Marius Burtea Georgeta BurteaSimona Anton, Corneliu M[nescu Awam, Cristina Ligia Dan, Daniela Diaconu,Ion Dogaru, Carmen Georgescu, Diana Mihalcea, Elena Necula, Gabriel Necula,Elena Popescu, Cristian Radu, Cornel R6dulescu, Daniela Soare, Gabriela Stroe

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SI)

Cl tSrt aXll-a

MATEMATICAProblemeTeste

exercitii)

semestrul II gruPurir inele qi corpurir primitive

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lJndnr0

r . ( t " Z) :ezelncl€c es gS (q', € 'toZ :ezelnclEc os PS (e

'W){'x Aa'. 'I,| = .,7g eeuniPur e4 'I

E lli = S erec uI |nz€c I{i : -r Bcrldrur' s >,('r A pc?pe-c h[- 5 eru{purqnsg .

= l.o_ '"2

iri

#R

1. LEGr DE COMPOZTTTE PE O MULTIME

5 1. 1. LEGI DE COMPOZITIE. PARTE STABIL{5

.....5

.....5

... I I

Brevisr teorelic

Fie M omullimenevidd.o O funclie rp: M x M -+ M se nume$te lege de compozi(ie (operafie algebricl) pe mullimea:)[o Elementul rp(x,y) se numeqte compusul lui x cu .y.

-4dunarea Si inmultrirea modulo n.

rle 17€1\ qt Ael-.r Numdrulnatural r carereptezintdrestulimpd(iriilui a la nsenoteazd amodn qisenumeqte restul modulo z al num5rului a.Pe mullimea Z se definesc operaliile algebrice:o @ :ZxZ -+2, a@b = (a+b)modr, numitd adunarea modulo r.o Q :ZxZ -+2, aOb=(a.b)modn,numitdinmul{irea modulor.

ldffiarea Si inmullirea claselor de resturi modulo nFie neN-. Pentru aeZ notAm 6={a+nklkeZ}. clasalui a modulo n.. Dacd r =arnodn, atunci d ={r+nklk eV,} =i.. Multime a Z n ={0, i, 2. , ;t} se nume$te mulfimea claselor de resturi modulo z.

Pe mullimea Z, se defnesc operatiile algebrice:. rr+rr :Z,xZ, -+ 2,,, i*i = i67, numiti adunarea claselor de resturi modulo z.. rr' rr :2, xZ,, ) Z

^, ; i, = ;;b, numita inmul{irea claselor de resturi modulo ,r.

o Q submullime S c. M se nume$te parte stabili alai M in raport cu operalia algebricddac6 V x, y e S, implica voy e S.

in cazul in care S = M se spune cd M este parte stabil5 in raport cu opera{ia "o".

[oI

Exerci|ii Si probleme rezolvute

l72223293639

'.. : : :.......:.: ::'.. i;52

................52urue. proprietd\i................ 52

. ..,, ..''....'..'..................'..:370788098

1. Pemullimea Y ={1,2,3,,: r,yeM.a) Sd se calculeze: 2"3, 3" 4,b) Sd se calculeze: (2"3)"+

se defineqte opera{ia algebrici "o" astfel: x.-y =l r-y | +1,

o4.

2,(3"4).4

qi

Grupuri

Irndnrc9er€ip^u1 op Iippun ed nircrexe rS eruelqor4 'e-IIX e vsylf - yf IIyl\ jlyl\',,o,, rerlBJedo elqEl pJsErrugclE es ES (q

'Z"t'E"A'f."7,:ezolnJleJ as !S (EIeJts€ plrul]op ' W ed erfrzoduroc ap ue8el rS {t 'Z'I 'O} = n otg 'f : - \n')Y gc lnuriqo e-5 (c

-_1 ro'\-f , o)-loz t) ['z l]

: ' u'l=(r)F'au (qrr; r)

tz 0) le 0) -ll-tr , -[o z) loz i]. i (.)r=(e)rr(z)r is

r n) ; (7)Y:Lue'tY (r:i 0l

a;in1og

'r)urur = { ox -(t

(r r) 'tt:[r t)

't= l+ tlz

_(r o') _1,r r')-[r t) [s ")

