: th inser UU ieee op eee ae ee ee * GOA ee B any well oleptred collect’on of abinttnct: On distin ubaidie eeme a well af med callactlon meon that thee exist a nule wth sthe help of, which, we should be able de * wthether any fm del belong aD on nok de the collection Under 4bme Apeipie yale The, Auk oL Atuclents un a class The Ak months dn a year Ae £4, 2, 33 ¢ y henenat captt af etre ABic. xv, 2 ee one used do denote “sets and dk elements ey fowercase Leer Abe age | - = ete: | = Etements op a ate 1 \ A={4,2, 3) ah objects th a ace ane catted elements ch mmembeu The alitinct cements mean ve element Sa supeated The Dott uishable ymeam shat giver any check em element fy etthed dm the set ox not in sh Ack So the clemente 4 Oo Atk must be dintinct ond dating wlaholele — The Ayrnbel € Cepatlon) 3s used so Encieate belong: to”? it A Tas dhe Elemente}: wee A then maymbeltratiy ws onthe Lea The Aymbol ££ D used to tholends ft does not below to SL 4 ds not an clement > ack A then Aiyrobellcalhy, wh watt - oo uga * Syrnbols ie Noe Alahunal Mumbon 9 = Te, -- 4d T or 2 = Sek of Engen = § weet Ao Ata} ZF = sek Of Pooikive eeage set op Negabioe tnfeqou = R = sek ep Real Nrambua Sek oh Rational Numbeu Guwhow wo q FO) See ay Comploc Rumbo,week anut mwene a ty amt pe yd IS AMY fsewryl fe bey “Id] @y (WOU Ga WO - me pxvamndne y ar Pr Je grou . youu) pre oT V Re ur que ame Te Haque ay : rs fo, ary pounpxny PUP YT rv ur Hwuay pune 15 requ ay “2 Py 4 ‘ON Pounpwo) Co) FAS to Pysjougo wns ° 4 TS 4, aad hy * { pqoydyo vege vp peon Oo ek EXP LD ‘pabauy annyol ure Uo op 1%} vo Lega yompou wma uo mR Dey L wo j ENO uwNEEK I xy ag | [hex anax: boy | {rovatoh-2 f= = a yieyp cg fewiy ey air ayy o@pywcs ay obs Rrodart o Puno re By pon ghar nepeee> PY eb Owe ou pou nur Uy ~j weet PTE Ve. i ‘oben amir pomp be oy dmtatyia'e y NY Mange 34 M09 S2qoydyo wes ~, Reon to ay aL hS, 9 ) oxen PmMMAdy I~ _UWAPO? tuner Ry peaponp HUI WI VE POWRUL MUI Uy weet sayeay abun mae | aK wag gE we wo ped be suanoh epi py « MIMAYL wo UoyOTRWTKY «Oo stuf semy hg ae wv OY Pedr no Re “3 ugyvzuavewday yw x,Tre ack \ aibou kn @ Trdia, ° THY te Se TH a ret wontch 2s nok frit dy called Inpinte Sek Sq sek of Makanol Mumbew: , Sing elton Sek f- 8 Ae which contatu only A element Ja called Simgetton set” { wi ycnee AL KENG Toy © RUA sek my Bwpty tory Voth Set A aet which contatr RO om Zeno elements eg 8 A is catlecl Meath Aeb THE Ack Bs cunoied by h wa fb. Condinalt ey null ae Ay qturaye Zero * Equalt ek, (oR) BK Sea i by A ord Bos quo by 4 8 {oR Equal est ae we Mee gory, Aement of ack A ta Om element ef Board every clement ef AB & an elerwnk ofA” The 4 “hos ata Rend & one denold by '> A=. < “ree EES = i ns a me = © A B Ay and only % * Equivalence set tT Am element eft nek ican be par fae ene do ome maping™ with the olemenle ef anvther set then. wo Atta ane Cabieel equivalence sed TA Ss demoted, by wor eye (a a TL tne