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e> (pt 7p)t?e> Tr7€>T

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A J.LA-QA,L

,r-iil" &)-s*u.L

Q,,cs, - , -

LLe-lnDf!r!ou) ..LLc4<-4A,p^Ca,-.t-z la*i

trrr"rlC@t s lLrL

-A ,*i-""xo

J-9LLL

,ru . +or,/rt

c-@fitlu;^tr g dg* t l9 4

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e-p

\L2-./

en = CTQr-, + c2 Qn-rr " ' + ct ao-o ultrl'u Q;$ "

Lalt- ,4La-(_ rnumbzu . atd C*lo.

,,14- ,lhru- qu/tlDdJ

zLlatu;w atl '

/. /,b.tafuk*oO

2. aAa a c*-l-u-lai "4-a4fua^A

V-ura7,ri*C,4'u' t-\ lD, t4 12 :>aoL)

eo,at, ..-.-door.... -al A.< a-e ,rrr/ -b;u

.tt f-<tv,tc

--2

8) D,,(2

#a-^-d Y

Ye:'

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3 ga rrofliy faaa,<ittt

, Q<-a,eaafugt

lJz Qoru,.ralig /rt-cnfu /44- .1/z

o4?..t-r4J .

QCil. q6+q?x+....-+ an*'+ ....

/,ttclu;k^ S €zctaz&^- .

tnt x ar\d Y 6z Atao

-uiretsa) 4LL u't q X

f^,*+*

-.LZ c,CdnfltU-

*w)c2-

-& ak

kQ. k'

4d4e2,o-u- dtyaul

a

I ,rtryy

-^q a

ctl,a-^,

:'/xuY'l =lxl+lYl'

-8, @UrA L'tlt-

4 .'v a^d Y a,Le- ')@t d:'d"@ir urt-t't'

lxoYl =lrl +Nl '/xoYl.

a,r'.d ?-YcLta Lot- .

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w

Ot/utuh"ll -af(,ttVzta.rcn

a) 7li-ol' -ct n oadt td iar4f.ktIA*tr

d,ll

t*)ol:

atnazd.) -lro- fa,b,c,d.aXutl^:l^t1:*S

n=6; r,6

: ((n-t +r,r)= c (ta,s)

=roc,_ to.1.e.7.1 =prg.

t. e .2.+.5

tb) .l-ao L-xJa,t

e!,{nt O.

"rDfu-'cl 'o "'

C Ch,r) , C (n-r, r'-! ) I c

l_o,.or.a3...... .o.J *"el-r/nz^i

Cn- t. v)fN/t'oal

a 4.er' -.r"

. .7,fn q.un]rz.t-?, a apzQot

a-Il /-cnn-bLanknt i eCn,r)

./a,,.a.hi,,za-x)a-,-u , alt .f-rLe- f - cotu-bztaanu ,lre

c.(tlai cP*.ta;J ao a-td- A

i - a*tl,aaL"a Lt 'a* d'o'

..,y12 ,x-rrrrrtl -ol ,o?tr

NLn* A bz "c'<-z 4rr 4lz-

oi-ot

y-ctdir"l-iw "l^ li r,ulz '<ylu b") q,ao &-bcL y-t.zzztnuLt -{a-an- U-{o"l .;t c Cn-r ,r-t).

zlt .r-.<-z- f - Co/wb-baa'@4t

;1" ,-zt.o atA4") A a.t\d 8.

a.L. f - c8il4 \afu;N

ttLt, ,tLt -$ alt

ca*z"rL . Q,t .

,1o &tllct alt

ZAt qa.nbot ,$ ,<t*o,tok.uol e^?1"

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()./

,r,fu. qllrlbrl S *"f .-{o ,Z,b (i oq'^-

t.cota-bir,aniu ,/-torn O ,2 L.fut- cNt-unbe< $uza-f ,-/o zzkct y z.tirLe^7-t l*^ a- {"r1-Ltla,r ,;; c C n- r,y).

C (ntt') -- e (n-t,,v) + c ( o'r ' t' r) '

dJ.otD ,zja* 2o >oJoo"

,.t-td* oaa*' Lfzlu\a?k -

X*

on) o, utukvIA-.f *l/das-]

PCn) -- aD- n3 >o /* hlro

f u,t D=to) pcro) -2'o-tos = to2h-toob =uht

-.. pctD ,f" *",,,-t* uut Al4 U)'11- t-tl-q.t Pch)

lO o - f) 3 o ot .-t.2., 2 - h- L,u Feu,a^lz

{* 9D-or =k ---) alo 7low L.t^s* pto+r) a,cr, Voitiz Cb>to)

utiazx. )n' -cn*Da = 92D -tn+Dl

= I (r+nJ) -LnJ +Jn' + 3n+tJ

' , ' i2 - Jn2 - 3 h + 2 k - t : -+'vz

,.'2., Ptn+0 ,rZ o*o "//tt/L..'. 8y *irar.ai* f@/^

t pcn ) ;z ,1,4-tu .

-,tn d4/l1 , ,% zL

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N;,b .*te. /e"ua*?g t- "A4ei-afu -i,tg"'t'

= t1 2*-.t 3X' +i73 + -. . ..

\9v

4o,t JLo- "527'u'tcLu

lt't l6oc)

+ (n+Dxb+ " o"

o

%4t z-*eetdt"Yt

x,o Jn C r+ rr* x'3 x'I .. )

(a.o * La t-n?q)

-fcx;

fi-at^: ?T q |o'u-a'ttoo uttb-t ',-,2 ali'*tllt .,L.u a)v

,.Li-hJ 2At

a.aiilz" bV s a'u- 6'tu'ts''"'

w*t btsl b,Qn-o'*,d.

-- (oo-q

+ l = lq-+t a2a'5

eto6

s!,YJt-

Io*-^l

NtunbzU

a,LLL/tLb?) fi

J,1,a dtu -' 'lf{i"'r,9t,

Lo a 't rerr, d.ALt

( t+ lt-'* nton l ts '2'' ' ' ')3

9o l,:'"1. f.t/-t

- 3 -3

= ftr - xilJ - = Q- "-')

3 " Cb+7-r 'l-) r't'

bedl;;$ ,ef *'o,

5t.to *'Y=2

ng,rriraCCa+t-t, z

cn. efl'< u-e''*

) = ct4,2)__ )t cz

= 6.

..L8

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wI

te) .tt-L or.tunbtl -% a&t:,y"u.&,.aa*t"

qarun.za*i4t -$ q- &HtAt -1,, scer'tcr '

C A. U Pt_F >-ooq )

at

1!9!rt.t Olto

gl-p.r.p- La^-e- 7 Joatu ,-;e 2-('r' 2-E"s'

r-7 .t-s.:. J6rit u,ttz-Oz-< 4 flw'Afui^t =

= t2 60

Nilfu .4)t-

Qr, = 3.46-,

,bl:Llver

z-x?.c.f ala-nd q, ,2.

4n'3Qn-tf.r, Q n - Lan-, =o -1@

t* Qn=r'o be a- +ot'^ h3*

(0 + fh'-3r"-t=o

=s rhft-!J="_> y" [-y_+!J =o

o.'-'. ,Lit- cj\a^ atfu)'"'l 'uti 4 ^"1'4- "Zs t'3 -o

' "f'"',.'n'e

-,. bq *.AL o+-r-!tt,U

Qiv* , a, '& -l Q'

d =2 .-.(2)i3

,/atnu,{z 4"" Q2

(o., vtll .>-e"+)

Qn'4an-'>@>oCJ .3?<:2

* Qn'(*) cs'l

4 Cr)

iir rr tl.r

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NAot jt .rt-a- mtu''

5p, ,6o . D r4/ J

= DP3

Y.3

yat- .

tu",n/,

Cp/htufrl'-, 4q)

.-tlt-

- n. e ttAtst-

(p-nm"Lau-

,*octai*-t '

cor"k* 186 J

4 ,aJ*-t' a^-L

Apttcan"liYt a^d

q Qdt-LCa-^'a- J't*

3o l,

7,U 3o

zd/-LaljeA*e

eGU4 ?r,L^alctt CaA/

ir. Q ct Lut'/' '

4

n A-ol

tt- 4 Aoda't;*

Aocta'A)+ "')" lt cD

/5 O-L.ot -.t/@ ca'ft)

!lP@t'

a

ca-^- 3 z ci-aa'te*

.4/r/1 , ..CLt- /Lo- ,4

/at-. rr

Wa.r7' '-Lo

.'. Dq q*od/-c*o t,;h,- ;,

=Qca

- qt

3 | Gt.

U Crt

. rrlh tll

= Ih'23o

. = 2a72o cDa-f

;tr 4e+u-4 ca a'n = 3' 2n '' ry>' ''z ^*; "Z^^--v*49 -4/-c-t-u)-a^'t''',zla'c;'-

( n'' NIP,,!1 5 >aot-1)lon n> t

o Qn,s'sn

(r, ,2 F P,7 =$o

' (o't-l^*1-j r-oof )

,.b f* a'

,, aoctadtb '

QA'U, FIlg-'t{ I: e-eD6,

bz cl-aaE 'e't f*,

*"f

tDdft .

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a" t9Q n_, ,

=l Qn r 2

dr'LQn-'

rlA-L a.rao c;a-tad

qn " 3Qo-, + )n'

n>t.r;*L qo+rCo, -,). "l'*

dr2 Yi*le't'9ol:

Ln-anprArt-Dt' aaZul^i-

^'o'.i./O \oos)

Lo nu-a,uy

9/r2l-4;r-

I'lao,

J-ernngziz-ou'E agaal-ue't't h)a,- = 4,

Yi,ta &'t-a-

'loJ't'ot/'try ?s.ea,u/a cl-)

8o-lr.L7-La..o

auoda-t4-d

a,

?zuL@ 1/l'j'1ct'frN -l'n *

"^"'^1'o '< / 2o' 2'' 2" 2t' "'' g! >

.ttu-

' C A . u- t-\ / o, rvr {3 :-oa6)

o t"6 o, sz ti-.- 2rl

i,rnr-aln -Z Vi"'- 6V

J.ti g-e-il^a,a^' . / * -." -C)cx)

= t * 2*'{ 2'7'+ 937s+ '' "+ 2r2'v

= ;! ct{"D'd

t

a -L b a.u o c-?o*a cl-att'o I ztrzou't 2?4 atu'et")

4n - 3a-Drt =o

Xfu c-La4 a- eLui'| a/c

F(h) = 2-n)

t/)

Y'J =o

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Sa- ma"^y qrayolsv^

4as dcZ*<-& 5A a b4 7'( i'o * ta iooL)

wt .lo loo .//a.t

ruo . lz-hoezn

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dLo*tb,u b7/rh,u,qlzt q LL^'tLt!'L

&eltout-

"/-,42 %,e+

.bl:{*e

I -too tLfu-L

ou,tpzot;relV'

a-/\A A dt-uttc

*r- o42rl 42t,tz

tt t-

b-V

lAt

flalt* /et

- I-oo1LOJ

lAobl

.eB ) @'' {.EJ' +

_ t!:l =t' LJYTJ

, lal - lt' o8l:4Z*.r'*: 93 - 4

:27,

,U;ry gnaflrndfcal Jnaocsni,t , ?**g,f z2+ zJ .+ . , 4 2n ' 9nu'-2 .

So l,

(tt )

/ee Pcn), t-+t'+23+ " "'t2o

A,rrui42 pct) : 2r =zt"!-e, & ,-arr-r-

A aat ta,r-o- L.12

Atu,tt- --->@

P €R) :..g+ t2+ tJ + '

. 2k''- L ,)t

c/Ar:,. : PCk't t)' P(r+r) ! 2)22't e3+ -..,.-r z* + 2"'

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Nlut-2,

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. ^D - t*l-l -L .rLrr-r--

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r@9U-

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qJQt

LL-ar-i-

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frz,tz et-o.,4 ,p$"a* ,^ " N".g fLfa =oo

No.$ "Wa*-t =t, -- no. $ n-ottt -rz

J*n fi gr^u"t 7za V$rr", 4ob- N"f.

Y/J,,=lTl.,*,9 , U-'.aa-e. -t/g.Lh "2a, * r- -6arnz .tut@Att-.

&a- ^-./ a Zadzttt Dr4LjL

ta

Vaar.t .-a^) a 'ie/.

:fLe/ - o d.^D too pa.r,tt..

a-fu- LA^a.t a-d-z-Q.9t "+ou.

o

rlre- .r\ a c-fara ,r44,a1du}, ,-q.n co-? re

f' a.t zla/n , t/ -..e-xar

4 caua Ld/S o.t.2/.--,tooL.:z Jwre. arn&^f loz

YJ-n.L @4r- or .p,nillzB-V V,Xr-o,^--tat Fh?o,

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2 n'U/-d-u,tl/9la.dLafu ".cir./u- ,7k44t

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Lo l-alL crrr)u'v"^r-r* Loz,rLL L(fr,tt -rbett-d

bl :

Nn&-

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A1)

gA"t rrrr^. I " XLf4t

Q tvz* DPa = ooP,

ncD-D(D-2) = E hhl)D-L.6

a L fe4lwil't t'^ nP'..

DPr : DcD-t) CD-2) . . -. Cn-'r+r)

flot, Onata .Q e-un-Lt /rilrJ(t 5 Lt'tr- -lrd;ilJ

ABCDF€qH t-q^ra'L Lt^-'- "*U AieL'

Bzcautz t tJ,' '&'eg *ec "lvu*'t o'c u' a4 b/dL'

L,),o ca^. P .* vatut'Dr'4 bV t'!U ot'cun-e't

4 'P4/t+!,1-tatLoh- -Y in **ieoLt A

Ld2, AAc a^d' ;^d;;d'unt ukltt'u 0'E ' F' S I ' '

/*uA*oe . ctA't'<'t- az 6! ' +ao 'f e'z rn'u1ala)'a

' o " ^ca''P J-a "t':/'te* A Be O ('un'14 Lt/4- Jefral AbcDEFqP Lt-^)

II

DPo ' 6oP,&t>

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t=

Aq) 9Lve

sol;

t/^t- /eqtlh /ot o1r .

D Cy ,D!

(n -r21 t t

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fri"fi* .nt^o

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3D gAd, .h,- zt/uxf& '4

a f,,L,r1

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.8ta

drA/,,t" k/ i4 * *'Vil* f*il'{Vo4".

9!-' ^

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arluJ-a'4 Yuil). t. a6ouea- 4a/2x :'

,r-/, .4at) ,.al ,ll"t u74s- Patt'tt 'tztl

,0ltfu La,L a d.r7qe^al .

u

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ilo, oi//rnb* q /ht, Lrlz-

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o

o.lAr

Ltlr, q,Lar#&t 'al .,t'd/r ' 1

a,h//nbzt ,*t a;aXoae* >&r'1 =14

3D I t-/l,u-t- 'a.14 t 7wa-tfu*ta Vuat.d*

ln lato *"n c-e-.\, a- inr aal*e a^2

? uu haiz 2

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a-p*"Y q a'& uar' 2*'^ain't '25'

B ttt -A.i' .inut' a't 'Uj' 0'a'6 s ;t r''L"&J'

,t Jo, 4e 41/ 'tlr- Tuzaltbu "'/^y::, ,".. .ylo, ,fote) %)zrrtbut q ('2a? '4' '6oh't^!

.lza, e il . ,Z,n *t LlAt- 9u0'4"2 q *-/t 4

- ,6 -,

fafe" '

uQ.?--z-

z,allL v ot-t-g -Lt @t'

4a.tnT ci* .oL

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h ) o.

ltQfrun. $cn) . dGo )o

34) 6 d.lr.rz

'.ICn -'t) ' 6OD )'- t

4Lq)= (-6)

6co - rl = stn)-5

3cn) --'a scn-t)

Sca) = -6 ' 3 cn-t)

3(P) > 43Ck't) + ro I (k -t) -- o

art o i a,t'd- c'stlo9t2'\oLt) dqdaaLjL'u

- ruCK 'O + to'rCE-2 )=o

Cl,araAr,)*Z aTaat'b-'t)

12 -1Y 1to'o

y2 _ Lf -it+to = O ,

'YC'r -2) -EcT' 2) = o '

tr'<' 0'z)'(T 'a ) "o

a,at o tt''a*zd

ot-zl

UL,ScK>

JL,6

l.L., VJz 4e-L/^7'4>h'

qb*-o9e'qottS e?aafuil-,:,<rbcil . q,@u *4, cr)r

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8n) 9;^^

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uat Arr-df lu)(hi^ le'L 'rij' /x'u'u'&

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lr>o

> E2u+ 3x') "'--z I

=(r-x)-=;;,

lA,unrg^"uA+) =la,l +tnrl+tA'l+ le4l - I

! a,n n")-ln,naal -!At n a") -[A" n

+ I a,no, n %l +lA'on '-o aal + lA 'o

+ lh2oAa oA4l - lAtoA'oA"hAa)

9r,>

Lttu /".fu l* lA'oe' uAguA+

ul,lt a',a ocPoA"il cn a"tl%o^Lou't

e.n=DQr-,-|Qn-r+4h

Xtn a,$oua,&d JaocnXzat*t't

Qo= DQr-t -f 6Qn', = o'

Jtz daracA,.iat'iY2-nt"t6=o

I

n,AAz) -

nol - lAi

n son4 )

nn+ I

&eL/ltun^ f-a't-3+) ild

s:l

&*u aAz r'2'3o -

(h)

t .'. $alah,it*' & ( n

(,ct

._ d ,(L)n * d, ca)o

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I* l,'br^, r'ryl4hftErnhl1cA L

Suhl,a,-=a-

MA THEI.4AflCJ

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*T laa "at/- Wt

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t +i,* ,

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o

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,^? *t tlv- .Utru "r"", L" l.*Tt. )

val-,.,-r -& --' r*

- \ na =' -f :".+1----.f}a

= )o L=,,-.11, + i 1"- - til-'\)Q 3aE < = '>-o -<-2-

\-z-r 2-t > \*'-( = q

-r : "v +.>- A '>- ':-o

6.rive6

tu(ni'€ aoc.{

tr -..." @

lt .l

aldral r!r!' I rl t I

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LA'u F{lD-'r/tL')2'o'

-(!-@

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@

P..o rz

Jol:

&pt )e".

.;^1- er;* H)+22+ ' - "+ 2h1'' 9n = 2n*'-l'( A'tr' Al9 >--az- N I 12'>'ov)

?xrhc,i,,lt 4 mall-e-rna.tfctl or*atot o* t!* P

be

,fL

a ,P1^o/o4

t.L.r Pcnlo

Lt)

n ,.tt N toff*"

fepottttz.ei) Pc t)

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',1 ;,* /r/"^ad @r\- Va*fie J,.'tAt''

l"ku fo.t aaa

-? clt,oz;tt/, Jr-Lt ,

P .hat 'llt'

.f, ar- '

"& *rrl, utlt)alz,

,hu/, X* *^7

ttit

"aL

141/,

fecnl ,'k

f*n*'n

,hrL.

L,t l2fl?)

Ptn):Jo.r 2'+)L+'' +2h =rl*'',

Pcrl ' 9o72t ' 22 -t

llI =22-r3 ,*''

3 r3-o

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A +tLlt4-a. Ptn> 3 dLu-t- .

Jnh*t6?"+&t& +.. .12 =2 't

olo ,*a yz h(blt) r, .lr,Lt,

teo+&'+JQ+ . .. 14h-t2o+l

.[an*! ,1 u ii'r'D+t

' 2 f et1 ,J''

= 2n'')-t -- 2n*2- t

-'.PCn+O h ,*.rra . l1oni p(h) ) dtup-.

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0,

6 al,ug

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t,StK) -/ke1t-t)1*sck-a) =3t +2, 6co). r,sct) >/

Cn.ul alH :-oo:)J, lr"d $/-t' uldilAf e^Lout adltl tuh"

ttu- ctaoacn*i'sa| Q?^"hf- '1 nLe^ 'l'7

x2 -4d+ 4.o

-'.t =A.A z.L

-l o,n-o|eatoa -s ola l,tA^' = C A + Bk )Y/-L

t1)) Jo fu -ttz Vaxlta,Ll,at i*et3'<'al

dr* P1, -c+dx

-rfu-a' t,bzliL'um scl ) 'cld<lCK- t) ' CtdCr<-t)

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l&L ,ScK) - AsCk-D+ksCk'D =Jk a "'d

. 44,t'r ;t; rb^'b

o'e.t n* ttx-iJ asfc+dtr-zrJ = t*fa-+d r] 'h fc'+c

(t'oo)*e)k:3kbr4 * ?4 . "ut n*

o d =3'C:r2' :. P1,. : tL+38 " .

(ii:,) No

{nt*^^ cIe yn t-c't'&i'+ ' i"fir"at f '*'

?I, ' ak2 2r(a'ru A& a- Cha,4a or),1 4 tlc

e-oo* q.r"dr. r1

Page 68: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

6a*aga* 8(k) - ek, c*tt'

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411*4litu*Lt'')o 7*'q- rL zk - 4f+ 1x -, )'] zk

-' + 4 fo c* + tJ -tk

-'''o

,il* *a*" iau Jn tL, cDe tt-catc2- 4a.(r<2 'zr+D {') * t fucr2-ttt..+4) I <+l

. -2 k 2 k-l'.PI,=!Crt t =K 2

YJZ q 4,,./.Lal -Sdlu 'LLit ' eL'/- ,<-l

<Nk) 4A+Bk)2k + (rztat<) t R'2

Qtnen dk* 9co) =t => t = A'1 t2

.fCr) . t =>t =(A +B) 24 tL

.'. A:-r, , E = AZ

." scr) = (-tr + ] o) z* n (o+ 3r) +kt eE-'

6ck) .(kz +tx'tt-)zR) 1712aBk)'

&a "r-.4 = -:-

,-4^d.sa?.uth ce-

9" &* q(F,ocr42 3u,*

Lt'(e- |er$ah;Y /*^ . ^ # -o,'y'-L b o rUZC.a

(n.u a1.q >ooz)

z) ,h ,ltz Vzuaal-|'',1 6--c*i'" 4ti lo-nacc-! 4 e?u"-/t cL

/ot Qcr,il =! FCn)zn.

Page 69: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

P dh'p/,/ .A,4^tt- .*dq ,? ,'. aon t t<nmahb

J.a.a- o, 2 1o n =eo

., I f rrr, - Fco-t) -rco-:lf xn=2

2 f(n)z^- f Fb-r) zn -2 rcr-nd n=A n-L

f*Li or,, zD^ F (o,t -rc,tz-l - .fi-,rrn-,

- ,'fg Fro-z) '"-'l --oL:,, J

f+tr,rt -,-zJ ,19- Fcn-') 7n't 't,:,

t1

a

)

,rrrl

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=F(o) + FCDz + €0) {a-q" lr,.r"hb f,-*+

.qt-la.crir-.