(zz r-) (s o) (vr r-) ( .tt-t-t-tIu rrJ-[z s,] [r s] [

y repeSy 'l=p'l=;r'I- cl \=o eprinlosnJ t=I+pZ+) 'Z=cZ 'p=1+q7+n

' (e z) (t- pz+t )z l'rr,on*apearerrleBasurlqoog elrrlenr. pttnzar !s

[t r,] =

[r + qz _ n | + DZ )(t-pz+t .,2 ) (t+p ; ) (p+l ;)[,-r.*, ttDZ):[' , 1,o)*l.q*, ,)=

(r o)(p ;) (r o) (p ;)'[; ;][, ,r=[; -t)'' :u'le''lv

I o):r u (qr r-']*[s r]*i i ,-l ll l_f r r-) /c r)r- , ) rt z) [r- r .] [r z.J- [r- , ]'[r ,)(t r)_f z zl,f r z'l_[o r,]-[o e] [o v)-.[l ?1.[ I :) il ?):It :),[i ?) '**. E

elfnlog

r o)I I )T

7 9c puttls tr' Eaclr€ru outuuolop os PS (q

) (E r) (r z) (r o) lrl- -l '1 - lrl" " l:ozelnclecespg (e, \.r z) \l- z) U t)

'(u)'zu ) g'Y A'g +Y +g.Y : g r Y

lud atse lf ?c el€re es 9S (c- ( /,) f' azolncl€c es qS (q

lz)r ezeytclec es 9S G(,)v:(q)r r (')v^\ I"r)=@'1= w a$' 'e

'1 3 d oI nes

-x) rc eu{qo ss sr4Pmq)Eqnzeg 'J>{'x e1g (q

ft-,t)(7-x) :rusr'Y (ee$np5

apse 1 gc elere os PS (q

)= f "r E, e1ere es PS (ea,'g+,t7-x7-tk=iox+'21=7 eeu4Ptuo4 E

atErado elqBl

= c: [ :UI;)A$

esnps

*(r z) - [,- z)

(v z) (' 'tt-l[r E) [r

(t r-u-v

:1e3:1se ',, 1,, pcr.rqeSle ederedo elSeugep ".

(U) z7g eeru{pu e4 .€

'U=I=lrerinlos nc ZI = g + x 7 efience Ellnzeg' g + x L= I - x. E+ Z. E + x. Z. Z = x T 7 pc euriqo eg (q

'tz-=r-zr-s+st- = r-(r) t+z.E+(v-).2.2=(v-)r z :s = r-sr-6-oe = r-(s-).e+*(s-)'s +(s-)'(e-)'z = (s-) r (s-) i71= 1+ E- e + v= I- I. r.+ z. t+ r' z.z =I r z :ue^v @

a;fn1og'Zt= x 1 7 eflence ellozor os eS (q

(r)r z Is (9-)r (s*) 'v17 :ezelnctec es ps (e'v= t'x ;

'I- {E+ xg+ tx7 = [ 1 x :leJ]se ,, 1 ,, etftzoduroc ep ee8el elSeugep es V eau:nllntu e4 'Z' r = r + 0 = t+ I z - z l= z. z = (r + r) " z = (t+ I v - t l) " z = (v " t). z

rS g= 1+ Z= blv-zl= y"7: y"(1+t) = r"(t+ lt-Zl)=v,(t.Z) :euriqo eg (q't = I+0 = t+ lV - ? l= ?'? IS Z = U I= I+ I t* El= t " E'Z = t+I = V I t-Zl= go7 :rue.ty (u

a;fn1og

< -:-=1-1 +l=0+1=l: - -l-1=3 Ei

=lr-r *3r+31 -tr,

_ -j,=r (-3) (-s)*: (-:)+i -. = -16+6-12-1:-23.-:i.: -r 5-12 cusolulia

.1s,.

a)

b)

Solufie

Avem: 2.3 = min(2,3) = 2, 0.3 = min(0, 3) = O, 3"2=min(3,2)=2.Tabla operaliei:

(6

\0r')

t)=

2) (tt e\tttt-tt8/ (-1 22)

5. Pe mullimea I = [2, + "o) se considerd opera{ia algebrica "o " definitl astfel:xa y = xy -2x-2y +6, V r, y e 1.a) Si se arate cd xo-r =(r-Z)(y*2)+2, V x, y e Lb) Sd se arate ci 1 eseteparte stabildinraportculegea "o".

Solufie

a) Avem: (r-Z)(t -2)+2 = xy -2x-2y + 4+2 = xy -2x -2y +6 - xa y.b) Fie x,ye1. Rezulticd x,ye [2,+*), adicd x>2,y>2 sau x-2>2,y-2>0. Prininmul{ire se obline ca (r-Z).(l -2)r_ 0, de unde rezulti cA (x-2)(l- 2)+2>2, deci xo y> 25311 ;g o -.1, e 1.

