y

X

\ {x

}F k

Sj j k j n j o m d j n k n k o

Si D i

{0, 1, . . . , n

}D

m l m n k Si n o (x, Z) k d j Z X x ZF f j k j j d n Z = Y {x}, Y = Z\ {x} F h o j n j z j F

Z a d a t a k 5 . 2 . s z n j m

nr=k

n

r

r

k

F

5 . 2 P o p l o a v a n j a i F i b o n a c c i j e v i b r o j e v i Fn

x k j Jn o j n n d o l o m o l o 1n k d m 11 @ u A d o m n m 12 @ h A F x d n j m

j o k z n o Jn = Fn+1 FZ a d a t a k 5 . 3 . h o k d n j k o k o m n o n m n D j F m o d o m d o k o o j n j F

I F Jm+n = JmJn + Jm1Jn1f o j k n j l o d l j n

m + nj d n k j

Jm+n F o m m o l o n m j m m m + 1 F e k o n n j m n l z d o m n o D o n d o l d o l o m o m o k n Jm1Jn1 n n D k o j l o l o m l j n o m m j o n d j m o m o k n

JmJn n n F

P F3Jn = Jn+2 + Jn2 o m m o

n + 3E l o l o m l j z m j Q F m j m o m o o n n n n @

J3 = 3 A D o k l o n Jn k n o k l o m o m o k n 3Jn n n F

x n j d n n j n o F e k o k d n j d l l o k n j

o n d o d l o m o u h h @ d l o l l o m l j z m j Q A D o k m o m o

k n Jn2 n n D k o j k d n d o m l m m j o d l o o l k DK .. .

l KDK .. . l K K K . . . k l o n m o l k d @ k o j j d l k n j A d o m o o d o l k

D . . .D

K K . . . l

KDD j F m o o d k n j l o d l j n

n + 2k k m

Jn+2 F

Q F

pi=0

p

i

Jni = Jn+p F

h n n o z n o j o l o n j l o d l j n n + p

F n j l j n z l k j m o

d j n k n l j X k o m p l o k n j n l z o n o i d o m n D m o m o o d n

pi

n n D o l

n +p 2i (p i) = n i m j m o m o k n Jni n n Ff d d j

im o l o k o j o j o d

0d o

p m l m l j d n d n j F

R FJ2n

= J2n1

+ J2n2

o m j m o l o d l j n 2n l o m l j z m j nF z l k j m o d d j n k n l j F e k o m j n 1, n , n + 1 n + 2 o k n d o m n m @ . . . D D . . . A o k l o m o m o k n Jn2 Jn2n n F e k o o m n m j o k n o d m . . . K D . . . l . . . K K . . . l . . . DK . . . z m o k d m j

n l

n + 1 d o m o k n j l o d l j n

2n 1 k o j m J2n1 F

I W


20/51

S F P o l o n j p o n j o j Fn

Z a d a t a k 5 . 4 . d j n j d m j F h o k d j o j n n z o n z n o d k o o m j k d n o o j d j d n

Fn+1 F

R j e e n j e . y z n m o an o j n z n o d k o k {1, 2, . . . , n} k o j n d d j d n o j F d n j d o k z j m o n d k j o m F y o j

a1 = 1, a2 = 2 F o m o d d n j j d z n 2 n 1 F o m m o n k k A

{1, 2, . . . , n

}k o j n d d j d n o j F

e k o j n k A d n 1 n m j A z o j z d k F x o l l m n n m m o j k o d k o m o n d k j m

an2 = Fn1 F e k o n n j k A o n d n o l l E m n n m m o j m an1 = Fn F k l j j m o d k n o m an = an2+an1 = Fn1+Fn = Fn+1 z o k o d k o F

P H


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k z n l j

6 R e k u r z i v n e r e l a c i j e

6 . 1 H o m o g e n e l i n e a r n e r e k u r z i j e s k o n s t a n t n i m k o e c i j e n t i m a

s z z o l k cran+r + cr1an+r1 + . . . + c0an = 0 n z m o h o m o g e n a l i n e a r n a r e k u r z i j a s k o n s t a n t n i m

k o e c i j e n t i m a r e d a r F m o j n j o l k

an = xn

F j d o m o

crxn+r + cr1x

n+r1 + . . . + c0xn = 0 / : xn

crxr + cr1x

r1 + . . . + c0x0 = 0,

j z z n z k a r a k t e r i s t i n a j e d n a d b a F

T e o r e m 6 . 1 . e k o x1, x2, . . . , xr m o n o z l j n j k k n j d n d D o n d j k o j E

n j o n k z j o l k an = A1xn1 + A1x

n1 + . . . + Arx

nr , d j A1, . . . , Ar o d j m o z o n o j F

P r i m j e r 6 . 1 . j k z n l j X

an = 2an1 + an2 2an3; a1 = 1, a2 = 2, a3 = 3.

