Goras, Liviu - Semnale circuite si sisteme

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:bazear,&, in fond" pe procedeutr. de mal sus dar este nult nai ugor de ::

licat. ::.

Pentr:u incepu"t atribul-n cj"rcuitului inci o bornH, neconectatd, pe ;;
re o luEn ce nou6 referln!,X pentru tensiuni. fn acest caz
nplet6nr sistemuL trebuie sE i-'r_a:
de ecuaglr cu o nnuE ecualie care sE conlini
rentuL S,n borna c*::e anterlor era born5 de referinld (pEstrAnO :_ F-.-..
'lven'tia ca sensuL rczitiv sH fie dinspre exterior spre interior) Si
lu5m ln ecnsiderare faptuJ- cE avem m tensLunj- in loc : -,3rs:
de m-1 c8te
lu anterior. Mai::lcea care leagi cei m curenli de ter:minal de cele
iensluni. nag*crtate ra borna exterloarE se va
{s* sugereaz6 faptut cE potenfialul numi natri-ce adnitanfd
tetinitf
bornei de referinld nu
::ericpereacizaadLm, itaacnegaEstanendeetiifninditdlegeasttieeloectnriactrji_caecidrceuist)c.urttncirfocnudi,t
:tic-ulard" Froprletdtile sale rezultE din urmXtoarele doud
;ervatj-i.

&HWATII {1} Suma curenfilor care intrl in cele m borne este
nul-X astfeL inc&t curentui care lntnE in oricare born6 este egal
bcuorsnues,aoc"ru-icsderferffiasrchfii,nttbeant saiwcnuirleen7tailobrorcnaer.epirnintrirrrimnarceetreel_laaLlitae:m-i.

S '',. ( r) =E tH *t, (P) ) u: G1 =n {12)

trebuie e5 fie identj-tate in rapont cu tensiunile. Concluzle:
natrj-cea adm:"tant6 nesiefiniti are suma elementelor de pe coloane

rruid "

{2} Presupu:leni c5 an conectat o sursX de tensiune intre borna
de referinpx 'pi rinul dintre terminale, cerelelte terrninale fiind
l5sate in gcl" SsLe evident c6, pe cle o parte, toli curenlli de
inlrare vci: fi riirli-, neav&nd pe unde se inchide (borna de referin!,i
rilr este legar,i eiectric la circuit) iar pe de aLtX parte,
tensiuni-le }e t.c'"!{,e termi-nalele vor fi egare cu tensiunea sursei
datoritx earact*.r:rLr.ri. conex al circuitului (nu existi t"ermi.nale
cqmplet rzoiate u;lele de allete iar prin circuit nu eirculX nici

un curent, Fentru a *pere cdderi de tensiune). Se poate decl scrie:

ik(f) =iEn7 Y*,ir'r )u(e) =o: vkElr,ml t13i

J'L

unde u(t) este acel.agi penlru toate, terminalele, prin urfiare,
setricea ad:nitanti nedefiniti are gi sumele erementelor de pe linii
nule. Constatdn cE, dispui:&nd de nnatricea de scurtcircuit in raport
cu o bcrnE oarscare, putem obsine rnatricea,adnitan!,i nedefinitd
borddnd lnatrlcea admitan$5 de scrrtcircuit cu o linie gl cu o

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t3t {3)

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Fig" 1 . a . Circuit l-iniar, nodel posibil de sernnal rnic al
unui amplif lcator cucuLlnsturarsnazidseLotrenisrriumneonttraajnesmfigituorractoamiinin;
b. ecelagi circuit
snrs5 de cureut.

If^tJ- 0l (16 )

&1;r vr I
oJ

Pentru condensatorul c2; (1?)
(1E!
len -c.tl

,, , ",_l_ crD c^Dl

Pentru condensatorul L3. 0ltrl

io /- nlI
c,=[o JJ

PelrLru r:ezistenla "Rt : l0 0l (1e)
t rl ", =[o orj

Pentru sr:rsa de curent comandatd in tenslune:

lo ol (20)
igfLl/.lL sccE=l I

0J

suinand natriceLe adinitantd de scurtcj-rcult de mai sus se obtine

natricea adrnitanld de scurtclrcuj-t a circuitului cornplet,
coresFunzdtoare uodului de referintd (3i;

lYl.irlc-.Dngc-^cDznDc.*Gcr-D+-cc-o,DD+GrIil (21)

Tinand seama de modul itr care este excitat circultul, constatd:n

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TmT8u81 nE6 Tfuernc (rz)

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n3*, TnlTnort3 rulued FcTToqups saroJ ug aTTT{pnre 'eJpErn uTrd
(Br)
(sz) lIoec*oeeerc?'-+eqeb+-S1a-+6qEGJe+JE-gE+oq-zJJ=J. ,"r".'
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TnlT?TnerT3 EeraTil?Bep ryugerslop Es armuTluoc ug uetmdo.rd ag

(c*zcj+ (? ) rn6+
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7fu"r-{€r} (? ) €'t* 6fu.(a ) rn (b+re) -
(s.r*'J)

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e1e1{uereITp e111{ence

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It+ )'r]

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a111fenca 'erpum uT.rd
'o=(1)'T TB 'c(1)e=11;-1 9c
$fll3r{n3tt3 v t!]nr$t0 t0 ilOIil tttitl 6-TT
ltllsts t$ ilill3ill 'lluttls

stmilf, ctf,cutTt st stsiilt tt-to *riur Dr A ctnc.,llrr'r l#l'-[, (

'tr,'f 'rs'*r*i par'
gl pernit determinarea tensiunllor dintre nodurile 2 ci 3 si nodul gl-ri
de referlntd 1. Ecuatiile dlferentiale se scriu i$edtat,.
t
3. ffxaew Dori-n sE scriem ecuatlile pe nodurl pantru clrcuitrd din
FLg.2.a. care congine, pe lingd alte elemente de cirstrit 91 un norl

girator. grri

i' l2 I

lit,u1TI *I tt\ J- -,
t__t
E
(4) 1lnlar b b,
|! !d
.a_ contin&nd un glrator,.
;q:ri
FLg.2. a. Circult
Glr.atsrul lzolat. 0B*

vos consldera ci circr:ltut este obtlnut prin conectarea in r:: :, l-
paralei" a clrcuitulul firE girator gl a giratorulul. tn raport cu
nodul de referl-ntX (4), matricea de scurtclrcuit [y], a clrcuitulul - -L -
flri glrator
esteI:Y],=lnl 0t*nGt ra+ggt (22 1 les c
-G'
-Gl s e:1.

[ -n, -c, Gr+Gr+go F:- .r

l{atrlcea de scurtcircult [YJ, a giratoruLul este: JJI^L
i^--
'n'=[-t" f] (28)
-lvc t'
siSfntc1rturrO(iar3btrgsc)eollrrrriavcan6upt$oolntrcrotaEeadsgtumetirerlalaletafrtalocccnrelturoaoclduruautdldltemeu(s4liptuta)erinoLfgpganio,clrtdt&ito:eenlornpsnsdcaeciuule"pnlrtrtanlcoltaeepirtorcnjir.uoncttdietotruanearaja.Llcdcolmjutr_arlrtt(ecae1unni)gtLtl,EirnlJ(t.rZddueeeli
crrrentii de lntrare i.n nodurile "ealde ', ale giratonrlul gl seg
tenelunlle raportate 1a nodul de rfderfednqlnirta,itacar.repennutncloainrceidzoelvcau

-nodul de referintd aI clrcuituTui
.aceasti incompatLbilltate, care nu perml.te gunarea celor doud
oatrlce de scurtcircuit, vom adduga glratorulul un nod eupli-mentar

'F,lTu[nrc9ruTac?TTnnlumTpeulrTnnEo8:TpcrumncesJlueTppgterutupolTpuutpdpeEpeceTcrTlE:uot eul ura?ep 1od as IEI
prnpsom3 (Z)
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FcTToqqs eacTrl€u PtEp Tpirnu n3
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apTrcnq (z)
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gB rpTu un TTteXTcxa ldarp 'nTduaxe ap 'pugldopg TNI
1{uarnc "p Trnsundsgr lderp e11m1sua1 1€
11fe11cxa ldoJp (ei
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TJ lod lTnoxTr un nrtuad Txnpou ed a111{encg {Il Il:$uAulsso 'i It

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rlt(rs) nc:
",i:.i: ;;]=t,,,s"ol u!

t ii i fif 'q

:luns 'lplp=q e1J,e1ou
nc aTTo$rys pou ug esTrcs '1n11nc:1c nTrcsep erec afTT{Encg

Il'gt*u'o-*u"g eg-6- "6s--61l= ut
s-"s-(os)
qc+Es

I 6 z9*'c)l

:r[],] Ts reeeu'z[L] Eecrrxeu arqurp
Purn6 a?so lalduoc TnTn?TnoxTc E [L] ?TnDJTs?JntF ap pecTr?Pl{

lo 6- 61
lo Eo E-lJ"*'"ul
(6u)
t"- oJ
:TnTnrolprT6
p gl.TuTJapau ptruetTutpp eaoTrlEu a?84 gtuTralar ap pou Tnou *
nc ?:odP: u! TnTruo?e:16 e lTntJT3?Jncs ap EaoTriPul 'puoJ u!
'rolErT6 uTp
EqrFpu] gTlnndTnplTlsnaccrpTo'TIusnTs(uta)l eTJ Fs relncllred
fnpou BT lPXceuoc Inpc
nrlued gfulraSar ep pou Tnou Ttr PA ares

-+ mullllxils t ililm510 10 toollt ttrrr,,r_TT il'tIlfflJil

]t]l$t$ Is iltllilt3 'tlilti:

s[&ll$Li, ctRctltTt st SlslEit LL-n cf;rrvi fi[T''r Dr i c]flc.,r'*f,n sitt

de intrare gi r-tna sau mal nulte mirimi de ie$lre pr'risnc*errl*ini inarea !te
nndrimilor lntermedlare (care nu intereseazd ca iegiri sau care
reprezintd intriri nule). ln acest fel se pot obtine reprezentdri rfE
siurbolice a1e unor ecuatii diferengiale care contin mai putine
nec(u3no!scMrrtae,trilcnepaaartdicnuitlaarntod slngmrd intrare gi o sinEurd legire. f ':n
nedefinitd nu 6e nnodific5 prin
introducerea unor surse autonome ln circuit dacd, prin pasivizare, J^-
circuitul nu se nodlfi.c6. Aceaeta inseamnd cd sursele de tensi.une re-
trebuie conectate in serle (deci prln intreruperee ramurilor) in l- -
tinp ce sursele de curent trebuie conectate in paralel (deci intre
noduli). Cel ce se modiflcd sunt curentil lnjectati in noduri. :JJ

1,5 [egXtura dintre natricea admitantH de scurtcircuit gi afE

deserierile multipor[ilor ...

ncuatil}e care descriu un circuit m-pol se sjmpliflci dac6 3-
ternl-nalele pot fi Lurperecheate pentru a forma porti. (pentru care
eurentul de lntrare intr-un terminal a1 portli este egal cu curentul .'.::
de legire din ce151a1t tenninal). ln acest urod, nun5rtil de neeunoscute
scade c&ci, pe de o parte trebuie detennlnat un num6r mai mic de :-i
t.ensiuni de exenplu (tenslunile Ia porti, tensiunile intre porti :,.: ]
neinteres&nd) cunoscdndl de exemplu' curentii de ia porti. Legdtura
dintre aceste mdriml este datd tot printr-o matrice admitantd de 3:1
seurtcircuit, dar care este de ordin mai mlc ai nu descrie cimultul
dec&t in conditlile respectErii restrlctij-1or de curenti ale portilor. '::l

2I }4HTOSA I{ATIIAN D$ AilATIUA A CIRCIJITETOR CU AilPIIFICATOARE :_-.'

CIPENATIOI{AIE ID$A}E :;,

&nallza circuitelor cu anplificatoare opera$Lona1e ideaLe presupune :.:
utfi:"zarea modeiului nulator-norator al amplificatorului operafional
adlcE, in fond, utilizarea restrictiilor inpuse de scurtcircuituJ. _3

\/ t-J. L udl )-

Vom adopta urindtorul punct de vedere: iil
^l
Circuituf cu anpTificatoare operagionale va fi privit ca rezultAnd
iri urnia conect\rii anpJificatoarel-or {ideal"e) intre' nadurile unui
;:.;rrca;f fErE anplificataare, cu restrict,ia de a canecta bocrnefe de masd
a.Ie tuturar anpLificatoareTor 7a borna de referintE a circuitulur f,Erd
ampiiiicataare, Aceaste condigie este respectatd practic in toate
s-ifuag:,"r-1e dataritd dificuLtdtiLor de a realiza $di nulte,ra.se
dis*jircf€ gsentru anpTificatoa.l'eJe operaSTanal"e dintr*un ficnh:7.

TurnaTl$nelTonucnrcTcnpu lTno-rrplJnos ap PluelTuipe ea3rJ?eu 'aJu"l1:1 ap r;Jua:no 8,Spllr

sJPc €T afrrl€noe €T ?cPXe elunual ua:,nd Po 1n1du1 slBol
aTrrlsT:lsal Bp euuas pugul$ PJP] T
ap Tg aJBoXeTnu ap Jo{TunTsua? asndw: " ereolspTJTldure g:q; TnlTn)-rTJ
gsprn ap
Xgcep eldurs rEuI rrienca ap slJosep alsa eTeuor{e:ado e;eo1ec1;11due
tnun a
n'acJpoTTne?ATnrJsJBTaOpcgeJn'gc1lenxzoEp3es.:Pead?nluoseoruendoea'u€TruznnTTsJuuaol JapTInaJ]E?usm€uaguc1fp,q1odeeSl
ptr9?Tn
puTsolol (Tl€?oa[ur "c1{ua:nc 3sounJ as nu BJ€3 nf,1uad) tsTeuorte]ado
TnlTno
e.rpol€rlJrldrxe 1e6a1 ne-s aJ€3 €T JoTrJnpou ai{EolEzundseJoc a111{enca fPuorjE
e1 eJunr;a: elead as 'aJ'r?urn uTJd 'aJpolBOTlTIdtuE TSeJoTacE aTP TJPJIuT aundns
1e6a1 ne-s aJEO eT JoTrf,npou aTTunTsual €f,?uT a?elrTPba ep 11{e1a: ap
Tnf,gunu nr lebo s?sa TJnpou u3 aleuolJe-rado :o1a:eo?P3TlTTdure €TTrTgar

op t{e1oe[u1 rlncsoun3au lduo-rno ap TnJqilrnu EO ?uaprAts'4158
'a1ncsouil3au

E?epueroep TJofPn 8p TreXuauryldns 1{ua:no elsefuT .roA aF a1euo1{e:ado tuYOI!
JoTBJpo?EJTlTTduIe a1e r.rr€a1 1ebe1 ne-s
of,e3 eT aTTJnpou ur - ':o11J
la1ebe luns 1euo1{e;ado :oleaTJ-rTdurp rEPTa3e rnun aTTJE-I1ur
1e6a1 rlri*s aJB3 aJluJ foirJllpou e:eclgzundsa:or aTTunTsu"l Tn?Tn3J
: aturcasuoc'aleJ€olqluJll
ap pfut
raToJ€TpTa1JTTenoUT?TnefcTTrnJnoTTJTnTdlcufnrEpcaJpJTaPOr?e,a?aTcTdau€uIooccr-'1rgend1pcnoapuds'a€a:clrara1s5lnradpnTs€ana:iJdj?TuAut!efnloaTJTnaPclrueToosral?€cJJn8cdso
e-rn1?6a
'aTpuorl€rado .:o1a;eole3TlTTdurP ales€ut ebal :on
t{:od
elTsnecJJTpO' el;1nJg${.ue1p:a;E:alul pa?pTIunlpppoua3nTor?UEoudlpoJ uI EsTJos 'glncsounc PcTToqur.rs
ap srr3sap alsa ats?Jgdapug ap sTul
a1euo1{e:ado a1e:eo1ec1;11due nc InXTnJJT) EO aundnsa:d utoA
('roToJeol€Tnu aTe TPnllTA a?n0sou
Tn'+rr€Jrr
TnTn?TnrJTOl:nJs aJ€o?gzundsa:oc,,41r1{ua1e.rd" E3EJsTlss Ps TTlTpuos ns
BIaf,eo?€Jo$) "aJEo1€f,ou rS a:eo?€Tnu uTJd e?eiapou Tl lod aTBuor{.elado a.re3 n
. E3 Tn$uos uI ?caJoo qzeauol{cunJ ECBP P
eTaJ€o?eoTlTTduip In?rnp-{rt
a?nopI ?uns PzeauJn eJeJ aITTleJepTsuoe aleo; I$ ttn
pc ezalodT uf
.TJN
' e1euo1{e.rado alareolecT JITdUB
EaJezTTT?n urrd Tpumu P?suoTlnlos aJlu3 r.
n:1uad a?pnToAa qTuEaUlqaoT:edpoou Joun 'nu aJ€c 18 ?oeJo) Ezeauorfcun;
alsa ur (rol
Tl aleod aJpc
lrncJTc a:eo auJaosrp V 'nu aTa?TE 15 1en1:1n TnTnlTncrTslm3s aunTsua:
p1d1cq:d puglcadsar Tcap 'Jeturl azeuoTiJun; Es af eluotu e;luTp
eTaun Ec EalelT1TqTsod g1s1xa E3 ulql€fsuoc '411:16ar nc aTTJFJIUT Tsap 'aJezIA
€TaJ€o1e:ou nc aTareo?€Tnu ,re?Paqca-radu3,, luns ar€c uT Tnpou ap at{cun;
uJ rS aJsosJ"AuTau 13 a:eos:anuT aTeuroq a1e6a1 ?uns ar€c uJ Tnpou ap uTJd p
eTicun; ug apuol{e:edo areo1ec1l11due ne alTncJro alTnu reu apundsaroc
1od TT aJeole:ou 16 eleol€Tnu nc XTIFJTs Tnun EO uITluTuIEaJ au 'a-rTSar
gcep '11nu Tpli 'EJeTUTT au€ulgJ pA ?ElTnzoJ TnTn?TnJJTc ee:euo1f^lun; 9c au1{nd
E'1e:aau.rae6pauJIe'?oJnpboTusp nu lTncJTc rln-JluT aleuolJe:a'Jo TJP?uaz€
f,oToleo1Bc1311dutu aJEC np!
un-Jlu3 eaJe?Jauoc 'a1:ed
E?Te 3p od EAJEUTUII