clementi af At ane aame Aron tas eg uivolint — 4et ™ Subsek % det pane & be feo non. ernply acts | The ae A b aume g 8 44 andl ony th! Ueveny clement Set AB an element of cet B Subset ih denoted by “oe. sy mbottalty , ACG Heid » xen = eB, vneure ay As Lijas Be £4,2, 33, Hes, ASE A ds contatned dy 6'* Supersek i 1k Sa denoted “2" Tp ON da Aube o} Bo then , 8 ks aupunet 5 A B2aA froportly a Subset: * ik che “Acbiel o) 7 AGB & BEA " HY ASB Re BES e AGk a then ® A= 8B then ASC Ca & obo sulsset of CD. * Power Set tthe sek 9 alk possible subse of cet A VB cated? Power set of Set A Blenoted by Pad” Se A= Laap . Peay = ho, ay, gay, {13}, A= 4.2.37 Maye fg, OF, 3, g3p; tay, F438 fasabuei say» ‘ete s xq & Bet hay Sn! i Bee ee panailste » Se Proof % Lek See thas on Cements S04 Nor ol subsets Aki ng one elements then “2.7” sutssety’ of Aine element’ 2 my No- Sutsrety tatyG woo element = ney wed Subieh saniad Abnee Clement = ey No SL Sobek staicPng wo etement = Me We knew that og is subset of sve sek Se. ‘Tetal noe ay Saba obs FLe My MQ MQ ou ne My =n yy, fot Mey e My Mot - ae +My = (ery? = (yr. Pach - Finck fea) Yon the \piterstra seb SARTO Mme fo, gaz} Dona Le, Cory pele . cor do yc oye {lors} BD a= Ld} par = fh hb} DY A=W ay — fe sar} Syoa 2 & m= Sar* Unton > Let A and B be 2 Aeh non-empty sels the unten A and 8 Ss the ant 4 atl elements hihth oe ether’ in A on th Boon tn bbe A ant 2 Gord tie Unton of A are BA denoted 4 “AUR Tt Lb elie known! Qs Totnt on pegteal Se, A part Symbolic : “Fog = { xt xen ox *e8E. eyo aS fans} Be (usd AvGB = 4 4,2, 3, 4,53, @ Properties 4 Union of Set ie ‘| ae Be commat adic * (ommuctateie Properly Union d ve AUBs BLA- fnoo{ « det 2 © (AUB) > xen on xe'B > veg ox EA From eq” Q2@ > x €(avA [avs = 80a] so. (avs) eieva) ——- ©: du x © (BUAD ~~ KE B on KER D MER OX XEB > xe CAvB) So, € GUA) & (Ava) -——@ ~ Nssowtative Property 7 Onton 4 ac ¢ Absoctetiix Ze AUB = (AUB) UO fase} tek eC AY COVE) > KE A om %E CBUED KER on (CX EB on KHECD > (Hen om EBD om XEC > KEL ALB om XE CS ~~ x © (CAeRUC) So > AvCBYO & (nua vc —® het ow € ((AvB) UC) HK EWVB) on EC (a EA Ox KEB) OH LEC MER on (% EB ox K ECD ER 0m TE BOC ee AV CBUO Lis ueso, (Avs) uc & AL Cand —@ faem eqn © 2@®, Av (Bue A OB) Uc} ~ Ltempetent Poroperty # Uytton 4a ds idempotent we AUR= A Pasa het,x€ Av . > xe EA On KER 2 XE A. So, [AvA)S A —@® te xe A ~> KER on KER > x © fava) So, A & (nvAan —@ Pam ean @ 2 * det A Gnd 8B be two -ten- crnply eu ApLavaay BrP 4.2,4s) AUB £3,2,3,4, 53, OA ©£ (hus) >PB £ Cavey 4) Ace shen AvB = 8B hee RB XE AUB > ER om KEB > wen om HEB CH ASB then XEN => ve B) >a eB. se, AUB SB —@® We Knows that , BS ALB. —® From @ te. A= 4,03 e Bs £4,233Bh the set aq cements vahith belony te both A anol gb, femmen to dhe Aand &> : ; The Enterrectian af A and B ky dened by ANB voaliealy , ANB =dui xeR and x eB} (eA £4,253 Be (4,3,55. Ang = {1,31 ® Properties Inter ection | seb > ommutateis. Pxope tty F Gntowectsn of Seb Be Commut atria he ANB = BOA: Puvoft- Lee xe ANS => xEA and xEB. > mEB and XEA => me CBMAD So, AnB & Bon. —@ hee %© GAA > xeB and re A > xXEA ark xB > xe cane So, Boa cg aAnR ~——-@ From e9h © 2 @- a Assoctattye Property & Tntenreuuton of seb fb . . we. An (eacy = napa Rnookt- ek x © BD CBE > ER ond %E BNC SD xeR and (eH EB and XECD : D> XE ANB and EC: ~ me CANBAC so, ANCBAC) & CANN. —@ pag outatt ve het me BCA QB) Oe. > xe@asy and KX EC- > (we wand % EBD and we Cc. = EA and XE (ANC) > xe AABN S, (AB ace & Antenne) —@Faom 9" ©a® 7 . * AX (Bad anc Trteuection eg ket Js Tdampotante te. AAAS A. het Mpa XA and EA mM EA Sy Baa cA —@® ke oxen 2 EA and XE AS Sx elon) 2, AS ana —@ From er OO {7 Az l4,2,33 Boe{4.asy Aow=f 4,37 ~ ANB CA Ane cg. ’ AN > AnU A RSSEER AG © ROL ‘ 2 a ak Compltment of sets t- tee. be ae urtvonal and: “Pybean e a ak. Compliment % A da set ¢ sakning efements 4 untveusl act whith donot belong to ten AS wD E The compbtinent aa dh dunotud A SPV 44,2, us, 63 Ae hayes Avs $2,3,53 Symboottealty 5 epuexeU ond x FAR NOTES Cy oe LA AL Same. cog Ahose dament whieh baba, so Al Bue dome vot belong he tt as denotd »Y *n—B’ “oo A/B’. Syrnbol'e: > ‘ “4 A-e=fxsxeA and % ¢ BY ep AEG 12433 Bel 2.3.45}, ms MOTEDV Boe Pa BeBe 4a3 BooOW Fo Tee B-A = ayy}. {P28 LBA * Symmetric difference 4 Sets F- het Rand B be Avo aeks then aymmebie athencnce pL A and @ Ss a sek font atntng oll the etements that belong a A om B hut not both. Te db denokd by “A © Bt Se AKL 42, 3.4 Sh Be hous yet A@Bge A®s = AV B= Caso sured AN BEL UY f4,a,3e3. CAUBY ~ CANAD A@B= vay (AAG) = (12,364 % DMoasotnt beta ie let A and & be soo Att then. ‘ arew t d no Cammon cement ble A and B Ahern. J ‘ ane Aad 36 be dibporne acts ey A =k4, 23) B= fs, 8t H AmB sd wen A and Baw cis Joint A * Pooper Subset s-' Let" a ond B be Ayo non: empty acta and AEB I there Lr atleast Lene element Sn B whtch dow not bel to R hen , A 2s catlect prepen Subset af B end dt & Volrnoti ct bg Ser caces) * Venn Bagram ie Venn diogram tao pletoafal q Mepserent ation ° ats ohteh ane ward do shows nttodZonships EAST LAeu « The urbyeual atk La ruprtsenrted Enterfor ala seectongl and Ub subsets. are nepacrentecd Sy Giucan ateas atthin cba seeckangle.* fe feine Atk ANB Gb AvB au Ww “| GB] || | ine ine Hae Z EGY BL * -B a 8 uu Nilo shaded Grea means ~ &, A-B = th a R~A - Ack &~ woAeB y a @e Baw a venn © bh am ok sete A, BRC Ushene- Rand B have common Cement \ & anc ¢ have Common Plements Buk A and € ose disfotat sim & ACB, Sek Rand € orm diafoint hut E anc ¢ have elements fn Common. sv > ca @F)bre cwees son S S'@ve-ce 3 (Os) 2 Od= RY 4) n-B-c) D a- Cane $4.2,3, 0,63 2 (Ava) -< 62,34, 9 CB@e) = 16,8, 4 % Arte = fied D AaA-¢n@y = 44,274 2 macs@ed = fe, 33. x Algebra of Sey & . Tdarnpetert dow ft 4 4) AVA A yANRBA- * Cemmutabic hows. SAVB = BOR y ANB = BNA: woww ey an(s@ep> As{s2,6.3F Bed TR ee