.' Fc^) - FCn-t) - PCn-z)-o.

@

le

/5e7ua^ @

,"t LtrLt- Vil.o-ae;

-_o.

) zn -0.

-z ftt

cr,z s -l - t [o,cr.,]'1, -r -Z'*f ..oJ.

il t-zlt lt-z-t$(r,z) =

10,,,

Q (c'rz)

I

I - r-2"

"45 ,u.- .r"g"La 7.^ury l,*cta*

Page 70: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

4)I

.hlt, .,(-(- ct+L-A2./ur -ta-l4hdL + *4 q)/nLod -l - I, l)n' pn-t*tr-r, h)lco.ta-Ui+< . 1,., . lr, ,.

..lol: CA,t), m t) :gra r.r Ip racr.-r".e

2,tz .sarLlQrae q Nil,tno"ei ,x*tn*ent ,"hrQt-dXr, 4.e- e.u,a4irtt (. "r"/"Ir;*

l, = l,-, *tn-, -oar'd aaaT-<i-t -t<- n';b-aj /,: q f .=

doL?e

lSe4.Lac/ttz-

dtuhq,!

(t)

d4*

ayal,,ix

@

y"i&onocd

oJ,ot*i

=>/ -trn ln-t

Lv" -t .. q u, 2-^.,o

F'1 :o -rAU n -a

a aotl-,bia .r/ Lilr-

(D 5> rn -Yn''- f''" =o

.r, l- , rl =or L-T'71.'. XLt crla*act ^)t,i j"a*"" -P-e ,2-7-'"o

T: tj 'f C;FJZ:;

4)

T = ll 'lt+tr

*e* r, .

." rb.1UJ0n

-f , =,* f,,x,(tr),"(,*)",

Page 71: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

(9I

( '*6)a, n (t'76)oL= Q ------>@

=, *il,=r,(ff)1"o (*)a,fuGf ._ (prif+ o(, t t-'> '

-th

*, (t-,1?)' + ot, (r-G)e=4 -->@

(r-G) ->

( r+ rts)(t-{B) + (t-ti)t d, -- a (t-G) -->@

&t, @) 1t .) dr {r+'G)1 + *' (r'ti)L=4 --> @

(O -tc)'>ot, 0+ fil fr' G .l' t;J = 2' 2G')+

d,

(

(r+r)f'trrJ = -2G'2(*{,) GaG) ' UD(t+ti)

*,G =tIO( =-,ti

U) =l (t+ n) J- + ( r-G)o(t'4G

-tIt,

4vq,

! +r +(t-{7)qt,G

Q-G) d, ,-P

Q-,|-a) x,

:t =rJ,=k''JtkYry\.?[f)'

Page 72: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

%z

aL

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Va+hie J\rrlrrl

,@-itra.c, cttin , -trlora ,-ta-ab t*.

Ln+t n+) ^n, 3 +2 ,ri aa;lLtz o,V t.LA',.-) ql11 zoa?_ rtl>u<)

s9 ,'{t P(n) = B2'*' + ,n'" f, d.it;i,silL 3y r

fo tpanve Pr,) -Z ,tu.u'2.t t t+2 3 3

i"' + t'-'.8"+2" =ll-t8,36 .(6)tt)

urthl ,2, ao,nilL OT t-'. Pc D ,;4 ,-bj''-t- .

63!_:' j a ur,v utua-t' Pu) ,i" dt-u2' )K+t k+tt.e.t { t L , (4)ct4) fo< z a^.

{ At ! , Jo p<ovz pC k t D tt .l-i<r2-tt. _rek+t)+t l.+t)t2 LrtJ k+J|(.e.,, 3 1r*"' -, a + z

, e'k'' cg)' + "o*o c')

. k+2.

- {t'cil + z*" (')

i"lae-t

2E+ t

&*t*/au-r- .' Ba

ll,l-nd,uc,$br,

r LK+t ' '"" 1 +(it s)- tls '1 2- J'"'t-

Lk +l

= ) (t)(m) 't + G)o o 4r;dbt- b.,t I

J4t4tch u u o

.PC r+r) i,t a^a+ "/'<'l'^2'v.3^ P(k) ..''''.r^r- ?;""?o q *"-'+*'*"*YtPcd "2" v{,u,L loo a* V*rZd "

-tlfuXot n-.

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6) dLrivhd

uarTttn ,fanunttd cer"i^tt 4 a- &e,Au. ,r1

t&X, elflla,lat '/*ttoaza I'11'' W'""fl

{aL Jr+h

le fa-tnel.

U) J.t-e. -b.ta/- gtu-dl?, "af VaawAt JAa* ca-^-

q'tata a-"oad,t ,h ,ilufuU v'orCn.o . r-711 o-oo1, N/D a'aq)

3ol:

d)r, ct .a,u.t-4- La4z "L6 "/fM .^a t" "dnie

a,^-d -"- "A;U ee-.^- bz dlV ealtn, "'V L"u' foda&

tul-k- Lr/Le- q./"t ^a^b2,1 fr 3- cna.cqc*oL "pa*s u-'-o,sot 'fi

pg.16.to.d6ooco

6.<-tu/ axh Llrt- 91-,t Lru-bel $ 4 Cbazaca*ou

faNuoxd "tt Ae . to. to. to .a6ooo

1t "-n

cL dtt- ,Xnra, ,X,t on^bet oatr ?u4 t/o/d. Ll,

d6oo+a6oo6 = a861o

Ol aL"r* L.t^r. dr;;b 4'u' ?t'at 'L'|f z4r2d ' "a"-* 4* a+haJzt &.^" lL ,.Ab-tu f-tn "'

^;^rir c drirrh e-nd ao ot.y*t?YX44 Lbl]- ct*a,bza 'o$ s' chq'4achl' /b'81-;'

-& d6'tos =dJ4o.

G) X^t- '|hil-hlzt- *tdr_4L1,1 claza.lt

llt

-f,-

^1-ea-/h

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qAltlLa $ P'ut'n"a '1J,54o+tgXto zdto6o '

sol: o ..-. I o* t. o

|a.ooa.un' 9-14- 2^t""i2"*'b- f e'x--l''-- 'n*

t^j- f+e -fnk*u--o--z^t ,,)to ^r^icfi,a a" n iqz,*L"tfu p qrt rp- a'4-2' "-tf* '* ''"n.

A.bn- "rue- ,f dlz' 'fflta,buA.'* L#'l -, -ph,^''

flat be

fi7*nlnl* ' ''Ltr'tuU*rt-bor $.

a

*"ndw"rias eoen

n=tl, Na-i, ,re

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+) ff,J.ss=121 =""l_zx sJ

, =les:f _ ,"Ler gJ

* P.t -*

bl:LL "E Lr'ta'

qrttx)h -A a fu*da-h .

U

Ltie. O1.t t-/t\abz.L

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^f L/r- bq "2 .

ll

X,* e '/'L Ltr-'' 4t'tLx t'

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r'toty. tat =[jaJ=ttn

t"r =fLll - *Iea'd ffi'+t

Uoo) ffi: tv.t

agfiil-.l qu,.3;,ru,.,-,o

g ioqro ,**: t and

LV "n 4 Ltt-z- dra.tz-l-

CA.rl . sr lD roo? z.oog.)

q ato,rxbz.a. ,4 4./&4rut

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lboo)

lgocoo)

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.tu.urubot *td.;-,";/"lL

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ltE-o I= t--_ / "yt,r,1l = "

f Aueucuol

= 2 . laoeoc"r, = f-:."-1.,I -2'l' t 'il

,-a.^A+, aztV +4s Q

-Ldcout /-dso

L1" /n u au cuD)

?LA+/L q Xn ur-k^ -ErcL-,-t io*

= tA) +le) + lct +lot - /aoBl - lAoc) -(fr n Dl -lLoc t - (o o ol -k o t / -t (AoBoc/+ l1o eoa,

+ lA n(oD) + lBocna )- le o eo cn.p)

. fu Y, AS, ro + 3q) - ( 4 H 2 s + t7 + t64 tt+ 7 )(8+nta,t)-t

-tti +ts -l

-&SCn-r) - 3s(n-t) =6 .62

I

=a%= t93.

1) 3 Cn) 2 t,t<-tL 6Ca)=3

lt ao1=E*,ll =t 'lAncD'D)

6dbea-nd 3Ct)=l b7 ,,-f"1 2z,a;a'a-fy

l:' (n'' u8!r^ gcn) - gscn-D -J,scn'2) -

,. e. Qn - &.Qn-, - !'olr-" c-o

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wIo.,aJ

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Qivt* :

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(L) =>

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,r) dalue.otpt L

9o l,

uH/. )fL C,t t L )tzt1C2- .,t/-/'h-tv.Y,!o A)\d Ir. t'ct' .

. LA'U

vn*" - bY"r/lt Yn* tbl'

- F-t l_p >-oo8)

Qiur, ' , Yo* u'6u*,

+ 't.y4..t-

LYn=o

XA Uana*ut:frc e?u-*in

4 oe+l A)t- | 2 a, 'r . 3

'f2 -lt-a . o

(r -s)b-tt-l =o

Qn -- t, Gt) n +xr-($' '->6

6Co) -ggCD .'

4or3.

1.Q.,

a., --t )

1. e., a6 ;3

t.2., qr.lo

4" : *, ('')' +r(. C Z) '3d, -r 4, =3 ----> @

Q, = 4, (-fi' t- t<2 (3)t '

-o/, t 34z =' -)

@

Jrap = 4

42 =ld,+t =3qt = 2

.. 2C-t)n.t.Uf .

oro, dnr, -tQrrr+ tlfli-t -A4, 'o

Page 78: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

1a

1-^ t 4 .<oet

o

o' + J oti

42 =2 1Q,

Vr *44t

G)-tal => d:+ Ja, =-J ----->@

-t- 3xs =o -__> @>ot, Ct)z + d, ct)L I oL Cz)2 = -J+9 da : - c- ----) @

Olaaaetor?,.s.tz\ g/^er.r;L -Zf - tr" -+n, -6 =o

o.-L.t

I ll -6I

y, -5t +6 =o(Y -s) (.r -2) =6t.2,J

Hu.u LrlL "to a.lt A.12 t.2,JHLn t2- en = 4, (t)n +or, O)n* 4u (3)D _>O

9tven: Qo =4, e. =o.ez=-f,ao =2 + 4a = xttr)o 1*, (z)il t" (a)6= 2,

et.e./ ai +otz + a\ -- i __-_> @

Q,.o + Q, -- vr, Cr)'+of, (z)' aolr(2)t=o

lt -L-qe

Qt-G) ? aie +cort - -a > @Ct) x 2 -2 JoL + Ads . _4 __-_> @

(b) xr * ed, +6,<n = -a ----+ @Ct) -C+t 4 eo*.2

l*;:,1

Page 79: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

t'5yt-

(6) + Dr+A > -J,

@(D =Y d, 'h+ I -d

cl , -2 =)

"zlz ,tQ- clt-11hep-

n22 a^d qo>t,q,

()n: I ( a^-,- ar-r)

* Q, - oZ Q.n -, + &, Qo-,

'-. lJL o4a^'4c// bfuc

)L ,-aA+& 'o'.,1 -2rG-;;)

o^ = 5( t)^ + c -t'L'' t(94

6aA.)e

Ue4Laz

9ol :

,rz,tt*iw a^. a (an-la'-r)

( A.u M l= .>-oz)

--o -) Ao2t,-atfu'- ,2 Vl"e- 6At

2! 2.1

- l+t

:. A = t+i,r-i ( r-oo7-, a^, aitdr* ,eaay*x)

-'. 6ol,*,k^, ,,tt d-n =fr (r+i)o +o(r-i)"' d/4r^r- 4 orn d B 0'4'e' az't-tl*-'a-1 . W4't aa*^^-U - C J6 "fi4e'1' @ta- 'LLq' '*t'- taille)J)

a,t-a -<-e-al ,*i:*'", co'AXa^*)

rla"" 11>o -l>ats e'

{ ' r+?Y. ,r[c[, o +i oon eJ

Page 80: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

-lrer-ott t, ['ifi,O = 1"" (9)

a-r.d fi umo?vst't t*l'-o^'t'tu'

@4e4-"4."- rt

(Cote -ri 6b,o) = c^lne ai sia t>6 , /oT n >,o

;. nt-lTJ

b'" P'i'"'

D-Ll-t) -/l-- ) Itosnr- -ur/ l-- _ _t

LA

n (of l,'7.,,- TJ)r[Ll-

,l

u:e .r>a,.te , (,+i)^ =

b ,r^f *tr, *)l

A/\d

an =

te* C, , 416 , eL =;c+'s)4maL.

(2) +(2) +

.Qo =t . q, =2

Q6 =l rQrr?rq. -- (G) fC, co" 6 te, sio

. (G)" k*f -t ,; :t )]

->@

.C,

Page 81: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

a, = (a)' fc.L'

, a lc.)L12

,C,tc2

) a * =L'/-')

cotToprsh

, ,riJ

"

[*q)+tc'^el-'. ( t)

u^ia- L'ttt' tbtt

9'-, ,2 -t . 25'l ,

,,oa)il2 ""^axa't

, h )a

7a, tn € Z r

a-,\d m odd , g,t'ove cltaL cltu)-!- z)Lt-'/'

* -p oi-h;tZ i-lr{,^ n ;+*ch 'lit* - d)'i dt^ pn-'

(A'''sal:l >eo3J

lA+ I,

B1 '1r' ?qfa*"1* ,PaZ*otPt' a-^/ crr''-

'dr.,.,in;r' fion-caru' "1* *': s, t e X'+ Lo'bL

rlstttmt -t'llt''- )t - t and g€ - t u\g^ie L'l-i'"

4 a/r-;- '1t DL;"k^ 't'f o*

'b'tt?' t'e^'

,l(& -t )

^t ,s* 'a

argl 1mc-a- ft)

Itert u ttl I 2- -rhzt-s.t

Xt eaz-z Js't = X,m +t al'dqt , -'a m+r # 1,,1,€N a-AtDL -t - lt- ' oz yLrll

r./ a-^d

"'r;4e- la ^

-(ts-r) =(7,t*') - (1,

=. (?, - 7,) m', h t^' j.b't'- ;s6nL-"-r)

-t, odol u)o Jttu)L tut (t''-)' II and cbr' 'tzs

t'1t 'f o Ltouts L'2t +b

Page 82: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

/3) Urn/ \rtaltua6fi'*1D c2

-ZcXnoQ-enba,

n(h+t) C2h+t)

I

?*vu -Lt*

Sol :

6

,2)2: t 12 13 1-

cA.u At

J.+ D'

orlA

t k+t)

d* P (o)

(r)

I nt

.tu4/^/-

0D c-Latr:

ilt,o,

H ea c9-

8*suno.PCt) : 12 =

l.tsLLnn-L

P(rc) , r2 + zL+

,olLu

r(t+r) (l.r+r)

6t/<rtr. .

(2k+ t)yt U-t4

t--->(u

- (p+r) fzscz **!-Y-,--.-.,/l

- 0<+D fcCtcrnr)+,)

F(r+D h ":2,r: ncn+t)(2r.-t

tt?<r.^+ H :-

*rrr-a-u'h

,)'/ o-afe

"^/ u.tVot

P Cx +,) ou 12r,2

PCr+D : tL1z2 '1 ' ' 1t<e+ Cl+')'

^ k(x*r) (2Rtt) , -L - o- --' +lk+t) /turb9 CD)4'u

l) >o t1-)-6 12-

nco+t)(2ryt:

l:l b

Page 83: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

lh) l'..o, e

"It4<rV

bl:

oOLa+

tud.lad

\7rE

h- sn .ri a- .)t*r.z,t* 7,az

iad-z< ctz-da

F l r,,o

Xo* Ptn) t

Jo

u4tLdA

Hzn@ , A

g '''- 3'" = ,-u..

-/-rr.r- ,

t-,1 a

s3/-:,

,rurr*r7,t .-e{ ->

-'. Pct) -, u6^r- .

A +t tr-tu-e ,-*)rn pct) & Al,<,,.-z

Bk-r* =G)(rh) /* "y"- i*9a- *

Jo .bAi ue Pckat),i- *-r,,--.ok., k+t R. k6 -3 ,Ba-s-s.

=(r,n *,tt) I - u*"

-,(ar'n +ECal)-tsCtk)

a-

oru)*7tz ,4 s-.utr*u*.

"1,<,<,-4,t!op.b

{e* *t dz-.q*-tz- L-tat- cA'u' Y1:-n/p '>-eoL)

/4<ca.n rhi ,r* ;**:;#,ff*f* 'v'

qai,hiea..2 dr:r;:C'a

--ai c

V4+ z-.x-u-z& .

,4, a-

-'. P ( r+r)

-3 u

= E fem+ sJ

q,\aa-tV *t tt.

ts). ,7u^tI ooo

Sol:

F4zuQ PC r)

/A. d ".1

/D

a- fnu,la"7l- -sS8n- aD

-.1)

o-eo;+ )

')rg+

Page 84: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

{'* 0

oLat .*-:*

&12 4* ,4

a,* a))ibt*

,0ttt.al =uzYt ^.

J,ll

zx czt d-an

/, laLte) '

J,2

/669 '/'a*

ht = ft:9o'lL1)@,W -ltosJ =to

I n n el -- 119991 .fr t' 't) = r z

l?v tt --t

4,ur-ila *t v'*)e i"t'lza h'st

,*, U* -aLt

&/;;)'4'- ";/a'^ + o)t- tt

1 trL

J,U

wit7-

al r'"i'*IPa)"

et 9 nclrutu' - ExcL'naio-

tAl+lBl -lnnylruLtqo-t2t)o.a&o + vi .i,rl-2o.+ %9t z%'L"db

e' hA x @'L tt'

6 -dqx-

J4J,LQ-

a;',.s9tl-t4^4-

a

Y;^d uUr' 4 8-V t/2./1c!- -bel-aro A "'-'tla) l'/-cA;*

"i/ 6 - aq L ,r,rn )"q qPa*abt '/aaeB^ '

zoz'-iii v ' ,K u. V tj -oo\)6 -Aqx

' go*t'n z +l

C F4 x) C r -6"<)-->a

Page 85: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

\t/rlal-" ,

A-ac,Q'rr0+t)

n-+-1-57 t_6k

-A(t-Lr> +B(t'5\)

,e -ff = a(,-f)p*6 -as

*- =]

l.L't L "9 =26A

Fu* 9<' L- 41

t -AItA=-t tg

(t -st)

=A(_l_q/ ,- 9r)

t.9.t

4"ilr"*; #Y0cr)

t7) daLve ;/J2' tt e ut'<'tttttt' vttla't';'$" a' -'lQn '+lo$t '=o

4ox n >.t gt"te* d'14't Qo'to' Q ' ' ht u'snm6 A"'- "fi|Lnxrj,t.' (A'\r ' N(9 >-oa)U;, t"- yfu^ ..1*ct4^,LP-nct- ,4ilar-b* ;'=.-i u

J-an at-V*i^-d ae-7'-nc' -2' ce'?l4f i^* '4

7" L acr)rD , -.n8cn) =-C+6) ra(6)

=\ f6;:l.tl

o '^1 ae 9zt. 1 utt'- L" "

Q -et)

-,[,'r{ '' * aY+xJ-l

-, f ,* Et- 1( 5*1'+ " '-)+L-__1afr+ot +(br)"+-" )

-,9 tooln **) (e)nri -b

I

a;!3"-Jto4=z_. o -

Page 86: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

# o"'n -, Eon-,

zq,uAh.;h, -r,y { atd-r./*I

x'+toZan-a

aL oo

alnyg

,lD4

rttQ n lD -'1 2 < ao'-,r-D-'l. rox, ;! ao- ,

7n .o

l) -2-2 :on-L

lo t2-x7+)

to -29tGtr),(J-tiLr-ot)

G(t) = A +9-l-a| l-D,

, A( t - t z)'' -t rZ (,..nrr') -'

-- o [ * L.^+ c2 x)2+ " ] *'f' t -t+xtt+''

Q(*>= a 2 r" n'* 4;f lnn"n>2-

A^: b$4/-cli^r % xn i Gt>)

4n= At\ +8 6D.t n.Zz.

q, -- tb2 P?t n>o i'n (a) , ae--yaaU

Q., ACt)o + gcs)o

@ = A.rg

A +$ =tt

mr&rpbt**r"r""y -fan

axt+z

-fA,A

Qtl<n

=> ferr*>ro.-o.{ -ztfGot-<"] +ro*g(\) =o

=2 Gczt'ro-4t^ -azfba)- toJ + to xt(a]11 --.

=> G(D (, - lrn,or,2 + 2qz -,o =oQlzT ' Io'zq*

Page 87: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

te ., Cl lveo a , z/1r , pub h >) "- (a> , * o

. .Q, > 5cz>' + gcr).1

4r:LA+r5Bl'.2., Ln 1D g ''ll

5otvm6 C t> qCD , @e_ Au fi: g, B-1

1k4. e n,J.2a +a.5b

Q.+ b >o

a-=--q

'. Y)-Q. Aalu-tziY' J.Aj, dAsOAat d

e^arun tot r-o u-s

\

@9

1d,

datyz dck)+ 6gck -o =9.5r,).5.(a'rr' s 11 >-oa'l )

Qiru* 9e-crr.uu)ca- adt^-h-eb- k

Utrr) +a6Cr-r)=-9

J ht- a+zo ei of-& -A*angz*t-out ail"bb,- 3^

SCK) + asCr<-r) =o

Xtz' ctt"n acbrtl,sn". n? &i* -:,

(Cx) +6 8Lt't)='o

4L,l)

$or m tc )

Page 88: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

H zr.-to *U 1Lr* eTuaf,rbl^-J

ick) +EtCk-0 =q L<ctn-s

C+Fc -96L =9

\'-,

,.tolrhoh.-'. J^L f7.; cua*f?)