€r-IT\:. roulirllsir v iull[35]0 1{ l00i.lI Hillll ltltsts I$ lilltSu[3 ']lvf,$ls rlllllfllUl3 ! l

stfi$frtt, fitilJl]t 5t stslif,t 11-to cAriv' irr'Dr 0r 0[scnr[fir f, [rnc,,rrr,0R !iiiili

cri a$pi-ificatoare operatj-onale se poate obtine Cin natricea circuitul-ul F1g

cu amplificatoarele operationale extrase prin urindtorul" precedeu n,

(mefada ffatiran):

a. se suneazl doud c6te doud coloanele corespunzdloare noduriLor
l.a care s-au legat intrdrl a1e aceloraSi ampl"iftcatoare operalionaie

P-aj

b. se elinind llniil,e corespunzetoare noduril.or ia care s-au lega"t
iegirrle anpli.fi.catoareLor operationale.

Metricea adnitantd de scurtcircuit astfel obtj-nut6 este factor de
propor!,lonalitate intre curentii injectati din exterior in nodurile
circuitttlui cu amplificatoare operaiionale, altele decAt cele Ia care
s-au conectat iegirile gi tensiunlle la noduri, inclusiv cele la care
s-au corlectat Legirile amplif lcatoarelor operationale, tensiunile egale
ap€r&nd o slnqur6 clatd.

Dupd determinarea tensiunilor la noduri, se pot calcul-a gi- curentii
debitati de iegirile anplificatoarelor operat,ionale din ecuaflile
inilial elimrnate, ecuatii in care se. cunosc acum toete tensrr:nile.

1. &srnveyrr (1) ln cazul- circuitelor sLrnple, scrierea ecuagiiJ.or
se poate face f{r5 a apel-a la metoda Nathein, prin luarea in
considerare a restriciiilor impuse de scurtclrcuitul virtual. De
ssemenea, in cazul clrcuitefor ln care se pot pune in evi.dentd
bLocuri functlonale cu anplificatoare operationale a cdror
sfuenpctoiot nsacrreienuu6eosrtereinlafltuilenintattr5ardee-ieingteirrec,onaencatdlizria$sieppeonatrtue
utiLizAnd conceptele de la grafurlle de transfer. care
face

t2) 3n cazul in care una dintre bornele de lntrare a unnl
ampJ.ificator operational- este legatd Ia br:nna comunA, tensiuneo
celeilaLte borne de intrare este, conform prj,nclpiulul
scurtclrcuituiui virtuaL, nufa 9i astfel col-oana matricei adrnitant5
de scurtcircuit a circuitului fdrb aniplificator opera!.Lona1 se
eliinin5.cdci elenentele acesteia se vor irunulti cu zero.

2. Exmrpfu Considerdm circultul dln Fj.g.2.a. care conli.ne, al6turi
de alte elemenLe de cj.rcuit gl un amplificator operational. Iprim
sii determindn functia de transfer in tensiunl a circuitului (deci

forma sirnbolicd a ecualiei care leagd tensiunea nodului (1) der cea

;r nodului (3)) folosind met.oda Nathan pentru obt,inerea matricei
a*nitant,i de scr:rtci-rcult gi apoi prelucrdnd ecuatille.

Reeuitd ci adoptarea noclurrlor trebuie flcuid a5e cufi se
rep::ezinld in fig.3.a. Este gregit si se numeroteze Si nodtrl_ la

care este legatil borna + a sursei de tensiune cdcj, acest nod, prin

pasivlzare, coincide cu nodul de referintd. Pe Ce altd pirte,
curenfli de scurtcircuit lnjectatj. in noduriJe {1.) Sl (.) se

Tcap olse (A)[n ap p?rcEJsTl€s pleTtuaraJTp e1lencg

tss) ,rr"ffi=t.J'n

: (rTToquri,s gsTf,cs pp1{ue:aI1p olfence)

{1)a Tg {1)gn ue:a.rudntBsuuTeu?IJaeJ?lauprpasTaaTuntrTeTsuaar ?PuaJ!E.uraTl;usau?eaJpl 13 1n1nua1s1s
ea:euolfnlos ap e1{cung

{re) ltl ::X,j T;:;l=tiit:El
:1uns p-rn613 u3 leluaze:d

?aJ3uoo Tnpou uI ?plTaxa fn?TnorTc nTJ3sap ares o11;{encg

{€s} io o;;._.;r*Jrc=l ur

|.,;- :1euor{e:ado

:o1ec13;1dute nc TnIn?Tnc-rT3 E lTnc-rT3f,rncs ap P{up?Ttrpe EacTJ?EllI

ur€uTlqo 'g pTuTf pugrlTuTfo 1€ 7 eueoloo no I pueoToo pugunpv

lf+'J 0 UIrt16_ t J
(zt1 II o oc+lt o l= i,ll
a
Lb-t_ I T
o "gn'pl
:a1sa elJpdapul 1euo1{e:ado a
1n:c.1ecr311duie n3 TnTn?TncJTt € ?TIteJT3?Jn3s ap paerJlBH a
ai
'q' z '6T.{ u3 al

qlere as urn.J PSE 'gznec uI TnTnpou ,,TTJEJTdsapil pzEq ed pupualap eF

'u€r{?PN Tapoxaul E aJPcTTde ep nlduaxg 't'6TJ ?E

'e eI

-T- -r .IO

,i na

Itrton ;n

9IN0u{t[][rJ v ]ulrilrsl0 l0 l00lll fill,dl llttsts I5 t!t03il1 'iluflH1s u0lIrilr

TX

$tHttt, ctt0utlt $t stsrili l-1-ru frirTr rtI',I D[ t crtcr,;it0l stiHtrt.

'tsct!tnt va

-ott-n-dudlc|t!+6,G,u" (r) =G.' cd4d!Lt -G-G^s(E\ t36) f.

3, SRAFUAI DE $EI{I{AI,

ln continuare ne neferLn la o altd tehnlci de anallzX a eircuitelor
gi sistearclor, bazattr pe concaptul de graf de semnal. Un graf de sennal

a*te un nod slxbolic da a repezenta un operator mal complicat in
func!,le de operatori uai shpli.
Elm€nte1e din care usnuengttecodnosutlituniotedugrlragfiurslleennsluflnctEraopneurartiolertrJg-i
nodurlle. Orlce ranurd

{denmLt. transnltan!,5} care duce varlablla ata6ab$ nodulul de plecare

{Xntrarea i.n ranuri) in varlabila de legire (ramurl}e sunt orientate
Ft, epre deoseblre de ranurile grafurllor de lncldent{ care descrlu
ci:rcutta, graful cu ranuri avAnd sensuriXe gi sannels inversate nu este
sdllvalent ctl graful lniilali

l{oduri.le, p€ 16ngi senni.ficati-a de etrrmbol al variabi}elor,
reprazintE, atuncl cAnd se afli la lncldenta a mai nult de douE ranurl,

grnatoare"

tn Fig,{. sunt reprezentate c5teva ramuri de graf cu scopul de a
s{bate in evldenttr diferite poslbilttEgl de sisrbolizare.

Fig.

g-I Ai) e"...."k.'..-/ 6-Y Jttat

a. b. c-

r{d*t; e1*-- A{.}

g.+_€)t -ar -\X
d

cC.lgy.(4t).=JLea(tmldut+rlyg1lo)no;ddu.rly(dt)e=gPra(fD. )ea(t") y;e=.Ay(e=)c;leb(.e.:')=+ksei;e,) l 2. t?

u

[. Hxrolpilr tn Fis.5. este reprezentat graful eistenultrl i

alqebric5 de ecua!,l1: a

prinr,:

depenr
necun(
reducr
condur

" tn acelagl mod se conetrulegte gi grafuJ- atagat unul sllto$ de mulLe
*cua$ii dlferentiale llniare.

'Psms TJnpou e?tnut ep

Teu qxsTxa fon axuepuadepuT aITqeTr€A elTnu Tput BlsTxir E3B0 g TNTN

'Jauruer3 TnT eTn6a: pugcTTdp lnuTiqo TJ Je-s aJp3 TnlE?fnzeJ pf ecnpuoo [('
Epo?au aJpcTJo uTrd parpAfoza: 'puo; 'uospt{ 1n1 elnba: nps TTJaJnpar
uI Pep
spolau :aJeATozaJ ap Bpoleu o pugcllde acEJ aXeod as alnosounoau 'Tm
JoTTur.rJEur n€s qTTf,uarTJJoprf',EBsarpEtrfT,unulcanXrelos'auTT1q:edTJ'1EAn;eaJr9luT';pe:ro6la{uuanp-:u1aud1a:dp 'r.ola'

e au16eu.r o exsB
leluoza:da: 1g aleod d11 aclro ep TTlence ap ual$Ts un 'E nTrJsi
:elaadse enop ap e1eba1 luns JoTelTncJTc pzTTeuE Tg pa;aTrcs"p elP?u
uf :o11.rnye:6 aTTTlecltdV TtsHt{ss sO uo.Ilutrduuo tguqzfTlJ,n 'Z sJBOe'
Truo?l
'r4-t
Tlt BT
Exg *txr=rx
(8s) nx6+€zxxep++rxxgf ==zexx u! le

zxo+xe=tx TPUuraE
:n.Ilsou Tnz€3 uJ
roTalr
'(tE) TnTnualsTs lpSeXe TnJe-:D 'S'6TJ
(es)
'gluapuadapuT eTTctpTJE^ ap TS al1plaToc ap at{cun; s]Ifi0tilt I

uI E?nosounoau o a?gc Aug:dxe as 'alfenca af,EDaTJ pugz1111n '1odq

'gpsrns pou lTumu pou un es-TnpuTnqrJle aluepuadapuT TBTTqBT.TE^
'pou un aTnqrJle ae T aTTqETJeA Torpoal4 :11{enca ap rtta?sTs Tnun

?BfB?e lnge-rb a.rq$aln:1suoc as a-reo uI lnpour :oSn p3:eual es

g=7x>l-A lnlsts Is lllnSilt 'lluIIts

g=tx- €x;r+ rxr
ir.s) g=?x6+ex-?xp+xt

g=txa+'x-txq
g=zxc+r.x-xe

,'-TTmlllfilun y IuJrrssl0 t0 l0oltl uiilfl

siltili, ctt[l,tTt st 5isiili 11-re 0rr[{A Nt'mr 0r Di$crint t c*'urltt'r 3irt

setoda substj"tultei, metoda Lui Gauss de diagonalizare sau orice altd '-t
*totl3 de sal.uti.onare a unul slsten de ecuatii algebrice.

b. Reprezentarea prlntr-un graf poate fl legati intr-un mod
dlrect de o stmcturH de clrcuit. astfel incdt graful si fie un nod de
a repr€zenta un nodel sub o aItE formi. Astfel de situatii apar, de

ex€mplu, Ln cazul circuitelor cu anpllficatoare operational-e atunci
ctnd conectarea acestora este astfel incdt se pot pune in evrdenli
blocuri a ciror conportare lntrare-iegire nu se modificd in urma
interconec$rii".
DotrE exenrple tiplce sunt sugerate in flg.6" LdsEn in sema citltor-uLui
&tsnnlnarea constantelor a...d, suger&nd ca moC de lucru suprapunerea
efectelor {Se conslderi ctte o intrare ahmentati de citre o sursi
tensiune, cel-elalte lntrEri de
fiind legate Ia masX: tensiunl nule pe
tntr6ri nu inseanni cE acestea din urni se lasX in gol!).

R'

c b. c_
k=1/RC
i p'*r{*8-nlr4hI l.-i--"-r4I u'l .*"'-"k"I.{.-).d.rt_u0
{'o

d e_

Fig.6. a. Circuit care realizeazE suma ponderat,S a patru
tensiunl (douX ponderi slnt poeiti.ve gi douE negative);
b. Echival-ent cu o sursE comandat6; c. Graf; d.
Integ::ator inversor; e. Graf.

3. &rst'n RBr&fr:r t[[RE crRcufls gr GRAFLRT Atte 1egdturi intre ::{
circuite 91 grafuri sunt sugerate in Flg.?. Dori-m sX evidenilu
-':
faptui r:X se poate asigura o corespondenli dlrectX lntre qyraf gt
circuit ori- de c&te orl interconectarea blocurilor nu schLmbl

eomportarea acestora, chlar daci el.e nu sunt reallzate obligai.oriu cu

anpi-j.ficatoare operatlonale. Astfel, ln Fig.7,,a Sj. b rrrspgcti,, c,9i d

€?I?rTpr&rcT€'lrSrrTc8Cepsree3UATrn?fe,uo$Tpees8Tl.Trnpe{?tiTauJEupTnuluadFal€uJ3oTndIptnT[PEpS3?rsuerJ€nxTpn[BprEa?zpauaTlaTpsAnuEacorJ3TePcrusII Ts relUOUPr! UT 1E?3aUOC
ETinTos'TaJ?sV'er1€e1 pt
aaElepcJE1ptnT1TorB1l:oduoc
nPT L n9
Fnop Eq

',5er6 ep aITrnuIEr nrluad 1;1u 'a11n3rT3 nJlued 1c1u B1ue1:odul l3
puTT3au'TTrp?oauoc €suTpJo 'gpeose3 u1 3e.:6 ap TrnuIeJ Pnop aTac ap F
:o11tm1sual TnreJsue-r1 a1fan1:d
eo5lttuEoEzaacxuda! roTT?JuasopdT se?saJB 'TaJ?sE '16 EI
Tg ['f'6t.r u1p e111dern6T]uoJ
PzpeuoTlsrn; :( 'l

'1rn1e:6 15 a1lnc:Tc p^a?83. arlu! TTlBfau '['bTJ .t
nl

I

-rT -r
ZNr-IxAi)-
tnt

L.---=J

,U

'+

,"T1Ti"+: 3,,cU,/,!+HC=/l1..i,u1r'.',,
ztn -l Lfr- t'n 024;,1 X tn
in u
{o}t-t

'p D

'^rAl5l Jo<'" adr
tr
apF
u/,u+t+l 'H ru pele

( 'cu TNIN

TFuaaeItTpn?c!mTTaXrdnsroreoql€XTlll6urEnTXleutpr:zeT"utte€Jacga:a:dep:e6e1J€a1l:'oTt1a:1';dn6$!3$utlsuTaaaeao1.1lccefsueo";EluTdz$lcnapgeriJ:.Pa'i{aioeeO'at'l'rottu6Jbr'I6eoTrtnr1lTJeBr$Esu.Juu{le3seTad:eppTEuculITeovanTuncl)lmT'IcJaBTnJnodsTcOsuaJc1JcT1a3T3uu?nnen'ntsu':'1a6n6TstulTmulTcee-.Trp?a{posd1uepusPaJeuTda?Jcgpnn1:lussu;pa1eu!ITelioedTTlJBoulenu:oeln:ua3a-6azre11Tn1ans:TJu6r[uuJed1Puo1a:uTfcra,n:il BUIIN
g$uat
'a1:ed uT aJ€ceTJ 'eXueza:da: 1od as a:ec a{ualeATl{ae Pnop 91u1za:d as
aTepun'

apt

poul

91n

4 ffllrr*3$l r ilrrux$r0 10 t0otil yillru 6r_TI IrIlSts I5 llt0tulS'l]vilIls slilflrilr:

sifiltALt, ctRclJnf 5t stsl$t LL-ro 5tHil

ctTtvt trr00t 0t 0t$cRltIt I clnclJlltt0n 1

3,1 Regula lui l{ason l,

Solulionarea unui graf (s6-1 presupunem cu un singur nod sursd gi rqj
lin
o singur6 ieglre) constd in transfonnarea acestuia intr-un graf cu o
singuri ramuri care unegte nodul sursd cu nodul de ieBire gi 91
Cet
deterninarea transmitangei acelei ranuri. req
Dintre cele dou5 metode de solufionare, metoda reduceril gl regula con
dlioliarMtatrseotne,tao" von prezenta pe cea din urmd. l{u da denonstratia, ci
care este foarte sirnpltr. inainte vorn a prezenta formula ta[
d&n cAteva definitli: rc(
de
'.ljn
Prin buc15 infelegem o succesiune de ranuri care, parcursd in
sensul sdgetifor, ne aduce in punctul din care am plecat. ae
Prln transmltanta unei bucle intelegem produsul transmitantelor
rasrurilor unei bucie. a.g
Prin bucle disjuncte intelegem bucle care nu au nici un punct F€I

comun la
"
cale intelegem o succesiune de ramuri care, parcurse in sensul qi
.Prin
silgetilor, conduc din nodul sursd in noduJ rispuns, fird a trece de
dor,ri ori prln acelagi punct.

