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Set Theory

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48 views24 pages

Set Theory

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mainakb2003
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© © All Rights Reserved
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CHAPTERS : ST “TReory A Ad ing ctledvew of Bbjecbe Rick ore kmorm as elements oF tnembers. Etemenlo a Ack” can be Sack as numbers noo, shrdenke "Bickel code debe ads te te teak ofa ad ao w box Bok teleian cleitala i ejecta imikde Te box. Tuot ao a box a carpi» a dd Cam be emp ly 90 well. We dangle He enhly on gw in de ack tn ow Sele monty a whe rele ales irl aa Jton seb. Note Rok, a tugkton fn differ I ag —— a ee MEGS ee ae anal ig bbl differen rm Te book tat : ates we ent Ak Perereh. lading i ae “Tel tonal teat ww 2 Rosie Me Ties + In His onthod , we mtte a ak ty list ee i) B= 74,2)... a ii) C= =\ ak ee ll Bs-MS Shderda tn NSER-Ke Al berks vv tellSER-K Urey, w) IN=.? Capt. es ) 24> 4 thatie, y = are aT can be wile ao rslec. a lkdatlenys wast i) SeT- 2 Nefiep: | tite mutha ive dasoribe = by : @ Acsonb ug te pobutine p te leuk ONLY schisjied byt Tem i) At Sk ob real mminbig Wr bebweon 0 Ord 4 = ee y iit) = ? -f Py: pez, eM qe G94) wite S °= Ua i 224). Deiwhew 14. Set) Ak tot hwo hr wller Ie ab. We dunt He enbly ad : ata aA wowenbly cere i Fi ae ae nae FAD. Note Het, pb rottsame an fy. [oY Pail ab tla fone foe be eae pfwlaw 1.43 (suaset) BEE Heal We ae Reb A aodat of B dnl “4 ASB, 4 aN element of A ty alos an denent of 8. Define 4.4.4 Eavatity oF set) be feo Ade. We samp Feeney ae A= °5, tay CES We ay pnt of R aloo a DeRnitiow LIS CPreper SugseT) @ bbe A,B be too Ada, We cog Rob Ain a prabercanleack of 8, ‘i CB ASB, VR ASB ond APB, oI her rardo, Anleaek of ® chmark sf # ee ee fea woe aw A Exomable 1.4.6. T) ew sede Are Sa, 24, B= je Rb Bat2.=0 4 {aer/ 22-384 Dead Y “Ten, A=B, ize BEC. ii) tN oH We oe Seton 4-2 | Obermhona m Sto" ere porate alee oe San op B fn A @ le 4.2.2. — Pot, 2.8, 4,5) B= [14 & 4. i) AUB= 4 4,2,3,4,5,6,4 KAB= } 23,5) 6,9 i ee [4451695 ii) A\B = 42,354 Ww) B\A = ey feet Dk sh tem Anis BNA » ADB ii) Nee HB AAB=BAA, hue te Rame symmchic pene “Theorem 12+ A,B, © be tree sels, Th, i) Associakiwly . AU(BUC)= (AUE)VE y ANNE) =(4>8)NC ii) Dishtbukiwks? AM CBUC)= (ArB)ULANG) Av (BAD = (4UB)N uc) tn) Corsrnnkodnckiy [AUB= BLA, ANBABAA Peet Exereae DeRwlew 12:5 CDisoint Sets) israel ee: Wage ea hla eo eae eee t Detethcens LB CompleMeENT) bib > be The Sumivaal eb and Wb ASX. The umblmenk, at KChtakin, 2X) nokta d 2a cote ASt= XNA =f xex[agday Remark 2% De trporlend defre the universal det rian mm mde te be able © talk obmt $e conblerenk ttteg! Nae ses a heorem [-2..8 CBE Hera ate Las) Leb X be te vndverscl sb and ib ABS X. Thy, 2) WBA (Av By = ASN i) CAMB) = AX UBS Proof: i) bb 2€ (AUB) * = Tgeeea XN (AVE) . Ten, 2€X od AUB. Terefre, a6 A and x€ BZ xEXA od mere a Be > xe ABS. Tanne, ae X\B = (AvB8)* Ss ASoBt. DUE 4 ATA Q® Then, SS. Bo = The AUB. bene, 5 (AUB) < ts ei oo 2 =e. CAva)S = AS oes li) Usvig D. we hare. (ASU Be) (A) @D™ > G08 = Ang S[Atvayt | =@oe)* Bs ASU ge et (hab) > Gee (Pod. Detowbew 1.2.49 CRwer Se Pectin Ltt ee one EAS ntadty P(A, KK be asd. We in te ae of debate =P MA. Damark | [2 10 \) Te power tek Ab ale stefan 2A Wt) Note Hel, P(A) A fr any ad A, PEP. ® ii) be MEIN amd bE A be cack of 1 eleminbs. “Ten, P(r) hoo ony 2% elements ( BKERCISE). E le aa cs OW = 1 tf ; een Then, P(A= 74, 18 ie 1H Ay Aw ja; by. Ten, dips it ay, Ley, AJ Wear nds Fe ation of Me carbs eel lit) bebe porduck » bv wit the def” of om mrdeced pair, ‘ " be 2.2 Coimrena (oRDERENS PAR) pg a aren bEB The eetund pair a ba a ond by tw Hob order mark 1-213 ee is wk becanocit sortie Ont Oma bared on aw es cheng cf a. n ae webion fam ordured frig et Ope ae ordured four , G@W= ©) Podedy ase ond ted. oS een un wide He mcd of cre ordre pair con efor Aefinod » We skehe tle ome “Bod Yo ask crmmenly moed- The de v in due tb Kurd wa ket Cla2p + Dp 1.2.14 Coppers Pig) LE A,B We “nenembly ala ond Ub 2G A, LEB, The ached pair (ab) b oe aire ones te he teen ot pb a 2A, byes. sclet, Toa, (0)6)=Cr¥) Yard only ih t= x and boy Froor —— (oh) GA) Me clin Siok =X Gm: rt Gas 4. eT 1B aaa a as, layy =f Sars fd ells) a a or i saa ype ic Se afb ot ft a9 ee ne te > a,b4 © (a,b) = 2 x i n=b=2, Pa heey) ie MBE a Next ao | fax _. ay fay, fab home faye fa a "aly 24 ae ae es me gid i ta ety Th vemoing = cre yor We hawe. by ae ba (4°) eee “ baa yd povitig Shee Trrefa, 4=0 and yee. Groene. asd, to pore On left a0 aun &Kerese (Pred) We ate mane ieady 4 inbrodun fe wetiow of Me carksviny produc. Diieted ia C C Cr fesian Propel) heb AY best Gackt frobck of A ond B, cleared by AXB_is Ach AS, AYES {@ b) ach, bEBY when APA BoP p Kien Ae? o~8= 4 @ le 1-2. i) kee he f4,24, Be fa,b cy. Th AXB = t (44), (46), (6 2, 9, (28), (A) 4. ii) bb A = fayby. Then, Axa= | : (46), (bya), (bY. Sechien 13. Reddo Mecidstst abe hE KF FBI Ta A aky asbet Re ey IF XX, We A is a telohew on X- Ceven a relodiow R OXY, and & XE X, a Sat mts R- retell f 4 GER Ie cease, Wi wile Zz ample 4.3.2. DWE X= [abel, Wit 1 3f onl Ub & R&D) Gd, (D+ GS. “Thew i) aR, @ Re, bRO, oRS \) aK 3, bRL, bR2, ofr, Ka be X=Y be te ab of oll human by babs Sefine Re fluyjexax| ain a Tew a ace eB ee tae we Ry IPrand only baba fiend y cms XeYe IN. We ef. a rdobyr Re IN as Fells R=] Cr MEN | Ris a cmulte ble of om. Im oformerds, ne IN, ve hore mRn load colts if N= oa ae kb X+h and th R be a rdehow on X @. RO XXX Ten, i) ReFlexwe: Ris cald bb be REFLEXIVE if aRx —frall eX i) S¥MMeqeic: Ris a) be SYMMETRIC. if for all mex, aky > * ii) Transitive: R staal! be be TRIN SITIVE. if for all 442 EX aRy and yRe = aRe Examble [3 i 1) We relalew defied we Exambla ii) wb reflexue, symmebic and transitive ii) Te rebolen defined viv Examble ili) a rePlexie, transdeve . back on — . 