Σημειώσεις Διακριτών Μαθηματικών by Nina Zazety

~____+-____~~~~~~~~~__~~Oa路An~~ =u-

~~____~____~?-~~~________ 7~ ' _~fl~ ~p-

_______________

_____________________

~---L--------------------------~-~----~ 1

5 - , - =-

.. . "

-8

c..

Ql1'Y..~

•

::0

.).

r足

-1 0

()

~>q

1足

JA x61= fAl-Il') .. ' A~·r 1Jrca-10~

.hArol~"'"

IAU~ f:: lA

Jt '

IBf~ IAn B1

",,- t9t. ~fI t va E. ~ r~'":..:IG::...:..---_ _ _~ 1,0\ T'I" 8~'&~" nxl')

. ..

t·

•

...

•

Un-i~)~'o~__

If~'il~.t:'''''' ~.

• .1

~\o."'f"~4j s;c", J

-=I

4~ ,. ... .. ..

o.l4rlt~

;"'10.

,--. ~~

__ ::

-=+1 ..-......:;..~

9~Q

0 '0 . '

'3 1 •

fl."},.,,.>

~ ,,,o,).tl91d· 'cr"~'I(.~~ 0I.ft8,l~"

~,~ ~ -at,?Cf"" ., 0") '-lOt q~t~/t.~ " . : . ., . .... . . ..

fA, to1o\ll it \Jtf'V~J ,

4O. !

8 " 7[

:: IfJo

. ~1Q.lt·e.1;~~ "

o '.

, ,

, .. ,

'fit

. ., .

It:".

~) 00 t. {

(- '\'>j it'D I

It ~II') ...;, ~'!'''1 •

">(

t ,(:; 1· 10'-".1.

--,> ::;j'

.•

fo'1'~ Jdt 1("'; .

1'6

it;

• . (...4,

t .....

s· '3 :90.c. ~ . . -= 3 (<to·'! ".~

A}AlClt~:"}I~ Avt.IlA...,.'Evw" (Awf"ff.vt' . .:

1f()~';'~r)JrCl : l1o"t.~

~~r>

or~t~

0foifIJ'V

:路

o.itlUflf'f.'(I/IIY)

(/1~0i1~P~\ P'1~':" 5) fl~ -r.~.y:/lrq7~ : KAK" A-O '6u

~ ~ C( ~~,.

1:1)

2.~1

~~----.- ...........

~ (~~

l; "tv.. 'UJ ; 4. ~ 4t'lo'1"Q. '"I'll (

. '. .

. "1 d / •.

81 5.'·.... 3! . ..

Af')C"1"

••

33j ·~tJ . 1~.~ .. '. '. ; .

A: S (,jAto. .:

22 'ii 0. qDr f Clw" Itl;) ,'0.. ~-~ 13 : 5' - 1(• . ': G 4 -1/ ~:

-lI

t :" -// ~----"

91 "

2z!

'1':l. _ _'--

ftc

•• f

10 '"d""fl,/' 0 \<0ro tl'" ~> TJOfio. fiJ'1J

t '" . ~

10 ~

1<'10

......)

10~

-:B)hv5

~" "~f'~V","

y- Q UU,rU6\'/tei -= ~ ttl -. u:l6 D'.

6it. C'f ~t"

•

,~,~)

, e, ,

... ,

•

caoo . " , 61a'f"Pf"tI':'';' tAvt.&'U1""'~ ~=

•.

. .. ..

.. .

•

...

. .. .

. ... .. ~

'" I" •

•

...

'.a'

;1: ... . : • ,

. \

.. .., .

•

II;•.

((1/,.. )= 6 n- 1'"

.,;

; ,

~ ,.

. :. ..

~~--

C ("7 ..,., .

L. \ ....,~

..-

r·

4'. . 3.J

,""

' t o'

,,··t

"1" ·

(::. 5

,-. ~ . '~

1 t

. .

1T~Q.~uIf'" ?d~) I!l ~ 1 11

ew ~f"'l> -? rlon)

src~" f*,,7f'~t(/J

61" of'l.'c.

(Hf"1\'''-S It

_)'7." PI"1 (.1(f'

A V~ J3J' ~ ,4/ ()-1 I

.bI\1"., .. ) , > () 1~ - If .. - ) :rtlf'finr%(JII~

oil

I t! ",ri1PfW

· ..

8{ ~:rl IA(A...(. 0/

1.tJ).~-1ln~

B { ('1 "f) :- 1'l f:, A- (("~) :. 11/,

J' ;J; _

6.YQ.l

~ )t,I o.~i4

11011

171(I·f~J '-"--::C"'--C-1- 1-, -S)-

5t1j1Pf7j.~h" .,J~ al

7 • .., )

A !c'

~o: ~~

a"OtI4Jf..../'f"t-'

~"III7.Jnl.J .