1. BcclJre r,ur Hlson lransnltanla rezultanttr este un raport intre

dou[ expresii,

r(e) =4
B

care se scrlu astfel {se lncepe intotdeauna cu numltorul) :

ts=L- ( suma transnltantelor buclelor) + ( suna tuturor prodtrselor a cdte
doud transmitanle de bucle disjuncte)-(suma tuturor produselor a cAte
trej. transnltant,e cie bucle disjuncte)+.. .

A=(o cale). (ce rdmdne din nunitor daci se elimintr toti termenii
care ccntin transnitante cteo'triatitnegrrmr ecanliei ac)a+r(ealtcf,onctainle)t.ra{cnesmrfuitnaAnnteedcine
nunitor dacd se eljmin6
rratinE'r caleai+..,

lnlr-urr cicniarcr"euami-ptornceotelan"educatnartcnluunreatci'nacvteanplterdrn. nuVliacdriieaonltaianckiierscgeairrceeoanipnuo2rtttdvaavlnanvl"oe(drosifr:llccnaeu

cinicracrucdit
cemportarea grupulni R"C.

'eTnu auEoTos ad roT€lueueTe euns T3
TTU;I ed rolaluauaTa eums a.r€ PcTToquITs P?TuTJePau g{uelplpe pecT-rlPlu

ATlou lBarE uTq 'TnTnlTnJJTc roTrolxa pou un alsa TunTsual nrluad
pisTraJer sp Tnpou Fc PerTqssoap no 'PrBoT.ro?,ue eso nJ a:Po?guPutasp

BrsTrcsap o p?uTzards: pcTToqqs P?TuTlepau Plu€?TqPP eerTrlel{
'luaJno ap roTasrns P?iTJo?ep PluTrsJer 1da:rp Xeldope
fnpou TF Trnpou eTa3E ar?ug rede aJec elTunTsuel Tg lTncrTo Tnun
oeTTBrnlpuo[uzeuJgd1a{reppao[Tu5loq1E1{tusa?rnrcncerrTluo;upneorsn1e96pe1EatuTerc?sTaqptpEE ap slelTTeposl
BecTx?P]l

's pxsTdrroc pfTqeTrEA m lTnooTu! T] €^ q {nro?€rado aJp, ug aceldel
plpu.rolsuerl pugzTTTln eTTraTrcsep n3' pTEuJoJ' parPuguasP uq3f,Pulau

'Tluelsuo3
lfuelcg;aoc nc aTsTlueraJTp TTlpme ulrd PareTrrsap no 9rn1p6a1
rtI pcTloqtrgs plue?Tupe TS PluBpedun ep alaldacuoo snporluT rlIV
aeoffr?anpuofnJ/uJT!yy$?eTwer'cs^EaouraTep?eeueTgJewaupJTaanrFl1laoS-aur1llL-redIJT€'d:lTuunTTflnlqTucaIapa:oeluudeTexreoeurAEnulTuTaT7zrg9auJaTdtuearyerulTa1Tp5

'Tfe;luecuol t4ewezed nc ac16o1eue a?TncJTc nJluad aletTJap aTapou
roun € erauT{qo sp TlplTTgporr EAe1g3 ?e?ueza:d ue 1o11dec lsacp uI

[vHnzffi '!

{r? } @Joq)+ (ceby+69+ep+ag) -T =*ex-

(ot, (6Jcqt + pa6y+61+ap+cq) -a *,x- e

=Jr r

{Gs) (6Jcql + Qa6y+63+€p+rq) -I tx,x - UT
I-'ffi=rr
Er
{86} (6Jcq) + (cetiF+6t+ap+tg) -T zx,x_
= plr To
ET
(rsl (6lcq) + (ce67+63+ep+ag) -T rx,x-
=Jr IS
o
:nglcadser '1tms rillx' Ix a:1S41 eppTf,ouTlTTJrnapIoau.raaJpNolpzufnfdidsat{:IoXc X 'Z
Ts
eTeluelTusuBrl, 'E "6Tg uTp 1n;erb
l01llt0:
,r_IImllltfiiln r lHlnst{ l0 l00nr rrrrrr illtsrs rs llilltun 'ilvtxts

sifitf;tt, clRcljirt st stsItil[ Ll-rz cirryf, frlr',r Dr r crnc,,rrrr'R

Orice circuit poate fi privit 'iscn]iRr

ca rezultAnd din conectarea in
paral-e]" a unor circuite nultipol mai sirnple. Matricea adnritantx de
scurtclrcuil a circuitului rezultat, este suma matricelor
scurtcircuit admitangi de
a1e subci-rcultelor interconectate.
Prezenla anpliflcatoarelor operafionale ideale intr-un circuit, in
cazur in care existd conditll ca acestea sa funclioneze 11niar
si-mpl1flcE anariza circuitului datoritd restric!iilor suplirnentare
introduse de nulatoarele corespr:nz5toare rntrdrllor in anplif lcatoareLe
operationaLe. Metoda Nathan de analizd a circuitel-or cu amplif,icatoare
operationale se bazeazi pe aceastd observafle.
Grafurile de sennal sunt o modailtate de reprezentare a ecualiilor
respectiv a legdturiLor dintre operatori. Regula rui. Mason este o
metodd de det.erminare rapid6, in cazurile relativ simple, a I
transmitant,el echlvaLente. I

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roleluelsls AJ
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trn$rrqs oqsa Brrlsunfsll[

tcE
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cp
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JeTqc rlBrsnu ed* JoTTJnJJnT elmds
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rm ?luTu63dal 'Enop aTas€s?€{s:n trs ea 143 e1 'aun1{c8s PlsBasV

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LL ' Flrf,{m* etP'r}aa T!{vds 'Aqone7 fJ,L4rg

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:lo!:-l:rl"ir:l 'jq: mat+:l'r+,lici .i.iri::orJuse, ulilj-zind a br!:liografle *ciecvati,
*sle a3:rs'r]ut i:+i:*'*;.r6 p{ifttru uiliiz*l:'*F" acestora ln apliqati-i eonc;:e'Le, 1n
mulSine
1. SIJ&T:H3, " nlil&TTi, &FX,If;ATIT etre$errt

i,3 ffiu}$**i i{i atunci
linl;.1'"1;r-i,i.c ri* mi,tlLj-in:c: gi *1e'rreni {ter.m*n, punct, mer*br-li} al urr*i eu €i.er
mul!-inrj" ::i.iir'!: lr:';iur:i prl.mare'. ij mill,!ime X. inclusd i'"ritr*o mult-;:rre in;ri
marr: $" i-:i,n-;:.* il:" 3" Ex
':eiei-;e..gin,iezntir*pIreJ'"rtic: ar* 0 cornilun {ca::e pct fi ele i'nsele
-.i:r:i.:iii'ii:,tr'*a unl
pr(
e:riL!i:rr:' l ; pet
- L'i-:,,,::i-:.:erlea i:::lprier&tilor r:curune el.emr:ntel"o:t: pe care le conIine
*au, eci:-i-r.:l,tail:... furn:.-ze"rea r.rr,,:i niecani-sm de qenerare a e-l-enel-teior'; aa:
"a'.. {"tr i .rr €,.€, x are propr:lef,afiea }. }
{1;! ca:

oSiniL.nl,r-:";. LLlit, i i!orceirrt;!& i,.llitli eiemenL x fa o rnultjnre 4- Hi
c*csilir,rar-F; x*X ri*c} *,.i- nrp+ai- d::cX x *ste aii element a} multlnrii X"
sui
:iil *a:,: r';r.,rtr;?.1^ :.ip ;;l-:-ti; y;SI""
tut[:'o:
iliirr.l:.i.lil,;'.1 a"r,.'.i":: r'.1 rr' .-.:, J.nf. ),ir- r.t eJ"em*rrt s* lltrui'regte nrl:itj-rile virlf,
nul.t j-r,t
gi $:;'r :.:.,i,..1.t;:'ti,;,
i1u.i.;-t ^,r
1" I '.':ll1",,lt { ii :"'r.l.i j.ri:"}e i'*prez*ntAnd domeniurl g1 codomeniul unrii Li-E 1
;I',:irr:i,,r.i selnttaJ-elor, muiti:n:L cle semnale care
ia:r:i.i.jti!*a ti.iir-1,i:L:! isemnale dlscrete, semnale perj.odice, seu a
g,r.,f j-sfi:,'r
i;jr;i!]!.ii* pl'cpr j,*titl i,i Re

s*jlrir,i.i,::: i;,ri:"r';i-il-uer '"ie ;o;*r'l-cad& pr,ecizat.a, serutale inxrgj-ni.i:e, Semnnis pr;cius
sullsri precizat" el-c. ) *t,c"
cilri:'dl ':r ii*i!?;';3.i..1 cit i-n:el
if1,el:ru'i:;r:;;:l,.rti.i,l1",.'ril:iL,)j.rrji':rr;i.;iial-;a:Vls::l.i{,5r.tt):ei=ufteliteiimiti),e],ta:ae€viRea'sl"corrqi"itlsque.rl[rjd.nueenaeU*siuapcfaiuriatim"i.iediLoi"rfrueurpenennu{tiriausilsectaedrseei
1:r
ul: si-s1..*;:, *gi.r: stahi.i. e";c.
p:r+ch
2" f! p r{ MUtTIilI Att . li=I,2, . ..t}} erte muilimea rioil,e:

f {*o:-ll;sttl ' Fi:'i'i;dl'Afr llJa IL
tUi.f;r:,:r' il "-:;;.: i4.;;.q.r; r;f iltn ate :

I.xr, x!u " . .Jf,n) , x, € Xrc, -l=Tld {2i

r Pi'-,j.t:,. .:,rr;:.,;.'1::i.ie-r-4i+..:ttge*d*inIje:;ir-i;li..,tq.iea <,r gt:lpare intr-un tot de
c"rbj-er:t.* djr- i
sa,; de Edndirea rr<>asLrfi.
{c. Carii: ':ri

FXa6le??dasTTerdnpXaunaTe:Tens?€guuuge?a€tlur1Jles3afTeTuaeI1xoarTaT"t€g:e3€ToloTas-adJE;c?aPIaexrllasgJ3€aoeBuTlcam:oeTru1onoauXu-TsnrIu.1rerdEuT1"tlari{rduede'"11cdFoxq:uFfeqIe'1aT=u'PPagw?sHrT:lsaeTra1rnesJnulToucpu1oTg?Ioat{eE3Xluadr€EeJuauTuadTnepqzl:eao€?aJTTT'eurnie{pIou1Ts1TJooEa;tpd)ff "PJ:
ap
'Agx pzealou as 1€ (A nP u e1{e1a: ug
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elqcaeu:ratldnuTqanr$F:s:T:ao?stsaTfai?{-lrasuxalgaT'Tf,u:sr"grl?ruTruEs JaT;?puPE';P,xlTxumTlaaTpzau?raptJrdTen1Tan:sInpo*d eaurr

TT{€TAU T'T aJec
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o*rtuTp &?uBuI6Ia no sseJ lod es eJ€E ?13uT1sTp :oTTTl€uTqlPc Jocnen?
earu"IlTnl$ a?sa TuITlTnuI Taun JoTTIJBd eoul{1nu a?uTnnc eXTe rt3 eTsu
'a-'rT
erP)
Tnun

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eaury{1nu "t
X ${r,gffu{ rssut ucrrlugd vffir$ult aur.rlT

'HX'"xffiHl{=xtt EsTpE 'H T',unTtlnu N Jo{43 TE ueTza?J€3 (r)

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'2$=flXfi IIETZaBIPO
aurJu
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TPU E
'6?red tl1 ?TTq€TJEA orEJaf; rulued IAUn
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s[**AtE, ff*tl,tTt 5t $t$fgfie !S ttit iilu, tli*iti, tpilftllx, $In$frlip3 sE;tit

I "d" "-,; 3.

locul al dollee ei.r*rr*r;ce di.f*:"'i.l*, relat.la se nume6te ap1lca!,1e sau un].c
funcgte.
aull
fnv€rse relagl*l Rn -t'i,:v-xX, se definegte prin: (aE
$."t-,{{y"a}; xeyi
suh
DffiEnlul r61aSi.€* &' {*lui[ime* de rJefinilie a relagiel R] eate 4,
m*.gLnoa elementelcr Eeg F€ntru cn"re axist6 yey astfeL ineSt xRy adlcd:
al
dtrmx={:r#x; fuev lxR.y}
Codcmeniul" rela!,i-ei R aate n:rrX{i*ea elessntelor peritru rul'
Wy casa car
sistS seX astfeL i.nc&?' s&y, aeli"e,Sr
ol
-X& Dl*a Rlcxxy Bsre o ffiffi;Jttrl- Yffi * ffoJn.r-* esre o rei.alfe rle
de Lai l"a Z cere se nffie$t€ b1t
Y la Z atunci A,?(}&slt{Z este * rela}le
congnrflsr8a lul R1 eu &; $l ar* liE$Elri.*tet€a e1e
Rr*R1={(x,ri; fiyev i xRty gi ytbz}
5.
t' fx"*t# (!) r-in s€mna3 analc4nS,c urtid1eensi.o$a} sste reprezentat de
eonstltuftX dJn total"l.taten perechilor {t,s{t,}}, ti?T" 6.
relagla cus e#61"6i"a61 val"or::. t, {t ruger&nd tfunruI} i1 pcate
s{t}es.

coreapunde o slrgiurX va-lcare s{t,}, r*.l"atla este o funcgle.
prj{*2u}l Un eiet€m S Bst€ deseris rle o reJag:e sc&xy tn
Loc se af i.5 care pe
ee*na3e ds lrstrare (aparflin8nd unul
doneniu
dasESpinosrtpueatemarncrlanecelt"enirzamfnetLaueects,"auici,s1seLiRastnepts,caiftfaodsfsiiurrtreeos|uLlrsntsntrupe&leiupelmJ.nnsi"ndctcotanlerSe$ueed)candasalruaetzaenelurgrrlrec.le:psli6t*nei"mnde"eaeer3s**clo.1srrade1rserolmgelpaepls"lneslrlec&peeearecdIa!croi,*lezl"oee,cainar,hatspsi,Ec.lledlxe.ognmXe)re-,nr-r[:>ia.iiadlnngcygtelrr,lf[ausssEsrEsnteecad]cmu-unr!ianene,nigdseaagil,l.eitrddldL)eeneee"
eorespu:rz&to*re unul sistsm"

2, k**o B&RImCUI$BH {:s}"e uml fxprortante tlBurj" de reLat,il eunt:
- reJelje de ecfi-iv.nJangE gi
- reJagJa de crdl.nc.