-,ok -;

J.l.t- Au.L4 a)(h)

$(x)' = Sr +

aabb&-- tj)I

Llt

= q,(z)k '+3 4)

fn C"> tDz- 3x*

=o(t (-a,o* + -4

i.u., q, ** --4

qt. [': .3, --t@Su-b+*.fu,cay (Bt -Z^ tc> c,:z g&

&(x) = 3, c-r r* **

dctve'. Dct<) -1.9(l-r) 6 p(*'l) 'o ueib''' Dco)r&'

DCD=8 ot Dct>)22 C A. \, . \-t tl ,Nl P >€' d)o

' o 2a uAia'aa' 4J.ttp- Ct>a;nZ,ttr., a lte k-7" -

y t- 77+6 -o '-fiY" I ; a '<oe*

t o -l L

o t I -g

y u* K'-o

': Co )

tq)

-+ - I -<L

Page 89: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

rlr-(=oy '.t 3r - 2"r _L: o

Ic.a+l) -&Cr+l)>-o

Ly+t)C,r-:-)=o:,.--2.-3

Jlt- aolph,ot* utD&) -' d , (, )K + x, c>')t< +d s t' tK -->@

Put R=o , DCo) = X, +- "(, +<L : Q

put k = t, gcl = e(, r- dr'L * oU-+

pu* k=r,. Dcz) =4,+4t.4+ Xr.1 ..22

d. +.(. +o( = IJ

d, +d4z - 343

d | + !+4a +rt d3 =

4oLomg Cs) ,Q\) ,(5)U

41'= fr t o(, =2

----) @

- s ---->@a> ------) @

L,ta TAhU

dA-t-rL Vd'/.<'.'J ;4 12)

=n+) tD4 + r('3)x

b" A"L4 t :-l{-i .httt njU

DcR)

,tut'in^ ,k

&k+l t o1

= rr+2 +t'-5/ol

.tlt- rt-?uxzq zal--t'h^ '

Page 90: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

@

S4^0,,y ttak 6l% arrd-,t*r- 1p" SrbrA ?U^upf

*^rrlVrrw,odivd" i^nl&-" ) ' l6e1a l\o'y o" ?oziliw-

ara;r+',s) lt *n I ;6 '2;+f"14 4 ?xiwru owtttr4"u o-l

L Lortifi'a. -q4 ?aodr,,* 1 lai'n'* n+w"Lan'

t'i f; 'en\bJ

$

YY"i-F lAr'L

BuL}'", ?fdA'ft ; ,\ot",n#vwvrs

l.t-r,,Jr,f,;* i"Wd'o's;' Pk)

ptr) Pt e) P(s)"'fC()t,

wAv;"t'tbr+* 6

0L

.AAl VLUI

l-Lb, ru7ln-+'-

)-a,\s@tdw

tknotrt-w ,,q^ lrr<-. r ^!.

/L^OJ.u^-Ldr'

o-l1

6', -

@, A*r-* le.F.r lie. firl*,-* O fu y

Te* tu&ttiv ?rra7u

Po,rf& "tr 667 "^"n*-'a*A

t\llPx '* ?r'l'ou at Pad""r)-+ eOl: t't7& 'il '

fa;nrvs ,

F@)is M.L a . a Z ,o PaiY'ne .

ll+nu, Pfe) \ ffi. .

S zn n 4.\.

Page 91: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

T, fD t?ft+t) -R k,q

fu 1?, 3^,h* u+r ,\ trnrio ^ ?ri,wu,W,

A+ak-".f v 1il, % k+r ,ri -nt: a Par',* ,tr,rnk+l f,rm h, WhHt^,,u Pq,, f. ,*nys a*-psk

-A/rd\. A<q/1k.

Atu-ry1;4 5 tr,- rurh-- l1foNfw;^.pq0u.t

?,t;,^r) -ot ta ?aodu,r $ k;nn, . alt" q/ L3;'lr,<-1 "a,

xt'ttut ,,a

Y4, i,n

41.r4nn,b<l

{il A,b,c',a^udn

Paimu,a Yad.* "6 ?<;,mn. tunq,,llfP*;; 4 &,*,* l'

PA$r//^L,te +fv /yu,wyk< _tl lea^|,a, i,t,kl,;.;;*:;:\;xff"'ll* - ,-o.,, 'f

&'g'uf

^"*dr 3, 5 ,ail'

&a,wr -dr^D+^ it<-Lrfr,,- t- &ow fou* -au-

d:"rqLZ '7.

Page 92: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

@

lll . I dooo -lL,J' farr.rJ, arr

h.>4

n .o anl at"l ,

---

[ananc nD).LH*-; = Eff:]= ['r]-1.

u^rX t"^onr^to-t i" '

" 0n-5An*r

SoLb;,Qturn '

f,nn'*g '" o

"Oad

.lDl -lFn: 38r-9.

: &'76,

anc n oj

, AorP

fi", a,,4

lff***, ,&oL- #s. *c'tt*"u-

vn"u,

v&7=l ,

7- an- 5an _, I f,an-p.-o 2 e, qu

Page 93: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

q,,ib,^tX._Sv*L=6,

[rr_a; (r_s;

eAil,td)^ sokh^

A. grd.

qnr.4t.Joo {, dD

?ut n=, .rL (*)

@

--o

----, @

46 : a(s l,( p..

0 -_<, +4-e +

; [*)

=41 8+D(a,L

2 84r +d 4a

aa --sp. *7*r'-

Pu-l. n=,

tul

t

(o.)

t- - 84a

l= ^4a.S&, 4,,nm), 44 ,,tn

tdaa-4a "*t

(*)

t> n* ?"*- .

Page 94: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

?.r} -g

A o-r*tr \,) qrve.esr-r y a"n?r*- o. -J Qu=.oo".r..D 15

S-l^j=.gcr)

gC\. )

inl+l-.t-an

- +SC\4-l)

n n'-lr+.}-+t+ . tz-' +;' -- )- -\(A,u.t{o'r/orc-z-oro) ( r' '\a I rG)

( ^'u AEaiMA'l- )-'".a,

-+ !+ s are-1) = u\. +J' , sLd)"| '(A.iJ qse /u+r-sg) e'*o rta)

Fi^a,{F{-L q,rrul,.H^J ^ *.^ ,1 f,-Ve-r'.tr-ci seavLnJ-Q'.tujlcwY _

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Given'.t -ta,

'Io rto '.c :

d*

t,

+ a- e q.

b -- b * a- laV"UrJa- e- te aq

+ ;t = A- +-, to't = b.

a,,F b .qL..CGt,*Ia-+ b)-' = a. ,k b

=cL

be Q*d

,1.-r -1*^f

L

tve:-q ")'*"'L !-- a

II " ltr ,q.,r.s^ 't*t"'19 i

qn ng 'h 3h eu't^ tv^'el%;

o-4b

-' [a r b)-r ' bn

b* a- - a-*b

LCa, *) k1t-',r-1.n-

Q"-"F

Abol;"u

Page 147: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

+?l.r.,L Lz, , *") k vd- nil;" hn\p ',

+5- tol L,1 U1 L"lt'lL,l

1"1

Ls1

L+7

L

!- + -I = 2

l2

z34

LL

ILt

t\;

1'

=L+qL=L+qz

,r

3

h

@

I= LtCL

,1

=L +EL

- L +q

=7+--

tt) 1iu.d AllH ' ft'-rlryuoheu

/r*A -F""{'uilh fhe

" l9 3""8

lhei*Q

t-he

,-i !f"u

cau.L4 4.ftr-t,t

H

W '*, {r.,; *He

itt qHtL) = ft,-rr'uHfi = f-t't!'ttu ti), f r.-ilHt-i)= 5-i,i!=uiH1 = H = H -L = ft,-t4

.aat the ,L d;^frrut nLq-ui La^Lfo * "

'4w r.o:+14

aui. u; = tt-i

el6

--i+tr-t

Page 148: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

230 fr q, araclt)

Jho.o t-rr.^ t

Sot Jai," ':

(o!

b oi-L lj^.e- eLe^.e^Ls, -l -rta r.bJ = b * a'_

o[ a Qxcup Cq: *).

C a o "r r"P z'otJ

LeE t-, r: Ae .

OeO",eget-l rxb] x(b'xa -1

rt

Lle a *o- =

-t[o. *u)

b'*be _6)

t'.' .* b= bl

xa J =

=

o- * [b* v'y * il-la-*e_:*a

a- r. dt ['.' . * a' = j'Je-@

= 161 x a.''J * (o. rzb-) = e.

-ta xo. =eJ-! -l

= b'* a-'

B7 cL,o,t.rsr.o a.Iie"ri

ti.vor.Ee e-larnerJ3

To P !to\.€ , (a *b)Prtoo5 : 81 cloz.rne.

Cov\ri drrl (b-l x i') *

ALSot

1o. *r.) * (fr

8.6

o-vu1 tro elar".,entJ ob o- ?'to.{p ( o, ,*).o-.X\, e & .. By IJ^-a- Sx-utof,ca. oi

o ^cr en''' , i' , b-' , 1o x rrj ' tr^k rh G .

-t- .-l rl- b *o-

o-rj sv'. 6' ,it e G --> f,'x".-'e G.

(q-*b) = b-r * (d, *a.) r. u

L B7 a,rao aoti",' l*1]

b lF e>Fb

Page 149: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

tw:^1 c,pb e W-? o[ orcdor n riLo .t'\e,

? xoLrp ( zn r +^ )fAu A/.{ 2-5I)trl

Let th-e c!,li. ? r*r,f" tc^ ,x ) oN o>td-er n X"nnrq-a.r1 .gr.,yant a. € G .

.lo .' tr.rq.L ' . th4. - dlrnenl. .( g arte ^,a-Lc_,..J =.9:eL;9 , 9, 2.. --) G -{trc}L t}L^ t ltrl = o.

Note rr..ot LrI i^ b,\a- g e.n orratox o! L z-^ +.Jlrlr = aj Jo, j = o,r..., n-r and t,qr.ca *t ia

U o rno:tphizrn -

e-e) Tt t L - l"omo..o >tp ki:* -! o- gxo"p crr crlo trqI ,a",rp &' thar.. Prrove tr.La..L ! xoup rL L"or.-orn-rrx4hirn?to-teyvu L"b,.,litllr . Leg_tg"_j]Solr.tlo" :.: Let o- a crr., or"A t ,] 6 l^.o"".^o..-o-pr,.,-a'- {rrar'.

Cn .;lo c" ' 1,,11.e1''_ f, t-: e cr/.

+ tLo) * e' = {to; = d ta r.e1 = t(a) * t(eJ=) e' = tr c.) t uy tnlt cck,.6 r, oLbr;n L^,.rJ

'4) Shou., .er,.o-t

1,.1o rYto ,,i.Phr-6n-\

Sotr^lzt-:

o[ o. m-o"roid- uJt LicJ-'. r-r .,ot

Xyot?.

.r,-ovroid. t -rl,uiCX

o-ea, * 1^1.

L noh a- frcup

Page 150: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

L1r 9+; - rt-r;X ar.d Xwe ar,. Ereo^Ph.

=eb:-9-tdn of4e-rov.o-ir- xryh^. ( S, + , .) ; cc.rro/. o- rrrn7,

+ r*\.e b&svry opero.Lrirn.t -1 onA . gh S -s^L;zt,flru ".$o.t:I.o.o&g. 1*"- .prtopenhor: !

(. (S , tJ u: o.. alrqt^o-n qrlo"+.

3 . The o pol o-tion 'u &b<-i6rli.e 617e, + ; .o*-oL i,Jorr o-, b, c g S.

i + tb+.J = o-,b +s-.c a-r.d. Lb+cJ-a ab.a-+c-aI

l"r; Stob r o ?xor"X.l .bueo1-crn frr ixotfz 14 &1e-

I c-."o",ru 1.,,^a? fAO Nl) zotz\ try_yg!I Sot ^ht" t

I tt G u a [i*]" 7'<ou+4 ar\d' H t.E a lula

I q r-"+, ob G tlro-n r*tc oxd-u o$ H r'r a diwi'rortI

I ok orrds:r of, cx '

| -

- 'It^-e- clhverLse o[ Lo?.''?e's th'alru Yt ;" J'l''"'

l,I

l '"J p".b;* aat ,Llxoup a-nd .t ortoi4 . crive an € xar"ndeLI "b -- -y.r'.;XrtD.+ rrrt Lir.r-L ir not a mcv"oid.I

I SoU^ Uo" : Det. Serniqrrouat:'I

cet s ba a- none^."fu Jeb q.^d o lo< a. bi^atud

I "P"r.oh;" sya S. Tte- 6d-?el' -qic .aVtl.,., L-g,") ra cc.u.e.{

I a; 4eri1rtrru+ 4 I,}.. oporo't.io o u a.rroddti\.e.

I

I In o.tra-o:r rr.ro-rtd- ( s, oJ i,a o. ,aer lTxo"p ^i j"r .tr^.

I "'V x,9,2 € 8 ' (cr-.,g) 6 z -- t<- o lloz)I

Page 151: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

Honoid:

4 aerniq xoup ( H, oJ trr ibn on iduhQ gLer.,1e,ot qJ. rr.\

lo,lpect to frra- opo:ro Lon o .: CQI'"d\ a, r<-onoi.d. ?nO f:..'-e, tr)oxru , g^ a.l{€-LnoJ.c .ag,, hr.," ( x , o) Ls Cqlr. .{o-

'-o,ro ta * t"" ,,i,ry e H..

(rc; Yo.) o'ze-nd, t '.el_q ,rcr4tl

:. e H. ?ox

= xo L!52-) '

oJ 9(crnent e- € H z,-.<-l.r ll,.ot /or.

I

t tu] ,1r;1 = { tt 3, t,* = Lrlr u

t tsl, lsiJ = ltst,yzlJ = [2J + H

= l3J+ H . H

=[q]+H, [2J+H:[sJ+H

3vO"*.nod..,, t.,'r ?r,.oy ( z6 , +b1 .

-foLr^hir":--Lc-b .t =1 [o] , f ,1, [21 , b1 ,lti,ls).b] J r,z a-

?t-{l ona H = { 1"7 ,\il) bz o. 4til. nr.4uf ob zt"crr\.,ter +b ( ooldilin mod 6J.

Tha {qtt c-oaet s ol H are_

[o]tH =tt"l ,LilJ =n[,J +n =,[tr)., 1rr3][2J rfi ={yr1 ,lsl}[r] +

tr+l +

a'r^du

[s] +H =

[o]+r[,]+a

ai{- I11o drlhr.,"F t_"it "r.4'r,lo" ol H o" z 6

I

Page 152: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

-.). tJhan

Jotr^trbn

l-{ l^o.,

t -r\ll a- 4ek ob e\er.ae.ruta {os<rn 61 r.ro r.oid. 2

f6rl ar.r rqql]:

4nOtL

alqefarto.ic

iole,ltity4V6$(H,.) & mcwro'rd 4 aa azl

..-lr-1r.. ,l-o-rpec[ to I*-o oPejlot;.'

3t) Det; .:"r t crive o-r. 9x o".pta o$ 1 ru;gL\a-! z at\o d.iJ, zoxl - -'lfrv A l*.-z-oS I

-So [-,,^hir.-r:.-

.,J t "icl^

"b

An afq e5,.q; c 4W6,n LS, +,tl^r- ta1nq'jy o pcrrotio 6r l and

.) t)

. qr..d

q,- r r^-l

onSo. ,u-rl?

4.\-ts;Ll.l

(- (S., +) & o-n o.betiqn Wu?.J. ts, .J i o- zennL ?e"oy.3 ' Tlra- o pqr.Ll-. itr . i5 dr.rtNi tuut ra. ovex + j tnc.t d, Jon

4rb, c eS.o-. (b+c,l = q-b+b.C or..d

[b+ c,) - a. = b. o,. 1 c- a-

txor'^a1o.-

+t.e- jo[o,- &g propert_r]* .

'lhe- :l-,i.g (zrodo'-or^ -(Gce- g x ro L

, +to, xto.)

=o, Yet IU hoL ofi

+o, 2 +o

u"bj ".o

t

"i. Zr o.

y) t\ 5 &notar l"l- .<oL of, poai.l.i.,ra ,.l..t"gn * 4 too ,,"" ec,9 eS, dr&;ls axY -'nL{r,Jt.v"-;h

uJt re.U^crl ( S, x ) .a ^ rnonoicl arr,3t tra1i;1l ,17"-11.t * b

o-yrociqAi.,<r. f At-:-@J

.9oLtb" : /oo ia r*,- i&^Uh s'te''€,"L ;' (s,x):

J ti"co x t< too = -.-. { x, too} = :' -rGQ.

x alroo Jof * r€S.Thotejoxo ( 8, ,c) u q -*roi"l. .

Page 153: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

s) Cai re a-n 5.ro,^r+.e ofi x.,-to Ze.^i7xot p. f6,., nt ts2a!-SoL^U;,., ,

FoY tl'e Ze.".i gxouj C "l ,tzr€rn - na{a-tl-ir'Q, t"t -9or4

LE, +) o$ [ rvr +J,

bJ S1^o.r, thqi -l efY1i ?xoqp l,uo,-_o_uapt uia n- p tn-te:.'.\tet LnzPe.roPenq tao-.eoluc,7. f R u n, rp *!

Let t =(",*) -)

4), b\a aeL e

r) oc ,aq-h -ter.n r

L t1 , A ) 6e- 4e,-i17t<.ot 7

dtr.; a f, cr_;

o[ o-u

? "tD"1,

SoL.hbn .

ho".l"o"'.b:ghi,rm.

I v\a./r

.

x Lnd.ernpo kent €ft u.

r- xJ( = r_, t[x *rj

t.*j: tq*;atc*.lt'.,**:t? pxn*xve{

1,. ,. Ii

H.

i ) t4 H is o .acabqr{Dup o[ +a.e ?* r+, ,,'r r arno,rT {rLe_

x{ht cn+e*^ of H tn Q, Prtove t}ro-L -t"er.o ." o{rfgr^r. -4Ldoq r{!up vij . . l.l . ft ,-, u lp z_o-Ll

.foL^trbn:: Lok Flo- * bL 94;?.,1 co,t* ol H ord. c^ q)ha-:rc

a-9 G. ++ Ha &- a- -a,. r^nrr-o"{, .ob U, U4er\ € € lt o-

"uhqia e ; fhri ia-or.ti{t1 et-a,-n eJnt u,,. G .

H o,- [ o.rt ggcnvofmcq Claryi cDn torirtir-7 o. - r..t

ort 21Lr;-r,rctle.6o 9teto.F,).r1. -!o e e Ho. =) Ho. :He .

B,rt He=H Ho,:H

Page 154: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

31) FdrcL or zul. 9 rlouf o& o>tdor L,,o oJ .,u 7,.o,rpo r L. L":^':-=::{ (zs.+s')

S i t.l ,1,.:) ; tnr onlrtr 4\rb 1r<,,.,f ofr o:rot-erg.

38J Qirn on e](enrga o[ aat',- ?e.oul,. b...L ,.ot o. mo}.,oid.C n" srs ",- "ilJoLhi,;: Let D= t.... -\r -.t-,6,t,\.J t

'rharL t D, xJ x o. aerli g rtorp b"rt hot a monoid-

3, P>t-ore "^-t

nt oup a b ,r"- "r1

e.Lqrne^t. C ao e rr.t -.-:J

fd-cr.^fo tlrrt €(ernent , p(^ ater,,ent a € G r-l ,1c.id, +D

bs d+e,*pohnt t^: itL rrorpec[ to x + o( )E a. = cr.

acfor*ib r.teqcnt : An C(errq,,k e ob c{ rA ,<cid LDbe idlttlr\ ebr"e.nt, .s.T. e*a : q*e = o_, {on eve:lla-e(n.

Trt a ?r{oy lru. Ld"rrtj-q elor-ent e orrk -irrf;; f,; e,s

th4 Ld4hpobant cond,bLbn e.*e=e.Llencq tlLq pr.o%.

4"J L,ho! d.,o yst! c,olr to o' hor..o,,.o np hir,,u ob a 2",...i -?r-"f irto it-rerb I ff" o,, ".,-g

Sot-h.in..-

A h^o-"-o*,,or,aolia. o$ a ,lernllrroup J*.*""tb n co.u!.I- a- 4ery\.r-?,.-oy e..do'.,"or.pt^",)- .

Page 155: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

Eva,.,j Cacltc- 9'"*F 1.o o-rt ,t-\,/-aa4 <ae"F

Cge9i a g-tts(F Gi go ge,rata-E r

\---co&,

+-at Cq,+) Ba

- \-dY D(. ) e 41

Vr.!, :a" .l : a. Ser< ivrtaaa-t6 r, t .,)

=t.4 g = o-o *^t = ^**t = ott *=

ot* ^o =91.

.rc*\, . 3+1Ca,.x) -'-'' a'r' "^!rP\i-t 6'i':ar'

{42^tits &l,nt'.Ir.- 4 o- eu-io3'r:a'"'P 'l-

!l^a,{; ^T .r\c 3--*r'

f n' v r-rarloec "-atc- l

-sa,t".Etor':l-e-l

G,r----P q'

e-t-o-.tA-.-l:-=

H \.a rr.c- ='-'1-'3-o'-p 4. =:,,.-a d P'e + l^'-o ;'&*L'"]

--\d e- a,.s .z_.aigJy.

at q ^'"*i L1 'te's \

* az\^.=^':..^t::^:;,

e; f \-g 'i-U *L; \

r-lzY"-o

V>e-" ^ , \-r8:

"+I

a,2 = a- e-

--A-!-r^a"t

-/ A g.a-io ae t-.1

RL LAl.t:Ly

I

)trl^o.l a'e' +).,--, e-z zJ ^i

1t -^ -g.5+jc h goza a 9 o-:- d '

Page 156: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

L& f : q-- d l>e a- fl'arwevnoYri*-^ T-;ti,

D ke-, tte,!- K ' -rFren- P..e" e ttr& t<

r.rs-,^-l "Sf-*f % q 'il e"to -:ire''.e..pfu. {.- tr. \'"tf "t B

I ll5se\utrbr' f Au /5 'o\u ., tu I

X"'f:

'-,{

we, kJlBuJ tts a_{- ,k* tt) = l(.- eo Igc-; - .')J

sj^ce I te) = er -i,= a-\rra-qt {.,ue. , J\e--.t

e e [<e'(-ft)Tr. B\r'en\rro:A:- )<", ( fi) ; neL eY-f\1 ; q

LeC" &e -l-'.r,s ele'. e-nt--. o- ; V L k- C E, ) ,lFeae{os,,e. , tt"-)= d o--J !Lu)=r'G*', [[o*b-') = ft')x B(E')

- f,t-) r fttul' f6 ,. t"**-*.p";{= et ,* d

tLo* b-') = e,

t<en ( [)a-,Le k*Cg) :) a>3'rrex.,-,6gl:. n- ( E) :.D ^ ""t-l--T % Gl

tet q .^A ct be-iJe,rbdt l ele-.enL e-

-^X 1.,,e X"-f w.itE

-.1 et

=-f t : Cn -+ ort be ct'6b\; it) ..- a r\o'''*'.\

A e' :.e-y..tc.el"hernerno-r pi[iorvr , t\i er.*1---r

Page 157: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

Qi.ren e -; a.n -id e^U\ ,Ln q ^J JarL -iAe.,tjtq .Ln c,t'

Let !c = ke.rt€) .1 zeG \[t-) = e]*-: - , lce'. L6) _,-, o. S*tfl*F s[ o [au.e"d;1

r{e,.,, , ,,.1s }.o5 g +s pao.-s tt-, .r.,,' ke. ' R..,

"a Itr.ce"..^ o-l-

I uteY€ lh c1-L t" i g) i-,=' r

TD P-ro".,e, o_\C _ l<, q_Foor , lr-L >. 5 6., orrd h g r.,* [-* Lx >.*r) = {t*) x E(a)r gt."_,)

= t('.)*.,r11__,; [,t-= Er,) * ! r*._.) f. l'I]*T"-!= tL*r :--r) ^"*=x*f,)= [(o

!