R j e e n j e . n j m xn

an 2an1 an2 + 2an3 = 0 d o m o k k n j d n d

xn 2xn1 xn2 + 2xn3 = 0 / : xn3,x3 2x2 x + 2 = 0.

s z n m o n l o k X1, 1, 2

D j o j n j Xan = A (1)n + B 1n + C 2n F d z o n j

m m o X

1 = a1 = A (1) + B + C 22 = a2 = A 1 + B + C 43 = a3 = A (1) + B + C 8

= A = 16

, B =1

2, C =

1

3.

s z m m o j n j an =(1)n

6+

1

2+

2n

3F

T e o r e m 6 . 2 . e k o j n j k k n j d n d x1, , xm k n o m k1, , km D o n d j o

j n j k z j d n o o m l o m

an =

A11 + A12n + + A1k1nk11

xn1

+

A21 + A22n + + A2k2nk21

xn2F

F

F

+

Am1 + Am2n + + Amkmnkm1

xnm.

P r i m j e r 6 . 2 . j k z j X

an 7an1 + 15an2 9an3 = 0; a0 = 1, a1 = 2, a2 = 3.

R j e e n j e . u k n j d n d X x3 7x2 + 15x 9 = 0 = x1 = 1, x2 = x3 = 3 D d j o j n j an = A 1n + B 3n + C n 3n.

s z o n j m m o

1 = a0 = A + B2 = a1 = A + 3B + 3C3 = a2 = A + 9B + 18C

= A = 0, B = 1, C = 13 .

P I


22/51

T F P x o m o n l n n k z j k o n n n m k o ( j n m

m o o j n j m m o an =

1 n3

3n F

Z a d a t a k 6 . 3 . j k z j

an + an2 = 0; a0 = 1, a1 = 1. Rj. an =

1 i2

in + 1 + i2

(i)n

6 . 2 N e h o m o g e n e l i n e a r n e r e k u r z i j e s k o n s t a n t n i m k o e c i j e n t i m a

s z z o l k cran+r + + c1an+1 + c0an = f(n)D d j j f n k n k j o d n D n z m o n e h o m o g e n a l i n e -

a r n a r e k u r z i j a s k o n s t a n t n i m k o e c i j e n t i m a r t o g r e d a F

o k z j n j X

@ I A x l z m o o j n j d n o m o n j d n d aHn F

@ P A m o k l n o j n j aPn m l @ k o j l j d A F

@ Q A y j n j j an = a

Hn + a

Pn D k o ( j n o d j m o z o n j F

f(n) aPn

C bn@ A

b n j k o j n k k n j d n d YaPn = A bn

@ A

b j k o j n k k n j d n d k n o k Y

aPn = A nk bn

p (n) R [x] , n j o d p = m@ A

1n j k o j n k k n j d n d Y

aPn = p1 (n)D o l n o m n j m n o d n m k o j n m

@ A

1j k o j n k k n j d n d k n o

kY

aPn = nk p1 (n)

C nm bn@ A

bn j k o j n k k n j d n d Y

aPn

= p1

(n)

bnD n j o d

p1

= m

@ A

bj k o j n k k n j d n d k n o

kY

aPn = nk p1 (n) bn

o d j n m C

Db

A

d l j j n k k o n n F

P r i m j e r 6 . 4 . j k z j

an+1 5an = 4n2 + 2n + 6; a1 = 1.