0 rala$le binaril ps nrultj.maa X, R,cXlrH se numegte de eehi,valengE
d*cd*ersetef:lexivd: xRx, VxtrX;
- stnetricil: xRy l"mpltc$ tsRx, Vx,y€X;
- tranzitlv&r xF.y 6i" ,vRz impltc& xReo Vx,y,z€X;
Relatla de echlvaleng5 se mal- noteaz[ r<:>il,

O relattc bl.narfl p€ t{, RcXxX, se numegte de ordine dacl este:

-. EeflexlvBl x$x, VxEX;

alsa nu aoerEoap euTpro ap aTlsTer'paclTsJele(u|ts)nTulrrs'.>TusTPuT|TPgTBrAUTzxaTter

'(n lgJap uoJEu TEus lcTrls luauaTa E?sTxa nu)

{r} N=x oZrx'I{<xAln
':,1 Tewww lueuere -
'(r[ lEcsp uJTBr Tpur ?oTr?s x lueuale ElsTxe nu)
Pt
(s) W=x (= ZrX'WrXAlAr
:w 'y, ad Te'/'FuTu lueeFfa - :1
Ef,roro xo trfglxu
t EJF
o nc fr{rrJ,sxzus x utrLrofl Ijlxn r$llm6t rusltillrsf,tu rlrs8r! '9 ATT
uT
'TJnlsef, ap os"IO roXsece TT{uEluazeJdeJ pUTTJ I-H""'I'O
elaroEnu 'n oTnpou TrnlBaJ ap asETJ alIunu 'p{ualenlqce ap eepTc t{ 3p
Iatlsp TuTJap ld aS 'ptruefe^Ttlc€ ep Es€Tr o ETEzPaeGTraoJraT$gPuTa0cEfliellgso8! ap
'g
n 1Bp rFrnu un PT TTrTtrpduS Tp lgor ror*e '(a

'ptruaTE^Trlea ep pseTc o plrlzerder lueuaIe ep
ereoaTtr :FiueTeATI{ce ap eTssETc ap sluluazerder a11u1{Tm nt (EcoATunTq
Ftuapirodgeror ug gsnd 13 e?Eod erEc) FJrosozT eluarcTa ep artiTnu o nTu
tratrtiqo 'Tasplc IE Xueluazs:dat nss nT?pluezardar luosels utunu T! erPc
'ptruaTsATr{ce ap (E=ss)Tocfletpreoc!elTglctrE[psulurteruIIrUoT{ a'(u=n) ue6aTB FJPq 'Tlr[tfnu adt
"edp
aq'pJpu o olsa oTnpor x TT4dInu P e?,Eo
lgJ psmFdTnu pzEeurot <=> BTtpTar ap oXEJauaF (8 uTp TE[lTnsqns) '&E?
't
pluetp^Tqra ap roTaseTr Jof,nln? BauT{Inn <=} OtoGOil ryJ ffiffTq{ aP 1

roun uTlr'de) alualealqcaeu or''(prrlTnuqn" t:3$:i::"tTillffi; a16er

oelu-rslTuBI^TpqlucasTeETT] qlrcmasaaplr[elT1n{etrTeErecop uTp alueuela Fnop erEcTro 'aurlTnu sp cT

FlTuTJap Pl"po 'eluTAnc elTP nc :POTP

{sl tPuntx=x e?se
T3 0=^CtJ*3 uaAp Aix Xarl'xA
pcTp" 'cordlcer Tg Ttrt?Iruqns uI X TnI e pcTun nPS e
ltfll3nil
eT{lrrpd o EuTura?ep X euflTilr o-r?uI gluepnlqca ap eT{ETer ecTro

{xgAf xarll]=-pr
:Ecfpp x nr aTiEfaJ ug eTe?uausTs alEol uTp gleGrol
luns arEc X uTp 's
B18s X€lx TnTnluauaTa e ptuat€ATrpa ep psplc $xstuatrEfl xo lslftJ

'rt'. zpze€?ou TEI as auTpro ap sT{sIaI
lggz'I'xg 'zgx gclldrg zUA Tg rlgr :pa111zu€r? -
JXe,{'xA 'ri=r gclldup xUd 16 AUx :EtTr?elrtsT?rc -

r4 illlsts Is utl3il] 'tulxts

lmlmus 'ilil:tut 'tunlt 'ilnl0t E-zT

$nlt, cncilIE 3t ststilt LZ-o mllnr, ttmnr, rnrcnu, slrrcTnr tililtt,

- tntlnu tnf (x) (cal nal anre nlnorant): r. Co

tnt(8 s x, Y xe tt tnf (t) >b, (8) Po'
D€I
un& b egte ori.ce rlnorant al lu1 x (b<x, Vxex).
pentru
- sruplriltn sWE) (cel wl nLc mJorant): grezenl

sWV, >x, Vx6$ sw(tl s B, {e} ?abelul

lm& B egte orlce Dajorant aI lul X (x<8, Vxex). of
{l gt x apar9ln nu191n11 X ln tinp ce tnf(r), sup(X} pot aparglne sau
saU Pe
nu nrltLutl x). sectteo9

7. ?t8frcf o nultiae lnzeetratl cu o relatie de ordine astfel incit "egtoDrls'I
orXcare ar fl o pereche de el€nente a,b€x, nultlnea {a,b} fomat[
fgnctii
db cele doui elerente are gl i.nf Sl sup se nunegte latLcet.
Pal:
1.3 lplicatii
:are Pol
o relagje fcf,xY este o apLlcatle (func91e sau relagle funct,ionalH)
rb la x ln Y dac6 pentru orlce x€X exlstd un slngur y€Y astfel lnc&t z. SEr
:fy caaa ce fuseuri ci daci xry" gi xfy= atuncl yrnz. Se noteazi f, t
f:X -> Y lar elerEntd i,eY se nrnegte valoarea lul f tn x (lnaglnea lul -eg€-a d
r prln f). seSInE
o aplicagle f:X -> I 6gte decl o reguli prln atrlbule unul :gregpoll
dmenlu, un care se :egula c
elaent x din nrltlnea t nrnlti y aparttn6nd
elilent -cr l.E
rtltlrll Y nunlttr codamnlu. o apllcatie egte astfel o nultt-me de r: vor I
pcedrl ordonate (x,y), xex, i€l adicE o relafi,le partlculard deflnlti
{o eubmrlgLrre a produsulul cartezlan &(Y, care reprezlntd ;eatnr r
p€ xx!
rultlnea tuturor perechllor ordonate poslblle) avtnd proprietatea c{ l. h'L
flecare xe! apar6 ca prln tsnen tn una gl nunai una dlntre perechl.
Bl€oantele y clrora le corecpund, prln lntemsdlul apllcaglel, eleuente -5t
rel, aparttn fuenlului de vaforl'Dsi.
tn particrflar X poate colnclde cu l. orlce apl.lcati.e €6tE rslagle *Iica'!
dar nu gl raclproc; o rela$le pentru care aceluiaEl eienent x i.l pot -i)
corea;runde nal nulte eleoente ye! nu aste apllcatle.

I l,lultLrea nrnerelor reale este latlce fui raport cu relatJa 5. Cea
nai. slryItr latlce este forraH dtn elenente ordonate, ttat€
exqrlu cr doutr n de
0 g1 1.

. V y€D, il xer I rf (x).

(II) zx ="x (= 1k1g * 11r1g 'xglx'IxA :erJce{u7 -

'(tr, tot ,,9d,, alsa e1{ec11de

(0r) i=(x\g I r sx E 't, a.,{ g

f, <- x:J eTPedTprret{7n=rc(uXlt)tIfiI:ua7rlfrc.atvdtJnrsld-u '€

'(fnTuauppoc 16 lnluauop ATsnTcuI) eTtB3TTdB aluazerdar e n:1uaC
(x)J eTtrelou "T;gTegzT('T)TI ln'(uxI)oJA'eaTrlzun!JuaoTriruaTp?sTTfFp?rTrTsrTeqflsFosd e1slxa roA nL
p$gr 'Troaun i
1ue1:odurg lcap
eaa'1l:pse' gTceT'x3uln,T€ebt!Tl1an{Teruncl1pue1alTudueeTreazTBTaauaJTeoduxarfosSr?aa(xp'()xITT)UTJTt{q''TxTintluanul]FclnszeoTtTpao'ax(aTu')JaalducranTEfcentuelanauJplnuluuaaoTtdudptsarayFfra{o'uaceerPepauucpon€ddp1esanaa66::ooaac::: !
1'a91{;ec1ln1ldueauaocplrogc'1I tnITsug(u')toIp :aluaurafa Taxl ap Rrl?uTulTlfJcepaaltXlsseus'tr
'z I
'tx)J UO|rIIitsJ,Cllt I

'u nps z 't roTTttrt{1nu eul{rBde lod ere; C

TaTpuT TlTnu TEU nps lnun ednp sX€xapuT T] lod TTtcunJ ap B1TTTTIEB4 p
alelcads TlgtaTrdord nc 11fcun; T

ap l1feds ed a11u11ep e:e1nc11red aTeuoTtcunl Xuns aTTTtnqTrlsTc T
Pl
'TeuoTsua$tp-u ro13a..
?.1
nn'gpess'ao'?1T€eJulneolspe:u'oxTtngSldTn'1u-6runeueeqppuAT! eTromsTaTPrtaroouumlsEueaJreeeazuTagTTdpPlrnaauuXeaT6UdTPPJxU?uTTTuTTzJatslr'pd.iollanJTfTf{iclgnrnsP:ltJaueodAcnepss
(!

TJoqe.redo aleuoltcung P1
1
'1rgu:ogisuerl TT13UNJ
TTlcunJ
rTtrunl ep TTTTUSI 9raunu

TT{}unJ 6JAUmU <-TnTuauopoJ \InTuauoq

'I TnTaqPI {61

'I TnTacie,[ ug axP?uozaxd
aoe} P nIlued
alltr{ueauoa ezTTTln 1€ eldope uoa 'TTlcuTlsTp BlTunuP (8)
'1tn1o5 'aTloun} ep PeJ nc ewtuouTs a?se aTlPcTTdE ap EerTunuaq
'tllwTTdp
afnpTnuPauso/Tpgocor1e6unTunreupslTlourq.rlTsnlug,gulTopf,rst{lTunclrT[lsfluottc.fE1l;lf1foodC 'I
11{ec11de Taun

r!0l3iils'tilHtlat'llltlll'lgllllls t'zT ItSlst$ I$ 1ll0tilJ 'llulls Iml

!n

l

$lntt, CIlClllt Sl 5lt:irtf ii]-o 6iii,rf;l, tritlll, tliletlII, SItU[IUtl sEnifll

- =_---\\ nenu

'"'"t''^' inc

{rMI .- . --it - ! -- sunt
z-
+.-
3.
I daci

\f nult

"> A=B;

Ftg. 1 . ? Eleffe:rt.e1+: t:;rv e dei:*r m-=r-lla g'.r.'m"t.i1 J 1i:at ie . Doneniul ,
gl codomenj.ul poi: f .r. j-lc:J,use ir: mui rilaj cjuprlnzitoare
ro, Yo. aceir

- bjjscfje { =sur-'.iE';t1,i++.1"i]-l*cLi€ i j 4. 1

vlr'4 Y, l l xi..;:" .'1x, =; {12} elemr

o apUXniceai taiepl3{f"ucan.*l!tiie{f}ufr"r--:!tlifl}*}bix-;eccalnlv€ee;i::ilXE .'., ! :.i aroclem in mod unlc el"ene

fotniat& dttr toate perechlle apart
apllcallel {fnnct.ieii f }i:ate in i rdj.ne :nvers6; r este bljectie dacl
gl nunai dacX extstH f-'. ise poate r*vedea lrrversa unei re1atil,) {Acr;

Conpuner€a apl":eai.iei,';K -' Y ru airijcalia g:'l *> Z este aplicatia 5. i
h=gof rX -> ? dsf1nitE pr-ln h{xi.eif iai }, Vsex"
Grafln* 6 ;l ;rne-r. apl.ir-:airi f r:'r.'irul.d ciltr tct"alitatea pereehllor
{xry} pentrl! ca}:+i r.r::f 1x,: $"e{{r",f {.n:i};xex}.
Aplicatia f ;:S'r:y *> jY.i'rileftj.i.ne),xt.,i (x} pentru xggl se
5,rrin f f'{,rxf}:"'f
nuregte re,strj{,rgJ,3 {f i}(1: acii,c5 multimea perechllor
txrf {x)}, xflxo err.e :lrr*lus! rr m:i.i3rir*a per,echil"or: {x,f ix) ), xgx,
DacE fo:X,,ci{ -> Y esto * i'.mctte dati, or':lce funct.ie f:X -> Y
pentru care f |X",'i,. r:epraz:i.rrt*. o prelu,Rgrre a iui" f., La nultlnea X,

l.l Huliiri {ffi}

1, Cennrnaaul l;xxl #$s,Ti*lr, MlrL"J'-i}r{ sSHf.yEr kj.rrdE$AbrlE, xnwxi*nnrrr
Caldlnalul un*! mulfirrii erlt€ cr rro[lrrne,*16)* ,*p€J]e ce"*a despre "cit

de srar6 sste m;.ritrmer* res"i.*:.:t ir;F,". iri casil] multi.miJ.or finlte,

cardlna-lul rurei *r*J.tj:ni c:str ri:"tiii.iil ds elsml:r:te itle nr-rltinii.
Xn cazul gul-1in:..t..i;r !'':\ni.in&n{J c rni:i.i:ri*te rie rilenente, daci exisi,i

o funcgle prin care €i.s$ent-el€ icii: se pot. irun€ ln corespondenii
biunivoci cu nul.i:L}nea rrus!€r'e].(-)r r.!dt udrils i{ { sau in corespondenti

blunivoe5 cu nr-rl.timea rumerelor:rTt.r'eE.i Z sai: iafj.crnal.e S) se spune cE
nultin.lLa sunt nusfir.e} iI e.

v-Yur Jv**fffi pJ aunds as (

tetuaqrfrwepg - gtuePuodsa:oc
$dr.[Tfn{ m uoTrrJ.Euf,do rN r,Lffi5rf,doud 'E
lluepuod;e:oo
t8r ) lv ,z ,'n-- i-x-- Ir r-(-\J=ts \ H ='{'N'Y)-- plsTxe E)Pp 'a

t TS n :o11rry{1nu e{ua.ra3;p e?sa (HrE) . TTq-T i

g apnTour €J€c H earrg{1nur nc lroder u3 c TTur,[tTnur BJe?uauaTdwos - 'elTuTJ roTr!

19c,, eldsap e
rlrstuv?{fuux

(rT ) {v tx '8 )x I xi=s \a

: (grq pa Jesarau alse nu) u TnT uTlJede. nu Ts s Tnf uTlredP
e?uaurala Te.re;1 eIe paur,rlTnu else Tu-tttftu Fnop E U\S eiuaJalTp -

(er ) {qaxFfv3xlx}=g.tJY 'y eaqflnr
uTp 1-.xga:x1
:g TnT TS U TnT TgS dtu".r1 115uer1,ralclneuqEnuolpJreedeegeJJc€aestaaTiaqlua-ulaTa I arJc

F?euuo; peuTlTnul a?sa 1$ v '( (x)

(sr ) [g a x nes/.r€ v 3 x I l:exi =8 f'l ? a.rso1r;gqcpax:end:1euaar

uTp FlEnrJoE eau'"rJ1nu :g Tgn'f(TxeSlTdtJaVsT'TWrn[aTtTTnaiJTlJJrPulrnTro$lJprpnEdfel ef,BJ efeluau€Ta rollqcered ea
a?ss
eaunTunal -
oJ rr$s$sa0 'n BTteoTlde als

t'r) Y3fl'g=V (=)g'=V ('11{e1a:
:FlIpIBao uX psntorlT alse aJEceT] P3ep eoTpp a?uawala rseatrece pcep aTtretTq
gaep Teumu TS Rosp ale6a Xuns 'TCiurTEtTpnTtuTPdunTopC.':B8a?TPS79y7:eVba'c- a1lq:a:ad e1
utr{uoc eaw.rJTrdr epnTsuT argtlnu e3TrO jE=V
'gpya sTun po{r u1 E
FJTIdnT Urq Tg S:ts "q JU:U 'e ;a11{p141-:dord ace;s11es EaunTznIcul
(zr)

(€r l ig ;P 'r 3eA) s =Y :g 1lurg{1nu

Tp ?r!eu€,Te TS dqrl gfelaae u! alsa E TTurFlTnu fp ?uouaTa a3Tro BsEp eJEOIPZUfJt
leunu g3 pcsp g eeul{Tnu u! aXsa E eaut{Tnu :PatmTznTJuT - InTuauroG
RsnTcuT Eurstr II.Ewlu
fifirnr{ 's

'aTTqEJqsnuou

?TuTItrf T€ ailquxg@nu ?TutJuT 'oXTI4J TuttInE aJ?u! oT{sqlsTp 'C Itltr\.

w€r3p; Fs TsTiuesa TJ BA roTaTeuues ETtoe? utr II.[uAEstt

'Tn1nnuT?uoo eerelnd ap lmrg
gc eunds es 'g alear ro1eJeunu eaql1nu nc p^TpsgTq Etuapudseroc uT
as arps eITqPrPwIueu €TTurflTnm're1nc11red uI'PT1gsf,pnmruou
ound 1od
eXgaunu as RTTq€JFmru nps FITUTJ exse nu a.rP3 au'glTne o

tr0liltis'tttultut'itiil$'IfltlTtg 6-ZT tElls!S !S iltl3il] ollgl8l$ Imlmlts .lllUtri

'

$rs6${n, {i${$t?E $i st$I$il{ 4 s': *$li:$!, nglfilru, npiteAT!1, tIfl,slult $E;;lr !

