Suond Cad hangues

homsgngs Recnene Reabon

ho Corside a sulaon
n>9

Ohe c , Co- and Cn-g au Sua! Constnts hG+o.

Aonelabeo g Hha hpe öcalled a Suord ofde inea homogensuy
usno aelahon oH Constan G-ffiint.
Cnk?
Thus an Ckn o Boluhon He guacbati
equahon (s ).
Thás guaabahe gahion
choatoske ezuahon fot Hhe orelabien ())
Case I The hoo noot; kl and ko egation (8) coe Saa
and esnct. Tan 9e
voe fake,
a, Ak,"+8k
Lohene A and & ae Condlanty as tha geno!
Hhe sulotion ().
Case 2 The
ond uoJ,9h k as He ommon Vale
Then oe take
= (A+Bn)k"-)
Sual Constante
ohene A and B c9e obilbany
Soluben
cguabions( )ou Conpee
KI and ks
(ase 3: Tke too sobs ohen, s Hhat
Then kl and ka each
Compler Conjegata
KË=pt i hen ky =P-ig and oe take
an=((A Cosne +BSinne )
 oho A and B ae anbibay Complex Contne
He gnea
he guelatoo()

hten
1, Solve the necusor ene 9,given Hhat a =,
a,-6an- t9an-O fot n
a, =l
The choateshes equator fo the given elahion
-6k +9 =0 (k-3)°= o
Ashese noots k, = k = 3,
(A+ 6o)3h
Thogot
an
A9he A ond 8 ae
5- A and
Cond'tons a, -5 and a,=13 hCgaton, 9e ge 8-
Sofrg Hese Hoe ge A=5 cnd 8-)
1a = 3(A+ 6)
puttig an -(5-n)3
This he boluhor the given selakon, Uhe ha givn
Conen
he necueno nulabion
ebli
) Solvirg an = (n- an-), fot n>

Hhal ansl and a,-9

Gafren tguabon
sakon, he cheatoihie
&olubioo : Fa He gieo
k-k+9=0

K= (a+-&)=ti

Therjele
 L9h A and B ce Consant!
r= lHil=Ja, aun tano =l,-T

Ustng He given fntial Condehond a, -| aned a,- 9 1oe gel

l=A nd
= A+B

A=1,B =l puttrg Hess voles Aand B

an
4
eabon Undo he gven
Condehens
,ay= 4 ane a, - 37 Sahsty he Gecoenle
3) 4 ao= 0, a,=l nzo
ulaton an+t ban+ t(an O aelbo
Conskants b aned c anl huo bolve he
Determine Hhe

and nl, the given slbhion,

ag tba, t Ca,=0 and a,t ba, +ca, =o
cund
9e 4 +btozD and 37 +4b+C=o
=] b-, and c=-) Suioene relaß
and c, Hhe given uuorenle
L9th hede Vales of b
ala,=o Bof no
ant9

n
The cheoatohe
Lshee rcots ard kg=-3
 an= Ax 7n+8K (-3)h
Aand &oou anlibncoy Constanb.
Hhe gen Condlbos a,=0 ,a=l ina his 9e
0- At B ,l=7A -3B
=) A=-B =L
,anL[-c 3)
Q) An age a Mongo and an erge coe lo be
a bonno a

fon aoys 8, Bs Bs , Bu, The boys B, &B, do

hot sh to bave appe , Hh boy B, clos pol ant banana 6)
boy eBargl
dEbibubon Can be macle no
boy
Bi B Ba Bu
A
B
M

rly,x)= |+3x
l,x) = Cu, x),9 (Co, X) (G3,x)
= (tSx) l+ax) CI+x)

= ((+4*rh*) (Hx)
(It 4 x + x x + h x x )

OKT, Rask polynamal ü given by

Compooig
 Forbiden
by Hene n= H

S, = (H-)! xr, = 3<x5= 30

So = (u- a)! xr, = S<x8=16
S, = (4-3)!xr, = 1!x4 =H
6', N =94 -30 +I6-H -/67

P,B.e e. koho auve lab fon o alionen

J) Foun þosons
fine that pive kables
one chai al each g
pany
an

P. oill hot sit ot Te oe Ts Find Pha numben y way they

he Vacant chan
Can cccupy
beloo ,upd enhra thu
&ol": Constdo
The &hade
the bocmd shonon
fockat Hhat pepsusoionl., T&T
tbles Ti6T