R j e e n j e . j n j m k k n j d n d d n o m o n k z j d o m o aHn = A 5n F

h l j D d m o d k l n o j n j m o m o o l k aPn = Bn2 + Cn + D D n j m d n

k z j d o m o

B (n + 1)2 + C(n + 1) + D 5Bn2 5Cn 5D = 4n2 + 2n + 6

4Bn2 + (2B 4C) n + (B + C 4D) = 4n2 + 2n + 6.s z j d n j k o j n l j d n n @ k o j d n k o o l n o m A n l z m o B = 1 D C = 1 DD = 2 F u o n n o D n j m an = aHn + aPn o n j

1 = a1 = 5A 1 1 2 = 5A 4,n l z m o d j

A = 1F j n j d n k z j j

an = 5n n2 n 2

F

P r i m j e r 6 . 5 . j k z j

an = 6an1 9an2 + n 3n a0 = 2, a1 = 3.

P P


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T F Q w o d l n j o m o k z j

R j e e n j e . u k n j d n d d n o m o n k z j d n j x2 6x + 9 = 0 D m j E n j

x1 = x2 = 3 F h k l D aHn = A 3n + B n 3n F d d m o k k o m o l k m o m o k l n o

j n j D o d n o n o aPn = n2 (an + b) 3n F n j m d n k z j m m o

n2 (an + b) 3n = 6 (n 1)2 [a (n 1) + b] 3n1 9 (n 2)2 [a (n 2) + b] 3n2 + n 3n / : 3n

n2 (an + b) = 2 n2 2n + 1 (an a + b) n2 4n + 4 (an 2a + b) + n

(1 6a) n + (6a 2b) = 0.

h k l D a =1

6D b =

1

2F d D n j m an = a

Hn + a

Pn o n j d o m o A = 2D B =

53

F u o n n o

j n j j an =

3n1

2 n3 + 3n2 10n + 12 F

Z a d a t a k 6 . 6 . j k z j

an 3an1 + 2an2 = 2n; a0 = 3, a1 = 8.

[Rj. an = (2n + 1) 2n + 2]

P r i m j e r 6 . 7 . j k z j

an = 2an1 + 4bn1,bn = 5an1 + 7bn1;

a1 = 4, b1 = 1.

R j e e n j e . s z k z j d o m o bn1 =an + 2an1

4D o d n o n o bn =

an+1 + 2an4

F j d o n o

d k z j d o m o d j o n o j k z j

an 5an1 + 6an2 = 0; a1 = 4, a2 = 2a1 + 4b1 = 4.

j n j d o n k z j j an = 2

n+3 4 3n D d l k o n l z m o d j bn = 2n+3 5 3n F

6 . 3 M o d e l i r a n j e p o m o u r e k u r z i j a

P r i m j e r 6 . 8 . x k o l k o n n m o m o l o 1 n o l o l o m d m n z j 1 1 1 2 c

R j e e n j e . x k j an o j n n z z d k F o m o m o l o F k o l k o j o n d m n z j 1 1 o k l o m o m o o l o n

an1 n n D k o l k o j o n d m n z j 1 2 o n d o k m o m o o l o n an2 n n F h k l D j d k z j

an = an1 + an2.

o n j a1 = 1 D a2 = 2 F o j n m m o z m j d j an = Fn+1 D d j j (Fn)nN p o n j n z F

N a p o m e n a 6 . 3 . p o n j n z j d n n F1 = F2 = 1 D Fn = Fn1 + Fn2, n 3 F o n o m l

z p o n j o j d n j

Fn =1

5

1 +

5

2

n

1 52

n, n N.

P r i m j e r 6 . 9 .

P Q


24/51

T F Q w o d l n j o m o k z j

@ A y d d o j o d k o k {1, 2, . . . , n} k o j n m j j d n l m n F@ A u o l k o m n n n z o d l j n

nk o j n m j j d n j d n c

R j e e n j e . u o d o k z T e o r e m a 1 . 7 . k o n m o j k j z m k o o n o d @ A n n

n z o o n o d @ A @ d k l D o d m d m o d j o d @ A o d @ A j d n k o d o o A F o m o d z n m o

o d o o z k o d n o j m n j o d nF y z n m o an n o j F x z d n j m j o n z m o m o l H l I @ l m o z

n! l m n l n m o A F k o l k o j H D o n d o o m m o

an1 n n D k o l k o j o I D o n d n

(n 1)! o m m j n m j I D m m o an2 n n F h k l D d o l m o k z j an = an1 + an2,

o n m j m a1 = 2 D a2 = 3 D k o n n o an = Fn+2 F

P r i m j e r 6 . 1 0 . x k o l k o m k m l n o o d j n

d j l n n c

R j e e n j e . n m n m k m l n o j D o m o m o o d n k o j d n l l n d

n k o j n j k o j o k F y z n m o an n o j F u d n n 1 d o d m o j o j d n Dd o m o

nd j l o n n n o o m o m l F y o j

a1 = 2D d k l D m o j

an = an1 + n; a1 = 2.

n n n j n j d o n k z j z n m o D o k m o d z F t e l e s k o p i r a n j e F

an = an1 + n = [an2 + (n 1)] + n = . . .= 2 + 2 + 3 + . . . + n =

=n (n + 1)

2+ 1.