0 i iti) ee K]eY=R and Wb Ree ](yWerRxR| xy= oF Ten. mi symmobric bok wet Age and etre, iv) bob X= faybye4 ont Riz f (4/6), (be), (4a), (b 6) J hn | Ris Symmebdc, transitive but wt mfexue . V) bb X=Y=IR and lb wo defeie- R= t @peRxR| az y3 Fo Gers ee Nia, i) Ry is reHerwe, transijeve but vst symmehic . W) Ris transitee tub not re Plyxwe and symmelric. vi) Envaldig Rulabirw bt X$¢ 2 . hebw define = ACW = | GA)e Xxx | aes G,#)2€X]. Then, A(X) rePlexwe:, symmebric amd hrvasitore, Vit) Gnapruence Modu a: fob X=Y=Z and Wb nGIN jhwo define = m= { (bezsze| des (ba) } Then, Sn is rePlexwe ,symmubic and transifeve Defioow 1.3.5 ( Eavivarens Revlon) luk X44 relodow von X is said b be am EQUIVALENCE lahew on X Ub ~ is reflexme, Symmobic and transite Futhemore, #xEX, Lxqre Cx Ree) he ees ane uivalence x por ~. Ip ae fal, We Hok @ b la re é [e] - Eoaly ta wett 7 Xf = | fay xexy bo dante the tet of all equivdeuce lasses ef X Remark 1.3.6 DD Nee tek, [14 A, for oll EX. The ig becamce ene [x] as ~w js re Hexwe + i) Net the, & Lel- le] UY see]. Indeed if te a dee a cid “da Ve aah € [we 7B x(. lomlarty | ove “Can yee ia Ea Terefon, — [a]> [el "d Examble | 3% i eee het X4 A and UE ? a ACH = [Owe Xxx] ey. “Then, an ulyrlence rediaw on X. fe Her , ees onan = Cal= fgexl yr = feexlgny ~ bo Thefor, inj io te qu mente tas f hy fer all Ke Oe amd hou = 1 Pose x]. ii) ms Neduto a hat me IN : on as Tales Wi og Hob fr che eet , ne fy ny =dlivtdes-Ch— a | hen, Sy On equecclence relobiew wu Z. lel wo wad le eq udvrdeuce classes. beh al Z. We [a] = jatkn [kez “t. cdlainy G). , le te S:=jatrkn| kez]. bbb ag ee as _ } +kn| ee cal a= 7 a dindes b-a , i Pot, ewe k CX, (b-a) -kn > b= — os Le] as: Gonvurzely , bE S. Then, b=attn, (es Armas ved whch imptes tok, n dindes b- Se ALS = w. be [a]. There Oise eel aleveer. S: oe is dock ii We ww Find tle dichmct equi vrlence clases. We chain Hok (*). We GD w_ twow [> Re fast cleb we claim We pee @® lh fut skh sTeP 1. al per 2), joy @)} Lemma 1.3.8 be X4+ A, Lb w be an equivrdence rlehen on X and Wb HEX. Then, exahy on of He followers, hotds eee ee, it) Cela lyf Freer: LF AD betde , temty 1B) lcs nef hed. bbw spoon Habit) doo wt bot. We sholl chor thot hotds ® Simca ii) deca wet hob, — Le +4, lew 26 (zl o[y. Then, ZX and a= Usurg the Sym ana hamnsitely ets we have need There’ se xe [y]. at i) Remark 1.3.6 htds oVed 5 eae A at igs yw. D f teow |-3.% CParh ION OF Poet). hak Xt 6. A ib a otlechow ee Hisjotrt —nen- fee es : Sythe t xy jAifie 17 vabads ft, acid d be D Ate Ae allie WD A Oa = Ale jedan 9] er oie km wear ae B 2/310 |b X=R. Tew, Vy ie TERY wa eee f WR. it) {Er o), [2 2) 4 parkbeen R. we) Ve 0, at ey 2 eo) 4 9 arto § KR % Vay a pabhon f R. fee a cf equivalence relohens lies vv he lang bee Theorem [3.11 (Eauivavence Reval 2 Prd Tion) yee: Ter - 1 wv is aM tre tlechiow uirrlence celabiow on X, then X/v parkitiew of X ib dichindk equecrdemte claseas fx -ferms aw 1) Coverzel Jq Ai fie 14 vo beohiicow of X Tew Hore exist de rare = Ben . on i. ® = F- prifie zy Foor bee 9 fait teeth be te dt & all dihkinck equevalen eee ce gh \e 3 i opie ie ifyriet): m4, X= UL], i follos thek x- VE] Lemma (3.8, can Ca $ frall Jer vib if yl ae ee le am