I (t 2o\ I6. Fie M =|.q\)=l -'' I aelRl siooeralia "I" pe M. definitaastfel:

. \o l/ )'d(a) L A(b)= a(a) a\b)+ e(a)+ A(b)-21,.

a) Sd se calculeze .t(2).,a(:) si t(z) L e(z).b) Sd se calculeze A(a) L A(b).c) Sd se arate cd M este parte stabili in raport cu operalia " l ".

Solufie

10)

t)(o 4) (r 6)

'ttz)=[o rJ'r(3)=[o t)

G) = A(2) A(z)+,a(z) +,t(z)

(1 4\(1 6\ (t\:t I.t t=t' [o r,] [o t) [o4\ (t 6\ (t 0)l+l l-2.1 |1) [o t) [o 1)

:- :;. -.-ri ecLraliile 2a +l = 3,

* = - .\radar

=min(

Rezultd cd A(2').A(3

(r 10) (t-21.: I l+l[0 t) [0

Rezurta cit A(a)-, l(b):[; 'l) [;26\ (z 2a+2h\ (z o\ (t 4aJ-[, , ,J-[o ,.J:[o

4b\ (t 212a+2bl\I

= [, I

'.J' Se observa ca

a) Avem:

;i A(2) r A

o=f t)(r t)x, !).

2b\l+t)

+ 4b)t.1)

(t 20) (z o) (t 20)Ltttlt[o : J-[u :.J- [o t )

b Fie r(,,)-[; 'i) r,r,:[; ':)rl 2a\ il 2b\ (l 0) (t 2a-

'l l+l l-21 1:l[o t) [o t) [ot) [o Ic) S-aoblinut cit A(a)L(b)=[;

4a+

0 0 I 2 3

0 0 0 0 0

1 0 I 1 1

2 0 I 2 2

J 0 I 2 J

Grupuri

lrndnrc

= f '.r oururratep es PS (P

'- Eer.urilnu gr riBlErV (craue:edo ulqu] IInIPIV (q

i-a),, €oz IiBIncleJ (e

1;p as sV eaur{pur e4 'S

,,,,((t)r) [e1nc1e3 (clrolact4eu

>urilnu gc rfe4suoueq (q

erienca .,4r u1 rie,l'lozag (e

iF- , = 7y eeurttlnur etg 't)

:wd else 1,tg gc fle1qtY (c:rte:edo elaqa 1irrugc1Y (q

ts r T (€ T Z) IjetnctuJ (u

_ 'l'01. = 7y eerurilmu e4 't

-r: rp lntuelsls tie,LlozeS (J

; I : l eFrlence riu,r.1oze6 (q:(r"s) r (s.z) (e

'6 = '{'r4

ul.f as g eeudlnur e4 'z

liE'r-trZJJ l- = , rulued (J

r; .-. \lLrzel I = D nJluod (q:'*:i -=p tleutrureleg (€iu!,:p as 7 eauriflnru e6 'I

?llsfirl

k= .:r ?i:nlos n. 8 = .rZ g]Inzal

..:s:r-1ns auriqo ag (l-.:

-- - - : = :' :li Llll.\l (q.

eruip,tul ep rip1run ed rrirc;exe rS eruelqo-r4 'B-IIX B VSyTI - yIIIV^1IJyI I

'{s'e},,eyrrinlosnc Q=gI+18-zr'uocnporpdnp'nus E-x+x=E+(E-r)(g-r) errcseseriencg (e

arinlog

g=1.({-r)l 'Z v! ' I lnuelsrs o^lozeJ es ES (c

, = (I +,,r )* xJ'z)x A't=o"x lgcuJIeJtrs? 'v>o euutrtelepesps (q

'x4x=ror €rlencs z ure|lozeJesgs (e'Z) i'xA'€+(€-rXE-") - [ ox

rS g*[+x = (*x ecuqeSle elrderedo csouuep as V eewJluste4 .6

t: Bp.*g = re:(, ,)= + ::0 = 8por.u8 = gpour(7 .V\=Z.t

:L -- qPeY!: qeYgg =: e (q

I = 8_e9-I : 3potu(o + r) = g* L:? = qqgurgl = spo,u(r r +o)= U *g

:0 = qqgu! = gpo-Grd= 5+g (ea;fn1og

'z ur !.2'(.t'S.g @''Z u! 6+

t'tJ+g 'g*g (e:ezslnclec os PS .g

'8+(8-) .0t= zL- ecel€oep's = 61po*(zr-) = atporu(s.o-) = (s-)ooiS+0t.€= 9E ecar€oep'S=0Ipourg€= g1po.u(t.S)= fOS (c