Z a d a t a k 6 . 1 1 . x k j rn o j d j l o n k o j d j o n l k o n k n o n! o k d j l n j o n n j o F

o m o d n k o j Q d j o n l n j k o j o k F x k z j z o n o m l z rn FRj. rn =

n4 6n3 + 23n2 42n + 2424

U p u t a . h o d m o n

! o k o j m o k o l k o n o d j l o d o j m o F o d m o n j n n d n o

o k j m o o l m o j d n o j d n l d m o k o l k o m o n o d j l o d o l k o m k o k F

n k z j j

rn = rn1 +n3 6n2 + 17n 18

6, n 4, r3 = 1,

k o j l k o j m o l k o n j m F

P R


25/51

p n k j z o d n

7 F u n k c i j e i z v o d n i c e

h o d m o o d j n j m k o m n o n o l m l n o m o d z m j l z o n o m l D n Fn!

z

o j m j l

nk

z o j k E k o m n j k o d n l m n F x o k o m n o n o l m n m j

j n j o l k z o n o m l m o n F S(n, k)

o z n l o jk

j k o d n

l m n D Bn

z o j j k o d n l m n F x k o l m @ o d d o j k k o m n j k o n n o m l k D o l d j P r i m j e r 4 . 9 . n n I T D o d d o j m j o n l j n j m k o n n o m l k D

o l d j P r i m j e r 4 . 3 . n n I R A l o j n l j o n d j k l j o n o D

o l d j P r i m j e r 4 . 5 . n n I R F u o d k o l m o d l k o m o n m f u n k c i j e i z v o d n i c e

k o j n m o d m o m o k o j n j d n o o l m m o n n k l z j n j

k o m n o n o l m F

7 . 1 O b i n e f u n k c i j e i z v o d n i c e

P r i m j e r 7 . 1 . x k o l k o n n m o m o 4 n 4 n o n o d 20

k n k o n o l n j m m o k o n o d

1, 2 5 k n c

R j e e n j e . y z n m o x, y

z

o j k o n o d j d n D d j k n F d o l m l X

u o l k o m j n j j d n d x + 2y + 5z = 20, x , y , z 0, x , y , z Zc o l m m o m o j l j n j m n l j o n F j l

zF x o d o l m o l j D

l m o n m o d k o j j l n o m F

o m o m o z z X

m o l o j o d k d o m o m k o j o j k o j n k o d o n k o m z n o l n o m k o E

n m o 1, 2

l 5

k n F t d n k o j n k j

j n j j o j o j n k z k o j d o j d m j m 20 F z j

l j d X

f(x) =

1 + x + x2 + x3 + 1 + x2 + x4 + 1 + x5 + x10 + f(x) = a0 + a1x + a2x

2 +

o d d k o ( j n a20 @ o z n k

x20

A F q o m l o m d o m o X

f(x) = a0 + a1x + a2x2 + + a20x20 +

[Rj. a20 = 29]

h o l m o d o n k j z o d n z n z (an)F o n o m o o l m j l l j n j m @ o n j D o E n j m A m o l j D n o z o k o m k n j n o j j n j j k l o j d n o n j z o d d F

P S


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U F I y n n k j z o d n

D e f i n i c i j a 7 . 1 . n z (an)nN d n f u n k c i j a i z v o d n i c a @ k n o p s A j o m l n d o n j

n=0

anxn

F

d n z m o o m l n m j n z m m o n j k o n n j k o k o d l o o d o F x n z n m

j d n o o d n o j o k x k l j o k o ( j n F y n o n o j o m l n m d o m d n n k o k o d l o o d o F