Tica,)= lt31+ 'r(c9,x) = tHX+3x

 Nos, by Proet fetnua,
río,x)
(I+3x+x) (l+4X+3x²)

= |+4x+ 16 X4 3 x343xt
Raok poynomfal a

N= -S+S-&+s
So =5) =l9o
('n=s) `, = (S-)! X,=,X7= )68
S = (s-a)! xr =3] xI6= 94

N= t90 -168 +96-96 +3 -5

t The number y eys ohich Hhe fou þersons Can

the chain

3) Eve teaches TT,T,TS, T,,Tss ane bh be mace cast

teaches fer Bive cassu G aksCu Cs, Gne teache fot caek
dass cdo not osh to belome cass eahes
cund
teaches beury asind the sotk
hos many boys Can fheHe tahos
L9fHhet dipleasig.. teatas
s
FP
 beane c has

r(ci,x) = lthX+2x?,r(a,X) =)+?x+lo+9x

. rCGx) = Y(o,x) Y(gx)
= (It4+3)(+x+10°ax)
= l+4* +lox°+9x+x+ 92 x HoxxH+ 9x

= l+ilx 4ox+ s6x³4 92xt+ y5

8, = (6-1), =5X)= 139o

S, = n) =6!=790
d, = (6-9)! = H! XHo96O
Sz =( 6-3)! = 3 XS6 33

+46o-336 +56-4
=6
N=790-13 90
Colaas
4) Agi'al shueat has

. Foidays
Dcps Can she
 Mocdule- 5
Tnboroeechen to
Bincey opena bon:
en-empty
L! G be a non A unchon ti6x6
Set, A G
©penaforn a

Anon-empy Bet eguiped

bpenahord bcalleJ algebraa sbuuhot
The agbraut staho Conkisbirg g aset G and
binay openahon G a diokd by la,4,-)
Quasi GYep 1 Greupotd : operakons
A nen
-emply Sel cuipped oith chigo binoay
Calle! e)

Examplesi (N,t))(z,+) , (a,) ,(2,-) o a

A
Quasi in

axkeakve Called Seni gop

all
(i) (P(N) ),(P(N),0).(P(N),a) all aoe bernl greup.
(ai) (a),(z,) n nol semi group.
Asociasve birsy penbion
A binay opoahon caleod atoahve
,C EG at (b*c)= (atb)c.
 Cenpnutahve Bircoy poakoni
* be a bincoy opeakon
taibcG
Cenmubabve and orny anb =b*
i.e cach ament G Commutes 9h each othen

"Idohty Elemeot a bincoy opeahon

Let 6 be a hen-emply Set and be xc6.
ecG buch thot *e = e* =X
Gn G.TBen the ement
Then
nespet to t
(e *) b datl o be menel i!
Monsidi Abem? cPoup
has Jaroent

& monoi iH
I, (N, ) o.
menoi 9tth tdlnkt, alemnt O.
2. (2,+) öa
rmonod oith tantt elemat ,
3, (R,) ö

Lnvense Elerment:
sel G an e
Let be a bircoy
biray opoahon
opeatian The
be he tdenbty elamunt in G fe biray
lement a'es &baid b be an nveoke g atS ÷ a*a ra'ta e

A menoid n cheh eah ement has frvorse

Greup
hen
epby sel and x io aUnd
oporabien on G
(funchon Bom GXx6) , Hen G Galled a
perabon * Gondekons hdd:
akbeG fot all a,bcG Chat &
) (a*b) * c =qk (bkc) fot all a,b,c EG, (That io,* b asloahve).
(3) Thou
(Heoe e i called an alla6G,,
(4) Fea ewey aEG , the ç an eG Such that
=aka = e, (Hene at Caly un inVese