'0+r.S = 0Z ocoreoep .0 = SporuOT = 9po*(eZ+E-)= EZO(S-)iy+t.S- 6I ecereoop .t= Spoul6l= gporu(1+8)= IIOS (q

't+Z.L = 8I ecoJsoep '!i = tpotu 8Il€+t' g = 8 ecereoep't = gpour git+Z.E=L ecareoop'I-tporuL (e

ruwsv 'w > (tz+pz)v = (q)r t (r)v

alinlog.0r=u pc"p .(S-)O6 Is tOs (c'g:u exep'EZ@(E-) Is IIO8 (q

'lporu 8l 'Spour 8 'EporuL (u:ozolnclec es ps .L

',, 1,, erfe.redoncgoderuJgllq€ls eyede6e yg

lqz*oz\v =( , o

[tqz+DZ)z t )

: = i l; -]b)e M. Aqadar

-l_t

l-l I -8 )r 8.

.:,- - \.i--15 = 0 cu solufiile

b) Avem xoa=:+(r-3)(o-3)+3=3 sau (x-:)(a-3)=0, deci a'3=0 $i a=3.Ix+(v+l)-3=O lr*y=6c) Seoblinesuccesiv: ].'" -i. , - -=1"-' "- ,;iprinmetodareducerii[(r-y-3)(l -3)+3 = 5 [x-Y-J: -t

rezultd 2r=8 cusolu{ia x=4. Dinprimaecua{iesegdseqte cd 4+y:6, deci y=2.

#t Exercilii Si probleme propuse

1. Pe mullimea Z se definiste opera{ia algebrici xo y = x+ y + a, Yx,y eZ.a) Determinali a eZ pentru care 7 "3 =ll;b) Pentru a=7 rezolva{i ecuafia (x+1)'(x+11)=19;

c) Pentru a = -l rezolvali sistemul de ecualii {, .':' -lI(2jr)"(r+t)= to'

2. Pe mullimea lR se definesc operaliile algebrice xa y = x+ y *17 qi x L y = xy + x+ y,Vr,y e 1R.

a) (2.3)r(:"4);b) Rezolvali ecua{iile r.(3r+lt)=S ql (Zx) t(r+t) = 19'

I x"( v+l)= 0c) Rezolva{i sistemul de ecualii I )' .l'|.rI(y+1)=8e

3. Pemu(imea M={0,1,2,3} sedefineqtelegeadecompozifiexLy=-in(,,y),Yx,yeM'a) CalculaJi (zr:)ll ei 2r(3rt);b) Alcltuiti tabela operatiei " I " ;c) Ardtali cd M este parte stabili in raport cu legea "I "'

4. Fie murlimea , = {rr"l

= [f i)''. .](t l\ (t 3)

a) Rezolvali in az ecuatia X.[, ,J=[o ,),b) DemonskaJi cd mu[imea M esle parte stabil[ a mu[imii M r(A) in raport cu inmul{ireamatricelor;

c) calcula(i (r(t))"'0.

5. Pe mullimea Z, se definegte operalia algebricd xo ! = *+)y +1,Yx,y eZr'

a) calculaJi 2.r ei (r*i)"(3*a);b) Alcltui{i tabla oPeraliei " o ";c) ArIta{i ci mullimea Z, este parte stabill in raport cu legea " o ";

d) Sd se determine x,y eZ, penfrt"ur" ,o y =i.r ._-:

Grupuri

IrndnJcOIoreip^ur ep dplrun ed rrircrexe 16 eurelqor6 .B-IIX B VSV-IJ _ yJIJyI IUJyI I

.t - -t repeSY 'tx-t7+x7=-tr-Z)[=xl{:ule.\Y (e

a1fn1o5

' .i-rFntx'Z= W (c-.ty+rg={ax'O=/{ (qti-Z).x={Tn'U=71r (e.L\p?lnluoc ozelpnls es PS 'I

,, etitzoduroc eP ee8el .