D e f i n i c i j a 7 . 2 . x k f1 (x) =

n=0

anxn

f2 (x) =

n=0

bnxn

n k j z o d n F d d ( n m o X

(f1 + f2) (x) =

n=0

(an + bn) xn

@ Q A

(f1f2) (x) =

n=0

nk=0

akbnkxn

@ R A

ddx

f1 (x) =

n=0

nanxn1

@ S A

f1 (x) dx =

n=0

ann + 1

xn+1 @ T A

o j o l l j d n j z n k j z o d n o j m o l z o n o n o j X z d n o

n z o d d m d n n k j z o d n z d n n k j z o d n o d d d n n z F

l k o m o d n j k o ( j n k o m o o j l o o d o X

G e o m e t r i j s k i r e d @ k o n z |x| < 1 A

1 + x + x2 + x3 + = 11 x

j m o j o n k d n j k o j j d z l o o d o X

B i n o m n i p o u a k

(1 + x)n =

n

k=0n

kxk

O p i b i n o m n i p o u a k

(1 x) =k=0

k

(x)k

D e f i n i c i j a 7 . 3 . x k j R, n N

F d o n o m n k o ( j n

n

d ( n k o

n = ( 1) ( n + 1)

n!

P T


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o m o n k j z o d n m o m o o d m o d z j n j l n n k z j k o n n n m k o ( E

j n m F y d j n z n m m o d l n j k o m n o n o l m @ k o m n j k o n n o m k D

m j o n l j n j m A F

I n t e r p r e t a c i j a b i n o m n o g p o u k a p o m o u f u n k c i j a i z v o d n i c a

nk = o j k E k o m n j nE l n o k x k j S = {S1, S2, , Sn} nE l n k D A = {Si1 , Si2 , , Sik} S k E l n o d k o d SFx k o l k o n n m o m o o d

Ac

l m n m o m o z 0

l j d n o m D d l m n m o m o z 0

l j d n o m D F F F n

E l m n

m o m o z 0 l j d n o m D o d n o n o X

(1 + x) (1 + x) (1 + x) = (1 + x)n

h k l

f(x) = (1 + x)n

=

nk=0

nkxk j n k j z o d n z o j k o m n j n E l n o k F

o j k o m n j m o n l j n j m n

E l n o k c

k l m n m o m o z o z o l j n o m n o o 3

f(x) =

1 + x + x2 + 1 + x + x2 + 1 + x + x2 + =

1

1 x 1

1 x 1

1 x=

1

1 xn

= a0 + a1x + a2x2 +

h o n d n z m o n k j o m z o d n o m z k o m n j m l k {S1 , S2 , , Sn } F

P r a v i l o i n v e r z i j e nk

=

(n) (n 1) (n k + 1)k!

=(1)k (n + k 1) (n + k) (n)

k!

=(1)k

n + k 1k

m o d m j k

E k o m n j m l k {S1 , S2 , , Sn } X

f(x) =

1

1 xn

= (1 x)n

=

k=0

nk

(x)k

=

k=0

(1)k

n + k 1k

(1)k xk

=

k=0n + k 1

k xk

Z a d a t a k 7 . 2 . y d d n k j z o d n z o j k o m n j m l k

a2, b , c2, d

o d d k o l k o m

4E k o m n j F

P U


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R j e e n j e . i l m n a c m o m o z 0 D 1 l 2 D d o k l m n b c m o m o z l n z X

f(x) =

1 + x + x2

(1 + x)

1 + x + x2

(1 + x)

