skripta diskretna matematika
Post on 14-Apr-2018
309 Views
Preview:
TRANSCRIPT
-
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
13/51
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! .
I Q
-
7/30/2019 Skripta Diskretna Matematika
14/51
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
-
7/30/2019 Skripta Diskretna Matematika
15/51
m j k o m n j o n l j n j m
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
-
7/30/2019 Skripta Diskretna Matematika
16/51
m j k o m n j o n l j n j m
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
.
I T
-
7/30/2019 Skripta Diskretna Matematika
17/51
m j k o m n j o n l j n j m
@ 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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
21/51
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
23/51
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
26/51
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
-
7/30/2019 Skripta Diskretna Matematika
27/51
U F I y n n k j z o d n
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
-
7/30/2019 Skripta Diskretna Matematika
28/51
U F I y n n k j z o d n
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
-
7/30/2019 Skripta Diskretna Matematika
29/51
U F I y n n k j z o d n
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
-
7/30/2019 Skripta Diskretna Matematika
30/51
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
-
7/30/2019 Skripta Diskretna Matematika
31/51
U F P i k o n n j l n n k j z o d n
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
-
7/30/2019 Skripta Diskretna Matematika
32/51
U F P i k o n n j l n n k j z o d n
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
34/51
p o m l k l j n j k l j n j
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
-
7/30/2019 Skripta Diskretna Matematika
35/51
p o m l k l j n j k l j n j
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
-
7/30/2019 Skripta Diskretna Matematika
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
-
7/30/2019 Skripta Diskretna Matematika
37/51
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
top related