fed Boc=f4,25 Aub = 41,2,3.4.6 B@c= (Be -% wa (a-9 UU BC = FB. * pa sounece how F- YAVEBLC) = CAVBD LE araceng = (anayne. . Tdlentity law f YAO @ A yA We= A: Bound law * yA UU eu On pA Ae =o fies ° Compliment hawt guts > ap Oo su a Avut= U yAnAt > &. Swveliatron baw & (AD SS As 2 Distribute haw & @ Prove that unten > oy ae and vice - voud - ‘ AvCBad = (AVB)ACiAveD Ag caved = HOB) VL CAND Pasw Pas Av (Bad = CAVD A CALVO. kee 2% € CAL [BNC)) = LEA or KECBN) — KER OM WEB and VEC. deb fs dbtbubde oven Jatuuect=> (ken om EB) and (KER (ee ABB ond CREA LCE) = x € Ava A CALCD © Se, ALLEN) E (ved a Lek x6 & CavBy A CALEY => x € Cavs) and % ElAvC) > WER on %EB) and (EM om x EC) = on “ECD ER on EB and LEC. >» KER om XE BNE: *E ALVCBN (Ave) mn Cavey & AL (BAD —O trom egm © 2 S>)AL CBN Cavan CaAuC). Se, Pamve AN (BVO = (ANB & CAND Erno hee ox © AA CREE) => xe A and %~e(Bve> > KER ah KREB on HEC (xen and xeB) om (xen and XC) sce Ne) om RECLAME. > ee Gand) uv CAN Se, Ao CBve & lAnsyv CAdc) —O bes (ANB &G CAALD 3% XE CAND 9% — KE CANDY (re Biand KEBD om (HEA and 1EO xmER ond (%EB om KEL) REAR and WE(BLOD . xe A ACBL) > fanee (R08 a An cen —® Pom et OLD” SlAn CevO: = CAABL LADD rbiUuyoes me gee os 2» Hows = Prover- Avex = AS Mm Be Poshr- La xe CAvBzE- => «x ¢ (Ave) > x FA and nes > KEAS and WE Be: > x ELAS 0 Be) So, (avey® a tatgety —O- her xEAS my Bo, Pe EAT and EBS => x AR andl x EB => x ¢ (ALS) > we CAVES So, (AS ase) & ALHS—O ' From ar O 2G >| one = oe] « * (Anas = AtU Be: Lek mE CAABS > wv ~ HARB) D> xe w oma HB. D> xEA on nEBS > xe ATURE so, (anays © (AS vet)+——- © Let, mm xe ab oR’! => ne AC om WE BS: => x ga om “YB > x £é CRAB) > xe (Anee , So, CAtu'pes clang s -—-@ Brom ean 2a |s sae shat (R= Be = A-CB UCD xe (A-B)—c xe CA-B) and x © &e a and x ge Bdand Bee ‘ HEAR and (x fe and «¥ 6) . HEA and (% ¢ (Ben) x € A= CooL) 1 ABH S AR Cnve) —O fee, KE A- Cove) D> we A-and ux ¥ vy xen and ( EB and Me ¢) (ea and x ER Ord HEE . 7 x © AB and UFC Dx © CAHB-€ : SS, A-CBucy & (n-ey-e. —@ Prom eqn @ 2D .. 2 |(a- 8) a Cpve Prove thot: 0 (A= ByACB-AY ~E CAB A CBHAD 7 KE Cars) ond x € CB-M > (x € a ond xf B) and Uwe B and eA)’ D> (een ond HF AY and (xEB ond x KBD = x CH and x eb. ye Let, > xe. * se, (A-One oO, —@ fet te We -Knovs hak is subser “ab every Act e © e-wAatB-AY ——O fom @ 2G slla-ma Co-m = 8 S Paowe that A- cans = (a eS iss, xe AS CANE wen and “x € EOD 5 xen and Lx ZA Om 4 ¢B Cen ard EAD or (HEH [ard «Ae «eR on mae Ca-8>? ue “bv CA-B we (R~BD Se, n- tao & (A-8) —O ye ousHN RL andi nt ES me PUCA-B) Xe b on * ECA-BD. med Om LEA and x GB- (% eA and w AD OF CHER and «£8. KER and Cx EA OA HEED EA cand 2 CANS) ae AH CAND, lea-@ © An cAagsy —@® grub bors co > [AEBS = AT AG]. Per OO. & A ond B ane two Ack, shen (ANB) V CANES Nace nue) ane equat so ? aBS, Lang vo CRO MAD ‘ > An[eo 4% _T By visribubie Lewd m~ AO cey complirnant bow J 7 A Tey raentig, Lav] AO CH AvB> : => (Aa wad -¥ CAN®) [ Diatat buble Law > Ww vu (ans) . Tey vompttmant rans] > Age {By Tdanttty low J. at Ack then prove & tL Rand B one Foo subset on uuntvors a gortowing - 4) AHR = BA YR ony 4 2 A-B = A Ay mdeb g. A+B Hoe A= 8B shen A-B = BOA ha ET SB Xe AR ond HB 2 ~ee and x FAR Pe (BAD So, (A=B) & Bea —=O- Lek x & CB-AD . 