C (",)./-;. 8 'f

,:; b

, •#It ~) (., (1p. 11 h ~ I4f i J

4i~ C (

. ---~c;.. ,",<

9, 5):: 12 ,

.. "

~ In~ it rli

c.«((l)r) (n-1)!."I. - C . .

• .,IJ

't "

«I •

. ••

. . . ,.... . ,"

..... ,

. .....

•

Cn-j+r, r

• t"

.. . •

, . '

.--.-

•

- ---. .

)

•

. :z!.1 . . o~

{~r ~ -:.i. to 1 0

for j'W j

C }?l./f OJ 2.)

t~ (,

:: ?(0 1-12-1 ~ .:. Q~

_ 1JL1-1-: S' < ...

\onof)&~'rJ ~tlU'( ~;

uno op

. LJ.W'l )'\ "~&It f,J p.,

•

~ "..1"o~ l.5

&,'4Z~-t) I

--.

••

1 •

0.. fJ

• .

. ....

'" . (t 'P,S,a..

(0..)

'l'y . -to

(r4. t»)

,,;' :

. C

. ,en,"('

) . .,f

~ ~-r)l r!

0"'; ' .: '

(i) ~'l,~itJo~4t

,.t"r~

t\ .)) ,

CJ~ (~J

t'l

1ft' lJ I

'1 h)

1\ ~~

A0.

P.Ol

12. . :; 66. $J1o!

:::I

, fc_

piJL"

.. : -(

. . : ~. ~

JQarro~

"'~I()'

·B ..

r) .:

VOfJi

obtl

·,fJ

z: oil

.. .II~

.. P-

r,t}V'

;S?

UI&t4

•

.. .; JJ1.JI

"'Cot)

/AI"~

6 ....

tJM'}i

......

-- ...

~) (\0

t.\

1o...e)~)

( 411p

C. (~OJ 5) .

(~) AfJt'k~ ~'l« ~ ~ . 2 l'~ ).i.lv,v'4

-q

'J/

19/ j/12

. .

,. J' '" 3.20 ~.~, I ?· ~~I ~ . ~~J ~ . ~~

(()-tr- 1.)l - (r1- !)f

p- (q - ,) t' .

~t ll .

(err'" '"t )

~ .,.1) ,,(\o ~ • ...:r~.1

.. ~;, ~o

~ J4 '

~!J"O) ' ' #-

'e 0tJ ",,-,'"

(~- .,) ! (.n;') ~

(kit) "

k ',l\A.~·~

o~ \

~II'

l..-lb

31)\.. f1 nx "., : ~. • 1._ q n:::

A V Cl~f~r·v.4)

-Z~l)o': Q".: 3

-i

'1o'() ....

-:;

n../1 .

a. n = 0n-! -IlJ.

0 ,::. Q 1

Qn-"2

rib.

QC,"'

':. 1.

f ?Jill.

lLf~Cl

({~.J.. )

lio l

"" t~ - ';l e&.(o

020 , . ~.s

"l "

11""

II.:>

PDf' "(.0

;

7utJ

Jc:~Ul:Vo •

..,,1 .

r",t')·$ •

~ep <lot "1 -t

16f-4

"'f v.1",A

T.v J " rl

()(z + . ..t

~,,-

c,!)

(0 ~ ,)+ '0'1' '2/~ .. .+ 0 t (n1l1 - o· qo '+ ( 1 r ~ t . .. + (.,- J.)) '. . G• ..

·n .( n- i)

\oJ v

iTf

'1 ,,)

/J] ~

~.:.

'-->-

00 路

Z 'V =7:. ~ a... - 颅

W':: 0

(~)

1w v J ,.. I J/.O'

~11

(:) ::.

'\l

C( )

. oiO

1/ 515/1 2 ,

'-' cJ +(4)" 1 z..

5) 2 f ri~

l~ 1~ ~路 t)

路 '..

.1. \ ~'~t, ,, .'4''' ;

' ,en

W"'1:rT~v'

~ao1 ; ,.. ( ~~ ~"2.) ~

::;

K .Al2

'7 f{ .

1'1::; )

0 01 .

Z m

ArzJ f f (3 Cz)/)r <,

o fl

_.",p

A(z) ..:-

_ 1_ 1

( ~)

O f} : ~ ()

+2 - :-:Pt ~ Z-

A(~) =: 3 . -:;, -2. ) 2 +- 9

1 1 -~2

'1 ':>

•

-f 1- z

3 <

&-2)'t

0 -

0 1:=-CJ

. ~ ::.- 1u

·2 -1-5

"2.::'

'1> = &0

-J.