Page 158: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

C|,aj-r.l- '.

="il^-t1 +

et I r<, 4 c"t

: $'.9\r-+q

+ (K+ t)dT^"d

q

(t<* a-) = BCa) j €ear o-\ ?eq

IrI1

@

--> {(-) = 6ru)-' $ (*+ o-) =

* tu N'tti- orp, - onra_ :

-ro p.,,e.,- S txr_o) = f [r_tb):) h-r a- =V.t.x.r, t t"r a-) = Q [r* U)

-> t(^) = trb):> f;t-) + {[u_,) = ttq la €(b_,)= { tu* r')

- tt-)bL^r+{tu-,) =ef, Ca,+ u*,) = ero.* b-l e lak+ q- = K* t

.i- eYra - ghs

iAe'."hl.g j. Cr, ):- hsme.,e.Fli.d

x d(b)

k+t

Page 159: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

=*

0 ; e,.Ls ,

Cl-co',.:f;snLtsLeL Heq'

SirtC.q

'ltsd-t'B .i= ent_o

f,t-)=,+ (k* o)

"- C,^!s .

, ttse1e ex-,=E a-e q s,t -I-1

= !C^)

-t .i,.- o He,'.e".sll\"r-on .

Ns.,c t (** ox k+ L) = 0 [** o*r1! C'+ u;

= fc*) r Bcu)

H eh ca-'+ -t

A{c.-.},e orl-J

€.\*f"f ,

Leu (c^, *) t=

Porgve. L*q_o,._, .f eu ,.,rrr *,".;l;1'^ -rfiao.:'e,^',

nu rvie ""i"')a"''rir';iijtA 26,

oLc-) = r-

a- X-t*f w|.e.e e-Ae. .i-. _r-

Page 160: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

reL (H,k).L,o rn

L= o., Sr)'a"-f T

H

Co >eY. H * a

O[H) = '^LeE h, , hr, \,o..., \-gl-s.. e,.uta

L- tF. ..r,,, J;g6o-e-.,b

}"- Richta1.* '' otl-1 * o- = f h,+ a., l,i* a,-.., L-*.1 , a€9

.S,c'rco ffieoe j-E, or-..e1.^-,e... tn- -r=-,-TL- Q ,*:;=Fil"'-T;.e\e "' =n1.. -E H+ 4 *= o a-au'.-u '

He^ae e.acL .:ccr}didrnsr ;;;-;f"r

.eset eg r+ ;'ru q h,- tv-

!'I ' r< . 'r- -dfl |Lt ces et- T H J.rr q a-zr e eiih!,rd,r'j.r\t J d:".L;---" 1

t=J:*L=, iA,:., Tu'* fi;;,"DJ-iE_o- u'ie. y the:e \< d;.Jrn LH j,.! q * t.al* _t__6 G1

co=era 1.

Le-L \Ee,-:e \+ .l-: ..[,..,--g .j..!qtr-t cese.F-= teH* qr? l-.{ t aa, Ht 4a, -'.' . u*ooaEq.' q= (*t o,)u 1ux ar)ur....;... o(_<,) = oL** o.,)i- (r+* ac)+ ._. -r-

'u(Flx 4*)) (rrx o-t)yL = r-'r -\_ yn{_ -.. t }a [X U"^ e,=)

tr = k,tw

rrr , -;- = \< t.., oj9 = JC

o (".)

etH-i/Lq-rD

l-^r\-rg=e: €j^J.LJl'

Page 161: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

lsjruce \<, -no a'L l4Leqen ( {i.ne) . rE .ji> d-t-J;..,.iuosr sp rL

6- t.:) o L r_r\ .ib d- di.ri,-rsu e,p

--r o I* ) orr*= . (G^) b

TRi.'D p-,re.w e,: ltr e I oq- o..,-t.,o -I-Fc eE ver^. -t -l

t'l Lr) .&5.-- E,ej::ll=__J1gp! :aL- - ^ ' _n

i. .|]a'h;\]'r dt* clssu" P-y,dtJLr) .{^rrooial:cwe

c"rr*<l el .z

t Cq, b)

Alo^,

(o, t)

* j-,x)) * 6.,4y/ Co*, or1 *L) x C(,J)l

f o.rc: qld ,r ( o,] * g)l-''-)

L ac:<, ed,q * o1l *t] =-(,)

* fC*,y) r (.,d)]

o Cc")

LeL S-exq, tre tFe seE q a-le. er.A_.J

".::,T ;Ti H;-;' ?'Lwe'.I-H

i1 c!'elr<, (s, +) :,- o z5 errLi-q.4 6.,-P " r_,.., iEca-,-r,ruuL aiIye 2

ir'; /\l^. 4t^S rx- -,J,e.LbiIr1 el".,errt -g s'=\5, LAU wro z-orz)

Page 162: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

= fo.,u) *_r'

Lo.-x-L 1

: I o.-'*' c- ,- L alx-.,

f ,.c, rA 1 Xl.- f:.A + 1)+ blo--J

J".-l?cn^ (tI a*d @I *- l_cv-

f C-.,u1 + C,(,ti 4 (c, d)= Co,D+ [L>.,t1 )*

* j> 4r>scfaub:ve_

ts,*) j,? .s€*,,ie^-"f,

t.,d)l

+ o.-r1 -i. Bl

*1il * t] - cr).

(a-:c, trc+\)

t*-,1) + Co,t)Cornrntr-Eali>e

L3) C"..r^*t*l-"= f-ep*r,t,..{

.(^,r) * 1*,[=[]l*.1(t'U) * Lq,t) = C.,ca, r-t+ y)

F.''ot^ (B) *A G)(o,t)k Ir.tr\ -l d/ ti' (s, x) jD rust

) =Je'.t;{L{ P-r g,oe,-rt Eu

Let f ., , .*) te- lr=. +Jerrh.{g g[er_ er,!' { Cs,+;*T 1"" ^"11 f a., b) e s ,(a,t)*(.,:ea)=Cq,t)(o., , de2 *t) = ta,t;

er:Iarde2=L+=o,(a*_)rh. id.nL:Lr1 el1:.enL _ Cet , e.z)

. (tro)

Page 163: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

F

P:reve t\a-t the n ece+dtul d-A s u-UtJ:cL e,LL- t-l"--L-oh-dJ-w 6yL -y O, a. nOn_ ernDLd B\rL.+ eL ll eE P?--^F { o , "+" 3 +.6 te <)- )tJ,qr^t1 u a,L e H

<9oLrrt-:s,. .

- Le! L{5 . q'*\.ryie_ thd-E H- -.L_. ia_-. sul,4-r o,_,_c,

T q s,;h.La- H .Lto.U .ii, o- qjl c,. 't I

{* a-,LeH -> q-- "1

- ,lr-.LT' t^.,e Lo'vs

6inco bel ) t-,e r,-.. Fo-:r ct,b efi _| a_r

'-t eU

%,Let a+b-Lr_r , I* o,be,lMs, L\e hor- f- p:r or,s th a-L H%cn

t't =d en}-tu

rr:I

ele-'.en! -t-O, J-b

i-,- a Su11.no.?

Le-t a- eH:) o._l e H

=) o- * o.-, e !.1:) ee uHeh{.o r-r-tie -tderublg

Lii) Tnwer-,e

ele.ne.rt te'a ,,

o:e O Ha- + a.-l eH

enq?l

i

=

f(ro-'€H\a"'

% H La..- il5, ir,.r.ro- inH

Page 164: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

1ir\ ) clexrrrte :

Let b e tt :> E-le rrFBq arteH -) art-'e u

-> a* [b-'.)-, e u -> a-,rte u

H'.H

:- c!,+>eJ

-i:D a- sutq--*7 % q

('-..,= lEdL ik. 6eL =+ = Ifol ,f ,-1 ,Iat,[.1]j.., o- Cst"nrrur-Ldi5/ e er;,L1 L,rith o-r +peoL

IEe b-tn-.or4 fer a-tccrn aiAihb^- rtoL{r'Xe

o,^A uuliy.ai*,* r''aedrrl& -r4 o-,.'J x*

sstr-Lhb'1- ' f fi\) NID 2a \2-)

-111e c-r-.re,--iLu o-' iaL\e- qonUr:d-u-l.o ',* ^-A. f.a u,,*..p^.cJ=",-d'1e X* -i^ +4L1e

+-

aAA)*;e,rLNoAl^Lo \

x. - --)-\'-

f"l f,l [.] t"]

[al

{

f.l ['] ["] f=lrd

r'l

f.l

[a]

ooo olI3l

osZO6<j

Page 165: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

,47

-*r41 "",qd( b+a c)

( bxa c)

Q+\o--oB*q\\o-€Z\a-x\ I =a- {"- o)\ o_e z*

iil (o*u t) xa c __ o*4Fev ;:.>c-.,..o\ -\'-

Corr-:1dq,. o-=\1 t =a,, C=Bt'\s',r-

(r +'* r)-ta a -- 3++ a=e,At",' I+qta++B) = r -r-+ \ =aa^d ( txu a)x\B : ax43 = Q,

Al^- \ x\ Le x\ 3) = \ *\ r, = a,TErrJ> , a>r."o cr:atr3,g

^-4*, i,2_ ,s.lifl;eJ gor-t\ -^A *u tH zr+

Ll*) rlR,J , a\'.cgo r-q o =

I x,. o- ='t

\.r:e h-q'.,e- s .1r:

ar.d t -i:D ttr._

A^L th" erulaAe-,- -i"/t te-iEr ttie+e z\ . l)a\ce ) z+ io cte.=A

a-,V,c ez\, it ca-n terl (a-r4 t) -F4 C = o_ -r4

-TF e e.ntrli e-r- -.,- th e {&.st -:'rotr: <\rteo,, hh s-=- T .H= f!*+ col.r-,. n j*.tFe-se .E '*- f.:i"U ce .u. n ;-l+tl e-- "1..1=r,',o-

=? o .c ey'v. rn ut.oJ,:w e-7re:Feot +e beth +4 a_J x4

-tg.t \e- te\"r.,uJ.l ++ oJ ,.*

tts e -sar'!e-tetr q--teti,, 1ts"

t litts

-, tfh djc

tF.- Jd;t6,-- -t le,n

\-{tr\U:pt!cah:ve- -#T=r

'E =+

Page 166: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

L5) Fge.- the

inrzerr:oe %5ebfiic'e\

+ob\e , we k a-ve oA);*;u ao, 1r 2'3 atl e o'=2-2r1

o-r-. J r< r.rLti p,l,icativs --f'nv ez>e-I

e.l-..rerrL t, a,V a,te 112,4

,++ ? = 6-{- h i.a >."

C grn rnu-! qilw e

a- A--*f her*e,*erpL;*1(,,' , A) tke.n Psrs.ve- -trid-t

L--') = f 6c.>1,fe,r * 19z"rz-)

Y)-ora-;aene

sre-' pe-et-irelrl .

F:n crr- ltr e *,a|,1 q , * c<'+1 te e::4;,t'4ve+g;eA tF'dt

^ X\ (t +a c) = (o x* t) + 6o>.ao'.4 f b** c-) xu o_ = tb** a) +-fc^,t

lrex Ercor. p\e ,

< *t* L 3 r-,., r) = ax* o _o( a*. >\\ -'-\ -) r\ (c x* r) =.G, d:oL',itubwe Bve.r

) L<+ , r-+ I X+) i-D o-r^-{ifh ;d" e.n-L.{-u.l

co")

''+ftenco

.q.irr-o.t

-> q' ir=

,x\ re to eq , E

ftntlr,'

^^a

=1

F"-.-Ps'rB

SeL'.,"tie.r

Clrvert , f, :

Vo, t

t t'i' u)

Iq,x)€q,= EL')

io a- l-\omsmo"3|i s,n-> (qt, .a)tre- \-a-v:

^ tcol _ L1)

Page 167: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

Nols , I to.) A.t'-

:) ffo-). c-:) ff a:r

xcd--') = E

EL=) (et

a f,t;') = e_

) = (g to)-'

(o.+ o:r) (u,'i*.g r)

'- aX o--'= O,'

t -r,= ke '-e.ne'ry!.i,=t'){te) = e.

r-e-L ( S, {) be a- 6''t?-''Y -T}i-orl fue'we

1.,.& tEexe ex -i>B o- 1s'r-'g66npLr:-r^

2'. S --+, ss r 1a.fuo:-r e Cts , o) :', o- ,5.-.^r-q-*f

% tu,-.1r6!-,: E-* s +-o 3 r'-nA e-r t!ic.

.p"-"^tr-"- T LyO) co'n|e>'Lon

se\utien . Uu NID zor\)

feL o- e. €

8., : s__> _'"; ;3; = JJ.d.-*6(') = ["-xt)+c

- ar( tu*c)= f"- Cs+ c) f"'*' ;

axro c;_alt_-1= ut f,oL.)) - t €* o Eu) c ") :.r

.-P to*u = 6*o tuNe.r;

^t*, o_ |nof x: e -*s3 u3 x(.) = !--ggt o,l e e

r6-ah {('-*. b1 = T-* u

Page 168: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

-- t*otu= ttO o XLu)

-.X!^,ts; = iL.)o2Lu)' .t'' G.q- lte.ns tn s,ryl,-.n -gie.;. 'S i..{s s,s

P=.or- tH aL ev en:t {+n,,Ee ?- --t, T Bat Aea

rL iD - 'terrus-r;, t";c, *.e c!- p"" ,,-"-go-tio""-

ff'r % s'r J-e'r rL cAu Nlg 20 / A( ,,q1r."t31

<Se\uLgru'.

S:te1 '- t

v.{e- E;h-a"\i Erove 1k!- tkee.=,e.'. ;* = Sb7,'

v^re aho"\\ tJcvrt 1-i^l a- seE "' %

Pe-,' rr"wE a-blevt

.S-reP rl

sLel .- -1 \^J e Pleve qr j: -- ?--tS.lLeT: a F-!,;kt o.J.l, =--:s.^"o-rp}.-c:;; g , q_> r.J

Lel q Ee fi-^;.!-= _q="..,p ef e,,'J.-, .1-b, '\ | bIel a-ehDel.r-r.'e t^, Q + q tl {,-C-) = a-:.

' 6i'tqo t*aD -- E-.

t'{) :> a-).- = -y :> a= y-r J5 \ -r6a-

6in-"c., a X..n, tEe., lo(;,X)=d,.tr=1t"- ia ev'"Le

Page 169: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

-rF,-tt>- , ftz- -.,- o- L:-f,e'J:'-eru

6fin ce q I'a.-- 'r,' =]*=rtL , f,^ ;,: 2ij-UPe"-,"n glaJi sn Eh ' n' -tgmleL-.

!€; c'.= t f,_ l-ec,l!.Srep.g

Le-L

+ba-

R6e

Q' ;.- a'

€-'€ob f,, C"-) =

P o P.6G bb

= q \-r{>

?"-"F

{. [[ut,)) = focu,") =attE"U H erL(q c"' j,- cJ€tef,

iAe."ti\

= P C t"\boL '

*fF-o- rEtrv e>'r>, e .B L.-Cn'+ * .1--f

i- q, o E;,

.STEP i 3

v{e P-ro*e q o-,'A qt a.,r -- ,>e.-enP!,-ir-

-?n- + : e_>4 b? 4L-\= E"-{ca-) = + tb) => f,-: !r. =} f,*C.") = tsc-)

:) a-:.- = Bz- :> a-.t.Hence_ q i.> \ _tEi.uco-

t- o e rd-s 1 .,b BrLoar'"' tLo-b) = !-u= f,-.Bu= pCo-) o $tu)r. + ; Itet, e,"j\.i"=m

-' G,\ '1 qtHer.rl- 'p.ror\ .

Page 170: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

a-!'1 =

NOu>

-n-R

(n,

-r-d

?=row e tHe tkeo.:r e" . \et t(a, *) l=e- a'

fir,jl-e Cgcbc .151..,? qe-''-q-' --LJ bq an'

.-t*=^u j' e q ' =1 ca .io % e-'J'- '' ' ttsa}

j.z I G'l'l =n, lts en 6\-n=e, SB rho'!

q -'t - , @1 , o-', ... .I = . I Fr-r:rttre-l r'! e..lq hj;l ci &r--t peti b:r,e =ntetr e-' f€l rp)-.r& "a-n =e f 6u ".il ,orf

't 5

So\r,},-'en ..

=t ffL .i,.- tFe g:rJ =, T a- , tE e.ru

EPE

*i€a'

o, -t=o.l-5

-* u -t 2 z-9,ta

t] F-trLd

iil trrr.d

(i I w\ocl-rlM-o!

e-TFerL F L i;

n -- !c"."

a-\ = o,Y,',,

_- [.'9*=ek

=e

eY'- t:F.e ,SeLtd ,<-rr y=

e.xL,t=

b-ir''--rr1 -7:":Nr.r-mteruo

"T*

*) -i- a- Ser'.i,q.:tot7.

;ud-.^trd| eler.' s''g tt

'rhvQrrEe-e\e.'r,e-n\; h,-ale !H21 3

fRu nla zoll /Au rrlszorz]

Page 171: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

i

So\uUs'. -,

(\) e, x) -;Li) c L-s:rr,le P=repe"rLlr

"t: x' ! e e' :> a-t-x e ,

zr-t+ a z-ri e R,

:k s-b-6g.re>

6errLiqs!

2 xrr C D

>

('r*\)t z =

Con;ytJ.el

-t (1+ =)F.-tgo.,

tx t,

t (-*,

clerll,rq E.f*l:l

= :c--1_ t _i ar<_\ _t' z -r . C_*: * , *y ) =: )c.F \ -r 2 + I r_r1 r_ azr \-r-z.t .-,uJ +. "- *1=-J*"1-;-\

"x(Ir>) = -+ (-\ -F >-r p!-\' l-)2-ry-r -=* a\= ra_ L1 *=* a,.., =.,-+y-r=* r\= * r-y * ar.r-i

)

tr"r

= -:** z-t a>cir- oy=* ,=- _ n*, j 11qtLd (.41 r \=-,,

tlr=)' -iD

r) .:.-

- (=r\\r" --ctrrbecio_fy

-q- 6e-rYlt:ljrro?

Page 172: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

iI

I

!

l+- ea_

er= oeR=J:*^-+-'+q elernsr.t elx j:__t:

LeE a-r te He .{-,r\/er\ri g

a-ela TE!J1 4x o--r =

;' ( r+ aa-) -- o-a-d- = -tt

=o f --'ttaa+o]

elet', e.rL-

| 1- 2a-a-*

f .=-lai' q1-i: l:

*:T*- .rB

| -r e. ca,

oe{r."e rr-.e sq"J-qr X"-T-G?lL.- -iB C.or.fs>jt,:crL .La-t\e

TRc .-de-^tiXX eJer',ent a,-.d

e-a-c}' e}o,-er-rt

4nJ

,k) odHe4ag p&l

6*i-^/eiDq

G2 \ lro fl^l i J e,.ti\ el-e,^enL r

-

> .OJ

ca- - a,

(3)

=t ;/-.k;q e\e..er.L e:c.,>L , .lsL it t-. e' r-\o'.r, o-e R.

&>E e - ^e -{. e_ + e a_e-

e ( 1*a"-) -e= o

fAu * tA Lot)

Page 173: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

-q

Se\l\igq ;

e 3Ll\ \ J]

.3 ul,a.,1 *_la 3I+l3 a,-_]A 3 lF -fa I \l

[=

f-. r

/'

i :l ',.= [ ,]

==:] .*=f =

-1a =

'+ -

2 3 ,_l, a =_l

e 3 ll3 \ rl

L;l-rL"

1

sl<-Tl L-qr = J-rIr \

rra n5 ng

vt3

_rl h

55

5.

f,

q8

ql

5g

5S

Jl3

.5.ll

.,\3

q3

53

5\

.x-T

}lB

5-1

:,\5

.\3

5-I

stg

5rB .55 qr+

-t11 st 5

ns 5rg

r\g jr

5, -nl

5tb

5lE

q-I

tls

5lt

sl\ \3

$r srL

-'ta .na

q^

ri+

Page 174: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

H€1e ttse

-e1+ = [- r

LrHeaq I - \-\

L "'' J

C o+)-t' = .n +

Lqr J = .r\ --1

:A-rtbt-.J e\et,eoL -i:a z r+f, 3 t+]

, \.\,qB r L-a) = ora

, (ts)-' = -'= , L- u)-i, (:d s\ -\'- 5s

LJrSJ = !r

- -rE ,

She.r:' tk dis 4- qn o'rPb'tr?'-f ( H.,

69\Ar!On',

ke"''e r,.e,1p\-;o1-^

a-net}r e.:'l

TG'

tFe !<e:r n-er\ eP a-K

t,,, *) i,.t-- " a.,.

A) j- cr- tCX--ff6o ara =or1)

w\{.T

- k",'(t) = {-..n lgt,.>=-'].Sin c- |

t t.) = -' i,, o),-,.'ay,, {r*. , a-ll eo--Le e ken (B)

Tvt s\t-.e,,u, e o-.J,= \.. t8) :-- neL -,-ptH *t^ gLet rF.e +,.r-e .\.r...8 arj e ke' 191 .*t=',= , j to-) = ei o,-.d f,Lu) - er 'E) 2

Ne.,r, tL-*t-') = f C^) r Etb_,)= t (-) + [f r":J-t

= e l+ el{totb-r) =e'-) o-; 6-t e *e,r ( g;

i.', o-,be !.-"-(tr):) ax6-r a k-({).'. l<en tg) ; a-

"*q:".rt,o T q

Page 175: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

{ [c^, *)Co* t) "

SoLuh.ort ..