,5-d*i * 0

* rd:trBr*sgir{dfafe," FtrT
consl
"*"j#*S.i& d'b=Sle t2s) muitl

- as6tr"{at-{vitate"' { 21} proPr
{sJD} ti61""*",{g.,tsl {.db) f }FgI(glO rapor
t?2)
dcss &€'s atunel au8=8, ntxbs&" tn partieu.lar c
AJs-A
l[b=* ola
.d",Ls*E C]B=...8
defln
* 6&m& .&*d..trF dis-A propr

ecdJs frd1p*Js i 23l ecr€8

* d.isfribtJtlsjtats f:,nfiersecfi,a $: r*€{.ll?:{Jne6 .rii*1f d:.strjbutjve una 1
.f,49# #e 4-ILC,1 :
tz,nl algsb
,s]{&,!S1 * {Sl-s}U{dW} "dJi$lg) * {ffB}{l{S.,}s1 qener
Preclr
* .f;or:serjple Lu:. D* l{arga$: _
2,1 I
.d.,|:P*ffi {25)
1. I
^d}s*ffi-&
2. I
6" ffiq:*gxgr sfi &P$[eAfJtr o apli.calle de Ia c inutr&1me X in a].t"d
wuLiime Y poate fi prLvi*6 f& un element a} mul"tinii. tuturor I
pot fi defirri"te S.rrtre elementele nul$j,mii X gi cele
nnp3)e.i*ma$u1).:$i"1h.:;n: ife&gr.e ilulbXsea tugu.r'i:r apllcagfti.or din mulglmea X in I
mu-ll"l-eea 3 &re sardj.nalul En. Acest pu:let de vedere va fl utj.llzat
f$fft* *rrlt, Ln prezentarea un*r eo*eepte dln teerj"a eemnalelor: un 1
ssem-t*3 vn f3, grrivtrt ca ua ot;)uflet't dintr-o mr)"gise eare. stracturat$"
ura de'""enl spagiu de sesanale. I

}, -qTRIJffi{3RT A},#ffiNIEE (

:, Srggr pg ce#oErgl3, gffn$clrltftr o Jege de camwalg:e jnternd ln valal
**].$me* X {operatie j,rrterni bfnarX"} este o aplleatle g:)gxX -> N
nfit"std a<l:tiv {+}, unultipll,eativ { "p}*rseacr,hr lcu{xu,yn} alt senn cusi ar fl
ns'Ls:::=*err3o car* asocLasi fleeErel e}enente dln
de Xo

*l.fiseentul. S{x,Y}=zeX. Se scrle, de eaenplu, X*Y=r..

5 fl aFlteatie X -> H sste operag.le unarX iar o fip}{ceti{r .Xn *} X
*$ts *p.era$i.e al-ar6.

"rl!?mTtdSj'rnu T* aT%T€ E.r3uE R?RX0'.I TJ elsd ET{rreal0 i l{ <
ctr'€ ETt?,n^rassCI'6r rlEcrlr.;H*"ffi;*#11"T:"Hlfil.ttJTTq'FA
.'x
(6)$'ts)J,=t6'"1r. '2,
pe?qeTrdord nJ T]
N<
rg(sls <-> oger
uT!
osoe:-'rrg.frt?u*aeTugrdTerX"usrrctdrrldnjo.zSEpguSoouo$adddogenJaas:p?:6lorTu1aus.m:Trlotl]spiBsTrT?pBeFnd{Iettron"fTZtenred'aonne}3ro:ilETlBznn?aosrtasdrftofuetiroT3rerT$dpueTeggJTtrrlTTOddrunlrutuJ$'rrs6oopp1JleTrrctrueprcaeefxluttrJePro,lozulstE{eFuZ€ruc)a&."TrtJoXalt
,R?
'BeJT{Ttf,rg nc 1:ode: ug dtuElmr
un:
UaF€&lof I Bc r"r€E TEE T6e"rtug rcts$rmru €atr[t$u{ {I) mlml
1EZ
{d,z} t+'Ggl3 (r+zf) +XaE+ ("rf+xi <' {+'EglrSE'rt'2t
llATluTcosB - uT!

{oe} l+'8*\ S {r(+Jr) q* (+ '59} $;f'x aTac
r+ ru lacTqo Bp ,mBtrou {tsrsuTq} Bure?q eTtsrsdo o -
lTETSap B-s e.rrlc eU npfrre'u wp{{nu o a?se ,(+'6s} mdnpnus "I ioJ
nTe
eiue?uT TTlurdo nc TrmJruls I'u
tsz'
'{eir€lnedTTdns TfgleTrdo:d ep eTre€ o 13 uneerd
(?z
Tsu?.EIunFA'BpoOsILr6BTrrId'rrse:o{'rrqrTyoTlBr'dnCo'+dEr\n)?rr8?orTTAeFf?ErTTu{nt5Ee?pe'o4TTsrddro3uJCodrngu5rop'usTR.npmPr?on3uunTJIlxusaoca!rT"tsoJeorlcT@!ao3'TTec.rrrqIoeubaTp6
etm
ps(Tq&BlB.ronr'gn(ssBl'*.xr?sFts?s3Tr{lB:J6nO$}6uTT[ EetTaBajsllsFdTesx?rt'u.a.torudo8d?'epee?ucunus{ePF]lsTss.TIxxeaBCato8'?terexoludsltnGxtTeeoed?perultenTTdusrrdetoelolpdJ
s ic.:€11u1 nt€ercda nc xrodrr ug 'pcTrqefTs eTlTzo&oc ap e6el o {sz}

ussrd T6s tTuTJep E-F eraa ug eurgdpr o F?sa gJTJqabTe gmxonrxs o lzz'l
'EITE nc lrodur {rz}
{02}
gru1'6Igc31*r1aTt?rA1TrslrutslEBeAr.[fxapedtWneo'apece'}-*Baredr?n,ntmueE1I?Eao/updgtoofz?fJBreprTFsAeaSuTuarAoa'XTxTa?:*snEt<oq$eT-va;ra*tq.oarscpdTxtnpt:rjTqoxTFoegei?lrurBss:J.??orgWaxIldsuToeTAapcTeys?eg{ecTyu1TtzTc{?oetftasfTriBpeclddgesooptpoa'TlalteeSfnl?ToetpeTTf Tl:[OdIsItuo$o:uTdcg[
IHlilt
h i*tteu$ 'trutltt*t 'tlrtrI$ ':81fis* 11-aT ilil5ts ffi tu$]ut']]uillt

s[HAtt, tiEe$t?[ $t $tsTt;r H StRt
poal
1
9eff
L?-n ;il.Il;r, lcr.rit, [p*en*, sTnr[Ipn! i 6.
7.
{opera$ille s-au notat nultlplleatlv). orl.c€ s€n*Erup are doul 8.

reprezent&rt sl*pl.e ranarcablle nunite r€preeentArS. rogulate sau u

nuLtlplfc&rl la atlnga gi la dreapta. De exeuplu, nultlpLlca:rea la

st8nga ascclazt oricf,rui. eLmant BeSg transf,ormarea doflnltA pe Cq

care comettr dln Lnmutrflrea le st&nga cu I a eLeoentolor dln sS.

3. SwM,a ($o,+; este un semlgrmp pentm cane exLsta elenent
unltate a adic& pentru orice
pentn! (16o,+): x+e=x xgpo. Acest elenent se noteazd

o Vx€(efio, +), :r+Q=:s

{28}

gi. .cn tr pentru {Ho,. i.

4. $xwr.s ill Mut&,i-mea H a nunereLor naturale inzeetratd cu
opEra&,la de adunare.
{21 Huj.glmea lntrcgii"or in raport crr operalia de innulgire.
mel{!,3i;c}eFaletuotunruol"rt,ingetrfulnrlitlo6rSf dlneltseiaD(o'rlc-uuvrlin(te"al"1ofra'obe)t?nf),osrmtr efltee ^

S'

cu
ei"ubolurl dln tl, Conelderes cE mui"liqea Sr conglne gi cuvdntul vld
av&nd n*0 elmbolurl. D€flnl-e pc s" operasla & de concatenare
(Suxtapmere) a cuvlntel"or prln;

structilra i{"s1,,.s:}"seoeltae(enq1oon"o..idgkci&=(csl1o,.p..esr0a,t9ia1,es"t.e.91e)vi.dent asoclatlv&,
ter *1sa$tul nul. este cltvgntul vtd.

5. {qUgru* iG,+} Bste un nonoid 9n caro f,lecare olmlent aeimlte un
eleffisnt sj-setric numit opus {-x} pentru (G,+}:
Vxe(@,+) 3(-x) lx+{-s;=g (aS}

Sl ta'lere {x-i ! pentru {G,. } i

Prln urmas*" grupul sati"Eface propnl*tatea de -fncAidare ln raport
gu o.pern&ta internA, asoctativLtafa, ultata
eri.stenSa elemnentulul 9i
a lnverswS.ul p€rrtru eir"lce eLemEnt.

ttn Crup comutativ c(aa+reb=obp+6ar,aV!1aa,b€eGst)esenonutamt€Igrtaulgtlmplp.icaahotltvlang. i ale
Un grup
flnlt in
c6rui elmente sunt gansrats de un singur element, (generator) prin
r{dXeare }e f,oate puterltr-e se nus}egt€ grup grup
conllne un nuruAr finlt de elmente. monogan. un t$.nlt,

tln grr.lp Honsgen ftnlt sa nuue#tE grup clcl.tc. tntr-un grup flnlt
{rulti.pltcati"v} va exteta clr certltudl-ne o Elltere (l,ntreagli nr pentru
qqre uD eS.want rJ"dieat la acea pu&are {multlpldcat crr el tnsfiei de k
.or1) va furnLza etrw€ntuL unltate, €ea mai n!"c6 valoare a unul aetfel
dp S.ntrag s€ mr.gn@ste ordtnul erE&ontuLul, El.@entuL gensrator al
&rs, ev*dent" ordln-rrtr oga3. enr ordlnul urrul
grlp clellc gnrpulul, Un gnup

'-rplT{nn s}se rfrTFuT pTlErdo nJ xrdpJ u!
n:rlnffI SusGlfB B}s{:le PJ$C 'P'q = q'p slpp A"T'uXEtl}Ioc a18e TBlrF un

{r)d TnrctrrTod'y' llTxe€cTlx.:leEpu=T{xs}d3TTl8aTTrBlp?TJpExrEEczTEnq[oeudrTo(PdAeT€p-nep[uTTquonuu] YI
tTtetuoedlEerelr?odc soTFF?JsTT:lrielFdf,wg€;:rJ1t#u"fTe'AleTu?1€p?nldnuoaolee(lJIeeoJrFn,uenleeEezoeuo&llorod]
I Qt un-4uTp s?{remTa n$ uxri BilnTFuaulp ep etreJTrlpn {E} l

'sxeArF ors nu Pa.4lrnc{I {

'I 91ss erEa 6x TnT TnTn'iuriTiTlao3 ETlderxe tl3 TTnu TT{usTcTtso3 'EA
TuT'tirllot)rlTiipncJTSpAleBrdTFTqcrrrcoduso{pu.rrTxT''Ntgposdatr1u€+1X1ue6;63n1nplnrceo:TlaesnXrdp'Foeou@X€ftsg'psaawrsrTrsn8uTpTnaoJce:edolp.IuToeTdBpTporeTduun!tTuspenArpreTdeclnplfqlrdcToTletpulslenoTdnop
ceJ e, sns TnR ap aTTTtETa:r ug .rr:11{uaTsTJao3 ETssnpord Tt eTerrrg E

c-f o.r pT1

{8e} t .Kibrd 3 3* ix)b ixid n]
3s.
(tsl pnirF+tg; H.,o'S S*r*td3'* (.x)b+ ir)d 'g
nc
:xoTreo{ryTod * *rTXTTwriI sp ;S a.:vrmpe ap afEnzn eTTT{pJsdo
ts llrw*zuT a:!ss {x}*,9 totr'r.i X rl,s?e un-J?rg T{uayclgeoc {ei
r}r I eTTqpIrEA ug JaTsilF{rqJTT*d TB (x}*t nTleqnffioe Tnfaul (e)
PzRl
"H JoTnp{iei olTuT}ap luns EerTlfnm3 Tg ?uet

ErJErmpp o.Ilc w! H sTnpoffi T:fi'r?8a3 sp :oraseTr TnTauI (I) lld{lxl 'l

{sl x'E+r('"4*x' l.z+i1 "d y-F"{'x* l.z+Al'x (= {" o+'I:) gz ':{'x 6gr
r6"1Eunpp i:q: qfeg g^T?nqT.r:lsTp a?se eerT{fmmf
NlTfls ETT
:'i1."i.s3ttdTlTrm dn36Tuas e?sa ("I) - np8
uETTeqs AT?Tp€ dn":6 o?se (+'I) - Pnq
gcep IeuT
ele ("+'I) 'L Itilt
6{L{e1?(iT 1"r'$c.:r.+,:}i: r/ir{xil ii;r *;rgrqa61e €rlr?JrrJlg ffiEXI

*;:i#s%ffifi"ffi*
URoldr]p'I.dT: n$IX6In'lTHrsJr€Ese;.}ra+^eAl,o:{1]rir$aJnr€rTm€Fr:gra*:ai?}puepI?Xr{orucTiIlresd.TrrLaert$rarulE'e"pxsenar6uyrlrfolfTsJnennTutiprl)EJepdo'eIIrEed?utsnUepcEel
nc '9
eD

'ro?ereugD

terlrTe [m a?B# s6r 6v$par; 6?sreuaF 63TT;]{e rmdrubgns utp plstroJ aleod

;' tlt3ts 'tttmt;au '{il?ttg 'rg!t$* t. t -t'[ icilils 15 til[rul3']ililIs

,&

sEffttt, cttcutlt si ilslttr t?-tt, mtint, ltturt, *uctrll, s'lucl,ll stillllt, t

Dacd a,b€I sunt elemgnte nenule gl a.b=0 e6 Bpune c{ a gl b sunt Un corP
dlvlsorl prcpril al lul zsro. vn lnel cmutatlv ff,rl dlvizori al lul
uero se nuregte dmanlu de lntagrttate sau, si-nplu, duenlu. llulti.Dea de I).
l1(x) deflnit6 in areuplul anterlor eete dmenlu de lntegritate.
8e rnngte caracteristlc[ a lnal.ulul I, cel na1 nic i.ntreg natural calolB
q pentru care q.x=0, VxeI. Un eleuent ael este dlvlzor proprlu al lui
c(a€Iltdfeaclf,, bex=lcsaqa'l).b€EIlmaesntfteelleinacAgtl cb=ad.lbn9I1 a nu eete lnvereabll ln I 11. tE
gunt in r
prire lntre ele clr
au ntcl un dlvlzor propriu icnonfuna.sttnfeal ciensct&ctaaz.ax+9b1.bpl.satlsfac
dactr nu 91
lu1 B6zout: Erlgttr x gl y
toorffia €x

9. Ipftl, lntr-un lnel conutatlv, un ldeal Id este o nul!ture nevldi, ruJ

lnchlst tn raport c-rr sl[a gl cu nultipllcarea su orlce elenent dln dir
lre
1nel: iL, tzeIdo tL, zzel *) (rrir+rri, ) gI'4 tssl
rz. Po
Un ldoel- proprlu nu crontlne unltatea fu raport cu i,nrulttrea dln
lnel clci a].tfel fd ar colnclde ctr f (orlce produs al orlctrul elenent pol
d1n I cu rutltatsa (dln Id) trebule sl fle Ln t decl f Sl Id coincid.
Un ldeal nu sste acelagl lucnl cu un sublnal sl deoarece acesta dln in cal
gatlEf,ace
urcf, nrraal: r) po}1n€t
(s.+sr) , (sr.a"l e9l {3r}
a, sz€fi{ :oeficl
tltr
8€ spuns cd un element r2€I dlvlde un elenent rt€I dacd exlsttr 13
anter!
estfeL inc$t:
'€*tentr
rtrr ldear eate pnnctu:J=k* lli'.T'* elsrent dln idear care
dlvlde orlce a1t eloent d1n ldeal. Cu alte cpuvrllnnteln, neulelmtlernetecleu Po:
ldealului sunt generata de un elenent al lul
celelalte eLem*nte a].e ldealuluL. prtarl
tn
fO. tn nruI. (K,*,.] egte urarplonret lculntr*cra)reseeolermgaennltzeeLaeznl efan!utrlede(cen.letl
dlferlte de unltatea ln cf(x)
ca grup:
"'r€f,{o} -) 31 | x.L-x, 3x-t I x.x-r=! (3s) rrcducl

Ih corp 1n care anbeLe operatLl sunt comutatlve se nrne6te corp ud€ f,
c@utatlv sau c6qp. un corp, conslderat ca lnel, est€ donenlu de tf1(x)
tntegnttate (nu axl-attr divlzori al lul zero); singrurele ldeale eunt {0}
aL {r}. Ca1 mal nlc lntreg natural pentru care }1.1=0 egte PE
caractarlst:ca corpulul. eor?urlle flnlte au caracterlatlci flnltl
regt:
(reclproca nu egte adevtrratf,). DacI un corl, X este de caractartstlctr
utde g
I'1, atunci tt egte prJ.n. tn general l{.a=0 VaeX. lntr-un corp de factor

caractsrlstlcd H ssta adevirat ctr
(a+blr=at+bt, Va,b€f,"

e18e (s)6 EsBq '{x}6 TnT 'Tnu alEe (x)r Tnlter ('r(}I)rITnTTnTrnTpP3rr6oeltpsmt

TnpEr6 ?Ecap cTr TEU alse
(x)r+(x16(t)0=(x) I
!1S9J
nc TTrT{rRdq ulmroet cTros Blcod es (x)O Tt (t)t eoTro
ru?ued (
"(x)I TnI TE Tufid TTro1cpJ ceillu e3
ulxol ep TTro?cEI rBT'elT&rtTp BrTqTpnporT otrBolrTTod ?uns r(xrl)Jrxl)ar;prtml !