d j l k o o z m n o o o m o

x4

f(x) = 1 + 4x + 8x2 + 10x3 + 8x4 + 4x5 + x6

= x4 = 8

Z a d a t a k 7 . 3 . y d d n k j z o d n z o j k o m n j m l k

a10, b7, c12

o d d k o l k o m

15E k o m n j F

R j e e n j e . i l m n a

m o m o z o z o l j n o j D o d 0

d o10

Db

o d0

d o7

c

o d0

d o12

X

f(x) =

1 + x + + x10

1 + x + + x7

1 + x + + x12

= 1 x11

1 x 1

x8

1 x 1

x13

1 x=

1 x11 1 x8 1 x13 (1 x)3=

(1 x)3 =

k

3k

(x)k =

k

k + 2

k

xk

=

=

1 x8 x11 x13 + x19 + k

k + 2

k

xk

= x15 = 15 + 215

7 + 2

7

4 + 2

4

2 + 2

2

= 79

Z a d a t a k 7 . 4 . x k o l k o n n 24

j d n k o m o n m o o d j l m 4

d j k o d k o d j

d o j m 3 D l n o d 8 o m o n c

R j e e n j e . k o m o d n j o m o m o d 3 D 4 D F F F l 8 o m o n F

f(x) =

x3 + x4 + + x84= x12 1 + x + + x54= x12

1 x61 x

4= x12 1 x

6

4

(1 x)4

= x12 1 4x6 + 6x12 4x18 + x24 k=0

4k

(x)k

=

x12 4x18 + 6x24 4x30 + x36 k=0

k + 3

k

xk

x24

=

12 + 3

12

4

6 + 3

6

+ 6

3

0

Z a d a t a k 7 . 5 . f o j j

p1 (n)o d

n z l m n d j d n k j o j j

p2 (n)o d

n n n

m n d F h o k 3

R j e e n j e . x m o n k j z o d n z o o l m D o k z d j d n k 3 m o j k E

j n z n m o z n D n m n k j z o d n l k o m o o m l j F

P V


29/51


o l d j m o j o d n = 6 z l n n m n d X

T a I C P C Q a I C I C I C I C I C I

a I C S a I C I C I C Q

a P C R a I C S

a T a Q C Q

p n k j z o d n f1 z n n j p1 (n) n j k o o d d D k o d m n d m o m o z 0 l 1 D d k l X

f1 (x) = (1 + x)

1 + x2 (1 + xn)

=

k=1

1 + xk

e n l o n o o d d m o f2 D m j m o z m m o n n m n d D l m o m o z o z o l j n o m n o o X

f2 (x) =

1 + x + x2 + 1 + x3 + x6 + 1 + x5 + x10 + =

1

1

x

11

x3

11

x5

o j o k z d d j n k j j d n k X

f1 (x) =

k=1

1 + xk

=

k=1

1 x2k1 xk

=1 x21 x

1 x41 x2

1 x61 x3

1 x81 x4

= f2 (x)

t k o o l k

1 x2k

o k F

N a p o m e n a 7 . 4 . o n o l o n k j z o d n z o j j z l m n d z o d n o z d k

d ( n k o f(x) = (1 + x) (1 + x) (1 + x)

n p u t a

F h o o z l d o l z m o z m l j j n l j d n n Y

o j n

n z l m n d D j d n m o z l n z D n l o n o D d o j k m o z l n z

d F o z m l j n j j d ( n n o o n o D l n m o d j s v a k i o d o d a b i r a n i j e " j e d n a k o v r i j e d a n " D k o

k o d n F o d k

E l n o o d k n

E l n o k 3 m j m o d m o o m l j k o ( j n xn k o j o d o o j o j n n z l m n d d o l n j d n j d n n n Y z k z d o d m o x

@ d k l n z j d n n o d m o 1

A o m n o m o F d j l m o d z n = 6

k k z n j j d n n

@ z o D z k n N, n 3 k k z n j j d n n A F o m o d k o d n d n m j z

n = 6o d o j d n o m o d n n n k o j m o m o d o

xn z n k j

f1 o n o X

1 + 2 + 3 x1 x2 x3

1 + 5 x1 x5

2 + 4 x2 x4

6 x6

t n o j d o j d z k n N

F j n j z n k o m n o n o l m o m o n k j z o d n

n z l d z o l z 4 o l m 4 z j n j D o d n m n o n j D z o j o z d n

x

n

xm

= x

n+m

o z z l o l o j d n k o k o d ( n d o n n k j k o f(x) = (1 + x) (1 + 2x) (1 + nx)F

P W


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U F P i k o n n j l n n k j z o d n

7 . 2 E k s p o n e n c i j a l n e f u n k c i j e i z v o d n i c e

z m j k o m n j n

E l n o k X

ak = Pnk =

n

k

k! =

n

k

=

akk!