ALelan neup
gioup in lhich biran opeahon io Cornmulohe
(ot)
Ageup G bod o be Coromutatve (a) abelian a*b=bta

a baid to lbe a finle gpreup q ohls n

AGroup (G,*) 19e olBe D(G)=n,
Then
fotte seB, b calle

The nunber olament ina fntte pronp

Hhe

Eranples fot Groups

) Peve Hhat (zt)
fngintte abelin gop
sol Th Syslm <z,+), oh Z---. -3,-2,-,0, , 43
and bircoy Gomyposihen in2, kao intg
a, (:
() dosuou Axion; y .béz =) atbE2 S
: )
i) Asseichve laus: atlbtc)= (atb)tc fa,b cEZ,
G)Exstne eist olament oeZ,

(iv) Exsteno Tovede :

4 aez, Hhe exaat an nent -a¬zGuch hat
at La)=0-(-a) +a
Thus <z,t} b
(v) Gornmutahve lau:
t a,bez , atb =bta
abelan
vi) Sine the numbe
inginie sel.
Heno lz, +) infinite abekan yop.
 ) Poase Hhat es,x)
are the cube ools Pale obelan
The &yshm (s, x), oher
8-i, 3, osh ,w,
and Hus | cnd 'x' He binaauy openabion o 8,
i) Close Aion

and ox = les,

LX (Ox) = lo)= Ix=1

an
1hus
Assoakve Laus bolels

Tlankh,
Undn idanbly cund leS
Idohty edmen!
(iv) Exstene nvese
|xl=|=la]

Sino wxw=lho.
?nvense
SinG
', he ?nVerde

3
ament g 8 esals
a

(V) Gommubakve auo

 No (. eah-w )
each)
an
Commutalve lavs bdds
Thus s,x)
x} abelan
Sine
Hena g, *) Aotte obean group
fintte
(s, x), dhey 8 bo Se
J) Prove Hhat <8,
i.e $- fl, -l,, -ij
Sln: The Aohero g-f1, -1, i,-i}
and xx'& the binay operahen on .
1i= icS
|x-)=-1 es, (-)xi- -ics
|X(-?):-i¬S

) X-)=ieS,

Josed
(i)Asoiahve Las
X -1xi)= lx-i-i,
(-)Xi=,
(!x (-) ))xi =

Simf la cny
Asoahe Lauo holeh.
(ii) Exsbne
Cunl le S .
elument i.e
 Sinco |XI =1 =| in Vese 1.
()x -) = | =(-)X-) -l

Hhe nvoe
lnent
Hena s, x>

3) 9a Hhe bet humbens R

a, beR-i} by aeb =ats -ab.
biray opercken * foi all
a

Sol:() doa Asorn e R-fi

t a b e k - t i ,thn anb= atb-ab

ü) Asoale Lao.
L ab,c eR-li
y Then (arb)*C= (at b-ab)* c
at b-ab +c- (atb-ab)
atb-abtc-a-btabc

a+b+c - a b - b c - a c t a b r

Cnd tas (bec) = at (btc-be)

= q+(btc-bc)albtc-bc)
atbtc-be-ab-actabr

atbtc-ab -be~ac+abr

(axbc= alb*c)

such tat
3 an elnent 0e R-
 sheo 0¬ R-ti}.
(iw) Eteno Iede i aeR-i 3 +a
Q-)

Such Hat
a-l
-aey)-o
=at

Q-1 a-l a) =o,

Thus a-l a

a-)
ohee ER- Ca)

Ihus

(v) Commutel've Lau:

bta-ba=ba
Qxb =ath-ab =
Gomnutahve Lousbol

Thus R-i is an abeln

 heoten
lat Ca be a frste gpop and Hbe a Subgroup g G.
Then o(H) dtes ol6).
t & be a pnite g
bor

Let a,4 =H aH ,.... . auH be Hhe diskinct byt Cosets

H
Then G= a,H UagHu,.. UaH
olG) = ZlaH|
aieG

But hoo dgt cosets HPn G have bame humbe

elements
Sine Hto obo a det olets Hin G ,it gollad Hd
each yt Coset q Hin G has oCH) numben amene
&o, ol6) = otH)
=k ou)

note! ConVede