We we Hk F:4 Ay lie b a parkh 4 x. oe Paci aes oc aa we S ee “sy i only it for ome iL, xy ge Aj - “Then, a “a reFtex me and symm To ter. Tove ee 7 7 amd 4g F Ten, for Fe v pe Ai ood ye Aj Terefore, ae And. Sina AS an pairwiae jel ee ub lozas Senn, 9 2. Aj WRtdh io were ‘fe Rn se en oe heb 2€X. Ton, Hore axils exadly oe 1EL uk tok xe Aj. Note tok I= fyexleyey ~ Tgexlae ay A Tecfor, Vg afr] Ai [iL yy whrde bres le —, @ echo 4 Fonebiena ee widton 441 (Fenelon) heb X,Y 4 raobrw ROXXY between X and ¥ is sacd i EX, tere exist a vnU eY mart 4ayeR aa we SA Taé Kis te dpmasiv of Rand 4 Y. ¥ te Examble $4.2- heb Xi= fa,b,e4, Y= fd, 2 D Res | Coy, Cha, (4 9 i) R= £@D, 09, Gs it) Ratt furckow 7 2), (4D), (ez ot a Guchrw eB e Ie 2B atrw . a Coda a tw i, Tee + Feschiow Notedveny bA3 hE Mea A amd lk ROXXY be a . We mile Rix Ye donrtt th fimdhew. foreach xeX we mele R(x) b denote The ygue 4e for whi (4 Yn)JeR Ine Pets nude, We cam wake the fimckons in fxample 14.2 as Fellows i) REX—3¥ 4 defined as RiO-1, R(b-2, RW-2. I) Re XY bh defied as Rw- RW-1,R©@-4 Definition 44 XY ¢ and ll F2X—>Y¥. Thea, we say Rot DX is fe demain of Z = i i te = wy) ea h AE x) is y fw= {4@ say a te ee E- Remark aa Not dot F(x) CY. fice ella op Example | 4-6 Dd eva —_ ON, heb XY +p ond ot PCY We define #@)> pf , For all 2ex Note Hot f (= 145 - i) Ipenrity Fusco: ht X4d. We define DA: x—X as Ta, Ke @Q=%, forall xEX - Note Hed, ra ® + x. 1D Iwcrusion Funcion: he XY¢ 4 with XY. The Panchen C2 X-39Y defined as t@=x feel nex: So Glen te clwcuea Seasttey iv) Mepuivs oncfion: Define Il: RR ty ik x Bo, Ini Ay eco vy) PorynomiAlfoncion: kt mE Iv {ry and lb a, with A, FO hate define - ps ROR by p@= Fads (rant rae , prall xe. jee We say Tok f is peal ioe C hE X,Y; Ub ate aan AB beter Fmchons We say : re 5 te toe fis pad bg BFE iF iB 10= fe for all xEX Example 1.4.8 i) Defue fi N—? 4, oi Lee et ® 7 Am EIR tat ae eh degree wn:= N, forall ne IN ey ee ae Then, even themngh F(n)=4@)=n, frrallmeIn, ffeg we have fta as domains apd eae dif ferent et it) Define #: RR, gq: R— [ee by Toga? For oll ER eT tg as erdomains of f ond aoe dif ferent, Nowe Hak F(R) = gC) = [2 . Defewhon 14.9 hob X¥4 dit PX. We say Hak fam ; f =. ee iL frall zi ex ) Cue-ane A ie te cne-cne /inje (a) #_ Fl) Shanever- 4yF ao D On fade Fe ante angele sh POD-¥_ a fo each YEY, Here exist 4 such Hoe f@-y. iy) ties aden fi bijsclae F is both we hig ouhs Remark 410 het + RR. “The, Lama iF the raph off imbersccbs ene hevizental dog bie. ii is ove-one i fle h imlertecte hwvizcontak bme : ae : a i ie » sc haigbe Ww is bij: rei ey S + b — pet ga rhe £ Je LAU Sh: bok be defied as P-L, forall E000. Then, Ft is one-one but om wrt onto. Note Hak f(G))= (1,00). @ W Mb fe RR ke defined as f@:= 23, fore llacrR Thea, f is a bijeciow i) bee IR R R— [e~), h: [2 2) |e) be Pak Sees Le eal fare x, fr all xER, Gis fr oll xE R. Sin = fr all xe Lys) Then, Fis neither one-one ner oule t Is ons out snob one-cuc and his a bijeckow iy) bet ¥bER, itha#o. and Wt f2 RIOR be defined as F@:= axtb fr al