(1 + x)n =

nk=0

nkxk

h k l e (x) =n

k=0

akk!

xk j k o n n j l n n k j z o d n z m j nE l n o k F

D e f i n i c i j a 7 . 5 . n z (an)nN d n e k s p o n e n c i j a l n a f u n k c i j a i z v o d n i c a @ k n o i p s A j o m l n d

o n j

n=0

ann!

xnF

D e f i n i c i j a 7 . 6 . x k e1 (x) =

n=0

an

n!xn

e2 (x) =

n=0

bn

n!xn

k o n n j l n n k j z o d n F d d ( E

n m o X

(e1 + e2) (x) =

n=0

an + bnn!

xn @ U A

(e1e2) (x) =

n=0

nk=0

n

k

akbnk

n!xn

@ V A

d

dxe1 (x) =

n=0

an(n 1)! x

n1@ W A

e1 (x) dx =

n=0

an(n + 1)!

xn+1 @ I H A

P r i m j e r 7 . 6 . x k j S = {an11 , an22 , , ankk } m l k F y d d k o n n j l n n k j z o d n z E

m j m l k F

R j e e n j e . i l m n ai m o m o z o d 0 d o ni D k o i = 1, 2, , k X

e (x) =

1 +

x

1!+

x2

2!+ +

xn1

n1!

1 +

x

1!+ +

xn2

n2!

1 +

x

1!+ +

xnk

nk!

P r i m j e r 7 . 7 . u o l k o m j d l j n 4 l j n o d l o B D AD ND AD N Ac

R j e e n j e . 4 E m j m l k S =

A3, B , N 2

F l o o A m o m o z o d 0 d o 3 D B z l n D

Cm o m o z

0D

1 l

2 X

e (x) =

1 + x +

x2

2!+

x3

3!

(1 + x)

1 + x +

x2

2!

= 1 + 3x + 4x2 +

19

6

x3 +19

12

x4 +1

2

x5 +1

12

x6

a44!

=

x4

=19

12= a4 = 38

Q H


31/51


Z a d a t a k 7 . 8 . u o l k o m n n n z o d l j n n

D k o d m m o n o j n l D n n o j j d n

o z o l j n o d o j k c

R j e e n j e . n l z m m o m o n o n j D z j d n n n D z d o j k X

e (x) =

1 + x2

2!+ x

4

4!+ x + x3

3!+ 1 + x + x2

2!+

=ex + ex

2 e

x ex2

ex

=1

4 e2x e2x ex

=1

4 e3x ex

=1

4

k=0

(3x)k

k!

k=0

(x)kk!

= (7) =

k=01

4 3

k (1)kk!

xk

xn = 14

3n (1)n

n!

= an = 3n (1)n

4

N a p o m e n a 7 . 7 . o k o m j n j k o l m o z l k o j d o j m o z j n j m D o d n o n o o d z m n j m

l j d j d n d X

ex = 1 + x +x2

2!+

x3

3!+

ex = 1 x + x2

2! x

3

3!+

D e f i n i c i j a 7 . 8 . h n m n j m j z ( k n o k F

Z a d a t a k 7 . 9 . y d d i p s z d n m n 3

R j e e n j e . y z n m o o j d n m n nE l n o k dn D i p s z o j d n m n d (x) F d j

d (x) =

n=0

dnn!

xnF

k m j Sn k o d n l m n D m o m o z d o j n d j n k n n j n

k=0

S(k)n d j j S(k)n

k m j n

E l n o k k o j m j k

( k n o k D o z k k = 0, 1, , nF h o m o

n! =n

k=0

n

k

dnk

o m n o m o d o n z l

xn

n!D m m o

xn =n

k=0

n

k

dnk

n!xn, n N

Q I


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n=0

xn =

n=0

nk=0

n

k

dnk

n!xn

1

1 x =n=0

nk=0

n

k

dnk

n!xn

= (8) =

n=0xn

n!

n=0dnn!

xn

= ex d (x)

= d (x) = 11 x e

x

o j z d o n n k j X

d (x) =1

1 x ex

=

n=0

xnn=0

(x)nn!

=

n=0n

k=0(1)k

k!xn

d l k o d m o k o l k j o j d n m n nE l n o k D n m xn =n

k=0

(1)kk!

j

dn = n! n

k=0

(1)kk!