3 eB ard “x A” > wen ana x & B > xe LA-® Sop (RAD S(AT DB —@O- Frm © 26. 3/078 = Bauy % (A-B) = (R-AD shen A=R. x © (BAD “ -> ye B and x A> , —@© xe (AB) : > nen and xn fe. “© ty” © LH can be equal’ only voten Aas. (2 Sons shea 8-8 SB ren ANB=h Se Pe : > ee A mond xe | . coe kek, An B eo? => x © Caney Tee : Xe CRB) AB LA Ba Ay xe A-B and wes. Wen ond xg Rand eek. HEA land Cx ~-B and rE BR wen and xP: xe And xe ey, : ant srdngcttels oun sauretion Sco, ANBES oo — kes oR oe Bak, we Enow thet > is auth of oveuyiat 2B Wo “o¢ ¢ AaB —-® trom gs @ *G- Tans @ Med AOG~A hea, ee AWB ¢ > new and xFB- > xca co anos oD, S$, AS SA — QO. wk» X EB : > nen ond HHL. fe aos =e]. A xe CAH BD so, ag ars =~@ From © & @ OR) H troB) =o otshet, % & CA-BD xe care oh Dwele) ex xe CAOBD ~D (4ER ond % BD PR CXEA and % EBD 2 eA ard we Bom CHE A and LEB, SD xe A and CxER) aed KER) D> new ande KECBLED > wxeA ad xev > meeny) 5 4eA s, a-B £A —O- 1 * Cartesfan Product + ‘axe = f Cay: «eB and yesy Lt eg ondeed Pom + A= 4033 Re Labs. ARB © Cavers cy, Coed, Cad C207, C207F BxA = { Caray, Carn, Casa: bdr, CO» b29} j Oaderd Pol i Tt doo pow ah obec ne using Ane shoo . componedbt Cin Sper Se @acen am the ondened pal CY) 2% ah tre, LY Component and y ts se Aetond component * Comiden stwo aes A and B . then cantestan proce Aand Bk trod é aun? ard Wb abe Act ey a possible wrdened wth Gay) Lotth WEA anc yee Symbottealby » AAB= & Guy) Boece ond gees MOTeT Axn we PRRA ® Mulbt sek ft muted ore seb nihene Ort element appee mone than one- Se A= L454, 22. 33 masa fe wan alo be wonton as- ES aa ee route The mulupiic® an’ etement, tr a ¢” multiioet ipl do be the nee a Ema an elemant appears tn rncuttto e€ S tye muutiepltoty Yt goby maaeeng | don , » 3.NOIE Apel ype ef multaek dn chit the fy fy fueaay ethonent Ek ene coe vraltty ¢ PMecbartatt Conclinality a Mubsieet de equal : te te conciina lity 4 corvnetpondirg (oR Candin Mubicoct 2s olethed as the Conotinall fu Comes, dead Att: Assuml Pot alt the clennende La istinct th F the tel mautioet © : Se candinaltly of muilixeet A = Caxdinattty q cosmmpondiry sera Sf A= {45,4 2,3, 5) Pe fF 4.o3d rece Ae Candinality a Mutieet A= 3. 