A:r5t -.r, e

a-t E *-

r\e.r, a' 4 t' ' =

o-be-X;,c-n

E2 fn,i, o-rt

q

= a-tX.=t.-f , S!"sr-r- tt-t

N l D 2.lo /, A rf rl I: zo f3]

tts a-t q .ie a-be] rq'b+ o- , o.,t € c,l _-_e)

(-a rs o-) + (ba t)

. f".*blrfa+ulf^t bl_

. (a+t)"= dr+ b,cot vq'.,--e-\ , ory5u.ne 1..l

(o*u)a =. o-2,6 gc-

o-t f ax (u* t[ f r. a^.ooc^F;-)4+ I-(a*u)x tl

^8 f tur o.) t tl By L,)

f I-1L co',cefrot tr-tar_,-]

[u* *)t* o.

C{ .i-

*b = (ox__a,Er tcUel,h-r-

t){< tREqha

<t Ca.vLc4ltd-L-o,".r /a./-.r]

Page 176: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

.S[gtrr tka-L (2, e-, ><)

r,,r..lh-o:t s > -i-r-

fa

.-i,> arL -f;d.-ea.z-L

rhe seL e] a'Lt u4*u *19 =.ro)

Jo,,,or,-

Selrrb'e,. .

=LI

FelO.r..J

Fe.''

(2, -r) ltreoe7-

Fes orr-1 o-rte> , rue Lo., -ar-t = tr- o-. V qrE

-" t=, -F) -iD a..n- ol el;o-r,'- ?-r"PMo,r- , c-or".:;io\ o' L=, t)

rt -&- oB"w-i-o.r:o LH cut t x ' ,5 <Lbi=4jeJ

Az*eciaEte' P",re7ez!1d.JFo.:r

\LZ

--nA

Ze>to Jiv-t-o-.-=,

--tnYeata-l Ss.t ait -

where z- .:E i[Fe .3e-t

o{E ''b tEa .Seb

tt-.qb tFle {r-to

t r , -") cul.o- -i-s.^s',*.-a.

4*socra-bve P.-rf .,j.--d--'U.lq e-t-;e;t i- o

ele-. ertL c,- Gz , & -&1ve-r-'-->a

-a-ez

\:-\e:^ssva lA , >.) . -l,oJ--^bhttut -= €)ve,r t

aAo- fz, r- , x) har>

L=, +. x) j'

=t L=,+) --i tE,+)-E dl\ -i,rJLe€e'b o^A

everu in!,e-qeio , El'o'r:

6."-,11ore-1a f:,-t) oJ

rLB

a-n

e[-,.r.eraL i:rr're n eeJ

C-ar-ty...|-La-h)'we. artl x

closu,,e-

C eo Nl9 zo to)

Page 177: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

l

r!.

j:I

ii

LeI t. C=, +) --+ (r, +) ba .le#--t t:1

{ L',-) : a-

.. E_a-.-X) = Bcx.)+ TLy)s._. L ''€ Hern o rns,ty!u:,no'. t'. (>,+) -> CF,. i> ... u1..Ue .! H s mems,, p1.^;,,,B .,: he.-e".ery!,.i,-"n

G: -t) o..4 f E , -r) ane =e,.,e,rplcc -Lo eo.h en

"

6oLrtien-.'. :

Page 178: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

jII!f

ii:tI

crrieci-q}I-\e .s'- g

Fqe?

req "^4 cotwiltra-u

nt q-LL .tJ-''-fEe..tStrb*sr.siJ

cer.r"ula-k-e *o'"er,J, (*, *)el=" =-rL,

(au

, tFe Set

€€E rn -D a-

lp

r-1

2ott)'P-ssF

-r,t*

C. I,I) F= o.- .A-L\.<'L . rne"siJ -^lrJe'^pete-'rt e].".erB

.ir= o- r5uLtenetd

-[-",.nLo

-Jenh''.-{4 6;1e-'' errL ! e' -s.tzog,'.-.-

% s a,e -l-a-fter-t fl-.- *,

tec-cr.^ e'- rJ er,tPeLe a t' .c z: C-

ub [a,r:n. - 4 : a.-r16 e 9:] a-+Les

A*+,,,,',-e. a,t e s-'. e,Fn=? q^-d b*b=g --_- Lr)

l{ow, Lo+ l')* C-r u) _- a.x (ua (a,+r,)) f .r_u,_= a-r(.->st6t)= o. 1.Ca*b)

@*..)*g= 4_+ t

@*U) * (arb) _- a-r".t*=Lr"--

sAe-"pEentCo+ b) j6

b eg

.S l-e,t-ltF'. ae.t

cl-a;^ '.

(s, *).S.i-n-c-e, tF e

lrJe }.ot. S

Suldot m -r :

<S.r)r'-c-e- tHe

e+e =e, e

o-+.

- *ec-"f..{L . X *-- C-em-,-,Lob"e.-l

tr=it-!o t,\d-'( + -.- a,.le.r:ot;.,e)u.oin1 Cr)

Pqeve 'k iDe-O>',\

Page 179: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

-. S -i-- Srt*-'sneiA

". -T6,e 6e-L

ot- SlrrL'^et"eiA

i< ib

\+. 'ffi* c e*

SsLutisru

Q-1- j> tHex{urnte>,:--

(-l )

La)

% d-ll "rAe..petent e\t"tnt€--

a.g"t-.A, * ) -ir-..o-t1 aLeLta4- q-n W u..rhe,,e.

bu a-x l.- = aj- I -

-a -----=,tb,lalt a.,b e. grL>( p.) M tJ 2Dl a_]

He>re

.r{ o.fisrLdlaet { o-rt 1-b>-.rEwe

d-* B = at. eq-

o.bc

.- c4)

F-re-

Lox

(A)

b)x c

+[B) r."3e-

= ax (b+ ")

Let'e, t-

.- (s;qe-L'l

L3 ) =de',bc*.,------------'---l

-lEal ,

T6n o.U o,L, c €gffie- iJehtd{! Je,,,s,rL

d-+ e -- a-

?=4e =a

Ar'aecl..*,lve B.egs.t.4'.

Cor u)1 c = eb_+c2

= aLclt

a- x Cb* c) ba a-bc

=Je,..tcLq ele.'.ent ; s:a€e,

Page 180: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

4LL!+ ) JnwQ^r>e ._

-Ttreft ,

L€-t be- 1tr- ri,nve:u>e, of, cu

(=Je..t:!1)

a

Leo'' IInv er>,e BE o- :r- cli.t -I> eq-a-

L5) C€rr.^u,batir,e :

No.!...l

B1

a&t

' Iq*

= sL_b

-- aL"';-

. t) -r-,

-Eo,r a-tl

o-tL aleLiar.

art

f".

eE*

\s. Shou, th a..L

ur)lLere * icD

6o:lr a-U o., t

S o{-snbyt :.-

(-t-t'\,+)a-+ t=

CR- l'\ )

dbdiqnd-+b r-al>

geL

d-edl-.J +tX-'Y'

ER

Hene k - {- ,1 rileo-ru.5 -tH et lur"a te,.r-, exc-lL _l T o.-t

() cle>t-- P-..?=.Lg ,

"l*\ o.xb = a-+E+ o.t e f- t-'

Page 181: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

P=repe, !q .

-.-.T---..4L-?)

Ca+.B)x c = (a--ptr1ob)+ c= o--f br of +- ct fa-+ b-+ e!) a= a+b+ a-tr- c--r a-c- f bc._r a-b c_

o*fux c) = a*. fb+ c+Bc) -6-r

= q--{-b-F C..ttc 1 o-f b+ ct_B)

tr *" (A) -^da+ fbxc) =

a- r-t+, C+ t.:_+_ aJ. +. az_ I q5--

('lLB) *- q=t

1a+t) + c ) Esl otr q,t,e e (R_

.ideyrtit-r1 eler.ent .

1'\)13) =Je-bi\ .. Le-L ,e., Le, tl{e

-,aEe>a-

--> 4-i e {- a_e. = q-

e Lr+ a-) = oe=

Hence .o, -ro

L\) Tlvetrbe. : LeL tfretTE-erl , aA dl= o

a- -l a-'a a-a-- | = 6

_o--, (, r- o-) = _ a-

a-' - - o-

tf . ,il enb{q eler," e.rrL --^J oe e_1q

l*.ver.>e d- be a'"B

)

t'\)-O-l+ d-

C.r/er.tr+H

E CR-

Page 182: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

Cornr''r.^lalr:ve

a-+ E = or-b+ ot: b+ o_+ E, o_

:b* a-'I o r. t- - 1

:.- -^ o ,E* ?] a.,b e[n- 1,11 + arl o,LA-Lq^L-

C B- i'\)?"':f

-]Ee- Uni-on

n-J h-pE

?nool ,

---t) l-eL ur? Paeve- -TqLt-

we knc'.r: r6-o-L L= , * ) .s

t.J=, c. JA; tc er.

o^A Ha =t-[r-=3n,".ezJ= { ., *+, .rU,. .\

e\--,r\ !* , o..A H 3 a:.e. A.rt1,roul+

Hr U l-t. = t:. l.r<- e H, s'r r e Ha-\

Here 2

". Hr uH.z-'. FrruHiHerrto \-{ r

Tc

-t L,,o 5"bf-*T- % '- ?"'f',u- 3''-.t?--f

the.eo.e-rn t-l .-- Ex.or"y\e .

o.- X".-.f T *t a=-,-

= 1o,ta,i5,r\,=,t,-..\€ Hr anA B e!{2 t,rL ,r_3 =Eg!H,uH&.io rueL c-\eaed ur,A..r o_dAib:o aJr-. ngb a- q-.*F

ur{1 j> neL q_ *"loq*-t T q

Page 183: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

-tqe- D^i-aYL % U" a..tq-,--f % - ?---tr ." +a- .ar-rtq-r-*7 one ib cr,ruLo:"-.4 .)-1L LF.e

otFie,' .

foelo^l ' .t<

C4 'lTLe-r-L

H c!9 si

sutrycr',p-,- %.:.j:D ^ 3"!?-r1f,: ,&

ole l-,r,e "dfl.-,1.- g e

- . l-.1 u t<. = K e'n HU}<" = H

Henc.g Lr u rc -i.o a- gulqja ouf .

Co-riwe:r-.e.ll , ;]u.pps+e- Huk -io o- SuIq \^re 'j-r"^ tL--" H c- k ., o .T-.T %

Ejgeae th a! H -iD rust Conta-irrc_J _; k- o.^JK .Eo yLBL corl"ta"UruqJ .irt H .

-TFar! irh e:re e><-l-L g[e'-r'enEo a , B +u4h bh o-E

a- ett a-^J o, {H -L')h'erc o.J b+H -:-(a)cAo.-a)4 a., t e. H utc . Sirrc-a.- HuK ..;.b c

dL cn, a-t g H uk, . K- Dt.'C'o- l-tuK jio a- s .'bq.,ttsr?

l-l er.c.o ob g I-l c'-r a-B e k-ca>e -t:. Le! a.b €,H .Eti^ cE- qe1 , dt eHl-l en co- o:'6a-b) = b eH , urhich ; e\ :>4= .t- (.)

C-a.->e - a : [gf oL € ]c..Sirtcs bgk, lc-, e u"

P-r o.1 ,

LeJc H

?-1rej-lH e'r

Ia eruca-

-l<o 11 1

-' . o\1,

!re- L-u,-e

!l ukK4H

Att-r-ne- t-r a-^f, |cl-.{ c t( o,1 kr u

b-r 1o.-U) =d-eK, t uhiz,l.' -;o a- co,t6 aAirt;e,a-

ar*u.r^plC.on .jD u, an1 :. H9 k e{ tC!; H

Page 184: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

''I-ka

T{

ar\ta.z-=et:xi,s-t -* ^ta -=.,,o ..at--{ =*S3 *p. -* ^ go"..

-- no".^--l -sL^lo€1r6trf .

(a-)u -^4 t- -r-r+ .t-a',r.^J -sulo3r'6ue< -{ ^ g -*ts q,!n+oHn)rro - rr-oxrvr<I :3Gbg"5srf

fAd M-y 13cae l-orRJ

k oz-q ,-r-ar.naf <u-Egaa.,ps ,a>tL € \^-t q-i-6r{!>< ...[-, 4'a- -<y.Joglrduf a+ q.

{,.4lQ4veq H -'.a+ H*4 r-1 \-{o k Xz,

t'lotO uJQ r^rauoe G p,-r-av e Hv-]. Aaw fo q-ar,,-!.

Let -C e q a^4 V\ e Hfl\4 ,r

{ cq. -.,\d h€H ar\.J \.\aE

')t +h*:lr eHd +h rs :il 1\La'"4

E .r H o-..d fe a.lz- ^arrno-l

,ai^})qtelP.J l

F.,-"--t O2@UrzgeE

-r+ h * "C e .Ho t' .

* une <2- -.rrJ 3v-t-q'r'6'{P a'f 4 '

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II

i!I

gnno" t!;ve's'9 W W,*, 4a

i;- w<2<t--u

l-of ]'. u -+ u' Le o Ao-,o' otPicr,l + 9n*^uJil6 ketnai K, k2o p*w t-r'dk K b q n'.'^o0

^ i cr\ k t-d* " U".- oryLic [; tt'1,'W*) % \rt ot'c

%^r % +. f nu "1r uot,-., Av {ili*,,-?ir,' o]'ve 4^<z lall' r q e lS Jte o'ern 'akl{ ,rd ' t lty or-,", ol,),.1-2'r.rr,,' Fp,,

l)*no n

Q)

(D

p'ote rr'a* nn,., naz.a-7u-b4T'*';*:-;.!*T,''7t(; *a ft.rv-\ EnpV A-)'ae-*

,.t _": ,:,: _ 76,,tla.o )-ot7 -v k 6- ,sJ,s'*." A o'bcP "> *a-'ft\1"',itiiil

6.& .r.u-+.u ,: L:*{*.X"'#";rqti,^t E 4t',^9 r*" fPo ',tD 'otLJ

i,n,r. 8,,1-' lTw w'' at )e v?-) €"i6 g@*? E *a"- n i^

Pt"'e fior ,, , ^ulo) r1* a.".p 'n erzLe'v n

f-,h,-aq)";" : .^ ^,1:' t',ot , @,t i4ti e-ot'\)I-co Nlp *n,,"fo*r.,o: ea)

a4 ^, ,, i ?,-"?-^,"ff*!nt'-*;:-,f-,:ttJ %J Nu-r'7be^{a d'c

ab * ii " ot* iu ' ^' 'nlr bt4 Pa2e !""'!

S [o-..>

tt, a-

ia o-

n'a* l<ezo"'!

5"L2,-^"1"

( hr a't

-7 (t

,lr* t6 a^ ., f

^ /)ia *-t''

t*?o+ a- Qo, u1"t-7'1"'a''n

i., k; a-no hon X",--P (a'' a)

I oo P:l- tott)

f P94 1"-o: 96)

^td;n g'F.P .St;,.s hJ-

6.r, *lo c-olo t Plg >oD)

f!*" t 1 31)Ly

V*r)a

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-F=.SI\D,J Yrek ( z, + ' x) Ia o.rr 9^kg.q! Ao,.;^ )

(,u'€ z la w aet "6 "lt g"lagot f e o ".Jlo z"tu

(Pqzt'rw: 9t'1

@

f\. o^A

q oJl

&>o-d-ltva

i&'"p. trn t'rrorial (Mr Q lto '9e*

e)o ^o nus % !!. N: a

Ca u N/D t'orta

t+-a-h (o+' *>J r\bd

o>; ar ia J-ab\"'d')

lO " $@9)1,tt !-oto-

)ul..,ore\J-

9*,6.te"h'c"-''N tn

i4 o,n o)*)"o-l

b7 cn*b'lL

Lf,f r': 4t)

9*"lD.J

+...J| a

.L\A, n*.-.1 *"4*-P'i q

L^"{' ;il*-P' le'': w^a) s'uo

lg^+" 16'. 411 ,

a--o19-

i

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-i-J!

Par<-r - A

Z- \ PslB\e tkai a,a (o"-.t D = -,

f"a. Lo"tGc-u- (w,

a\L arb tL

th. GILB

- Lr)

a,

n L*rlo\, weho-"e

F.rcrrx 1r; o,*"J ge, > \Ne !r-..ra

a7T*7v>-Z-^ - (?)

ot";eur{ a- L-

? rfic dTurEo,^- %

a 4 & v (a-'ab)BX Cz) *")

a,v (a^D - a.

6t"ar)"alr\ we c-a,!L

a. a(a"vt) = ov

3el-rrhior,-

.-.Jt",&

J-t =F.rt "^-* "e-.l_

e,llta|erLv

ce"^ple-*e*

fq u r'4i r 2s 2lSo\u!.bw'.

.Siruce a" b \^t>

r, /\b 4 a.

i'. -, A g-J'eor -

LuE , we ha-ve.

L',) , -. ho"

P.:ngve

oo)c-a,'\- Al-q&^a- LHU N lD 2- ot 2--)

I n u H 15 zol2-J fau "ro z"t1]

ce,,.upLe,nen1eA J.ioh;t"f;.,e )-dffL<e-

Boo)t-ax- )\a&ta-

Lakfce u,rffh ' o';^ wh,,;dr eve/t(t1 det*et l+ot'

a.- Esol"eaYu

rl

a-

er fgv-v.te"I! IoLeeL.n* JP a.

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te$e^,rruX cerL&fLb^,- '

Vorb.,c-€L

Lr: 6-,..o,- o-

L, '. a" /\ b = ba q.

L2 .. a- A (u n c) = [a_at)a c

eyeta.b-'o-'.o-tkc

o.\ o-, ' o-

sw \2 = tv a,

o.Y tL\c-) = (o'rQvc

o.v (@at) = a-

1 2 b*.a-,rasqr.-EteJ ti'

a,^(o".vtr)=^a- Y (b

^ c) = Co.., b),a. ( o,v c)

Da i a.^ Cbv")= (a_at)r,(a-ac)

-rF.e^-e ex.i'-t el-ewuerJSo ' o ' a.'rl t , ' 5, ,r-a[ i!'t a-t,

4 ,,\o = o , o. v o - a. t o- | = "' o'""d a-v I = I V

a. * o- eL , tkere exJ3?L ce'rt'1 e>pr'Arnq sl e-e'Lt

a,r -;,. L Swc,}" tF& a Aat -o tr"J avJ=t

t.

3-

A.

5.

b.

f

L\

b,

\n heJ'.- j: a-

>Btr$r-U^ - :

lo&rOce- AarL to be

f nu rrrlo zor{Le,t CL, , vl te a- A:--e^ l-tc ee

Lto-, tst!',, \o' eLo-,.^errjc o'--A 't' e'1r-r''n-e-n-E

;r, ar.;A +- t= tret --f,.J Xa-VC''u ' at ;t'(u,^,v, o, 1)

Fx.-,*g\- t (+(") , g) i,- o- s's'-'-.A=J \-o'tlsi<-e

L{e-,e rrc' o;-,"-...! ,i-e S t t,, e\ernen-E j-z' A

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Let .4 = f a.,b,.\ .-A f Lail V' -r-B Pcn': e:r -5gg

e5ars a- l-]d.-'De- Ai-Z.-q.^cr,.4 a (eG), e)

ser.$rb,.,. f nu Nlp z-orflo. lo,L,.\

? (A\ = [ t.tr , lu], {.\ , to.u\, {o,.! ,1r,4 ,lo'r',;5,S3

l-'

{,\ ( ^LL" )

Jio-gr..--

( e6"t , q1

-Ga^ te-

t^:h-o:n e A

Jtep:'re-re-CUeA

= l".,t,e]

| ^,u, "]

.dJ cevwpl{-te_trsrg zortJ.A tA c-eqas(1,f,g .t'Sulabet. h-r- a-

A lalx,r-:c-e i.>et i-t* h-crh e',.otub t.l

ADL.

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che*" \^.*te\t-.eT tFe

{t,,t,ar, ias).,>3

=*bH

TFe Hante cl!o-q-ta*'^ €- tke Ps=eL '

I C,, a,u, r\ , o] i-E

pei,€lto I C .,., b, q) , .3 o-".d

aae lottiee=r crvr n-eb .

fcu A IM ap\l]

Hs,e l.^,u 1u , eJ' . q-i,ve-n pe'=el

e ) Tfie ll ot-r-e-

Je.-r= "teE e>.ir.td.- Jld.tt^-'c'e

3* {Ct,-=2as,ras)PJ

= b vq;,, n-gt:

@

A4o

Fey I Fd;'1q<-.j-t

' q"i./e-r.- Pe'*:b

% n-sde-- te1t' Xl"u a^ d J"ul

)-*tE;c-e

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DetrYte s\no - ld.!wc--e t C., N rp zo r "]Se\^$,ie^^- :

L€)c f u,nrv) be q- Lo-tE<--e>

Ag L E= o- g\rtrae! T t -rF-!-r' (S,

o- SuLLo-\tic,e "t CL,a,v)rB s.l-E1^-.A=, tstq gp,e.'r a}!er.r' . .-A V

I

\'-'" Va-rb e g :) q , b egqrrJ a-.v b €3

^^A ta-t

,t, v) ;_c-te>-A

6Lrorr: tha;' & a- Laltj cee@b : E?- c --,-l f a+r-)

6 e\.r-r-Eb.,' . f nu N t5> 2orolL\ ) G1'werr- a- Z.b 4 c>

Eir.c-e, o- 4 B =y, a, v LE 4c :.> Lv 6, =qz-_C- :) q_V c =

trcrs* 6; a_^d La)4"v E -- t -- bac' a.vb = Ea C

(ar Llrs -- Co ^u) v Lb^ a

d- L L 4 c. , t\ e-,-

f u+ c) = ( d-e)b) +(b

-i,6o

o-.

tL

bac =

, a- c-

- tr)

-G)

RHS - (a"u)a1a.vc)= bac- = b ( u,"i*Xt.r) :- (a:) .- (=;)

p-a qrn (u) a" d 1 s)(a"atr)'(u^.) = (a\B)^ (o.vc) =t

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{3lteu, tts 4! ;^ a-

c-tr"^pl.er^e-'uE %r\\rDE l-e rr,\i-y-e

S€\rrtLb''\ -.

t-e]'

Cew.>\elr. en-tI

Sinc-e -a-^ z--

Ji'-.Lr',tulr-e ,ta-ttc<-e -, rLztrl ef et'.e'uL exi'?Y'5 |6en- i-L

f n u N/, zoro]

olb\rhe >c_ *A .{ z1j't =- trrog4

a- c-er^p\eqs.r-t q c^ , ,r., e }1o.rerB4Vz- = t .- O)

c-e.-*p\e-.^err-L q ^ , r^., e }r-c,.., srb

.o.w\ = I -

Lr)

f -..r.vo=z_)

EFb

Sin<-e \l.\o'^ t =

No.rc z- >

q = L{ \/o

F.:.tE,rn

7-

t^) ^" _A LB-)- -tu = 1.r."

a<-=1

) \,\Je Yra.Ve-

d

seVI)

'" v(a-.a{) [u-_^1 Cr)(x rzo-) ,\ CE .v t) (si:-$..itu_[-.,)( a' w a\ ,r (x- w t1) [ ce."",.,wr.ua,b.:-e)

l,rf-vq) (*^t L,))- tl G)

= U ", ( a- a '.; f r1,=i,-1 C.t)-- (U ., "-) n (t v -) L D:,+yit*bc-t= Co.vB) ^ C\r.x-) (ce"-".,r-t^.b:-e)= I L.t ., =-) f r:-r^X Ca)

\ = \ vo LB)

Co-r^yIe-^eruL io:- \)rl;:q.1st

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laak-a. n oJaeLt o- & -r-o-b=o

A lo,\ "--\il f nu ,rra.o,!6 / r.a zorr]

(n9tv Car4 c-q,ll d.b.'oyv

Fo-ra,n LA) ^, -A (pl)a': a-+-L -b

- a-:b

1P, a-V

o-=L

,; a-

feorBo

Lebe,.r+,orr) b- oy e,ooLeo,v, AlqeLro.