(r,s) ,,[ (x)t)'''q"[ (x) 3J]'.t (x) tJl = (x)J 8
T'IrolcTsnJTuI[l
lrlnB nu :BlJoI u1 lrgrd erndrccsap aTlJtolodDoTsTpquTnlon(pIoIJ.rDT n
eJeJ eTffiourlod
xueFte e
1t o ed T€unu TrozTATp Ec aXTrpc Bcpp fFgTlrnwJf 6Xse EouTIod un A
TrEr TEU TeTeo TnlueTaTJeoc ncsp eruor e'BleealPnrlTuuense(?xsa)IxTTnnuTouEIloTdrelnd
'(lErTJ uoutlod rm olnpoq Tenluena) c:
l.aro:pllmrpourmllodur.lnr scn'prord'lzn16eurmrorgr ul ATlESnroc feuT rm 'ro1ra1ue
nc ?rode.r ed rolercouglod eerg{pq (

1 prd:oc UTI

pu€rrE x T'nIT ?psgl3dr.rToEoU rucTle:redreelntdue"p'0le=?TuB)zetprdeTr Tei?usoeT'T(cnsTu)I8IeuoTolnuTeoTmrcuoTqoliatooudsT .
Tnp€re
?ui
(ec) of z+ ' ' '+xrD+ot. (x)J
,rrrct ep elserdre o e6epdug Bs I drot UT
rryrd {wumnrca .lrcr xn rm ed uouTTod 'zI
w urrxut ffIrE} (s
rd ilrwlod
uT
routTod un oTnpor elTu;Jep pq'dTtrJgaIn'eenepuoarudealir ep 'a1es TTqf?cnperT
'Ep
sms 'drtc un-rXuTp
ttruepTteoc nc 'u 'lsrtt ITuTJ pe:6 ep rolemouTlod EarftTnn (Sl 3€J
'T6er1ug roTerauru pesfl1nu
Sg n oppor TrnXser ep rofeseTc p $ueXuserder elprepTtuoc rf!u
TEJnrlsond roXsecu eTeXuarrTe l{dtru-?fir" e"?'sIo'0,}t*opneperuq[TlTnnsnfip(oetdl
TNTlTnr To IT'T
NodEr uf rtr.rd rlqnu n nr TEn
'Trndlxa o:rcaTt luns e4fpuul 10 eretmpe ep a1r;{eredo nt
uodEr rg ;r eralduoc Ts 'u aTEor roTerounu elTuttrTl.{ {r) ttrdtrxX 'II Paf
Tnr
T.&ftBo cserBu es (wtrd=t{} 3ltFfieTe t{ '(nldc Ezgalou as 16 e1o1ca lunr
puB F elTuTtr a11rndry'
'(r cp Itl:

lTrsJTp) nT.rdord drocgne rm TcTu eu1{uoc nu IcEp wgrd else X droc un

rnr$trs 'tilllrilt 'rrrilil 'tunit 9r-zT illlsls Is lltntilt 'llutBs

silHr.t, cticulrt sl slslilt L2-tu it,lun, r[*IIt, rptlctltt, s'tutlutl It:u

2,2 $tructuri cu operatii interne Ei externe (
{
1. H0[XIIfi (r,r)' (grup peste lnel) este un grup (tl,+) inzestrat cu
o lege de conpozllle eraxpteorrtnci u(reelperneeznetenlteatudnPulriinnealli(tuI,r+a,r'e)a, 3. I
el.mentelor) ln
einbolurllor (
operaliile interne gi externe fiind dletributlve unele fagi de altele:
a,bef, x,y,ef => z=ax€I,, a(x+y\=ax+bxt a(bx)=(a.b)x (reg

Noiiunea de modul este o generallzare a celei de sPaliu liniar EIc
(vectorial) care va fi definlti in continuare 5i pentru care inelul
egte inlocult de un c6np. Un submodul S- aI unul nodul t{ se definegte d{ cf,1
subspa!,iul linlar;
analog cu *yt E7€9ai r|t rael =) lrtsr+rrsr) egt ulxll

Un nodul (ll,I) se numegte de tlp finlt sau flnlt generat, cu (
orlce element poate fl reprezentat \
gereratoril OaeU dac{ gi nunal dactr
prln: l1g.
r=Eor(n)0r uFr
l.l
c, nefllnd ln red necesar unlcl, Elenentele {0',tr=1,"..,n} sunt linlar
c
independente daci c) cl=O, Vj=l,2,,.",tj (38)
dlrtr
Egr0r=O
tn cazul ln care or(n) sunt unici pentru toti ne}l, nodulul se
nusegte 7iber, iar submultlmea {S',0=,. . .,O-}gH constituie o bazE a
nsdurulul {s,r} . rnrr-un
ffi:i,:;"'=, Gr=. (3e)

0, reprezentind o bazf,. Un nodul ll"ber este aproape aeelagl lucru ca
w qpatiu llnlar. nangul (uinnucl anzoudlusl pllbaeliirloersltienleagrael cu nunirul de
utlllzeazi
eleente dln orice bazd Be
teroenul de dinanslune). ln cazul a doui suboodule s' gl 8, a ciror
reunlune este il gl a ciror lnterseclie este suhnodulul nul se spune cE
tl .Eete srma directi a lul S, 9i s=.

2. S,ssRttalfr (11 Dlscut&r deosebirea fundanentaltr dintre spa9llle
llnlare gl module. fntr-un spallu llnlar L un subspatlu So avind
aceea8i dimnsiune qr L trebuie eE coincldi cu L. spre deoseblre
de aceet caz, lntr-un nodul Llber {ll,I) un submodul poate avea
acela$l rang p$oiattoetuflglcosni gniduecraotlnmclodddulcpueHs.te
{2} orice cu H el insugl.
lne1

6 ge utlllzeazf, gi notagia l{ pentru nodul, L pentru spatiul

v.ectorlal., A pentru algebrtr,

TNT

tol l (.fel z*x*-x,r+f*ttt(rcR-)(z= +(ii+) x*x\E( r'E:e1*rtatr+efTz'*nyxAr=qzz*o'Ap('tP'rtc+1ExlJ ea^l
9rT(
' puE,
ar.r
B TlrqTrl8Tp
'ebtratmps TS aATlcTcos€ prrtTJ elTT{pxedd 'TluerueTX] x(ex'ta)TlrByrdo Er3
o .l JOJ
$3 lcal8ozlrI tx'+'t] I6ut un ex6e (dr!B3 alsed PZY
3p
'TETroloeA nTlBds Tg dJo3 'TauT 'dnr6 'dnr6Tuas E'I"6Tr
93
ppqerrftdr
{se
'roxPEm
e€
fnfgx6prgd uI TrrelAe: w)A 'gTalT€ m ?rodEr uT eletm BATlnqT.rXsTp
as
gnT(sIt6:e(l.TIoloatAddaT:eo?arotTt)utoFopIqBd(dErroqyfqpcsUcluaTolesTrpelJsdTsoo6dTonTJT:oT6foi)Tpm(rnorT3d'oroql)pr'yJ(E'l{'sr+vEer'ezXuruo)ell,rcn(ad+lpau'p}Ig)ucudlwTrrdtnJnu6gnlrnemlsruTroe8az?lruasd€arudseaT:'s)a
(e
arftrTtxrl 'o?pe(ErtTl{eryrepdeolulee1e'eTromnscgElsue.red?nxa:6aTeptEpraTdlporeEdlEortprpeTpsueoqc feulosBo
:eT
tf e.I"eTctnTflnrgIepW elsed Tnpou terepTsrFc TI eleod I€epT un {gl
rgeed 1E1I
Tnpon lpJapTsrroc TI eXpod 1au1 actrro {7} nc

llllsts It ilt0iln 'tlut$s a15r
TNT

r€T

: ar

, (.

gAJ€

nc1

lHl

s$llrt, cllclltrt $ stslilt L2-tg iftuir, lfr.rru, ttucnn, slruclutl I

il

tfr

{

illi

l

2.3 Wtiul liniu privit rai h detaliu; proFiet[li 2.
Datorlttr lnportantet gale deosebite in teorla esnalelor, voD
lnursra, pentru inceput, toate proprlettrtlle acestulal flnl
(x+yl +z=x+1y+zl (asoetatlvltatel
x330y(-=|ry)x++|LOx+-Gx(-ox(ne)ux-t6lae'tt(idevixuitnsaitttdae1linelv, erellltl defl
Yx'ryrze L, Varbe f -> ,3{aaa((rb+x+Ix!)lyrxl.-x==(.aaax. xxbi)tbax(teycx(((lsdadcisigdeoltrcurilin:abbittuulavtttilievtva)iit,ttaeatltee)),,
Iini
.dln care_rezultf, conseclntele: ax=Q <=> x=O sau a=O (12)
3.
a.O=0, 0.x=0, (-1) .x-t (-x), dacl

unltatea dln (L,+) s-a notat cu U, cea dln (R,+) cu 0,lar cea din t...

tx,.) cu 1. tn coitlnuare se va utlllza aceeagl notalle. O, pentru sub
.rDele unltltl'fn raport cu duoa deel 10 prinrl caz este vorba de un eler
alcent (raqctor') nu! lar ln al dollea caz de un gcalar nuL.
rD&
theorl un eFatlu Linlar ee noteazt ntnai cu slnbolul grupulul {L},
trect
gstatenga corpulul de scalari (c eau C) effnO subi,nleleaei. cela
a.
r. Crnrn BDerr DE sprtrr r.uTARt nnoncrnrr.E
- C, ul 1tt=nnEr prtu) : spalllle llnlare ale tl-uplelor (N eete un
nrulr natural) respectlv aecrrenielor (tnflnlte) de nunere dLn mulllmea
}.
"sIe*{0fa, c, o. . ,H*1} deftnlte peste cfupul {X,*rodl. .rodf Adunarea l-uplelor
elment cu elemnt, u.
nodrrlo
- RI, C : spagille linlare ale tl-uplelor (X reprezlntl un nurf,r
natural flxat) ordonate de cnrm(tenrerraepaolert respectlv conplexe, deflnl-te
etr adunarea elenentelor de
peote corpul R respectlv
acel-agRl.r',a(n|g :disnpNag-ullplelellenilalnranualllieregalrcuurislocraloarrdlodnlanteRdreeenpun€ecrtclvreCa)l..e
rflp€ctlv c@plexe, deflnlte pe corpul B respectlv c (la fel ca nai
ans);
- ll(n,n) ; epatiul llniar al natrlcalor de ordLn nurn in raport cu
admarea ntrlcelor gl irurulttrea cu scalarl dln c;
- -F I gpallul llntar al fwrctlilor deflnlte pe in ragort
funcglllor et i.nnul9trea cu gcalarl dln C; C crl
a4tAarea
- Lr: epatlul func9lonaleLor llnlare dqflnlte pe gpatlul llniar L.
Acest sp3$lu se nunegte duaLul algebrlc aL lut L,
- t z spaglul functlllor de nodul lntegrabtl deflnlte pe R cu
vslori ln C.

9 Elmente1e unul gpallu tlnlar sb rat nr[esc I'vectorlrr.

TBTuTT luns nu ,(O'T'Z) '.(O'T'O) ''{0'T'1) 'rJ u3 Tlu€puedepuT '] ncI
11ro1ca6 11; lTdt{xxfl
.T
E FrpTuTT aTlEuTquloc o €o Elr,rrdxe aleod as ,"t""rtoitt;H:::
'erluTp a?sa TrolcaA ap PxleTTuqTTlTeluu'rn{Tsnurxq nc
fnrm Rcep gluapuadap JeTUTT 'T4uapuadepul no

apun Teu
aTEe
('r) rq=te <= Txrqfl=rrot" {=*
'(
:Eclpp ecTtm xuns aluepuadapul rsTuTT e?,uetsaTa ep .
op aJpTuTI ag;fuulquoC 'EluapuadapuT .I€TUTI a?s8 glquq, a0ryllnuqns
elTu
TeerFoJlroaRAcpappF?(uFelpTuq€edxapguuFnuropTuunTeTselpsTaTqTeTrgEuTrnuTulI pp;lffeudfslutfr[pet{ft"t"T"rxu""o}
Jrrr--.
r.t
(s?) tr' ' ' t z'a=F g=rP 4- 6=rxre f roTe
eeEt
uelgue)puuaid("a"pZuTx.'rrexTuTeTlallumesuloxTeTeugTTc n1{eds pm :pcep Tpunu 1$ pcep
aunds pugul{rede (11u15 rgunu tmg
uo6 Euq.If,II tsif,goIXdSoII '€
'(1)
'cla tl-t[]dre'x3=11)x B[uoJ ep aclpolradlstAc rofTTros InTtPds tma
al€T{Ipd'neTlupeAuTorpepunm-r?TuT!trEscnTc?aTTTEauupn :o111fcunI rul
'expTuTT roTTT{n1os Tnltrpds
U.rP
lnlfeds
(zr
'areTuTT
,'{(
e{ue:ag1p nc nes aslelfua:ag;1p TTtBnca TaIm roTTT{n1os plfeds
lezlcard nTuauop tm ad alluggap ( It']
'd!1 lTunup un ep alTqeuns nEs anuTluoc ro111{cm3 TnTlEds
(e
'pe:6 aclro ap xoTeuluoulTod fnlleds
'lTuTtuT nes 1TUTI b/\

'duer un-r?ug 1{ualclIaoc nc ut pEr6 ep roTeulEouTlod TnT{pds fiilr
'gpeol.red lseaace ap acTpoTred roTTTl3unI TnTteds -
TT{Bds elTV

'=I nc apTouToo a?Bod tT 're1nc11red uI J =T <- 'T:J
erpTr5f roTTJEuuolsuexl Jornxnl Tp (uJ Iu)Tp-r€TTJuETT1E3ps1n{ecdrosT:e("aIr"l{Ip)Duttg-

18 ro111fec11de ee:etmpe nt Uoder

TI ld sns TPU eXpreunu€t arETuTT'eaTleTuTntaesdespaTlTEteod;s, alpJapTsuo3

xilt usy7

Illttt0lls 'illt]ndt 'nfillt 'lllllfit 6I.ZT IBISIS IS lltmu3'lllt$S

I tZ-ro iurlnt, rtltlIt, rpr.rctTu, sIIucIu* sfllil.t, I

$mtf, ottSnII 3t stslt;t r. Er
1q
(21 runciille ps
ho
elr, , )"re9 1,:
S€
aunt llniar indepen&nte daci gl nunai dac6 tr. sunt dlstincte. dln cel