4 Creation uta Muddvaets RAL As £34, 2b, 1c} Be {rea debs bdy. unten 9 mubtseh » CU) I (A vay-= faa, ab, be, HAY Tmtecetion of Mubltack CA) # ANS = {aa , sab} me diheenee C8 A BEL tea, ack dee} Sum Cr t- Bea © § 4d} AtBe { sa, wh, Ac, baz, ALB iy Ake rnuttingd yohow he multtplicdty yf a” Clement is rmaxPruim y de mater pier tits 4# A and Be eG AUB = {B.a, 2b, Me, od} Trlersectin preps AOR tn athe mubttoet whee the multiplidty J commen element ia minfmurn ef et rautkepiictttes 4h A ond &- ee ANB RL reg, Shy, . : Di}eente te Asa the rm alkisec shee the meliipeit ty 9 an element fs equa 39 voatatplectig a terrae aan A minus roulstpedttyy se net an B, eB Atpeace pootetue bux Seb Yequi te tro SY dhe Pehifquunet de zy and negatbe- ee A-@etbayte, Bey B-A = hbed} . acne Ate dg the multtoek vhene murleip tes: gan Alement Ss equal so Rhea mulel pel ep an clement dn both muleizes & and pS ea, Raw S tera, Fb, be bddaes ta, Bb, mad, find oy PUB .° 2D PAB, 2 PHO, 4) O-P. 17 PH POO = f wa, Bb, trey add. 2 P08 = { 3-a, 3b} 3) P-9 { ta, eked yy o-P = {a-d} =) Pro = § Fra, Gb, we, eas. * Set Inclusion Exclusfon Pobnetple Counter Prince le up EB) = “O aypencans) —® wLED = nUBA) + ACAD ~O NEAYED. = a fA-®), Lea) +nean® Por de values ef W CAB) Rn CB-AY nop, pom eam Ox dn er @- | RERUBD = nen = REA ABDSH NEED = NEAT x VE? REALE) = nica nee) = ACAOWS Coroll pace boca ne LAvgHE) = HERD A Aeay se NEO nERAB= NCAAD =“ NCHS 4 n€ansac) Pnoor kee BUC eB ACA BD = MCAD + PCB — HCA DBS = oneny * NCB = ACA CBLO) = ontay + NBA NL — MCEACD ~ H(A ss bire = pear t nape nce — MCRAE = + nflaceracanc) REA NIB FACT MEBOD = MCEAOBY NTS + 9 Cangncys& 40 Lecturers woo Intercofeued ma ivan , 23 woe ; fob, 2 Wen mathema- £ amd F whhe tether. Mov Aeckuneys vere masermatictams amd physToate $ bem be the aat mathemotidam —andé Poe the Ae Physics en OM) 3 as CP) = az find t nemaer = 2? : Given ie ! nome) = 4rd = 33 : MUP) = nim en Cry — n¢rmaeD oy fo wtmngy = nem + OCD = nEmopd pardors asana- "330 : i 2-33 25 dacmn ed . So, 2o Jeckuners wer both mathematyfon ond prytc 6 In a auwey sf COO TY viewers glen ths following Mformation, ; Poo ss roa A eee rmakches. @ ay . 25 AY wotch heciety mokche - ZB AS wakth foot wmokches - ®) AMY wath “ertcke 4 football makche- both. 5) 4A watch otacee 2 hoot matches both 6) SFO watch hecteey foot b' matchea both. TA) AFD dow wot voaken ahy of tht Hate game. etd D How many people watch ot thee Enel 2) Rov vany peeple veakeh exc tty ome port Fe, c be te ack ol ge Cdckd ofeutas Boke He Ack of Hockey vfwoa- Fobe he Act 4 Poot. yiewen Gin se ne = 38s nay = 2857 mem = ls wena) = 38S mw (CAR)= LO WOHAP) = 150 meconvey = 6, =ISB = AES - o waves nCeonvor> = RCO oR) eH ORD mcLe nnd nit ne) > =~ Wcnagy F HCHO BOF) vse -2 3as H2AS FAS THOR Ns mIse F MCEOND uso = BAs- Wes + NCEAANFD Uso = UBD NE COROE Cay 2 Sine: Bac WEED ANE? Pafnctple> = rol2» only ofcrer vferoan - me — AC CAMD— wCCARAH -HCCHF) |S » | = Btr- mo 20 - ME / = ae. A | on Houcey vrewus nen FE CAM) + NEC ANORS — CHAE) | eras — veo tr ao WP 1 2s only Football nley — PE ENED> HCE viewer ney — ROAD) eos = Ws 4 2E 7 DE = 60. No + of pPewoems exactly, wokth = n (exactly orceer) > NE exer woekey) * mC ex acrty Foote) = aor 5-66 = as i i & aymong four so telntegos - PP. ve : ty thelno. oh Entes Tohich. one not cdéviatble by > rex by = non ey te. a hich exactly afvierble bi 2 2) The noe op Ente vohich ane ure one & Tre mol Ent 4 fm a int B be the Ae. , _. * . Sate atufatbie by > bee; B bey Khe ACE oy. or de Saotat ble ud . ible &Y tbe that ol | fiis.gim atst nea, = | Eee) = are 49 nCAABac') =? 