Q P


33/51

p o m l k l j n j k l j n j

8 F o r m u l a u k l j u i v a n j a i i s k l j u i v a n j a

d d j n k n k D A B D z n m o d j o j l m n n j o n j j d n k z o j l m n k o m o d n j F y n o j d |X Y| |X| + |Y| @ z j n j k d n l n o j k o d o j n l m n j k A F j d

|A B| = |A| + |B| |A B|T e o r e m 8 . 1 . F o r m u l a u k l j u i v a n j a i i s k l j u i v a n j a

x k j S

k o n n k A1 D A2 D . . . D An SF d j o j l m n n j o n j

|A1 A2 . . . An| =ni=1

|Ai|

1i


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Z a d a t a k 8 . 4 . x o k n o z o n u o j o o o j n j m F x o n j j d o l o

n k o m D k n j n l j n k F x k o l k o n n m o m o o d

3nl j d o d

o l n l j n o d j d n o k o m D j d n o n j d n o l j n k D l k o d n j d n k o m

n j o o j l j c

R j e e n j e . m o n j o j k m n!2

@ k o m o d j D z m n l j n k

m o m o o d o m n n! n n A F x k j Ai k o n n j k o j m j iE k o m o j m l j n k o m n o m F d j |Ai| = (n 1)!2 @ j d n j z d n D o j m o n n n k o l k o m o o l

n 1 A D

|Ai Aj | = (n 2)!2, . . . , |A1 A2 . . . An| = 1 F x k j j

A1 A2 . . . An = n!2 ni=1

|Ai| +i


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Z a d a t a k 8 . 7 . u o l k o j n j k j l o o j n o j m o d o d d o o k (7, 5) k o j n o l z m n m @ P D P A D @ Q D P A @ R D P A D @ R D Q A

Rj.

7 + 5

7

2 + 2

2

4 + 3

4

4 + 2

2

3 + 2

2

+

2 + 2

2

1

3 + 2

2

Q S


36/51

o j o

9 T e o r i j a g r a f o v a

9 . 1 U v o d

D e f i n i c i j a 9 . 1 . G r a f j n (V, E)

D m j V

k o D E

k d o F k d o E

j

o d k d o l n o d k o o d V F o k m o d j d n k o j {A, B} EF

A d

e n d n n k o j

A eD j F k o o o j e k d e = {A, B} z n k B F

P r i m j e r 9 . 1 . q o m V = {a,b,c,d,e,f,g}

d o m E = {{a, b}, {a, d}, {b, f}, , {f, g}}

a

b

c

d f

e

g

D e f i n i c i j a 9 . 2 . M u l t i g r a f j j d o n m l k F P e t l j a j d k o j j m m o o m F

S t u p a n j v r h a v(d(v))

j o j d o k o j n d n n o mv

F

N a p o m e n a 9 . 3 . o D m l d e n i c i j e 7 . 1 . D n z m o j e d n o s t a v a n g r a f F

T e o r e m 9 . 4 . ( L e m a o r u k o v a n j u ) j d n o n o m G = (V, E) z o j n j o j n D

j F

vV

d(v)j n F

Z a d a t a k 9 . 2 . t l m o d o d d m o o k o o m o n o o z n n k F

R j e e n j e . o k j m o o l m n F x k n m o d l j j o o F x d l j D A j o j n o m

B k o m o k o o o

A

B o z n j F u d k o o m l o n o o z n n k d

n j k o o D j F d(v) = 3, v V F d j z o j n j j d n k vV

d(v) = 7 3 = 21,

o j k o n d k j l m o m o k o n j F h k l D j n j m o F

D e f i n i c i j a 9 . 5 . e t n j a j n z o (v1, v2, , vk) m vi vi+1 j d n z i = 1, 2, , k1F C i k l u s

j n j k o j o j j v1 = vk F

D e f i n i c i j a 9 . 6 . q G1 = (V1, E1) j p o d g r a f o d G = (V, E) k o j V1 V D E1 {{vi, vj} E : vi, vj

V1} F e k o j E1 = {{vi, vj} E : vi, vj V1} d k m o d j G1 i n d u c i r a n i p o d g r a f F

D e f i n i c i j a 9 . 7 . q o G1 = (V1, E1) G2 = (V2, E2) i z o m o r f n i k o o o j j k j : V1 V2 : E1 E2 k d j v n d n n d o m e G1 k o m o k o j (v) n d n n d o m (e) G2 F

N a p o m e n a 9 . 8 . e k o o G1 G2 z o m o n o n d j d

Q T


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W F I o d

@ I A |V1| = |V2|@ P A |E1| = |E2|@ Q A

d(v) = d((v)), v V1@ R A e k o j

(v0, v1, , v0) k l d l j n n o n d j ((v0), (v1), , (v0)) o k l d l j n

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Skripta Diskretna Matematika