5. Dnmtt r"ta ll@,Rtci a unul spagiu liniar este cardlnalut celei ol
ral narl nrrltini de elenente liniar lndependente din spatlul linlar hprop(trticc

oonslderat. ln caar
ttr subspatlu aI unul spatiu l.lniar L eete o eubnul9ine ScL inchistr
In raopfoarnt lfltleofpinerltadtilededvlencstopraiglliunl lLar(lsclnldreVpxe,nl€deLnSti i coresP(
a,b€K, ax+byes) ' ounegtr
anbspatiu care contlne toate elementele de forna: {xa} genereazX un
Cins
*=fr rrrr, gxK ({s}
pittratl
6. Bazi Eloentel"e (vectorli) {or}, 1=1,2,...n al unui spa$iu Linlar
L formeazl o Mzd intr-un subspaliu SsL dac6 toate elenentele deorr
{vectorl1} dln I ernt exprlnate'unJc eub fornf, de ccunblnaiil linlare
fintto de o.. orice attd bazi dtn I are acelagl nunir de elenente, Ot
acegtea prtfurd fi exprinate sub fornE de combtnatii llnlare afg
Un epa$lu Linlar este flnlt dirnenEi.onal daci -rY'(4
neluenndenrutellodrebealzeeinte{not*e}'ao.l (ortc5rei) baze este ftnlt. Acest hundr
reprezlntl dlmenslunea Epatlulul, E
rtrltlr
2,1 Transton'Iri liniare
(vector
o aplicatie T:I -> x se nuneste transfornare a nultlmii X. XullLnea
$r
transforntrrilor care pot fl definlte pe o nul$i-ne X dati esle senlgruP
ln raport cu operatla de conpunere a transfornlrilor: 61 eet*
(r.G) ix) -"(G(x) ) (46)
ti s€
Un norflsm (honomorflsn) de multtuti structurate algebrlc este o
apltcalie T i,ntre douX mulgrrnl etructurate algebrlc care pdstreazi tl
ln general diferlte, din cele doul nnrltiml:
opera9111e, 7.?&t*) * Ltl*t\ I f(x+y) =Itx) (17)
*tr(yl

DacI, in plus, aplicatia T este bljectlv5, ea se nutr€gte

izomrfisn.

fl ln1foinRiteio,arcslumbcspi,adllielegigdiLnbeanzseiuleneaacgaslgtoerbaricsdunat dueneflln'nttuelg(imini poate
unel etnrcturl topologlca) nr.uai foLoslnd sune flnlts.
lipsa

nslcnu r [rDueJ ",'aujtal Il €sdTI

rg 6 aXeod

:nc gzealou as T6 al6aurr

e=tgr fl t {*F a=Fslf g ( Lll

TS Is artuTp:tretrt|reddeeeqrn:arluugr)oTeolocmE[sstmTTprm[serTetlp1d6Eaq{n[*sTunprluPelJdaxFTppTnPuqnes " [g 9zParl

{ TE } nrluaa {o- (x)J I tzsx} r ! (&) re)J o oxga
ha:,i,'eegTfea1luoeuuBazlEaazaTlTpTolnl
0=(r).1 e-r€3 <- h:l ss '(h uTp rnu Tnrolce^) te'l
areTuTT TJBeroJsIr€Jl
ne6eratruT h uTp PlTnlTlsuoc eeu;ltnu druby
Tsrm Te neftnu ulrd PAl[tT

(os) :rA (ox. l,,J):.y,tg *ff, ";;fr;..''tc7.7,e* IErlrlu
PJEP
;erec nrguad h .- ht.r. aTtEcTTdP o olge WgTgiFTTIl.nr aTlecTTile o
arE a
'uxu aunTsusdTp ep g.replq6unlde.p eclrlcu o ep
'a1ua&
EoueTo1nssTTTeJplrpsxcrdepneueTdarupTuToTJTapeXlop€BsedsIR€rPT€sBtTueOrFJuxTeToepsJ&eoosrpr'toaprele;utru?rollospuelJ{EoEXz1uATEfT'uXrul*ltD=umoau[sf1E'cnqnecldpeacnsp1a'{pruxeoTxecXucleT=uerldTqlaDeod'ltrol)lPoe-Thatrmul=lsuTTheTlsu:re!doocruleegTIcnr{pauTuada{p!BopuTdo9ensdXlzgseueoarau1!rmo1urcducI eI"TuT
aTe?ue!
(sr) 6),rq+ (x)*6e= (rQ+xe)f,'/ gc eelelelrdord r€TutT

: (xaq'EA) h ofp A 19 r ecTro uluad (E?)
pugAs h <- k,.l aifecrrdp o etse (sxETuTT t1leds ap urglrouonoql h
rsTuTT fnTteds et h x€tul1 nlfeds un eI ap WeTuTt axearrc;.suer? o un FzE€
'(sarqr
{8r) r- [ {x)J] * (r-x)J F* (n\tr PsTrlcul
reTuTT
:pundseroc eE aTeEraAuT Tg ELlerd ere1u;1 TTtede Pnop aTao uTp Telac
a?Tral:Tp le.raue6 uE 'eTTtPXTun 'olsTJrqozT qns 'Fc RzpeJ?auouep es
'e?J
.((E.Z).(t.Z) <*l- E'e ac fftl rr5 9'z+g'z <- 9+€ <= 9'z <- I 'g'e
<- €) EarTtrTnqtr[ n5 XtodEr u! euttTnul Tgsei3e nrlued EET}.rorcooq IlilSmls
slre nu tep'eo:etmpB ns lrode: u1 JoTTDar?u! EeuFtrTrI[ n:lued
ap ETJro0lol[oq e?Ee P?upls'(u+oo'uo) nc Parpc11d1]tnH (Z)
Trndinrb
<- ("+ul :6oT ltrlTreooT 'I
ulfermg s?se usTtrolozT ep lTqPc.rEuer nlduare un {I) flfigxfl

ilililus 'illllndt 'illlr]r'ltulm lz-zT l$lsts t$ lltRiln'llillls

stilltl, clltull 3l slslttt TL2-2, murlr, tttttn, ttltctltt, sllucrutl s,,

2. Smrnfn (1) gtnrctura do spaitu flnlar are o deosebltl
datorlt[ poslbilltitli
lryortanle de a rsprezqnta slenentele sale
ln Fd unlc, atuncl c&rd aete poslbil' sub forni de conb1na911
ltnlars flnlte de vectorl apartlnfuid unel baze. ptr
(2) tle va prtea vorbl de nrue lnflnlte dupl ce se va introduce 9rl
conceptul (topologlc) dr convergentl.
ure{3l 1nlrnltttrr-iundsopeailrr.uenlnteft,nlcthdlalnrrnavstlondnaul,n lndependenla llnlartr a (r
cardinal guflclent de
nar6, nu asigurl poslbtlltatea generlrll orlctnrl elernent d{'l gr
ca o ooSlnatle flnlttr & gctlreinntneull91lnrehralndveepcetnoderlnltoer.
spa(glltu operattlle de adtrnare gi t.

' ecalarllor pot fl oricare alte operagl-{ asoclatlve e9u1ndelsgttrlpbruotdlutreee
partlcular
cue ed rsl$rre cqractenrl llnlar, ln
rodulo rn nutrr flxat.

3. $TRUCT{NI $POIOGICE o
d
3.1 Topologia; $patii topologice
s
Topologir are drcpt scroP gsnerallzarea no9lunllor de dlstangtr'
I
vecintEntaeteseenltli,notentbapLoldolgnlegtrondeetfrllnaitef,ucpleldolann{ulItaLnneufltlnrelpore.rzrei.cnatr{e.o
col.eclie de suhtltlnl ale acestala. €

l. hmf"fsr o sulgLuo de p6r!1 t ale unel nultini x constltuie o I
topologle pe x dacl gatisface conditllle:
este uultine dln I
- Orlce iniersectle finitd de multi-ltl din t
adicl:
,{,l'et -> A.fltoer (s4)

- Orlce raurlung de ml9lnl dln r (flnLte, inflnit ntrtErabilc sau
nenuntrrablls) aPar9ln lul t.

de s9uebntual9l fpuoral taeleEpulunel cItr o toJrologle t pe o ntltlre I este o colectle

i.nchlsd ln--raport cu fornarea reunlunilor
arbltrare gl a lntereactillor flnltel2.

ln 12 orlce topologla pp€trxrtiEloErteulnneciluususl$i,lnnlmxrl,ttrnoe=ap1p1l)r,tgllol nrulluti-1nexa.
partlcular, n$tLrea
cilpurl dln fuul elenente to-{Q,X} satiefac proprfetf,tlle topologlol
cp(guornnocgstuttleltudrl[en],dvelcdtaoeripteeoalo[loctpltsonlsodtglidlllctrorespptorfe,lozrglenl{gtlfop, ruenct tfclevalrtsodp!6.oI'lgeoxogttrl@ade1lrrft:idfn,t!1clc"nrdgeltnol

'(/.6) eTTTtTpuoc uTp TEU[ru Psnpep Fsuf TI sleod EB JaTlTuT;,ap oP BtoTts' 0b(
?deJp Trolne TTun ap P?exopTsuoc a19e alu[3a8uo3 oJlu[P EEIId at
uTp
't1:i-:t1 :x 'p?Bp flr5lTnu
r10
:"ra1a{ugcasuoc cnpap aB arec uTp
TOT
(rs) {FnfnTqbLmTra eale2Ffe6su|l. (Q!'?ql=Pe+ {(.p='c) )p0=, (q'e].p
(e'B)p P'reTu

: (ur€quepuT1) auoTxP aTareolguun rorTt
eop1sTles a:EJ +U <- Xx1lp eTlpcTTdB o a?Be (EfuelsTp) EcTrloll
'H.rcTnJTX.JEd aT6oTodo? o pugJeuao eclrleu 'ec16o1odo1 eTtf
TTfEd€ ep refnpT?red zBc un tsc aSyrlrd TJ ld acTJleu sTTTtedS
'xaF9rd nE8
scTrnpt{ XsoJi e '3?a afEoJ aretmu ap grn-r1s nes lldcunJ 'axaTduoa
eratnu 'eTeor aJaunu :dTa TgpTaca ep ecT?€u6?eu sXueuoTo arXu! (rs)
EtuBXsTp o TUTI€p a€lseeordsaasdFJpzlP€AszreTsTquo:aBuaa6rEoJ Tac '-u uETpTIsno TnTleds 1uT
pJTrlaE ap eaunl{og c eT
uTp plueesTp op
o P?u
acTrlau TTlsds iecHlan z'I
.AJPOI
"PJTr?au ap
'Plup
TTunTiou TnTpatl[laluT uTJd p6u; a?6e paraTpu[TJalppftanpqptsoou aTTaqlplnuauAupolscTTx€gu Taa
o TuTJep aEnpoJ
'X air;lTnu o ad a16o1odo1 'g ntP alueuaTa ap (PlTgprPunuau ATsnrsuT) eATlnq

aunTunar o ec aTrc$ aleod as r6)C psTt{csap arg{1nu TaacuTnrosFcgeaFlzeplaqlrOdord Tg Jo
aXss TT6oTodof
tl3 TetT6oTodoX E aurFtTn[qns o 'elt

A)xg '(x\Aao trFP 1I
ep ?u€
EcTebpocloppdoTl eTunnurtpTd6s pua1eppRxpceT1Tpnq€1cn'sT1acaptmTad€ls:lao)u1a1ntfdrg.1rfegTsurTu1'rc(ooaxA)lFunolxuonsamcqdXgFnerzp?eleosel?glaoEuturloFcdaiaT6AnaSn(aa?'Xs)s E PjreT

L)g 'A{3x FJTpE 'x lnlatmd aJnpoJ

euTluoc '6e'ra€lee1Ggue7sTsqacrsr aapl$1aqm{1runuasTa((rm'gE) ourgfgruerdns e3Tro ggx lcund TTleur
cT6oTodo? nlJeds un-Jlul ETE9 e
Tnrm e
'gsTqesap err|1r;w PlTCteS
b?Seurnu as r 1a16o1odo1 aug{rBde erPc X TnT B q eury{fruqns eJTrO
rml3illr
'rilrlilImlsfrurs 'trrnrrdu 'rrrultl Ez-zT ltilsts I$ iltotilt ']llHl$

sHtttt, cltt0llt sI stslflt L2-zo rtrnrir, tfuru, tpr.rcun, sltucrnr sttl

d(a, b) >0, sult
fdlld(((a(aia,,nDbye))gbaz=r)1dl:idldbt((a,aaat,,)ecba))-pdla<(dtcr(,ubal)' abItl nl!(
+d(b', bl (581 defi
defl
er u luil
3,1
Perech€a (X,d)'. se nuu€gte spatlu netrlc, iar elementele lul se
ilD€ac puncta, p€nl
cu ajutorul notlunil de netrlci se poate deftni o topologle pe
url9lrea x: nultlnlle deschlse sunt generale de netrl.ci prin deflnirea int
blrelor deechlse s-(x). B1la deschisi q.(x) de raz6 r gl centru x este ft.
fornati din rnudletimeleeamteugntut,u(rxlo'r)r€X-el:ale€nezntIedlo(ra,dxin) x aflate la 'in{
o distantE nai
rlci dec&t (rl (5el nect

spunen ct netrlca d induce o topologie r pe t, clcl se denonstreaztr exf,
deflnl pe nulli-urea x Eln
ci mltinea tuturor blleLor deschlse care se pot ditr

satisface proprletitile topologlel'-. unI
tn contlnuare notivele pentru det
fi utili. von discuta care o topologle poate
Fu?
3.3 Convergenta
rut:
de Topologla 91, ln partlcular, netrica permit introducerea noliunli
elemente dln spatlul metrlc l.
convergenti. spunem ci glrul {xr} de (Iir
(X,d) converge la Ilnlta x€X dacf, gl nr:nal dactr pentru orlce e>0 exlstd
cudonennftlEulnnrslurtitlraeernle(eedvc)eelidX,neaanp1stestncfuetdsrlulinvnaepcnrltlrlctrflnxcpletaecnrutetnnloapraungc>troentrne(vedae)ervegagiellanorvlu€ermirlliudxn{(.xgu-sl,ierxu)gv<riaerl .vCecadouencahfoyir)n,n coi.t
verlflcarea convergengei nu neceslti cunoagterea limltei. (tn prt-nul
cInazaplutd€nolvleearlpflcuatopevnetrrluflcoarlcefapputuncl tcxt daci este sau nu llnlti, 1ar 3,5
deternlnarea acestula put6nd fl
tn spatllle topologlce in exlsttr un astfel de punct, int
uneorl lmposlbili. ) orl
retrlcd, convergenla se poate
dcaerfelntlopdoirleogcltagnluneuspterlgnelnneterarntleddeiuol qe:
vec
a{ Pe aceeagl nultLue s(eX,pdo')tlelntcroondsulcdeertnrmetsrlpcalgdlllfdelsrlttinectdbs., d", Ett

etc., lar spatllle (x,d1),

ntoeptorllzo"agblllEl.leex)is.dtettrnrlvtcoaatpedorudlollngtelrloecrtlereilrlcesi.enmunadleeLrolvr dsudnlnt,tirn-ogennaetrralrc,tsr u{fnlLulcsnutent