IEL a',b Q > o'*A a- = L

ct oi* : a-E -+ d,b -o Lt)

Noro, aE =r at = a-.E-+ -a,. L

: o-. d _r_ E. q- L,*,.^9 c,:1= o*o {i, o [ . a.4'=o)

c.Lb _f o-b =<)6str".,'oe\ , A,.rs.r-,,. -

.gel-!b:s^4 r

=)

4L, + a,b = o

;"7 =-: =

7^,-:;:--1:i'*'"'(**a)'(cr-tr)= o f ar:t..,t*to,= r_*.,,)t 'f o-r L) = o" (o-*E = Dq.+ t - a- (a.t=a-)

( a L.:e,. i"bron

14r,, \Lol-)

( Qn-+'itu!:/e I4.,-)(b+E-D

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sl,\r.n^ *ke

lfre l-l qr.be

6,e\lligtr ;

-rFe Ho.r'>e.

t

^Jhrrt-e-a\&J

?4n\laL e.:rAor.in-q =rep-'"-en[eA br4

<Ii,o!qc a^ ^ . fA "A lM 2-o r{

Af4,g."1o,.v\, fl Lx , R) GD

R- 5It

(. 'u> 1r,

ta) (" , .D (.r,=u)(e,u)(:,r

{ra,e,.7 (r.,.1)

eEcu.*pleT o- ,!,o-ltj.- r,.rhic-L .-,:o

Ju-Eitur've fA.r atrr ,.,i]

q"i-, o\ 4

r^.oJurl_ol

SeLu&.b* ,

t.rf, N o-l:

(ixo,....ge-.l

dJs- ty :L

i

M6; e).,1 Dro.r,,raewv J Lo*tt -e -i,e a,r^

\..red-ul o-,, !-hic-e- tuf rLe,r^. -La-ErLe

%uldve-

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P--.tca,L es,Je.r

Bin-aln :trel"ah--ott

d- P.rt*FaL gtrJe-,.=regte-^ive ,

a-n-l!*qmm eb"ic- ,

9eJ.i"e Fa.,etbA 34 P leqatEe> uitE a- Fa'\trdj"

o:.r A s..,tr.g R b> 'L',u"A o- P*E d\ e-,d =''='l.Sc-L -- c'- ?ez€LNoEe ", ---1-t: i> ce^.\ve;\i--6'rLaJ- .go .1-- na6= o_

P,^,ni J elr d!^L13 U1 tF.- '=4' Le{' + TE;'S)\'.^t't d-e== n6; {e-e>*.t;\ rneo^ ,, ,l er,sl1, -.no, eau.al to " ,>.t- i''- rr--J- B* ,.o4L r.7,^,-te-.ro

.-r^-crt4.'1^lt Ld.$rc-e L*.)

9e&n^e -ruEo-ltr1 o-d""oA .3e-t

Le-U G .1\ Ee a-

Q,\Qru] L , q

, Fne-r" I v)

L P ,-'je

ca-\eA

e:rlh ezr 1-t tlPa-lt { e-^ d--'-J >-eL .

=t tru-ho"=

si.-f\e,t

t-' f,=-',-2 .n-

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-\!n-* O:nJerrna e'rL

a- -|gEalfq u;.AQ,.Je.:1 d eLa-ri'n

P o,^A t P,9sr ,brrnflt

z-_) l: eo-tf -Jg-,Ae---lr"

3eL

tb. 9e"ge^e gYrle.n\Y\^- .-J -- fi'""* ..

u=u tp, g\ be d' Po:'rtfa-\ 'e':'Ae-. eJ 8e!.,-':. ,,eE A aP Aru{ e\e"-en-E rc e' P i5 d-

\rt,Ea,l teu^d fet_ ,A\!hsn-e t .iD '""a "Yf.1 :t'"i B€-!- 'A@t , ** q;*-v '^-^.'T!:no' *'o;'l'-t^ ' +e' A Jb o,'* de'--et* 4- e P"-=;\;'; '*- b; .- -iewe, b''''^L ^-'A

iZ: {e;'r a}L xe"'e'' t-'^-'d-' q '

*T^--A

e-Ao,J

We\t - e"'A:JP-'.tCa-i\u e'-J"xJ s'eB

; L-=1 . .-"e"' --fL'l

XL-,-+ t\rr""be-t

j,> .of,f,,JEube& %

u-: elL -rt

18, gt* !-to*>e '[ago'* o..'r P'srt;allr{ s''J"J g=;

Aii,q'r.-* -

A P^'*f,* o''J'-'rn1 ^-' ^-* h:: "*:-t'-

-repgre-enE=A tq *t'T 1_

*_ ,-.L-^r-1 e r.,l-cr J

L a- Ha.,.,=e "iTT-. 1^_ *r*f'^

aLa.a>.,otv^ .

J&-ffi;% -1"'-7="'*'t^ bq a- s''a'lL

C;J1c-tr..- €''r 4 AgL

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-iFe C)r AP-cine\-e 8e.r.b

,ip clnq.rr)n-

eeveire =e.-

B- -- oP

NgP .tt-Jo.lr-ee-ru

-.1,2 &o rr ." telou- 'f1,oI O-

=- t- \ , a^,.!al a- /^-r-rLe

,..^-.-Au&Lld6d

J---- h-Bt c€v e,l a , tts

c-enn_e<EeJ Al3"ect\

c.- st^-\= Jtner l-leme-we-x , th e-,1 ane

c-..-.!Jl thr eueh ctrle e{ tY'l€ri e 'YTe-O TP. trt -ib pe,,oitie te gbtaf-'ru tFe .g:: %e.y,J-- J Pdrre ';^ , T* *w& a-

-l-Ee_ qLBEe &-,-eE-.\ \Y7e- BsuJi.e -, CrlLB {o, Ul =

Q*eoXe-Y. Lo,oe,,, BoulA ( cnua) e-lP-.,-) ( r"B)

.Sut>eL {*,Bi -c L ':'r !lL' o'*-A tF.e Le-.=Ea-6 b.

6. 16 B f .^= e'E e..r p-.edrrrt__ o"^A t )

aO t C d-'- e.:r "* T*J r)-"8-= Ala.t--,:e .5g.:*.ern

A L4lff.e,e .rL o\^^ a l.qel-' -ic -.SV+ter,r

( r-, * , O) \!ith Lu:s t''"'-'1 f :r a!;o-n5

* -/"f, @ er!^. L rr'#.t-ch-' an'- Lett' (rf

Cs*,r*'-uta,Eiwe o'.^d L'a) 4 ,,*ot-r;4!irr e- tt-'A(5 t,tBd1 lFe cr-Erc ts..rpb: czn Lavt= .

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ot^- Di'1e..!

Le! Lt, * ,

Latttc-e. -The A&et- a,i"c-

J}\^ wkic!' tE e v;^ ^^xt:t1. L >< g art e S,rel-" fFd, 3a- *'1 (o , , tr)o^,.4 (c".1oj\ ;" L-9

(o,,\-,). (oo,b) -- (a, + a., tr ^ti)

(*t 'L) + fo-o,b") = Co, O aa, t, \ ba)-i'- c-a.tteJ tFre A,i^ e.k P.:rela-u-cE .l Tte)a.*r<-E'o 1u, * , O) ^^^-A ( e "^ ,'v5

Eau**qroEontt-'A one - to .' one

ele."-^jE ef d- 3eED

YnD.rA,enal)-a'n

P=.o&roY

o) o'.4

6 oJl1't--}-7o."Jo..nc-e urttFr

-ib ' eo'lleA o'tv,

J \-anr;e-e-2.

L, , n, \) be L',o,5qeL-,er" Lf- X g,, , *)

g?eir anbr"= -r-. a.*J

EeL-cr"e Lall;ce He*so^o,*gk:"-

r-e-V fr-,*,@) .-l (s,.t,w) Ee L.,"e

Ldkjtiae>> - A -'fF-^X X:L-+s .:> c-artJ o-

Lqttjce H or^o-rno{ PhiDF^ F* iH- La-tt':c-e (t,*,O)

t- Cs, ^, v) n i- o^X o'bel2 C-x Q - XL"-)^ Xtq d.,^A XL.e b) --tCo-)vXtu)

A l6r,lir9ee

3s- alt'

[?-w\tr) ^ z

(t-, 't , v)z, \, z e

fr'ntJ-".].o"

-,b caltJL 2 z- bzVXuat"-n-)

:> zv [1a>)

lFe

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4 ecrruiJe, Co+ , a) .-t A

ft.o'.b!"- uY'eXe:t n, Dn

Eai!,:*e Jivri.o''o eR n-

J&-bit,U.tr ebLd;^ fk".\

Paat"a&rr-rr oct Je'' el =ebP-"# P"',b:J g."Jeur

'la6

I

.A(5:

Ae{rldev! '.

5\ =tr'bq = i. r,

D,r ,. sq =

{srr, 4) ,t>

P-e'e Ifidts Evql\la e drrla'' ttru h-Bb

6e\r"rbbn r

A Lo.!oie= r-1

r,.rh-cr.e-ves a- 1 cw Cb^ c) = fa..v E)

a 1c- tkea d-

d- w Lb^ c-)

> - L\,1> ,Z-t,q) (a,r),aa,?)>,<u.,,)Zt,:)(\,">J

(..i qr'a)

1r r'i)c'. )

Lr,a)(r,\

(Pq , e) , whs'' e Tt-Jergv.r= ttie ==t "E

dlfor-J L iF lFeHa>.se )t'1-T % *'

L = :D4 ). & rrar' dezr tFe

t,,n

,, l*ja, g\

I z- r ,r1a 'z

o\i=,LvibulfweConw e:r:=e111

La-tte ce

r'4eAlrl-orl

Lo,rrr.*Ld +o te

!.o' - tt<eiebd-

-iPtHeru c-e

\lre/\C

ve cg) z' ( av g) = (o_w r),^. a

L'^'o)

= fa-v

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Na,-rt e-d,w

Latt^ic--e .i>P:ne,e tkdr

l.4sAd.altF- rrsrr- <In-Evitulfv e

3r- Dna^r= tke H q+-e>< j€ tE- set %4Lnd the .rrel_d-f!'erh

lL.,h'.71 Q,A 2

&,l"Atr--c,n :

h+'rttr,eL .ii- ,awclu

Jco-g-to,.^^Z- ) t^,ha,rq

\e A ^ 6* ar..,:J..:5)]

% (x.dL.Leo,,=tFd.E j, T .*-s

=B A = {o, =\ e.x = t o, B,t, ta, a4, 3t3,auzt lho.E ,. 4 t .-ib'lh. .:re]-akiew L Gb

JilrA-= u r.r'.Ab-q.:lre.r! e>,t ele-r', e.r.L.tSa\lhsr. r

-TfteA_e )r,

ttie .l-ea:--+ €-t4rYter...t -,. JB*'' A

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Slto,v th a,b tFo .x-att i-e 'i-> n-eb Jr>txitu-tl-we

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vcl,!\e I u, * tr) o-The LalI.:c--e .i.E hlet

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EbLoi- tHe

t6a Hdli"e

Pa:r\iaIAi=q^...''i

B.:, J e-'\ no1 a-1- e-err-t-J \

C",'),tE,t),Ib,c)LE,a) tr,e)t d, t)J

gE\uJEei^ I\.dt

a = | [o,^), (u,u), Cc,.)C3,d),(o, c)(o,r) C',dL''t)

, tt,, 1) Cc,a)(c,J[4,.)

=:> tF- Lclltic'e T cli-s'.:'t: T =* a-

o^l.-D,'^" "&Jgo' ?

&ll!:*: t

No , \it- &vj:e;r:= -L ?' ib <r'- cAaZ'-

" -.A !+e-nr:-e- -iE JF rL et Dc-et. v)'c-t..'-*=AI

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geline 6- Jl:ot."-,tt'!:"' La-tt'"'e-e aJ Pae'e tgoj;b - -l-.,.1"ftu$;'e I6JIr-ee .

*-':1 c-na^'vt JE c- CAv ntts zotl

Se\lrlr'ea^: A l-a]fiic-e Lt,*, €) i,- c-a.\leL a,

d:'rL"'trl}'ve )atrE-e ,L 3* t^t o-,L,c e1q-+ CbeO = t-+)-)ea@ Cb,r.) = Co_o t),+

a>tlcC= c-

fa+c\(o o.)

9vet1

P-ro'eFverrq Cltq;^ ;.b a- D-t -.{-rAt u-b.--e

4

Le}- ( r- , z1 te a- e-l-roi^LeE arb, c eL

- -lcovrri de.,, l}le E>ll

owtnq Fo^,si ble can>e>>

t.-i: q 1L Gt, o- L c- -" JLn) a->L .-J o- 2 c

\^'/e sh-a.\l :.touo -sI" o,.u |E dr tF. Jir-bi,t.rlz.,= o"-."'

ox(E6O = fa*t)o(a+c-)c--a-re [t o- 4E gr 4 4 c thea \re l'"-.,e a4. ! - 6,-

c\O o- = c\ , a.+ c - a- c^r*d

s- o. + (b6 c-) = a _-tu)a^\J Co.+D O (ar c) = o.6 ct =ct

- (o_l

(t ) r- 1a; rroe. X-ta.jF Cb@ c) = lox$ 6 Co+ c_)

En c,o,:-e [i)

^2c !6r* toe h*we a-+t -- LL@c+o.

eh"i .i'. a- &t"'lorrlr'"e {^!tZe.

JTL

l-9D

-1

^-J

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so tlh aic d. r. f b€)c.) = be) c _@ c("LJ

C**b)@(a+c)-- u@c -L+)6:..,'., la; "",.J t \) \,r5 e ae}-

ar. (b@c) = g--xt)O [ar" c)

Q.o,er"rgr'- B

ffie

s={rthln e

1,,R=

e , - ,- lo\ o'*I a-

I t,.,1) l--r- 1 = r']% tfi e- :" e14'b-'o''

ote[o]::s^r'r R,

u.:halz or\s-'

R?

Gf .ir- o'v- =*o^,t p\. o- srdotr"'on rd"i-h iJ.tetf., .aXr^m<Fic- --.J onE sqt r"rteE r c- '

Se\^r-b;en :

Lel R1 Le 6r,,'-e\He'\ .:.el.qll--en errr ,,a

i'e- Ra 9 a\ A ?ve.!-L tX c.=tLr,'), L-i,\),t3,i(",,j), ta, <1\

Fe-:. Ra {e te *{rnrrtevu- e

\Heo ta, -) 5\.,'EnA te\e..1C,,O e ea o',-i f-a " | .-\:-.- te\e.2> t- R2

!.;-frr,ri=e ( a ,a) -,..J L=, -) te\e.1 {F R5 . -TRe, e.i,a h-e e\tre, Fa-i, L'., 1) uniti - o'T'd X A;yXeae+t '

Her.!c,e R2 -i-b -a-Y.r^ej6! c ' E\r-u t* O- {-5 t:

a,^t.:- ,aq",....=liie + t-,1) o^^^d tX,*-) I R.c,

lh en ac-:'-\ ' \-{ele [t,=> "^ a ta 'c) e ea t-ub e

o., A 3 a,1e- &=\,:..& e-\e'.''e-''E eB A ' Henco- R'

-n- a,*tf - agt'.wlelx; c- -fE'^-re? Re nD bettr

.5q"."'.e-hiic- ^J tuE*q"'"'etEc-'

, ,t C-,{) e e*+s Ra. l+ere

P-'epeaE'--'

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D

9a + t-\4

Cs's,:=S- "B F4fi> C - , X) S'rclu

= lo o^"-A >c, , \A e g ie ., 1E con.t>B(r,o) C*,s) ta,r) L'.,,O (s,s) (L.r)a) Cl , \ . .rle p.,r o;.e-ati:e-- are a--

{oltotr.r>

.:r erl e:<ive a-> it Aee-- t sL c-entoi^a6

hhd!

\i) R, -iD

Ao.-.

trf )

-B Pa}E

tr' 5 fe,Aeu

stb

neL c€"s.;I fr,D (-r.z\ --' (to, to)nie

R*-

tEe-

R,

-i,= i:r:'.eX\e;<i'ze , €iirlc'e -B e

o = ,- '- R, Ae e: tueE c-grLbd'':'r

I"ai'r= Lr' r) la.,i-)' - lro, rO

, B.'. -t'" -d

{E

\jii) js .sHrnr,'ekEc-

RrL .1,,.-) a-\>6 'ib

eR ; (r,a) te,a)

Lv)

ti!) R io? r'eE ^'".Vl -sq"''tt., e-E"; c-3." R" Csh-ta}n,=,

e\e.^ e1\= L-' Y) "".Af-e:,r Eioco.\,v!y\ e L.'o)

R.ia rret {-'oro'ru:i\ive 5

Cq.,D ee t:c Ltrl Ao.-,

, d.l- 8..l Pd; L :. ,1t) irt

"n'\ R i-e., lrrg) -.*d 1q,i;e R, o-J 30 erL '

( u - z-\ 1r-^itts 7- F uJLq,D eR t.* l+?

.}o.r ( r,e) e *, .-ln-eE bef. ?^- to R

3a. :>etll-1-e a-

lE-' s,el,a-F!'.o' " .te^r.

denerEe, \tte s,' elotr,-e^^

z Ai..j,Je-- S Be\t\He Se! ir, a,3,b\aeB -"^d Fii.A L nDD or\e- Ee tE-

t'. o,r,.t "t\[.,et

>te),a.-Fu-.'lg'' Le,E L detusrEe

fE o.,yu s- e?-r-a.l to', o, ,,-J S

" ->iwi J e-=,' .o!l-qr,-e Lly r^edrr

L a^d 'D d:re de4r^sf, s.4L

. \rrsr"r*-e L a^J =, o.t)

. Skolro dl-- ttro-E L o.,*Jsef,t-o-x\e o.rwE.tq.,^,r,teEvic -,. d

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So\rlden .

LeV a ' !rr2,3,b\= Jrr.D (r,r)(r,a)

T-( a.,u1 , (+,O La,\)

lz.rr\ a I tr>h e'' eY eti 2- i'=

tot:r)' $C,, D 1r, a) fr,3)Lr',u)

' 1+ , u) tb,b)\ si'. Q-

De- d'fxcAe-' t1

Lr,b) la,e\ fatt,b)\ sl'"*-

I'err= tfio'a'r e>'t e1'r.f

L, , "\ Cr't) (a,s)C-. tI g D tlh'o'r'e-"e:1

-TYren ,n

-rF,en LftD = {t-'', D t t,a) Ltr+)Cr'u)La'a)L''t)ts,R) (a,hD th'b)5

$\nce eL\\ B* Pd;6: f r '' r) (-a' a) LB '=) [-s 'L)be\en X 'rco i ,D , LnD ' o'\t \tn'ee X tFern

t.e., L,> o.,\e =reE)e,'i" a '

-rEene .i,2 Y\-E Pdi' L*'9) 'i' L \r.l.,:tts "+ to'"^A \lttli La , -) i^ L , E "- '*''*1oLe -,

.

( r, c) e u o^^A I + 4 b tr-t Lo ,r) Ae'-- v'-et

be\en-X Lo L F"l en c-e L "&5 a-^-tt - SqF \rneErlc '

Sitnr.\a-r\ D i':, d': a"lt -- '59rnrn et:ric

Lr D o\:A e t"o'""-'JE - z le"^ i L-- 'y) t'*d

tt .,r\ te\e.. q +€ ^*X % f6ea'e tttve-e

Ae! \k en (-'.-, z) a.!"e tr e\e .as !- tRe

,6d.^,\^e- .seg Fetr gx a^'*p\ e ( r 'c) e L t" J

La,u)e! al+o Lt,b)al

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Lel, A t- d- AeX nBln'a."r'r s''eLolc o e,^/\' A

.l,j.^ F" q- b<ro\ d- c-€bf'-

DPetr.re:r D;o- Haa-t' toq

4;' .vre'\e>ttv e ? 99t 'r'eltt e ?

'DJ3,ronz.,'\i.re "

Booft-: ' 1-s1 F' ter o-

<ltr-c,H fk d-t C c( , lE) -ED

ine,f\e -r''d C-srutar;u--\- . T-rL q er'.e7al , .ib.- -tAruE Iiq nr'.^. eL'Yt c. ?

Ss\.lrlet. .