,, erltzodtuoc ep ue8al .l:' !'pr^ou erutflnut o ,ru ell

)!pto4 'tDllatg

) 10 YrlYlgltrdoud 'z't

.- , i *.r utlunco tfe,rlozeg

, ad'{r}r(-'o)=e et{'IzI a r r'.t)) : 7 rfeururepglhu:1ep as V eelu:'rllnur ed '02

.?: arsa 3 gc rie4suourag (q-. -1IZ-l)=s riqncle3 (E

.- t- ,,ta-:a1 -

)l3iaunue 3 eeuttilnrued '6I

Ii -)Df =y tiuuturete6 (q

L

):::ued 7>rt rluurwrcleq (P::':oiai atSaugeP es Z ad '81

tl

.'1"uaisls e.\lozoJ es PS (c.

.:: t- i l :zallsuoLtlOp 0S PS ( f'(-'s)= I

" -' - - - 3.1 '....r31'rtsuoJ oS 'lI

'c > (t' z) * ('t 'x)go pt;rvil' D = (t, z), (t,r) A pc ezorlsuouep es gg . (2tr + I x, 7{ E + zx) =

Q, z) * (,(,x)erieredo Is {I = ,trt_ ,* lZ*Z> (,f ,r)} = 9 ueurflntu Froprsuoc eS .9I

'uerrilnurug nc poder ul u InI B p[qe]s elred etrse H ewc n4ued t? rnl BereolB,r rfuururelaq'{t= rl"-,r'O = q'nl gtq+r}= U IS N 3 rr et.{ .SI

'ropdcurg e e:eunduroc ep erie;edo nc pode; uI pllqels eued etse .t gcep IiBIpnlS

' {O * *'6 3 rz'al'tt + xtu = ""'!lUI

:a-:t-.-:l pe rry.

.'. - :r.>r,irl irlpdr{irii lui

, i. ,: 1 55 fie parte stabild a

-::. .-.1 G este parte stabild a

:. -.:::ea matricelor.

)::erminali a,b,c e R

-:--. --t^-. --.--..!!lul .

re :. tlncliilor.

--. :: :;port cr.r inmullirea.

:'- i

= ,i. ::zultd cd

17. Se consideripe R opera(iadefinitdprin rxy * ty-4x-4y+20,(V)r,yeR qimul{imea

C = (S,.o).

a) S[ se demonstreze cb G este o pafie stabild a lui ]R in rapoft cu " * " ;lr*v=z

b) Sd se rezolve sistemul l, -l = .

lzxx=Y

18. Pe Z se defineste legea de compozilie x I y = x+ ! + a, (Y)x,y eZ, a eZ.a) Determinali a eZ pentru cate Z este parte stabild in rapofi cu legea " I " ;rlb) Determinati A = 1a eZl3 13 I... I 3 = 52 i.

|

-

I

L soon )

19. Pe mullimea C a numerelor complexe se definegte legea de compozi{ie:, L z, = ztz 2 - :t - zr -t 2^ (Y) z,,zr e C.

a) calcula{i E =(t-2i)r (t+i) r(-l);b) Demonstra{i cd C este parte stabild in raport cu legea dati.

20. Pemullimea Z sedefinegtelegeadecompozilie xly =2xy+3(x+-v)+3, (Y)x,yeZ'

Determina{i A = {(*, y) €nxWl (r-t) L y = 6,x r (-rr+t) = to}.

21. Fie G=(0,0o)\{f}, Pe C; rdetlineqtelegeadecompozilie n*-v='t"i'(v)*,yeG'Rezolva(iecuatia r*t*.*'" t=e2,unde e reprezint[bazalogaritmului natural.

I, 2. PROPRIETATEA DE COMUTATIWTATE SI ASOCI,ATIWTATE

Breviur tearetic

Fte A,I o mullime nevidi gi "o" o lege de compozi{ie pe M.. Legeadecompozilie "o" estecomutativldacd -ro )':y"x, Vx, yeM.. Legea de compozilie "o" este asociativi dacd (x.y)o2 :r.(-r oz), V x, y, z e M.

W, nxeyitii l! un,o.!"^".*'o':o.::: 'iiiii,i

1. Si se studieze comutativitatea legii de compozilie " I " pe mullimea M , in caz:urlle::) M=lR, xIy= x.(z-v)+zy:b) M:Q, x Iy=3x+4y-3;;) M =2, xIy=lr--Yl.

Solu{ie

a) Avem: I Lx=y(2-*)+2x=2x+2y-ry 9i xIy =x(2-y)+2y=2*-xy+2y=-2x +2y - xy. Aqadar x I y : v 1 x, Vx, y e M, deci legea este asociativS'

10 Grupuri 11