'((ox)r)rs ur P1nu1{uoe elre 0?
xun
*;tU'rt0t^dgxnt#krftfruo('pcGTap)TrrfstsorgtoTterTdEePauterrlroB(ea(toruttxutglr?eifksfg(e?Tlepax'rtpxrs)uel<ce0-nx(1-iopnc1'Tlr)eo:I1rue0lTxut)rakedcTsgxTcgdelpPpuor1c3a0Ax
,ap
'BxlTn[I lTunu xuau8Ta m 8p llnu ap ?gcTJo
INTI
oTdordp as ps :19 Tnun aTBluoulofe sc Pa?EXTfTqTEd ep BtubeT 'AT?TnXuT
'alse TB (TTlcunJ) TTtrucTTde pT tsrolar aE aleXTnuTxuoo ap eesnTlofl oa

alulTnuTluoJ 9'I '1or
IST
'RsTqcu! elgeunu as I eeup{Tnu 'pTcuToJ Tnr'l
eaT:xle[etTtoTnxrsunerouEE1{nusooppo aaeTTxoEsscsEEacalguetFqrdo'T(tmXnuduaTe?Plsaaeolxct uTnncudItAeTupmnreoTrldaTtrxmgnrxPTgu_Trxop:nualTungXulunoo@lTlBEle{?lTtlst'II)
u'(Arir
: ' 'ox aP e1TrBlTP x earrllnu
ap 1rn:rg6 ulrd 0x lnlctmd ap lT??ntumgaPa?lsgeoT0:ox PTdoxdP uaXnd lro
utp axcrind ep
?u- Rcpp g eeugflnu nrlaad areT*unie 'ATlTnluI P1s'l
.0x=rx g3u4 ,0x*k_
rsFT Tg :aTT{p?ET:dord UTP elctmd eP sTrt
nt I
(k) :16 t;n ElsTxa Bcpp x nquod gJeTnnnrc€ ep lcrmd slea x30r lxctrnd un TTutl
:ro11m:1$ 1n:o1n[e nc aXnzlraXceres TJ lod ereTnurule ap oTaXotmd
a?Po
"'JJllllffi:fritfr3TeEuxnlmsuToJ'nwxTTraT€{Baoed$€s€X1cspTTwcpxareipe0'BTcnsaeunqTutlxTrTuerxoXducouassrupnrournlrTTrrcTodufelcndu€lsnlETJtTmrgTE-tnr:6T1PnluodugBoT'nEesTuenexsITT?uqs1uToas3louad1Sgr€e?nA8ldu9omoxeau
a'XsalTxTul'i3reeTT6raeoTTTxooedeuotn1luuFnTce€rxe[ adusoTuufnlgd-JT8?ouaTtm' pa6uTroTJTluaIno?opuoo(uuraeEfeZloB"eeJ1.EeplanssTeetcqq60-ux1'uu,rxI XE
ET pralar
rB +ft"ffp PzPa
Tlrrluee no psTqasep pTTq) 0x TnT e a?elgr[ceA acTro sxuTnnt exTB nc {6E)
TEE
- aixu ( {ox} Vi ) e?s
pTl€Tar coT era Bs e A alplguTceA esgro tu{uad eef,.l
?D€p x eargflnu uI aJeTillnl€ ap lound e1€aunu es ta0x l'und uO
ed
a$FmtP ep lcund l"[
as T
'EfTTtrnqtrlETp rL grn196e1 u5 '1u1ccns 'Tut;ep
TopescETatTrrlTcsl"lcB! ooEuqp€aTEtErxeperteTruugaa6EbrleFTnaTuTxobgcoT'.eroesdTpott??'c€TT€(?e6zrTopeTbqdoTdeuoodgd?op'g1ltzeTru{yweu)belre6eorplgeuAaeaau{:uremsacA6urEreoonoelulJuoTapgugTflJnaeupuru (8S)
rA
rt0$
€p

ar

tttllills'illxilldu'tltultl'ltlllfit ?z'zT illtst3 t$ tttBltH 'lllllts

I LZ-zu ruunr, rttrlu, r?ucrlrt, slrucrunr 5t:l

$nilt, ellclilf 5t stililt &nt

ercroocnnvipCetlrrnogonuotcnlltnt,audtcialtonaaotIelvtaa.arDdigmle€cpnlltladcasEln€lncndoastnng(vreie,naqrg,l)leoan-rnt>atu0llnnnraueuglztllaungltltleaiorardrpp@(rfei(nnxfvi)lfe,t(fer.(gx)€onap))eturi:rn->lWuutta0IIgtceeJA.arl cq
operallel de trecere la llrltf, cu f: r,
lin f (xr) -f (lim xa) q(,
t 61) bii
(r
flg"1.I Schema explicattv5 prlvlnd notlunea de
d1Ir
contlDultate.
cor
8e nuregte hryiemorflen {notlune analogtr celel de hmouorflsn de cor
La stnrcturlle algebrlce) o apllealie f:(X,d)
contlnui 91 cu lnverea int
Un hmemofier cu contlnutr. dt(f (x),f(y))=d(x,y) se nunegt€
3,1
proprletat€
c1!
tzmetrie (apll"catle care conaErrril dlstantele) . Izonetrltrle reprezlntl
o relatle de echlval.entd tn clasa tuturor ap}lcatlllor intrE spa$li te
rrtrlce: izmetrille nu pot fl dletlnse lntre ele prln proprlgtltt
topologice gl netrlce. @
8e
o mr1!ir6 I dlntr-un rpallu netrli: (X,d] se slrune cl este denstr in nu
I dacl A-=E adicf, dactr orice punct (elen€nt) dln X eate fle punct dln
A fle punct adperoeacllrBnuIrlasrpele(I"tns1ptaatiuunl uXl,l1arc)esdtlan f. Hal plastlc, pr
fllnd rerm*:nea se poate
epune cl A multi.mli p€

c{
8l

I:
tl

I eu nulti.nea punctelor de acrrnulare ale lul A. (De exeuplu, nunerels
ratlonale sunt denee .Ln R. )

15 tn spaltll.e topologlce oarecare, pentru care topoloEla nu
derlvl dintr-o netrlcf,, r'eclproca nu este adevirat8 decet dac6 pentru
orlce x din spatlu eaxllsutfl, oxclsaIr[ milrabtlS de wclnttltl astlfel lnctt
scelaspaoartef,orpeepcrtelvzef,n. tgapcaatuole
orlce vsclntrtata reunlune
nun[rabllf, de vaclnltf,ti dln
rctftce
satisfac Eceastl cundltle dsoareco bilole deschXgg c1l raa€ nunare
rallonale, forueaztr un cst nu$trrabtl de vecinf,tf,t,l pEntru o::lce x.

',,loldrcouTr else TnTteds
:oI"B eTe?uffiTe lTIp es le6rr€ecerutuIoTonTnlusd3s suIelP€olndtrtAtuqoccnes3leral3nuu1e18 ElTctI
TS md lt
.oanrocp aTftts

Tt rrt ?Flund axgecnu e3 ler Pund ep Te}lse un J(p'x) <- (p'g'I1)e:I1fscpuunnl
't=(rlJ EGroJi ep y1{mce Totm "areATozer ef Jnpuoc auelqord elln1

Iallcorluoc ilIdTcuTrd l,'[

'ale6nlpe elua*tTe eTTou nrlued' TTcTrtau

E T! roTTTfs:edo EerTuTIep uTrd TB oll€t6JeAuocau TrnrTg ep rofa?Tqf
.rlmod'mnTlndTl oeueTrerd6nug!pe'AuTllordade1ee:1eT1neTotpodE11u!a?luoXodEl-,Trl-er[TTdsuropcupTcTrpTerllauau6rnoTafEuodBc
oXse Aqcnec rTg acTro gtep leTdrcc elgsGru ss cTJlet nTfBdB un

(eel . (r)rr<f ,FA t7 (Fx'rx)p I (")ue 'o<aA
:TrEu ep lueTcTJns F'T rulued
orez ET epuft [x 1! tr e{rcraTa uFn&o-prluer!lurT{qpapn{eu3erlsFTSp lr selelaTxdoJd
pu?AE 46 rm cTrlo!
nllede elgeunu eS
'AqcnEC eTTrn4g rulued Tpunu
ITqTsd elBe nIJnT 13ett (enu nBs e6.reAu@ :EgJeqer?ul e1 gptmdsg: es
ls et$erop es Tcop) qTErTrelTlEzunfeTlTTsalTcaeotTtrumnaua6rueeAsIuT'oEcEs pErtpTJrtJlT1g6Tpnu1nn:T1a6lrcToTleuaatucol9cl

sarE?sal al6erop es Eerulsel EcTTdrt Plntsounc

erlutp TeimUlp t
tlTEtI o e1 rpt Tnm Ta{us6raAuoc EereTpnle '1e11re TEI up mc efV

aleldroc eclnar 1l{sds ',lqcneC 1fiu1$ 9't

psflecp1p11du6rop-qlcrIerpd) Tupe6l'c(uTeT'{dnF6rrelotTsrcuct T[e'(JpllsfTaaTTlqslnscetJerEtduoulrlounuecuarEe6'nuTsT1ueu1ooepgroTtaop6upIT'Tpeicflplreneupc'P1To6rotnpau.eurTllqlflqcuuuoopScc
I) pelpl3ffgeledas
EinplsTp) ea4uy6qu '(e,relnuncn ep alelcund e1eo1 eu1{uoo TgI l}
perapprlru3 ecg1d4 alplTcedroo ap EalelaTJdord pc PzearlsuowtP es
'roTaTTq
rffirTtmer u! snTcuT e?se t FJTpE S ed ug:edoce" (u""2'I=T) '(Ir)rg
o1rlTq lo EeleleTrdord nc 'k""Zx'lx'axueieTe ep op $uTt 1'el Lm 'r
sTlfErpsee.an1ceT1ro11'1nqlseTrtxsd€enacplgp clJaEpd@lncTatX5stert3eTerler uTncTllredeelvulnel:redea4neld\ceoduoc
'STlclelBunu Psugp
*gfpu o eug{uoc lc€p ffq"Jedsg al€eunu es (p'x) cTrlau nTtrede un

rnrilrrs 'rfinnar 'tlluut 'ttuilr Lz-zT lIltSIS 15 til0Jilt 'llUHlS

5tffifi.f, fif;elJlIi 5t sl$Ttrt L2-ru ri,rlrir, ttmln, tpr.rcr'Ir, sT*Ucr,,flr stiltL t ,

o aplicagie f:(X,d) raport
a€(0,1) astfel incit
d((f (x),f (y))sad(x,y) pentru orlce x,1€X. 0
yd:inptrreinf(axp)lj.eciarfeiya) este nal mici de a ori decAt
Di.stan$a f dlstanta intre x distan0a Borel'
functiel define
dintre x gl Si y se
tlcontractX" interl
" Sl mu-l
5e denonstra teorelna de punct flx a lul Banach Si anume:
poate 4.1 l{,

orice contract,ie a unui spatlu metric complei (X,d) in e} insugl adnite H;
un punct flx irnic. Acest punct, notat cu p, se obtine prin meioda
aproxlmatlilor succesj.ve. Se construlegte girul nultu
vr-f (xo) , xr-f {xr) , " ".xo-f (xo-) ,. (63)
Ia,b)
qnde xo este o valoare arbitrari. 5e aratX ci (x-) este un gir cauchy
intr-un spatiu metric coniplet deci este convergentr iar 1lmita sa este (
pmctul fix ciutat. (
Irceasti metodi pernite gi obtinerea unei fornule pentru apreclerea {

erorlJ" comisa consider&nd respectiva aproxlma[ie, 91 anute I

d(p,x,,) .fidiro,*r) (64) unde

Ipoteza a<X este eselr$ial[ at6t pentru existenta cAt Si pentru cao

unlcitatea punctului flx, Aceasti teorenE jcacd un rol important in I
teorenele de existenii ei unlcltate a
nult:
- solutiei unui sisten de ecuatll liniare algebrice;
- solutl-ef unor sistsne de ecuatii Si slsteme diferentiale; este
Faptut cE o aplicatie este contractie aslgurX convergenta unei
netode nliirteriee {adecvate} pentru deterninarea punctulul fix. Alegerea puti
lnspiratb s v6l$rii initlale xoBX, relativ aproape de p poate reduce
conslderabll nurn6rul de iteratii. gene

{, HA$IJRT nul

1. lhwne H6sura este o generalizare a conceptului de intindere nurl
(lungirne, supr:afa9H., volun, sarcini, greutate etc. ).
ge rlu.$egte mXsurd pe urultimea X o funclle pderevaa1l6o, rni eensetegafltnlvlit,i9)1 inc
nunirabit
nnErabil adttivi (Ltnlta unul gir 1u1
prln care se asociaz6 valori nunerj-ce unor subnulglnl B- din X, functla apar
u satlsface proprietX[i]e:
est(
tr (F) ro u p {a) =0, BlfB;o => i* {*!.ao) =p f (an) (55)

1ar colectia de suhurutrtj.trti B*eI{ se numeste cdmp Bore1 gi contlne 0, t,
reuniunlle num6rr"bile, lntergecflile nun5rahlle, conplenenteJ.e !.n

hu ssTr{rsep eITq Tem Tn?uBerTdEos 'r€TnsTxrsd ul)'(PTsaTTr6{3osTeopdoRl TuTTqtraJldspa
etln3fut6rlFx1Bert\Td€ctpeeuErIorTedao1orlsoutag1ornoapulzXTpa.lrorceFcdBluToepnJeulcTegeprETTa8cnlpuTJaotaufreTd[Tturdf'd4uqoTpciTl[rn'eseTrnToelrgdslusJeueouIcl€TeT'urudo.trTr{po1TconTouteuoTuao1dgderoFue?gTl T{fucrntefEz.deSrPecuTrnunnuu!I
TufaTEJ!oagElBArasoJTnTTrSu;TlfTlennJuougaT1€nueTr loEerAduosrolTudqnrrpe'flealTgJTeluslanvTcrEm'ceTu.6rIolTrapoXdcsoe?c11a6no:lof1egodd.nos11{uennd1g{legudcOeep1eut.ura1u{nad6

'ETnU elsa
BTeer eldearp d ep alcund ep eTTqerpunu eATrtJuT nes a?TUTJ TnrlTnu
Tsun IE '1crmd Tnrrn p:nsem FO p?Tnzal Ens Tpu sp a111n6er u1g
'e6Tq3uTTua8 eTEAJeluT Op eunTrmer o ec
sTras TI aleod og erec ulrd allqfpu a1eo1 pdnp e1 as [("9)rl]1u1 apun
?
{0r} [{g,*r]6="A , [("g)rl]grur*(d:)rl
(,u]
l-i r-{ r-{
ti, { ({p.-{g) 3=o* ({p-{q) 3= ( t*ql ) d 3=
I#
{6e } =(t"o,'"r ) 'i,).({{q'{Pl 'd)t=
IF

is

*1
a
4 'f ({s' rpl)gT.j{\t=tt-o,{pl n)dT.:l
-(oo'{st
s: )

$ ,,eo-,q((-e(,(PC,lB)dl )d ("u)
("u)
r3Q'e4
('u)
13e4

:1p6ar a1aJEolpuun gdnp (q>e 'UO'q'p nc (q'pl

eEerrogJ ap eTeer ef,oftru ep eTEAre?ut) eTeer laldarp eTe TeJoq JolfmiTnu
an6saqor
eselc ed p?TuTIep rt aTlounJ o alsa u ed ernspn

4

I.lii anfisaqaT PrnsPH

.::

'Ierog 1n1n&ryc ulfrede {q'p) '[q'e] '[E] puJoJ ap a114{pu 16
ReolppolplJEeue1s{u'uoacga'prB'(cq'fealropgurldou] g6pc 'caTlTuuTTbpJt€ruT'aascTqpcuuTITTrllssp3efBaT1p6eAuJ1egluaTp
es U ad ferog JoITu.riTnE e \ eselc 'aleor laldarp Inzpc uJ '€rTarog
nTteds algamu as lerog du€c un lTuTJep e-s erpc ad ew.rlTnu o

(ee) 'g'3{g-'E 'eale 'eatetf 'eetetf

:a1a{ue.ra;1p TS X nc 1:ode.r

mtill$'ttfittftt'tlltllt'tttttnt ur-zT lnlsls I5 lltniltJ 'l]nlls

$ilitt, cifffl,nf st stslffiE LZ-to fit,lr]ii, Rturr, irllcfiIr!, slt.lcrt,nr sit

{,2 HHsura Lebesgue-$tie}tjes Brri

Hdsura tebesgue-stieltjes este o mco{nstuJ-rn5uXEelnaerdartedadpetao{pfu(xn+ceg)ie->p r.*11
{x}y => p(x}>p{y)},
nedescrescAtoare 0) conforu regulii-: ::-i

ftxl pentru r -> F ( [4, b) i -P (b-0) -P (a-0) fu

{71) rc{
{,i
gi analog pentru celelalte cazuri de la misura tebesgue.
..,<{
conseclnta esentiali a deflnit:!"ei nrisurii Lebesgrue-Stieltjes este
_lu
aceea ci ndsura unui punct a poate fi nenul.S: p([a])=B(ai-p(a-0) dac6
p este dlscontlnuE in a1F. 5

Mdsura cll seltn este o mEsurd Lebesgu€-stleltjes pentru care nu se tI,

natr cere ca funcsla p str fle nadescrescttoare, ea put6nd avea orice r.!0t

valorl, eventual **r dar nu slnuLtan c6cl operatia oo-* nu este definitS, te

Mdsura conplexd generalizeazd misura cu senn in cazul numerelor ns

co$plexe: ea este o functie cornplexi al"e cirei pErtl rea15 gi iraaginari 4'
smt mEsuri cu se:nn.

4,3 fune[ii mXsurabtie, integrale
o fuixc$l.e f :{x,n) -> {R,BR)
este m[surabild dacE pentru deflnitd pe spafiul cu rnEsuri (x,Bi

orice mulli-nre B*cBR€Il, f-=r{B*}eB
(contralm*glnea Bri.n f a orle6re1 muJ.limi Borel dln R este nulli-rne
Borel" lgi x).
DouX fu:r*gj"i sunt egale aproape peste tot daei dlfer6 numai pe o
crulfixre de mEsurX ilu1i"
0 functi* -fn scard pe {x,B) este o funct,ie deflnit{ pe
de *ru]"f;:',xi disjuncte dln spali-ul cu m5surd (x,E) o nml&jxre
flnit[
in modul
urm5tor:

st\12E,.-8\ )-qE s(,,xr=fi u*' *'*' \,-t^z,r

0, xeOal
| *"t

Integra'la unei functii scari in raport flr p pe mulllmea X este:

p { t*i I =p { fr ta,a"*} } -L:-:np ( [a,a* 3 ; I =
=|iy* {a*3 *s) -p {a-0) *p (a+0-0) -p (a-01 =o (a) -p (a-0)