ee Nn ne OR Ease: fein d furdamtnrttols of Set théary trliodiclion te Soi Théany > A set 4am ungdeed Wleation of object. > Gbgectt 1 nB+ names, aman Ae- _, Report ation i 3= {2 $EF Ve gdexing on the collection GF HAE objects: Lloret of 2 5 aha cbpecte 27 9 * members of se- of pelted tO Oe ox o- £2,45F Mali? = g es ene Cie ase ara ct pushes Reps LE = 443 a 2 gofjall mtegau bls tt 107 Reuusertilion st 4 Ling corners eS oe aa Set buthlee supease7 OLE ae grew, n>! £ netooFSubsets £ tyalily of Sa Subset of a Set Suppare 9 29 aso & gore 7S a dubset BT 4 every element ¥ 3 ake an element 7. “3B Gy aes vx(xes —s ef) Paper Subset Gf Q Set Suppase SLT ace 2 20h > 34 0 peeapoe gubset ¥ 7 & every element BS & ahoan chment fF but tau 4 an element of 7 ushirh & met an tlment of 8+ SEF =( Vx (ees = xen) lan (“ETA xes)) Super Set af Q Bet Suppese 227 aco 2 goty ~ $4 a eupesot 7 4 evecty elament BF alee an Chrnent Wf G- cei taualily of Sets > Tass gets asa equal if they contain the SOAMé oh nent.Clcays thao Lscspoctive te Value hy - Meu, OSE bees Rake &o, xeg —s REO ee inertia. E&* any oa herpes, ¢cas.Re CR ae RR gc ge oa Fxwaocko Stak Tow exe Fate zs 4 fa3 - $25 © Sat ) hee ero2 ot ee @e set o ee $23 a fat © $92 F Powrere Sot + The pawese set Gag sen Gy aay set of all Bubsots SS ges. > ft & denated by PCs) example: G= £2,4,52 3-0 4, #24 83% Ssh $aa3 SaoF $2.94 , $2395? Cosidlimaldly 2 Comtesion Poteduckr Ccoscinallly af 2 ok . apa > The Candin atily of 0 sot ts the x ca As Cot Clements fn th0 sofDs: Ret rT yaa” > 1S] denobs casclinalsty of g. exam ple: S> Sa,3,52 S-£235,49 Is) = 9 /B)izak * lenyaty Set] oO % [Emouy Ser, Perpty S42) _ 9 x] S23] a Wx, rs) =, Gedened a - Tuple > The Heed n-tuph & Iehined as an asdegud) Mestisn of Clement CAD Sytem SG ea) as $2,534 Cactegian Pecoduct het GL bo 2@ sots TAO Caskesian prwciet EL7 % defined ag Sie es) ee bertexample ~ Go S97 35 Qnd 7 = SLYF SXT= Eli x), C19), (2,2) (2,4), (3,0), C58) 4 fixes t en, (RAN G3) 1g.) ly,2) (9,3) $ Casudlina bil, & Casteon Puwdot y (own: 2 829 as0 fiinth sole ther Isx@l = Js) xf Poceeh - Case (i) Let S27 bo mbn-e-mpty sobs Perce Ex, eee LAspagn = SxF = Say) Cita) a rr _ Cx, 4,) 1 (iggy _~- 6%8Gn), one bem, Hr), CXmy a) 1 ~-- (Xm, Yn) 9 pee yh Hence IgyG] = AA NC) Tl=n 4 /sx97) = (3) x¢t] seers, Nr eee eeeBra) =o «= Js) xi] oer sd 20. Ty yy, --- un§ USE vo Eg. ia) Be TD Lemon Pacs 2 bx7/] = 0 = Si ic oy Cacctegian Paco duet fn ee Casctesion Vscaduat of sole %, &, --. Sn? § X¥S9xX--- XS, See --tn) /£,65, , t26%, ---, 4n€SnhSet Opesationg » Vout 2 Ht Univesceal got, » contiine atl abjoak unde fo Cons deoxahan G2 i ‘ fe Union Opexatnn Oe OA Ret rer > ® thee union