ser. Thee, fix a biyechon Whe YEP ond UY YEY Defthe FX ty fw- Ye pr al ze X w) is one-one 1} and only X is swoleton 2 ps onlo oe oul inks ian 9 fis bjedwe 3 and ie ib Xan Selon Defiewhew [Ald (Guobuition of Fuchiows) kt YYZ Ff oe i ol op ne con if fonction ae f Xz chine * (¢pDOr ¢(#@) » af pee Examble [41S ) ht Fig: RAR be defined ac f@®:= Jey t@= x2) For all wER “Then, (et) = #(f) = ge) = 82, fraliner, Cre @ = Fee) = $U2) = 2%, fir ab xe Neti Henk, pe £4 fog i) ee NEaN a bret, Mia eek yp, 2a Alecia naz mT : (3 in f@-«> €e:=f, toy qe =1, aint, Ur Tee $f: X—> Zz iv #Ne) = ¢( #(al) = 419 4 fA f= 3-4 DO> Fe) = 4-2. Definition [4.4 C Joveese of 0 Bustin) z bok X,Y4 DB and Wb fF: XY be a bijechon The inverse A is te poli pon defined as met P'(i= %, whee y= FO, foralyer. Remark 14.15 Note tat, ae is well-deFin ed. because 5 is both one-one and onto. “The onto-ness of ¢ is required b define $4 ab each porné of Y. “The one-one of 7 is yequred so Hof 4 (ill tatiomataceeemcagh mall le xackly one poert in X xkamble Abe i) ht Fe RR be dered ac ae eia xX? all xER ft ioe. ao; 7 = Ppaye 1 forall eR hb abe R with apo. and Lb Fr RAR be defined ac f):= axtb, for all aer Tey, fT RoR i M@):= 45h, fo all yer “Then, Ti be Xs= fay by CY Y= TL, pV ond Uh PROT be Acfined 4s Hor, FOR b> HOry Sha) coaster St fea, F(prab Pee “Terrem |.4.1F ke YY$ A ond Ut ey be a bijechon Then flef.= D4 -on Xf fle tym FrooF. Byercise Theorem |AIe het X,Y#FA and Wb fF: XY, get, hi YOOX be Adv Rok Pr My i a Ty Cx) Ten, Fie a bijechiw and phe fel heat Bae prove Te tenem in Tree sleps. Ie the fact ep, me STGeEM, ; is ocue~ome - a a to Do é Cea) z Kye oe Ho: we Gi = (nef) @)- Tdy &)= 2, yn ax rae) = be iD) one —— as STEP2 . fis owls Indeed, Ib ger “Then, US ing (#) ag ain, flee = ( GDM: 3 tay (= ¥ Fe) Seb 4 and cheh 2 chow thot ic a bijechow . fence, rt YX a le the rien esr thot SsTEPa . Jp Sbne re Indeed, it Potter fon Tencmy b4% Hew and a that io Fe p= (T%> F Je c= fre me “eTdy= Fu Thesrem |. 4.1% O*) = hetde = Chel fe ft)> Chef) ft= Paty ft =f" There | AF a trence, we haye proved Tok fee Definition (41g eee Inverse bmage) fea: and Se ae ne we define Ing & op ®.BE i) the fete imege of 8, FIGs frex |f@cey @® 1.4.20 7 cS — FINS Y ond ftB)CX ii Hob, w Defewlion |ANF (ii lo. stand. oe a a9), ee 4 ad, te mre of F need mt fleay= ftE@a Fo. BEBE Remark |.4.23 Note Roe jim Ne awe cmrst say tok FAO 8)= Fo t(B) example is relerend ints coulexte brn @usides a map fEX—-Y¥ thet is—noé one-one. Then, there exist 2p %EX Such tek y4% and F@)= FH). kb wo Acfin i fu} ond B= Jud. hen, pz Ff(A08)> S fan *® = fPE0] Which ian|Ac. Prook: 1) As A, BC AUB, we home F(a), $B) S F(AvB) ms thok (4) U4C8) < an (Conercel dU = FCADB). Then y = Fle) for rome 2x6 ALB. [F EA, tan ve re if XEB, Ton qe FCB) . Therefore. ¢£ (A) v F(B)- 2, (ADB) < ° F(A FCA) - Stocefoe Flow) = HAV FCB. Sima AONB A , we have F(ADB) © F(A). Similaty, welone f(AOBD< F(8). Therefore, FAB) FA) FB). i) Ftcevn) = Ftcovf@) — (Bercie) Usiig the

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