1 CmtoB! Oct = ncsvRvc! . = W- ncavsvnAleut = © RADE RED) nces = REARS) — NCAA) — MrBneY 4 ncnaancy F Av ettce toe 32 s3-yodte =» 30C yoy SOC £m far) - ecAvBoG = SOO ~ 2¢¢ yas fa tints olivitsfete by a. "om = weAABD — HCARO + ACADBNC OS = 280 - B3- SF 3B tle 5 Ise erty dtviotble nny — n¢Aangd~n(Boc) +n CAneac) We BB Foe le = ug, Onty elevia Pete by 5 MLE WE Ancd— ne BNE) HA CA OB = leo 383 Boe le. i s | leviotle by pri a then Mun af nfegeri ea ble WP) +eC3) +7 (x? mC erautly cbt! = sorter 33 ~ 232 Qo AS chéldren went © a cheus whee ty attend a mage! prow , Comedy show and animal show." 20 a thtm ak three Ahoi amd sr ablended axteast too AhowA. hack Ahow Coot £ ks. The Aotatl roe ney colletkhad Foo S Find He ne Children woho didnot athend Ahue Ahovin. > ite | om Be the set magle “ Ahew Go ke the Aet al Lome Aho Poe the Aet ef Antal chow Ter al mrorey collected ?= ® t00- Geak per ® Ahow es ee attended + foo ~ 140 = Teak nas gL Ahowe nimacoay = 20 Mer Eh thilolnen attended atleast £490 phows = or Date Children attended. oxactly Povo Ahows + No > alienteg olf thowe = 55° mes of children atrended: => Exadty 2 Ahowes + FO FS > Mer gq children abtended exactly 2 Ahows 7 35~ i lees |PORNO Sy UMAR amencen EKA 2 thom = IND T USAR Tan E = JHo- (40+ FE. = \4o-130 = 10 Nov ¢} chftdaen etl not attend ‘any gy * Ahowe Fe — Cas +rorid tr Ceo) = lo * (ountable and Uneountable See * (eudstable opie St Fn inde ack, A Le Aatd to be Countable’, Cnlntte on enumebachle C sel Ik | I i | 4h te bk epguivalent tb wk Natundl no’) * } 4 thee exist a ene Ae ene maping wilh ss fa i i i fina eq? vel a Countable Set f A Ace A in Catted countable when ib fy eth sEmply on Nult ack Le A= db. on + Pratce | set™ “on on * countable Infinte Aek - . Widve ads . - “ ~ a4 3 gt bea tetas ¥ BR 4 & kM 7 eve natloral mois ts Countable 1) @ Show that fe det of | pratt. pears > prynde Www aoe ¥ . a7 a ee : eg ‘ he S/o War . | “7, 4. heat se eee} Lago. neay = Th aet is @untable Snir, Le* Relattsn 2 set a and B be duo non empty sell then . R wa metatton Jaom A te B RE AxB and SL 3a Be Act of omclened po " Cab) where a Ge A and wie BH th dendied y aR b w (aby CR and Wb rect os “a js nelated Y te b Tro ka ne sulation “bjs a and b then a Kb om LER -omd we th seat tr a in not aelaket #o b”- Syn betieaty, Ref caby: aene bEB and ARbL cr (a,be*}- va SSG Ae {YP | Begayey, & metakfon! ® Js. ofifned i qrom AP B fe ARb 2y “Ax = even “he - Mad wetation R . ee { 3. Re axe . AT HS,5,3N5 | Be 5,23. : Bx CUA RD C27 0D CV,Ca.2d-965.40)5 0097, i R= Lue, Cad, 02.29, C399, 040,08 D7 ,& Ae ieee wy BS F412, 34 Phra she relation R such thak a Rb mE are. > An gyuesp, Ba 43,2,33 BX Be L