%,o -i'z -i- t.e BeLatoen *''-A *3'ce,+tt F^orre a.^^A cen-t-al't'> tt'-: ?t1- it<arL

Booh \. R\ 3> n-et ;re{\exiv 5 ' gi..c-e C o ' o)

;'; "." u-tt''1 lro e'DB., ''-X b':o\ a' ' a->

; eon"Yo,.'*o- Ln--' ?'X.' T: bc'ofr cr-

j.+'aeAJ R' t > ^;

h-oL '6!\'6r^ etYi c ' sin-ce-

.-\1*. b L o.b) e. Rr Lb' o-) Ao- !!e't b etcrru2 tcRr

f seE- r'd, ,\ oTt l*"*kj:"T"^-"oT ttqen b*\ E c-a'lv'nev

o,*d .l-i-fte-r^rr-- e , iL bctoB a" cc"tt.oi.8' L*po.q.: rJ *^,". u ' u '3q u c-auaneL I-''"" 1 te* -"::

;.,- a) R' b azwt:- '+1'*r-ehi". d'':>

u*i. Lo ,b) -,: L\" ' a) a' :i" - l'1=[: .

reletl Yo Fr trJ :1..:? i"l; j: f -,.ts-' 1- t1".:-- ; c*rLboi*)

$ien L"e\ d c-e

-P errjevt Po4e'2 lE a\'v\ E ^'*A Eee&- t Ce-s\=

3."r. ;A'' Cendo'in:, B* faae= Yr cr"w

-,.A c- -so \t dE t*t a c*'v "rr-o"re o' A

Ce..to''i^a> tn*- F a- !r o-' 3 a"a"-A '3-s

C^,.) e Rr

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{o. 6Vre\}, \ts a-b 'i^,'

6o-r-D Co'* O3e\rrlfsr. :

rev (_e,)- ; . ;.o-.,A o., t, c e

L. H. g

e{':"eSe\rr.lie r. '.

_rrae

Relq.trb""r R

B-\eorr-

db r-tc*'...x

o-\qeEto-

TF,e f,ettou:rnla- Frt..tIX

r.uhe-ltsen -iE

-r/> !tre l{o+t>e- J-fr-r* .Eogtde-,'ed s.t.

".-nH 1)

.cl> a- L4l,Ecc-e

8-

Ss\ub)en :

DOe'c n-€E

a-(D b aee:, h-eE.neX-ie--ehE

e><bE

a- naJ*&-a, t..-re

o\qeEr-e-

t ) be B",o\eo-n

B

= Lo-*t) 1a.r+ ) = (a+t) or* 1o"r-t)e

= cr-cul r- bJ 1 ac- 1 be

= 6.{- qt to -r d-c- )-Lc-

= a-e-f at 51 bc = R' H 'S

Pa')r:all4 .J---J

9e}. !!!\ts, .rlts

.i- aa\teJ Bser-Lop )

Pd:&:d}. grJez

lurtti o- Pd:'}E J g"td"'ti"q R'

orn i.e-,r J set e.r o- pe=etA 3e*. P de,qetts-e--t

.,; aaltJ cr- f"br.X{

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4a. =u fu+eL h oo q- A.e--a-tL e)L,,,.enL ,,t-\h5-,- Fneve .,-v :D

:--t

ss\rrFr-:er., f cu AlPl 2-o\3)

Itl- e-E t Lt-, !) ' at r Qr on-e

e\-r e.ut .rin L-T'F.en ar l:._ aa 2 ea 4 o-1

Bg o.n-\i.*,rl r- r' e,b-r.-\ a: = et ao R.-o.ot e\e'..e.n-t

tt e)<.E t= -ib

4\ 'theBou6urr-n:[

8a-1trlc-e 4..,-,-Af, (E'(b ta)*E"

SeL,lIEen.' :

br kF) b3 ge-nc--e- LE, O tri t, = t,.L, =to

l+s G',.. a^/\ F'c-o-' p\e- % Lt^'e - etre""' e"E

Ec,c,.Le-a.^ Alqel-nt-'Se\rr.tsign :

(g,x>.,s>) luhs,e B =fo,rJ

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\b. glcole

kogearl r

faeyentle> $ La:e;e-e--.

4) te o- J.altjc-e

4 ,b,, c g L r^'a !"awe

a .^A a-6 o-=of -Aet^peVe't- Lt"t]

Lr,4

.f-..E.". .BLalt ice

l;rog

LeE

Fg.:r

a-+

6}.eu:rneeL

ane

hsP*rvrt4

hoFEr<r) Z '- g}rsrP \tr ajr \terr.ee-E *,,-J ls,ln erl o-

koperry \ ALae-ryt-'o^ \. ^^)

"\^^J4o LD-* b)

tk db ttreaore jd^ err

ai,rjgcfdjrve

tT':'t"* tB. o- La-ttr]ce

arl a - CB^..^rrtJrd5a,U:ve L-r,'/J' ' l'-Q ':

a+b - tre o- ,.-i oOE - 1>O o'

e.tLaOt):o-= o-.

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t-

Paer - g'=-=::===:

6Yr-en: trrdE tFe *"e* f:,rodrrc! ? ^^f +-u, o

Aiob;b.r$'"- Lahr"ee> 'i:- o- &iobctulct'e Lalr':c-e

fn u W' i"'4 (" 611'4 -o'fQg\uf,ien : -

' o.nA -v d@ Atd oJoe"t stfu1; cemmrltolive.,

4r.r,sci.o.Jiwe -J obae:^ybb. !o"jt' '(' L * \"1

Let (rc, ,\,> , (*,,.t.),(-., t{a) e L " l'n

1il q:<,r\r) + (.-rr.{.) = (zrrl'cc,H)Ott;)(*.v=,[email protected],)

= 1*a 'to) r. (:-,,\r)

5i-,'lail1 , (,-,,8,) . (>.r,\.) = (-r, Ur)' (*'',tf')

o.' A - 4^ e cs-rnrr,bajtfze '

\,) (-,,1,,) *[(-,,{.)* (-a,1">J= (-,.,"1 ,) -t (-'Y''e"1 a.vtlt)

= (.,cr \ (t, w"ci, \\,t'(S"'uX)= ( t*,v,(-.) w ,.a , ['a rv.t r) tt"))= lnrvz-2, grw\r-> + {>-},Xr>= f<,., :y,).r- 1*-'1.1*(*a'U)

5t'' i\.rttr .

{-','r'': fr:;:-i:,1"i"1 :a,Y:)

"' + o.,. A - q'\ e a:6-9galh-""= '

;

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A() Z-zr',Y,) r- f {,., . rtr,)

.(:. r , ttr,) r-'1u'(.*' 'r

,1">]z:r\t

{-,'v (ac1 Arcf rX., V 1.1,1.\a))^ \")

$r'^1,.\.t

1*,,. -t qnA.

H ehc-e

{- 2,, U,) ( at,'e'pb:o.

,[')-f(-',y,)-- (*,,"{.{ , (>.' r$,)

gd.h;Pisr atae"pbb'.sl.\LL><M,.,r-)rD

t aL B be- a l'cn:)ce Booleon a-tqet-ra- ''r.dq 6D a \l a},a'nn e[ B rneru

I€* A te tlie ge! % -:pasve l.'nal rte B-J; ^f€'ti ^- ,

* 'ED

-ir-..'"'.e-r phJ c +D lFre Botolea-n' d5et'l.. 'PCA)'

;; t ; a) i,- ttse Po"'eo '5et' eB '4 '

*!*:a' tAu Mls zovz-J.-

'.-t B t- .-X C.o"t ,

E,.o\to"'' d'Icetn '-\!e q-l,!\\ u€ce -i,.Arr &s"' en \ E \

\!e o.>b!^,\.,lre th ab 1t e. :ne:orrl'b i'= Lv \re-

u.:h"sn \e\ ry\

c loi", : TF e .vr e->rA,t; .,i.. l-y.r-e f o.r I el = n-

Le' B Be IEe Bqolean ^& e! i'- utr.;tF lBl =Ft

Le-t ra-' be ."{ elern-eruE ef, ta 'auuh ltsare L a- L\

goolea^ dJ4eLn* fo, a-] & Ba-

f o., o'fcJeulu..,-

)"o.r.^J

a. !a$i, c-e

62.

LeE 12 t

t- tF.

;

te Tt-'e

t,crDlea^^

B 2- Bt xB-

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rPB' -> le, lzlef "r,

^aSi' r\."-r*1 , t {.^ :Y \e"\ z \et\ = rr

- €f ;r.Jrt&st' o*s u'"^Fbi en , tkere e><"i>--E' o-

{,r^aLe +eU x 'J 1 s"r'& tHqL

Br tt (et-),U, o) ^-f,Bc :r t? L\) , !, n)

N. li.T Boo.f,eal' a.I-q-laro-

t e t*ul), u, o) =, e L.),u,o)- (ptr),u,o)bn 2. >< u ).

-TE-on

( e t=),u, n), (p[*), u, n ) - [et1),u,n)crB! >< Ba 1l B

" B .r (e(=),r-:, l.r) .3o" 4 suitat\e g^;Ee- *eEz

Nors , tY-re .srno,\\ crcL A"t"'t='t % e\-"'en]= ';"

^\ ^o1o.'*o *niil .z e\-""e."B c-orLt,ai"' eYJt o& t

=t z= St.q\'Len SeJt

1?t=>,u,O :: th' P'J-^' o-\qt'o i::;e]erYrerub

''' Bq SJt"otien Hqye\h e-=,': ^'. T" -'e'-'rl't i>

tr.-- T," --'). Ps-l=an A rX"l"'-

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Dsr d$r the- l-i a>r'e- cltd-q''rou ^' -T- dt^f

the p*^E:J o.'.Ae.t''X [Ca , e) '- A e e\ en

)He Po.,-r e,. .ae}; Pas) u.rhezr e s = | a-,t, c-!

F-"c'.J the rno.x.crn-a.L , tnfn'i'.al , 4st eaL*a-,) A-.-,,t>L ele,Y"enL. nE fu- 7e-*' '

+e\^&rer.. , fA I N19 uo\;JS= f o,L,.\

PCc) = t iq, 1"1, i.3, fa,b\' t.,J\,1b,"1,1",b,"\,s\

1",d5

t&

tet">-,.; ,^l-"- s,= to,L,"3l-leve

t r) !e---,1- el=..,..L

L'zl G:'e-o!er=\- e\e""ent,

(3) sao-xi.na& e\e"r1g6g

(\> \4\r;.nd\ elsr^ ent

to' e)-"'.er1b , F

=tr)e\er^enb-S

- s=t''u'.-3=+

- l',u,*

\a,c\

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I', o. Buo\oo'" Alqelro'

tgv cL\ arUeBD fAU6e\.l en :

o Co+tr) = aa +ot= ar- o-t= a-1r*t)=O-

t..o'e\e-..ent

Lei. \D ants\-Lt.n e,

Cs.n p\qtt-.en! eP, .Llb

, P.",cve tk<.L a(ol-!) =o-

Ni, .",4

Jr_t it.rk j.,e La_ttic-e {,_, v, a \ "X ..nq" el a c-gwr3\ern eiL tE en_ s.t "-i:

fnu N/, zot{

=. a...A H cr,n e t.^:o

f '.' a.o- = e]f s "t-t'i ttlt;we Lur-')

f *+r --r]

Ce.^p\e'neruE et a , rrre. ho'..-IDd.V -- =l -.......- ()

c ern p\ er.Le-rLL op q- , *a |ro.r=lF.a.'., t =1. -- tjz )

f '-' "" vo = -)

1 A (->cv 1) Lr,r-t .'Dt vt G4)

$i.'.e'x. .i-q ie =

\21n.c-e _1 !r->

o ,\t =

r-\s.p , x- >

ao,

o,

io,

,e vo

)c:

\- Xvo = y\ LaA= [,2 va) a

:ev faarl) f tei,,2tr))L="'o') a (-tI) Lb.:L.'ituhr".)( o-v =.-) a 1- ".1 ) ( ce" ...rr-ta.Fr_:v e)

") L*'-,1 t.!(X"*1 [ e"t-t*,!ut""")

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)

F-l-e.>c

+

= fo-vl)^ Ll1v,,-)= \ A t,1 .r"')

XV,. - LB)

["] cr-J Le) 1^, e

, =-vL\ = \v?c =

--\

f ce- rr..^-Uaiiv e)

L.]-1-X La))

\.o-., .

trd

Cer';\ernerlt

g, &''.f*t11 tE- Eoo\-o" Ex;-,e'+qsn

/ at. tt . c- + ..-' tl- c -r o'- tl- et tt-otr.cl Bc"oteon

oJ-A&aa- JAelllib-- f. C ,l ^

ll z" t {4e\^rJdo.' '

Q'r"e+ exPsrer>tla '-r:

o!.b)- L -t a-. bl .e + o.l-bt-"1

I (a.r =a)

( c+"'= r )( e-r =c)

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a- 5* Pf s) i:a lFe

aJ u, n dne

PJ\e."- \Fa-E LrCs> ,

Ps...,- ttre VreArrj'oYr

(t, +),vir t"(b -r e) 4 f avb)ac

.:(e\,-,lr:. t - ..

Leb A t-

Lo(o), q)Tezabco,t

l-lereet,l.b

.=e!oc-- jeio- Xattf-e

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rt ee-t\rru.€rYr,

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-inte;'.raecbjonHe:' e , t6 ' e-tet"et ir- +

t l' e\err'=.tL .ir.- a = f o

eat)- Fd.;n % e\e'en'b& l/^12 exi>d

( e C") , 9) :o o' la-tr-i'e

,r-" "\|.o.o belts

(- u, n , v) s:a d- La-ttrze , FFen ttt c^":a

c gL

Lc() o-v(tr,.c) z (a,t),rc

u)

vroA.u\o" \elua}i\q :

b

P-al ;

Ar+n r-" c-t- L t

o'-.v C =C

.l,;tr';tuLve

Brr.=..Lo\en

.{.'^-Ee eeic

a- Lallic-e'jt-

^6r1!.i^;1

c! Se.E g

.-J r-eet ,- A\>o ,

d ,lalrtcea-+ c- gi aV

-l:n-eaual[U'i

B,.t-t, -rrt€au.bk/ : 1tse- ho-t.

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a. vCb^ c) L (o.wt") r[avc)

( u-t^1 Lr)qec -> a"V (b^4 3 (o'vb),t6 _(_a>

e7v ea>-eli1 , a>,t \),r',-e-.

q\(b,r.) a (a.ru) na-N ort , Bq dXr.otc--

T L\rB ..^ l qls , *. io-r,a' 4 a- v Lb^c) z (ar L) ^. 1c

_\-/ d4 c-

a.v (u,rc). e (a-vt).a c :) a. 1 cF.,,crm Ca) o,,A Lg) ure !.o., e

a. Lc (:)o- v(o".) 4 (avU)^.

p-teve- lF.o); Bene:rrqanr-'a Lc'-ru \refJ q'J T"', \iz

(ow\,)r= Jatt o-J fo-at)'- o''vbl

\-",L:e,. f nu M/J gpt3 t ft: * lo z-",1, L\J'\r.\,n l- f' -\ Au ,.r l_D zoro]Clai,- - J ' Iaab)r= a'.rgr

L-

-to p:.cre tEe "Joe."el:t" , rt i.o e-ne u-q F to

Pc..owe tlao-t

t. loat),4fa'rt')=o _62' [..au)y(*,vb,\=rl. (o.,rt),r (a'y5r)

t

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=) (o.aJz,u) vf o.,rta1"')

=) q:vo-)o

I a,l t) o ( o.., L,) =o

a. (- .a r-) w 1a\i b')-) (e"f.'vU)) a (U., [ov u)+ Lo.\@\tDnl^rvtvu,)

(a,v,e,r;a.}:c"e. 1?"f.)

(aaJ = o)

[oa ". =s)

U)

(>:"*-ritut-'ve E'u\)

I Aa.,ocab'e nrtu-)

(_awo"'-- t )

( a.vr = r)

- La)

Clortn - A- I I o,vt)' = ,,ta tr

=t .i:, en-cr,.-Xh +-.o P;r, owe.

t- (-tU) ^ [o, at.) =. .a [-vL)vL''at') =r

I. (avu)^ (o,at')) (o,rIo'at')) v (Ur.(a,',.t'))

tli 4b

D'G$..Jbu-E-veRUL)

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:) Lo' r.o',rgr1 v tbA t a d)

_\ OVO =o.: (o..,r) a (dau) =o

( a^>e.rat-"e nu.r")

. ( '.' ". "d =")( ..' e ao =o)

( >iobli Lulr-'"e' **)

( e,*oc!ah.\e R"t")

['.'ovJ=O(

=Je.^pote^E t*)

- L3)

a. (onl) v (""'1 b,)((a- v b) v o,\) a (( av E) v sr)

(av",rvt) a [cvtvt,)(rvu) a[avr)

+

?P "x/1ett".-r',1

(o'RvtJn,t) =r

Frq.., (B) c^"^J t+)( arr u)' = o-,,'t,

.-L\)

B*\=on -d fr1=t-a , P'o'- \tr 't lt{e

a\obe-*e-n!o o^e e1"f--d*^L

G) *'b - cl- Le) o'\r- l' = I

f nu rsro z"r!0-) a-. br - o

Lrr a+B=E

.5ol.,r:r-'..en

Lr)

Le}-

I h.e-n

-) rfi.l

q1E

o-aL =o- ^^J a...r t -- t - cr)

a ^

tr . (( * ,^. t) ,l tr ) (u,=t^1 t'))

= (.rat,rb\) L-. ata,.r;ah:"-

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Herce

C: babr=o)

t', c^ao=o)o, z\ tt = bcr +t --y

Qrr :> 6ri;Le!- o..a, t\ = o'rafte comp\em enL sn- tsth -51da , r^re fi.re

f a- at') t = (o)'q\vb =l4^tr=^ -\

(iii) :) \-,)Let atvb =l

_: (a',rt') v [r-ar;=',

t ,'ffH:;r'

(oJ,.,tr)vo = E, (.. t,nU=q)

. !l6 =o-

a\vt =) *> L, I dC!) =) ti>

LeE L'el-tE en ota E =t'

rcr\e ce..^p\ -.,. er*- ea/r tstf SiJe-' .

( ". a U)' ' Lu)'

g! q-z- t

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c-crnv erv'e L"n 1 :r'*c!-l H*n/M a-o tf

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( t, ^, v) te in- g1'Lze-n &-t"i t,.rLv' l-olt"-'e

D, '. av Lb^O = (a-.lE) a 1o.vc) Y.o\J-- qJ -u)v- o.. t, c €Lu"- X o-4<, ,Iken a.vc.=g _ L.)

Lr) + o-vfbac) = la.,b)a(a.vc;-- (o.\tr)aq 1 "rr.,.X t.)

*L ^ 4 c tE e-n a"v ( bz. c) = gav b)a c

Ever. -1 Jx'"t'itutruwe L",-It-,:c-e 'ip MedllolBJ..b, c6h!er\r=e- -iD rLEE -I;n tLe

i'. , Eve,rq taeArr'\ ovr L a-tri <-e n-eef, rr-eL be-

Ji:.,$"lbun:-eF€v Ex a-.y'l e , \vs

1'1sAr-i o.7r t.,--t it

Pr.or- ftrdtFfgA.rLa-r,. =a ttieel.a^r"^ f n r',

J-t>drl tulll're J'a'H'tc-- '-ro

lot.,.",.en-A ) La-tl--,.- .Lj,. r.Eli Jio!^zitlrw*"e

D.nanr, 'lF-,- t--l q'>*e

I +, =,o, ra, a,.,]36 Lr)cr.nrJ 4, iD c! .:.e\ o-k.3-t

>"11 fnu A/11 203st,lr)"r:s"'" '

P: = 1',g".ch

,r]

D_tt-

, rc]4t "tt

gs. a)ArB.r{,t

fF ar aa

(t) -iFe Relah:on

C Z.z,lo) (e,rc>1a,aq)(e,au) (3,b><+,ra) |I (a,eLr) <R,3t>(b,\?>(r,,ar) (u,au>

[

[- <r?,4q> 1 ra. at) J

A;,o-q.>-.,o,"

R.

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(.r rt) (r, ra)(a,u..,4o,t)!re) (a,L) 1=,r:)<""ttJ

TFle AbTq t* C*' R) G

Re\&'ell R i-

< r,a> < \,3> (t,\)<a,

--ite

R= 5

I

Poer- ko.j,&,"-i=e,lr=

%a- B--[e--"^Sut dlq-t''a,-

D rr o , ht e seL T q-l I hlsiti.,ea pe>it&e .i 'ute-qe-r llo, -,t

dlq -1". .- ^,^A t,-J dt .il6

fo, n/a z-orf

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qSe\rrlfsn :]-:D,o = l,r.,E, \o, tr,r.,==,t,d

Jtr"c-e , D ,aa-h!{.c -'- 'setextve 2 an,t;+qn* e-h-i'

a^J -ltr *itt,=" P=nep -'.h:=--' ' D L> c'-

Palbai e.Ae-, Retoi.'en en D11o

f o"." , D) .iD cr. Po-seL

He;.e o- ,1 t = ?-a {o, t}-* to,tb \* o-,b e,D lro

\r) .i'- 4€x-bt

't' CDtto ,,1,-LE ,5 H d^r,e

Hene , _[-ru=t eler^enE [o ere.^ent)

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e.a.cY" ev=\ J-. enL

dio2>.a-.-

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F.:te.n

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]L C-e"^p\ernenteA .-LqI\!l)&-o

\He Hcrr-.5e Jtogrro_Fn ,, i-t. i& A;>.L,-,t,,rIi...e

iL iD gtvGs\r>l -ttr- -

Csrro, o) ia q- Boo\eo.. Atqst ro-.-The- Elr-t bo.ol-a^ eJq-t-:r.e,t ase

L') {", R = lrrrro}('j) f rre, s, ro,1r, aa, Fs,lroJL3) 1..,drr,,r ro\ ,r- {eS

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Sts.o tko-t

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tr'1 .n +- lffi- h?dtsrl 'e inteq.rn '

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r* >c e r.l

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rtB

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:> t - - X) -t (-'1 *z) -iz a- DerL -we i rutoae,l

-) =--> 'j'o d- rr-eh -ve- itteXe'r

*-Rt ,- t1R= :> 2c-R>1

R "1, Lxoaeitt-e-" R &, Po,rEi qL e:rJe-r R eJqb'-crt

h6n - Va

R ib el=[lexi\,eccrn>i Je-r

- RX & %R >c,

<Sirrc.e a R1 :) >c- - r1 .6- rl.,rL _ve inleq e>r _C.r;

tRz- :> t- E i.b n€ry, -ve \n\_*qe:r

-> - 1=-- g) S- rLEr! -ve 'nt eqer

-> =--y f r'o.. e) &(e)),g, -b Aif,i,:qrn- gbyic

Ar+ss"e *-Rt 8- tR>.z. R1 -> a<--r1 .i:o Y!grr

YL€YI

- 1-zr

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St" 4 La.lfrf--

t>cvt)^Sebhjen .-

C-lairn - I :

f u , a) P-e.,e- ff,qJ; "v C Y A >\ L

L-- v =)f au NID 2o\o]

LuR r'rt -.iD

iE .i: etvi€ub \ttd-

"fF:

,"., Ll n-) I [,c vI) -'r f rvz)F.rr ernr, ttre dqrr-.r,tie-*etv;]srr> tts o-U'

>c- L )cY t -

tr\}ail \n>1y42c-v.A

Fsre.n q1 o..A L.a) , :crX .i:o cal.,! \rfpq t.*.4 1{*-,ts^=3

\1er. gg acv vtr ) .t- w fXaz)Fv's..' \He. Aq.*h:en eB LuB ,

x- L '<-'v z- -

t?)o-'lJ 1u\= LZ1*-^vz

F.r el^ (a; -"J tO ) x!z- 3'>

te.,"J "L f -, .1 a z.\

e)

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ty^=) -_ G))

L.a,.r e- - -Ct1zt>) -i.-, d,J.e*.z te.,^.J "E f ="X , -.r=!