JB & the set contavning all eChments that axe ethe i ea ey V9, 3 luB= $x) xepv xa? Remask ¢ oO sp Spout uf both OLB tin fact, 28 4 the amollart ech @Oaee V0, 8 > (408 2 9) nave 28) cb ie nC Untesrcscation Speuatox [estaere set AL 8 thee mtoucetdo MoB « the got Contacning all emenk that ae Birmuttancgusly f 9 2 tn B+ YAB + AnB=Px] xenrn ves}ROM AAS ha. Bi piey mere ee lin fact, % the AO at eRe ¥AB: lang SA)a (AB uss gow 6) 2 a) ann dajaint % theix “H03 cotin fg enpty An @ = p Sect Opexakion, . Poser 1) Set Diheeance > Fox sote BiB he LY fprxaneg of Aen er the. St? of gece enOF ono in but net B. 4-B-= Py) *1% CP arg gy 2 Sullsiiea >xes)f OQ A > Als called : ee Cor lente Wy, Bia eee.Inclusion - txclugion Potin ople ‘Thegem - het 92GB bo 29 Minile wets ‘hon 1908) = 19) +/@}- !98) Poroot Suppese (AnB) = we 19-B)- p and /4-Al=9 'Al= 1A-81 + IAB) de ae tOC 181 = | 6-Al + 1978] = QO + [@vB) = 19-61 + Ie-efo 998) ah 0 Gees — ae PP a Qin} oct At or ee = Pt +4 toe ae = 191 + 16! - 1878) Sf Complement ~ he unteswe of da&cauue Can itself be GrB‘Vesud og get, call it U. bthon the contoset eleasly defines U, we eo goad ease ig em fA au, the complement & A 2a (=)Be ful 1dA4 Got Tclenketes . Red T Set Tdentities Ddentety 1Qu8B : dug= A Anu =A. Doméngtion lave + 4AuJ=v and -¢ Tdempetent Lous: yg -d ANA= A Daxble complement : (f) - A Commutative Lauss : AvG- BuA An®- BoA Assccrative haus: pulaycl- (Avaye ArB,c) - O28) nc Dis bubuttive louse: gylerc) » (av8)n lguc) An Buc) = (AnB) u CAac) i_ OE — De PI09G An's 1Oes % Chex ahs en i AuB = An® hes. dnb - AvB Lot Abu Pocouing Set Palerelilies e > fhe equality of 2 sot, PLB ca? PEOUae by & methods: en d Prove ACB L BLA Sepesately » Use a mermbesaship table Shous that Anbve) = Posy loc) | Recesh : we prove €h2 aslalorn ent hy housing AnlBuc) S nae (Bc) and Absume %e AalBuc) | We Knaus that weA and eithen eB ox wer | | (nB)v(Brc) < AnlBve) | Casel: EB , thon Ke lAnB) , 40 2c € LAB) uv CAac) Case2e mec, then wxelAne), do xe (AnB) uv (Anc) ihesafest, xe (Ange (Arc) “hesufase, pntonatees lara) wlAnc) 4et us thes @ArBe l#ne Ba lhucf ; e “xe Gna)e (Ane) We Kn eu thot Gees mi eS (Ang) ther wen g xe Go, & en) pg WE ( Buc) 42, xe AnlBveJ Cag S sea ; OAT as, then xen @ LEC 80, xen 2 xElBuc) za, Coen (Buc) Hence Go) ulAgc) co Anbvc) ees la ey, (AcS = Pn laec) Set Zdentbies : Pnst-DLet x be an axbebeasy ele XK wy 0x may 4 be in A x i ihe, « an B xen ZEB x~ElAvA) weloB) wea i T q Fp 5 2 ie Fi R PF G a R Fp D ee F 9 a AvB= AnR a enekalived union Un mien: 9, Aa. = a t= (CCA Ae flex P en : ° het rca a cone at t ” ” Se ee ee & ic WED |Gonexalixed thn Trt exes cation Bimasy intesscelion opealase : ANB - Qo nt osescokion: Pi AAgn— A Mn i =(C. (Anhao Un) Valin = 7) pe cer Pox infencls 3s Ng RQ . aang Sats act Bit thins > Sot 2 : an BM cre) ay 07 unbadesud catlealop g elemerr x. ecu Ney Biking SSP POR On wadese ox i: C40 Chmont “rivessal gor 2 > Elemente t9 Ln0 Bcght—ta-let wacdose tn the 4d Stungexample v= $a, Ac, de £.9,h$ A-~ $9594 B- Sc.ohj eG Rhona eC OF: vich ieee (ea fnaTng | le) ST] ref a pay eee ey >] offre) 7 (anita [RON ows / ata] ken fre] oR] > Set spemakons Who v,0 £ complement can bo Aten lithed by Derffaserninng logic Opeiakans an “ehe Cates ponding bt Cbergs Set Opec atang Lg Opesations An B A AND SZ A ORB % "4 Comper) A-4 A Awa (compen))Pp ton te Functeore rea ary fob o¢ a7 we te ikese Dita ns Wea» amenet ag £804 ofp mons say thata fundlion S59) ral Ravello elese exacth, one Chmernt YT t0 8. Some Function Fece) Ratogy Se ois pee Flad=y Baa some xeg 2 aa mec Comain sf £, eo a the. CO Coram ag ead See Soreage: gre wurden £, ox 4 the Pre-x, usher Lou) ( please Bx) mean: the Las 056 Lh Cp Coc The Ceébing £2 a: i es techy ose Ye i [a7 (cecbing Ox) means the Ermatlost ~72H0D ose 9G ie = 0) Sie a ea [3.9]. 4 Eafe eae Neate aes ee, ey cs oa by Lex rn CE ee ZB -x] eee Teoyh ath sms tla £ eb) poke all a bin & €qunabyt @ F(a) = Pb) implies a-b fac 2h ab ty g Dp PBF AK? 2507 4 inyective if L only y tla)> #6h) implas 2-56 Gee AY ab i 8.I —<———— hl ., ae Cer 0LP ck he > Only one ehment ¥ ee the mapped ta Any gitern one elayne Y Hange ca -o77€ > Infectwe 6" ava ee eae Suspoclive Functions yt yo 04”: a go £7 7 a ; on, olorent Pee bens por eveny ic nes with feel” ante I ee fn BO us A Bee bud rg oe Le burgection ° eee 2 quel @ =. (he wange g a Codema excng) ont Oe tne set Ba a a eee oe ole we the ger 670 Ca piece of it 06": 4 fre & «sar te te beclve b2 only 2 & bath ene-ta- eve £ ents- > tf 3 ako ed bijeckin ot Oo ia Coregscas DONAENICE «Fumcbion Opesatora : Aaset 2 Real- Valued Functions > 4 6” that bas oefthoc R wu one Y Lb tubgott as 48 srnge & clda sual Valua hi ere Function sgperatores fet £92 ROR C79) t. RoR. ehexe Grau- Fi +90) €xa) s RoR usheewe Gxg) x = ri x9 Function Compestl apeuatnoe cal Tee ee r BY 2) fg Bs) we ae ee > The campuoathan YL denated by Log Be 2 deh cd by regia) - (90a) fog + A 30. > the image BY Sundest € & Bmply the Set * Ca ey “77092 wf the chore ag ¢ Fumetion Opesotauy - Paset D Gdentitly Ferpteon het A be gq set: the tolon Life 6" on # & dehned as pt A> A ethew Flx)\ =9 VxeR > 1p & bath one-ta-bne 2 fants f7. ee f: 38> te aq ayedtie 4” > He inverse 4” of # i the 4” that Adiigne @ on ehment ¥ ef the unity eben! xES buch that #lidu=y > 4" donates the Snowe L£” of # Fy) =% whan tld =y @