Ca.ay, 212d, 49), C44 1 CH 2dD, ONT COA, (0.97, (0.39, ' CRY, Ct 29, CA DT REE Coay. Cady CHA EMA EBD, (G0, CH2D CED, Wn, © 999, 99% @ A= {4,2,3,4} A sutotLon & as + dabfned on Set A Auch that ORE Teo dis dfufetble by b AR Aaa, ENT Am C4, Ih AYR = Cad, CL2d, CH BD, CHM, C2159.62529 C2 9d ,C2PO BN, VB, C339, 3,49, C89), C4907, 8 cs Wd Ro gad) 20d, 2.29, CBAds C8, CHD CWE 4 * Domain and Ranoe oy 4 Relaton * Bematn ya Related s-Y ta tf the set cons tating fiat elements 4} conden ed Spabis "y arate RT bb derokd 4 BCR) 02” Bom tay Sura Ni y veviatiy Bca> = {az aen and ab} NeTe: Dead Ga-J 4 Aewond elements af orcteuel pide a} Ptigtten RTL Lr denoted My RoR om Ran( Rd 6 Wore, R= ob ° “ompliment. | Kelatibw ty a netotfon & delferd por fab & Stren Compliment ™ of sy aet te ordered potas such that = {¢o,b) 2 Ca.b) ERT Ro = Caxny = R. en on Aed Ro = Caxsy-R. Une}facd en Wo aes. TA denoted ii Ro an RY oy. A= {4, 2,3} B= fry L ve) tned R= Pomme { CHW, C,ud, G36) 3, AXB= { cas, Cusd, Ca faeces, a, Ds ‘REV =e (AxBD- RK. ace f Gys7, Oo, OF1 ne Rr ass R= f cab) : arb fs Boe Sf Chey 7 BBCY | fet, Rbe the mulation pom A ae and s be the ata tien i m Bape Hhen compoattton of sulation Rand s ib a | Naetetion: consisting sh, odes Ca,e) whene AER ancl | EEC. proutdad “imate thew ext gome Bb EB. Auch that | ARB’ and bRE. Tk in denoted by ‘Ras? Syribolteally RoS = | @.c) + FbEB, ARS and bse as fe Peta, Betauy, ¢ sts, 2535 eco Ome) 6? S=£ Can, C3002, af Res = § Cusr, C0), Cay, cred}. a @ © B. Rand § seletfom are deptned on Sek A= Sh2,34F ant R= £ Chay, Cm29,02,8), CONDE, 3,9, 04 3)F S= A cand, €2,99, 63,2), Ca,39, CFD CU2I}- Find RoS & Sof oS cyan Cay, Crd, C253) > BBB. Cay, 63,22,08.29 04 C43), CHNDF : SeRe (Cary C2, 3 C24), 03,92 C7 Hy 03d. C429, CD, 04 Trove that Snveue e, “(Resp = Ro! ose! Pe ket, ca,b) € CRUSIT 5 > Chay € (RUS. | = (bay € R on Chay € S. => (orb) E RE ON Carby ES" : => (a,b? ECRNUSD. O Se, (Rusa7! & Cetus) ‘ fee, (a,b) © CR USD D> (a,b) € RA da Cab) E So! Chm ER om CHAE S. Cb,a) € CRvS) Carb) € (Rvs)7! fo, CRF us-') & (RuSQ- ——@ foam 4" O20 > [CRvsy 2 > 2ay Prous thot (Ros4 = CRA sty, , Mek (are (Rasy? (bay © cans > (a @€ a and Wages SO Caby © RI and (Osbre So! 3 tan © Rast. Se Wash © Rt pel. —@ Sek Carbs € (RAM 64) > Ca.b) € RA and Cosy $77 > Chay © R and Chie s. > Chay Rosy > Ca.b) & CRAM! fy (@ os) © CRaY —@ Faom et © A@ SF RES Han, RT es -t. Glen, RES heb, (@, by © Rol > Cb,a7 © Re = Cha es ~» cae) € S*! Res) So, R-' g got, © Airave that compoattiin 0] sutatiow. Joltowe the auodahiie “ how. Pras} (RoS),T = Ro CSTD het Ro be a gelatin fom & A 4B 5 be a melatin per B toc. and T be a aelation prom C to > ket (asd) © (RoS).T. 7 = (Auch thot) Ac €C , Cac) € (RS) Now, (4,6) € Ros > BbeB, Cabo ER ard Faom eqn O16, (be) €5 and Ce,d) € 7. = (64) €(%7T) —@®. from 4" ® 2Q- forbs ER and Cd) & (oT) > (a,a) © RyotwT? —@ Se, [RoS2oT Ro C&ID --— © and (od) €F —@ cba es ——@©