1z:.

a~ ::: 'r

z路

~ -1 / 7 11 B- ) btv t ~Ol

/1v

1fQ~~~"" 3'

2+1. A(z)'=';z_'z+2

z.-t1

~ ~)(2-Z)

~ 3...1

oz- 2

rA L'7?r

-: -. A-I

:: -oz- i

-;:z Joz-1 ~ G - 1..) Z-:J..

n ::.; ..

2.'!...1, •_ z .. 1 ...

A(2.) "= z~ 'ut -..(tr1)(z.-2.)

•

- ..,. .,,-z.

g(~-.L)

•

.,.

\.~qPf4I~t, . Av'~.e l'i''''~S .r.J<'t~U.s f:l 670£9fPO~> J'\JYlf~"""~l (~ . Q.rt""f i"(t-i {(tH'f -1 + ... + (0 an ~ n-.. ~ Ii f,,: 0 7:' 7 i-: n ~av ~ ~; fa. ~o A v9 : fA v~ ();UJ'f~~J • • .• . Cl -'I ~'(.. rtl'J o.lC.e~",,9'. ". f ~ =. ~ !n-1.

'1"'1.

o~o -') f""'2 ... f't\.f1 -.f'l ~ 0 {n -f.,..,.Jt ~ f~-<::.()

(f 40 -) f ~J,

'U)S

c('(

Q,,:: On

-...

11!

~ (4t L)

,se'Jt/IC?J ~f·I'£.>

tf...

i .

•

Q."'Z

-2. ~." -'''. =(Alz) -o.o-a-t=J-.2i'[Mz) -,..J --:./He) ,

Z-2[ A(z.)

-'22..) -':2 z-1:(II (z) -1) - :> ~ (i:o) :'0 1=)

A(~)-1- ~z (A(2) "1) _~z2. (f+{z.)) ~ 0 =-> . AIL) (<1- tl z - ~ z.~):: 1 of- 22 - 2z = .L

. #

••

= D F>

Q.ttoAo,,94·A.S fi1 .....=t .. 4

("11,'Z ", /~),

/To.f o.~t{,ftA ~ ri;cro\ "'...

. J

1 0 - ~" . (\.J .. 1 o...,·c · -; ~~ .tl

1c" ':l?6o>

---t':71 ~t6.,

"tw"

J110~r'Y""'v ~f,9~~"~ ~~Ko")

....

." ,.V\ . ., • ,·. l«v )('1 Of;;J~+'t

•

. f fCw

o.el'ajAD~ CJn.At.J{ovV /Ai- "Y.~J::J~7}IIt __ l/ 1 'p~cS"(-' ( ~ . '

[ ~

•

cP", O.

rt '

,. " '"Y

£) n

. '..

I "0 t!

•

LI J.~7 /[97.. ---) j ~

t ,.:.. }.1

'J .~ t1 D ' ' \ 1q

Q I?

~/;;d/J ..

£1 [ ACz) 1 -10 A(z1 = -Qo

A(z.~ .~ ,-4 qo ~ ! ~~ (",y =.A(z) (1-1OZ.)

1-:~z

...

a.t :,~; . •

'1- .::1

.. .

10.

-...

- - 1.

_ 1

'aot

A (z)=

;LIe 0.

••

1-10z.

. .'

- -1

1-~z

qn =. (~o 1l)",

\lufJ'" A

J} (z)

::=. ••

t v

Av;

1 1-1 oz. ' L

# It'} fQ.'C.~"

et..,

~= A

~ =3

al1t1

Xf~61 fflfv'11f'~V o

~\!)o

1-]

•. .

e-- r

W... U

Itti'. 7.,

t>~OI4

r'/

Ii. 6k..yv

~(2Y/-l )

=-20"., l of

f' ~~7;rJl"'1

Z,,'VJV(A.q}l-.A.;"

Tfp;,'3"P~7~

5~o.tf;~,q

.. ~ n ~

C("I ()

'71 lIo};

(~ )

i "'Y4\f;r"".

1.4,.

;.~,.

, UiJ" 7,

. . .. oj 0 ~

o· . 0

~.. .

..... ,.-

<:-., Of:-'d"«

. ~o·~"C.t~· · ;1....."s

C ~tl/") (1

n if"P'~~ .o

(to.

nf0~

70.

~ 'c ("~,,-41;') '= t~;-4'~

+,,):;: i (;) 'I" r~o

..,

"1''''1'if,'\.)