-' . x- \'/ tt" =) . L-ry) a lxwz-)llenx-a- the V--J

Ia

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IE

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t" €y, .sq€

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6e\-,r\f en :

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6,n" _ t--\Fro \\or>e Jic'?=.,

He-t.e ,o, e\ e-...,er.L

t t ) ele..'e-.'-t

r-c-r,r I r, 4\G* lr,qa\ =

I v \1\ 4a

: t1 L

=4r, 4 4 il - {-

F)' = zr

\a,\

5- \eCe"^y\ e.nerrt 'gt ,B

i'e., qr)1

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e.,rt\ -b 2- zv\\ - It+

(-); I = rr,.

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6ih c-e , evei'\X ele-."e-r"E e'B E42 I'oe C-e..rp\e'-e-nt

C ur.. , =l ib a- 4ernflerne",t-A Lalh,'ee- '

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Lr) uelo (u, ,A,Y) Ve' qL"! e\L Lo-t"'c-e ' Fe-r '-^i1'

a,t, c gL, trc- !"o-'e '

V + c =) r) o.-,.1o 4 o'','.c

a) <x"tb L o.Y c

P"rte$ :Gil.,e.r bL c:. c,rr-e!b, c-3 = b^ c- : b ---u)

Lua I b,c) -* iovc- = c' --La)c.-[ou'i.na - r ',

6,,t8 2 a-as -Ca)-r:o P",o... L ?) , iL i: e.rs1rqh lso P..---

cnLB loz,,t, o..r.\ = a- ti. e -rb p-,owe-

@"U)aLc,.a) =a^t

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( a^ae ct,l.u-e- 1-'-)( ce.,-t wEa.V--.,e Lo,',,)( *.r.se cco-h:.re t-.r^-)

( =d.e.-g

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( r,rs,tna u)

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!r.re f ovt, o.r c\

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f ..v \.)r Lo- .) -- o-v e

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! o,w c --Llt)'rE 1- eto.tah

= o-v C

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C co"^.n r-boh:.,. I*.4

f a,,ro ocah". Lor.,.)

(_ rAe-mpoEertt*')

(.r^'r-2L$)

Page 229: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

(a) ue\, f r- , a , y) b' o- Xi'veJ' Laltri c-e " E-.a , t, c gr- tF.e.

Be\t6ur-r.-? -"Xy^tqti) a.vlbac.) a (o"vb)a [ovc.)$l o, a [u.ra >.- (o aD v 6a,a c)P-:teef, :

cLa.>,. - t :

lua c) a

Je-6rnili-'srt

d-4 o-vE _ (J)a",r-J \> a.c 4..L 4 ouv b

Fore^-r \rt c^,^d L4) , 4lt j.- o\^^-

{'.,tael!le.r. c-r.- crvt ! .- v(_L,^.)

-u\)Fr.e. tfi- d e1i.s-bbn T_ Lus, iL6bo,- L o.-* _

"fF"" tE^^"^J T

a-'"d t^" 4- c- 1 o-.vc-

Fs.o"- L3) o.-"1 Lr+) , a v c J..-o o*n wfp.n to...d"t 1"., b ,^.cfH e-",-c.e o-v c_ > o_ v ( br.c) _ (B]l

"r-XhelJa

l-srsrt'i Ihe

(o.rb) a fa.w c-)

% LUB - rt- J-b etrisu:, tk&

-i> eb.viou>- \i cdg

Fsre."^ 1n) -,.^Aa- ,Re.o.-, torr,-d

o,w Iua")f-lence tF.

a

( B) , \A)e hav e a-v I ba c,) i5"B f ovb,4vcl

3 fa.vu) ngovc)o"%

% tr)

Page 230: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

C \D^,'^ - 9- '.

a_ ^ (b^ c)

w.k . -ro" ;: o.ab

a.n.J Bv c- > t

Fs.e- L\) o.-"d

] ['',b vc\He-nc-e-, q- a b

l,^) .}9 .T

a > o. ^. c-

anl Lvc >:, c_\r bvc >

F-s.., LA) *J1q,bYc5

) o- Ac

a-Ac .- (t r(\), cra6 J> au/r

Uel-qa c\ ,1 c =:

q. AForern Cc) qnd LD)a^.! .^?p: te,.,^A

o^ C bv") >

) (o.a t) v (aa c-)

) a-a B

a- aE -

(")G) , a^b -rb a.^/! .!.o.rre,', berr-J

4 o.a fbvc) _Lc)

ce)

-!-e.,"e-, to.,-.J eg

( b.'/ c-) -- Lb)t re' !,o-., e o_ a f U v c) .;.o

f "'.tt, "'z..\(o-at) v6o.ac)

ct)

Heh ca fie p-:roor

Let ( t. ^, ,) u3

tv f ba a-)

twIoab)

s[ tri )

^-.: q,trc €,L fii.He\ A,>

L3) Cen --.. Ae, ,b

xt-=n

( C-ev'.,,vt.rta!.:.,e

LqJfuc-e. Fe.r

fmau-)n\

ta-)L*)

eeUB\^,1?

( ab+e-.' pLJun

Page 231: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

=bvfoac),(_t"v.-)a fuvc)= Lo"'u) a ( uw c)= (o"")z.fuve)= (cvo)n (cvr)- cwLcat)

= cvlctac)= il, (ca""),c

t=e

( Gl.,,="t Ce.J"rtie-')

Lo, - ce*'!bo( c e"*r"u&o-\i..e L*)

C !r-'.a qileil.Ce.'Arlie"1)

C ce.-^tt r"jg.t'l*.'r=-RuL,-\( *H -, - ."fuo*)

f <a1.,..' Ce .lJ.t!e,..,)

C ce*w.,',Lai.,,lr. L"^,( ate,s.pL,is1a Lc,.,*)

Ske\, thoi, n-La,tf,,'c-e

& 9L e> o--x tr=s€)o,'@u=,+>N/J got3 / Ao N /,

b' 1ol .

2-o t{

_ C,)

C ,t,*"a .,D

"Z..,sectali:-e La.r,,)

=oo .,, . -_ .=)

a- c e-^yle."^ erd;eJ --J diog"-,!.,1;.,=

seLubbn, fn ,r1) :> d,'tLet q +LrFLc-r! o- aE -- a- cr,r",J q-v t = tNe..': a_ zr1,) = ((*,rt)a b,)

= (aata!') r-L= q_,,1, o ( -.. u at ,

=o (.l{ err-ca" a_ aL =) q,1 tl = o

di) -> Cii{)

Le-Y; a_ AD.=9

Page 232: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

fat(e cerng\e.,r-r' er& €rL

( o,nbr)'= toJ'

tet'tt ,s'rAe=>, rr:e

f e'11 ca.a^c- eltolCevr Lo]-)( s.a+"ibut le r

".r..)C ...r^B'=o)

,side>.

!to^r e-

a.w E = r

aabr=o :) dwb =rtia) => L!)

I-eEat.rb=1

=> (.,nur) v (u,..U) = g

":aa_'vL =t :l L, e d

Li.r) -) Li)Leb rb 4= o-

-IEsn al n Ll : U-r-oSte

c.eF^d e+^ erlE o1y.' t e thI

(o,,,^,tr),= tU),=> o,w t - t

s! a- LV

lot4o'->a-LV

Page 233: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

t-ev ( u, a")

Xxb La.t)ffien 6\.ovsCe.,'',$r,tok6l e,

f Aewyef e.,rtt

Se\.r-^r^.bn,

!e o La,\tic-e

",*A * @t -

\tso"L te\ia x, drise qLd}iw e

La.rJ-\r-r .

tA, $/r 20\3 /

€L,

Paoeenl r : L=Ae,c-esuentt

LeE (-r-, , v) Le f-Lven.\ cr,t,c e L,c)- \t o- =* oJ a4

Psroof :t

a_\ e = LuBLq,o) :o,\/ o" z q_

Nsrr:, q,l 4 = Gnue [cr, o")

d.Aa, =o-PnopsRr) a..

C

Let (LrA,re)oY o,t, c

a, vt -_

).o-^r)

L alLi-e

Cer^r.nut:qhjwe

b e- Xrr.r"CL

t v o- o,"..J

P.:LR:a-vt = Lua(a,t) = Lue:(b,O = tva_

L .,. L\rB (r,U) ;o "^gqo-.,t -- F,v a.

o.At = Gur: Ia,U)

o. AL> = b^ cL

L

, i1 .r.,e defr"e c *.Cut Lo, , b) 1er. - ,t

^^,_J o *{t, atoe-ptre-- o-J

Ai-r N/D 2o\o]

- -rFo,l , Bssr

Lue(.o) = o-

= Gr ue (o) = o,

L^ 'Jr )

L o:t-'ce, . Tt\s-'r- { o',6

cd,^b - L^ a.

64LB Cb,(l) = l> zt q

" 6te Co,u) ;- tr-^".1d

N sroc

Page 234: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

ve o\ivlA o" a(ba) = (oaBac

hogee=1 a-. ( a,^aec,cah:--t-t I L,,\, \) !e qr_ven

*11 o,t, L e L

a, vC bv- = lo-wu)

f-arrr)

Lol["-c-e . -r6-o-n, Be,

Fa.$ :

LeE o.vfbv.)_J _- tr)o.^,"4 . r -Lo,wro) vc =e _ (*l

Fqe.* . -. .

*.rr, d .i,2 LuB

% (o,tvc)/ dZo._ q,\^A J>tvc _ Ca)rD krr bvc ji; L,

. bvc 2 t o^^J tvc )c- _ L+)F-;'rer,- q.a; -,-J L rt) ,.o e ho"v-

a > o,_.l .*. I- _!-,>

Io"d \ tE)dtc J

F:re'v'n [s), J i.r.s zr.,r,r ,* % 1a,l) "r,..J d>cd> 4vt -_.4 d>c

-\i,

Een^a 1e) e i-, LuB % t.vt,e] _Lr)Fvrewr (_t) o.,""d L.+)

d>e.Eit^r\.-\ , r,^r- co.* ea,,'\ p.,..,,,,. e > d

" A='u-ii\hq Crf .^"".A ta) , *- }t-^', q

Page 235: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

a v(\cv c) = f e,t)vcPaece-li- ,6\r..n\ dr\ , \r.re c6.^/r, ea€,\ P-'ewe

a. ^ Lbv") = (o.^ t) ^c

Pnot errl \ -. I atas-gt"b" !*)Le\, t L , zr, I ) loe qir,e,r.t I ":rc-=

. -rFo.rr tt" t"a

q,t,c eLorv f o- 71b) = o_ o^,'4 o- a. La!b) =*

R-.f ,

5irr.c.c-

A- ,\E z- a-

ct r-_

i: ilr:r_ qrB T [.,Ub_ L,)

a- _e)Itole-

.........-- 12\LJ ,,/

LuB, .,..,e V'.l-ve

_ Lrr)

(- r*) r^. e !'o*, e

LeY R t- o-\ ,^.oSrre R' =

P-ete tT-.dE

(a,u,) :--

Re\e_kje.,., grL q- geb A . TILeyL

IL^,u> e A><4 /tb,a) eR].*T, [a ,t) ;.. Pg--PL lFie-r'

alr-- o- Pt=.i'

etv.,eu-\

F.:re.,' f 1 .rJ La) !^)e-

a- rcaat) z oBq Jtr e d e1i".,-br-r" etbD

a- 4- o. v f oal)

% ce.-Li..i^a La) o.^.do,.., lo-a U) = -

ai' ,\-"^\ \rr e: c-ll,w Poove.r a la"v D = o .

f n, 6/n 20to;1)

Page 236: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

SetuLe n

G1'wew A i,,

R = tC'-,+ ":,.s-A

C.La-i-r :

R-'=iLtr,O} -;-Eivr-ee f o.,o-) eR"

.PiYite- S& '

t.' g o-v:o}" o..Je-, .gre\ aliot

=) c-r= tC.,q\ :- a- .=r{oare-Cliven R .:o Anl^!.5jr,-,,-,g!yi.

o- pa,,,\i-'a\ grA=v' '-rre\oL2rn

i'.., Co,t) enSr.'.. Lo , fr) g

[u,.) e

Co't) e

--.4 L u, o) e

R => C B, o.) e

R-r x Lb,o') e

R

-!R-lR.

X-' j.', A^Ll...t.. r.*t:i.-tCfivext e, j- d.o--jrht=

t.e-., (o.,u) e x, .,.J 1u,O e r<

Sinc- fo.,b) eR -) Cb,a) e e-l

-^f, tb") €R -) C",u) e

Sinte (-c,\.) e R:' S- C\', o)

='7 o- =L C.,,--.j.r.n*^D

:> (o,c) ec --(Bl

f ':) tc,o) e d)

['r -''.1t")

p:-l

e.

R i'- -f5 a"n:r;\wv e-

Eitr'co- R-\ .i'- .:rcrg\e>a v e 7 a;siltq rn... dv.rtr. "^){'<a.,*l;rh!q , R-l j.,= f-:rte a\ By' Ae-,r -l elq-ber,r-

-. (rl , R-\) i,- 4 ps>=Jr

Page 237: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

f-= lke.

Sg\l.r-t"'e,t' ..

6 }. e,rs tF qt

SeJE !'4'>

Set"uIO.. '.

LaJoice

nlM 2orlha& (=*, / ) o

f*uleL o- o'.I t .^l a Pe:;rtlae inEeqe.l

-Ih-e-n Lue la, U\ : Lcr-r [o,U\o"nJ c.1r-e lo,ub = q--lo,tbS\eurA ex;,=t Jiv, z+

B-r e-x o,v*fl e , Let q- -\ ,t - aorn-on Luala,t\ -_ )""^l+,:-]=1

Qua { e,u\ - XcA 1+ ,ro} - 4H e.nc-o- bel6 G te ,.^^ J Lug ex.,bt

T6-a- Pe=eE lz+, /) ; - LafiEc-e

e.\e'u-,i &-ir\tbe Pa,rt*:al- e-'Je-- ^l

a- -''l=t*h ,*-J rnl ro.^J el-n-,- ed-

fauAlM2oro]ld f r- , A , v) te- a, qi-weh La-\f,'jc-e

Ei-c-- L .i- a. !a-Wu'c-e tgth (r ua o''"J Lr''' a

LeJ, '.o" be qLB %

U o^d k b I' be LtrB

-]E-t:n, t"- .r*tt x- e L we l'o"e-

a- 4x-1L

F.:re. I r)erue[o, *j = o. >c- -- a'

tue lo,""j = a.v," -- t-

srLb

Gla I >< ,\,\ = 'cat -- x-L(rBl rc,t3 = .'c'tb-Lf,irriEe L,>fr.€c,e- j->'11 6.*"Jd

Page 238: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

+-

\.tCc-e

S\eu: triajt

Aj.r-t''.tt tc. ef Ru m

L)N\\'vER.5l-rJ Quea-rror\3

\!r.e- Ji"..l PaeAu"E 9E -U tue!-alr,ice:-- .:.o o- Ji'-!'itrrSS"e Loltr:c'e-

Let g te F:r-';Jce- e-\'o" o):\el$q- "'-J LeL

A Ee Yr.e ge)c: eB al! a9sYn'- s! B' -rE-eru

?:'leve \Eog tEe- c^!'Ft'!q- B l:D

i*o..e'r1V.rc +e, tne BoA--.crr. P[a) ,\:\ts:re P Ca) i:D t\i.e Por>e:n

- .b'e)s

% A '

-D:^o.,}5 the Hdr#e A-ra-4,lcu.n 're1ae---eru\ tFE'

pa:rt; oL e..Ae-i.yLt tLA, B) " a eBb ew trie Por"el

Ee.E PCg) to\s:re *= 1.o,t,th FinA tlte,r.-r.o-=f-.^a\ , rnrrrirrto\ > J-r e-a"be'* ^'"tJ Xeo':t

e\e',e.-k:o "L *. p-i'-u itf-9='! tr'p""'-}d

ts 2-o\2/ ryr-=:r'l fra.o" aEJ

,]

Ai,otrLt^\i., e. I oS.,:-e \ r- , v, A\ it6

@- ea \- cs^^y\e.ne"L \Hen \L. .i.: unr.T]-e-

fn, N tD 'zo\zJ f P_0 rj-1]s.*,t}".xy, tne g-J-or. exPse-sr.;'\ e,r\

dlXd-".'* ,A er*;be-- L n u

:lr-

Jn o\-

e\e,.-ewV

o- B<ro\eo'-n

o-\\ *, b €t" B f Ao ,v/p-oal.q-t=a , P ,9

L-.l

e \ko-V o f"-+l)= a,

f P.6 tto, z-

-.t. t\ . e 1 *.b\ , c *- o-\. bl - .) \-r.:\ rr-q Boo\ eozt

rrr /, zorz-l tPgli--3tl

5t PCS) .iD tFte. Pc'r-:e-r

.^i u.n o':r\e \-'$=n

fslova rs. d-E I t C.) , e)

'id^a- 5e! 6.o*'d .n..e.eL,

Page 239: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

Fsreve \t{& laed..r\o:r t."1dt"l e! a- J.altr:c-e-

L c- .*ae'_vLu ' '-)Iurc-)

viz Be,-y

(* a-v b) zt

\F o"t lw, L> '].'2

u:!.ori,e $\ .].:i>

-,.^A L

o-.r2 a,L,c-c f6unlP 20

Pgre)rre \Ha.-L "Der^eu\Xo'n\a L,-r-,i- h€\ALa-S*-eCe-eXc-"'e-"''-\=A Ai:l-rrtub:we

I

f o-r t,)\ : d,".\.' c^'-.A f o''r b)'f O" I:-::'3., '!!9-2e'1 )

5iu *l Bo..\eo.r' A \qe.t=ro. , ts,.en,e- Hrq-t tt{e

.Be\\e^.,:1nx 6\ra!e" "Jr'= 4ne el)'cv 'Jer't '

u; o.--r-\- - t C4) q-'b = o- ta) a.\u:r tbl c''or:of ry-Yq-*Zg,tJ fra n,, s+f

Pqe.,e YFo-t ev en{ &tot "t-"$;.r- Lattljce .i:MoArrla:t =a $1e cg h\rer\r-e- L'rrle I =**tr{u'.-- c-!o:-t" f tr nt,q_=:f'] f ?q.,Jl]

D=rr<r..\lr fie Hcr-'.=e

e,b, ra, aa\ L4)

L .1- o- .r-t e\crl!"Elrt

);o-4-r,u,^ t, Or R=1r,

"o= t rr a, ?, u, t, ra] -""J*o_ *rt Jf,t,.l

f6e AiMtu Ait'4 zo,a E" 'r'l EPino 36 IP,'sv* t+i oCL D r\o , -\Fe

"t % a'\l Fes"t,:ve-

&v.r:-o:v'- % a- Pto.t': ..:n-F:] t::, *

,*Bso\eov-. oS!.U,'.- ^,) TJ a'l\ i*-.- eul-oJ4el,-t.,,-, f A u A 1M zorr] f fO_:::_Z!

"-Wtg=.Je-,-J

g.-Lc)-"

S\s\^,geY >e\> 3\'ft,eqe=e: &"8i

t,Non - tl e.qal':ve inLee6;'t

t! rr-q-11r!

fld 1*- 'f u , zt rV),rr2

yi-3Jr"l frP-t-g!

a-\\ Peo:rWve-

lar1 rn 4 tu

tIft rL -tsf c u rv rp zo to] L m_::_:ll

Page 240: 0 JAfl 'tt^ tt-,- ,^rr;; *;%;,tf - Fmcetfmcet.in/CSE/MA2265_uw.pdf · p a 1D P,/e tpva)r7P 7 T F r f r F r T r r r F r r r r C-"oq,b-,.ct. .eAL /r-t/-t" * y LPr@,)v7P. Co^4r-uc{,

lo

t\' ar' a ! a-l6i:c€ 4 ) P-,'..,.- tti..t ,- w (_5 a>) +

t * tt ) 'a' (-:c'v6 ,.r N /) zotoJ [rqr:J"]

=t S\, G \he get crl\ ai.v j.:Ee:,\,-) q l!a.b

ot"A p' Jo': $te -zre\o'bbn " JiwlEE:l %"

Lr,

t\9

tb-

et!

S+r P:ne',e \1- d3o t s*" ' -b 'tY: c' C-o-f.. e-\l}

LoC*,2-e . f nu Nlp zorof tr€-1:JJ

lg.

S\,ers \Fal Ji^^ o-

a" I db-tv\t'.r}ejveN\J..> g\-'-6\'r I"' &

Cal',^"crdv,fi.:et\ )-o'D

6\evs ft q-:]l .^/\^. cr-

A;.r,+v't\rjtrve Lot\rlc-e

L€rL LL , 4)

i,* Lvv-e' EArr r.rlP ?:IoJfrql"'la

Ce. p\etlel'EJf nr., * l1 =:)3, t2-?!-!-"1L?o::.a!

a7:t1-y o- 1Er= o "->

;@ 6 -ra='., t ad

Le q- ldfit;c-e , it '..'e- Jgn.= a-r<t =

*J a-@b. -iub(a:S [u-.o,t.t,tfi*L teYa + ^"4 6 *Lj"tt

, drr,e ci-ohJv e , o.t.s s,',yU"e-^ q-'d

La*p [nu ,'rt:_Z:_'3., NtD_:::4 teOryJxJ

$r, c^,u)\Hen 6\-,sr.s

Cot-'.,rr.\- r\:i''=

iAet-yete'E

t-pL R \-e a- Re\olrE". e'v' t'- aeL 4 - -YF-on

^

d$.\o-

5.i*ts) e a*AlL\E,,9 o"\ "-- jt-,.(o'R) .i:. t-

?e=c.j- tF..r-n L4 , R--') *?^,Tffrf" l'r#":r,, r'e pebtrr l=n , \ I^;d,'rT.r.r_-oS\,e.,: IF. o-E e,\, €, q

B\ hiEe Po:'rEJ I er.JQ^=J

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