1Classesand Sets1BUILDINGSENTENCESBeforeintroducingthe basicnotionsof settheory, it willcertainobservationson the use oflanguage.By a sentencewe will mean a statementwhict in aunambiguouslyeithertrueor -false. ThusLondon s thecapitalof England.' Moneygrows on trees.Snow is black.are examplesof sentences.We will use lettersP, Q, R, S, etc., to denotesentences;usedin thissensgP, for instance,is to be understoodas assertingthat"P is true."Sentencesmaybe combinedin various waysto form morecomplicatedsentences. Often,the truthor falsity of the compoundsentenceis completelydeterminedby the truthor falsity of its component parts. Thus, if P is asentence,oneof the simplest senteqceswemayform from P is the.neqation ofP, denotedby - P (to be read"not P"), whichis understoodto assertthat"P is false." Nowif P is true,then,quiteclearly, - P is false; and if P is false,then - P is true. It is convenientto displaytherelationship between - P andP in thefollowingtruth table,where and/denotethe "truth values",true andfalse.Another simple operationon sentences is conjunction: if P and Q aresentences,thWUignction of P and Q, denotedby P nQ (to be read"P and Q"),is understoodto assert that "P is trw and Q is true."It is intuitivelyclearthatPnQ is true if P and Qarcboth true,andfalseotherwise;thus, we have thefollowingtruth table.be usefulto makegivencontext is1.11.222ClassesandSetsTheJSjgfionfP and Q denotedby P v 0 (tobe read "P or Q"), is thesenten whichassertsthat "P k true,or Q k true,or P and Q areboth true."It is clear thatP vQ is false onlyif P and Q are both false.An especially important operationon sentencet is impli"ution jf P andQ are sentenceqthen P - Q (tobe read "P implies Q") assertsthat"if P istrue, then Q is true." A word of caution:in ordinaryusagg "if P is truq then0 is true" is understoodto meanthat thereis a causalrelationshipbetweenF and Q @s in "if Johnpasses the course, thenJohncan graduate").Inthematics, however,implicationis alw4y urlllq$in theP+Oistruee P is truea lse.In otherwords, P + Qisthetruth table.The propertiesof formalimplicationdiffer somewhatfromthepropertieswewould expect "causal"implicationto have.,Forexample,"I + | :2:> z is a transcendentalnumber"is true, eventhoughthereis no causal relationshipbetween thetwo compohentsentences.To take anotherexample,"2 + 2 : 5 +4isaprimenumber"is trug eventhough thetwocomponentsentencesare false. Thisshouldnotdisturb the readerundulfor formal implicationstillhasthe fundamentalpropertywhichwe demandof implication-namely,if P - Q is trug then,necessarily,if P is true then Q is true.1.31.4BuildingSentences 23Certaincompound sentencesare true regardlessof the truth or falsityoftheircomponentparts;a typicalexample is the sentence P > P. Regardlessof whetherP is true or false,P + P is alwaystrue; in otherwords,no matterwhatsentenceP is, P + P is true.For future reference werecorda fewsentenceswhichhavethisproperty. .1.5TheoremFor all sentencesP and Q thefollowingstatementsare true.i) P + PvQ.ii) PnQ-P.ProqfQ-PvQ.P nQ + Q.i)'ii)'i) We wish to prove that if P and Q areany sentences,then P + PvQistrue; in other words,we wish to provethat no matter whattruthvaluesareassumedby P andp, P - P v Q is alwaystrue. Todo this, we derivea truth table for P + P v Q asfollows.PvQ P+PvQt"ft.ftttfttttThebasic idea of thederivedtruthtableis this: in line1,P and Q bothtakethe value; thus,by 1.3,PvQ takes thevaluet; now,P has the valueIand PvQ has the value/, so, by I.4,P+ PvQtakesthe value /. We dothe samefor each line,and we findthat in every line.(that is, for everypossible assignmentof truthvaluesto P and Q) F - P v Q has thevalue (true).Thisis whatwe had set out to prove.Thederived truth tablefor Q + P v Q is analogousto the oneforP +P v Q; theconclusionis the same.ln orderto prove thatP nQ-P for all sentencesP and Q, we deriveatruthtableforPQ-P.In everyline(thatiq for every possible assignment of truth values to pand Q), P nQ+P takes thevalue/; thus,P nQ+ p is true irespectiveof thetruth or falsityof its cornponentsentencesP and e.P,ttffi)'ii)PQPnQ+P24 Classesand Setsii)' The truth tableforP Q-Q is analogous to theonefor Pn Q+ P,andtheconclusionis the same. I1.6 TheoremForall sentencesP, Q andR, thefollowingis true:l(P + Q) ^(Q- R)l - (P + R).ProofThe reader shouldderivethe truthtableforl(P-Q)n(Q+R)l +(P+R)and veriffthat thissentencetakesthetruth valuer in everylineof the table'l1.7 TheoremForall sentencesP, Q and R, 1f Q + R is true, theni\ PvQ-PvRist{ue,andii) PnQ-PnRistrue.Proo-fi) weassumethat Q+ R is true,and derivethe truth tablefor P v Q > P v R.Sinceweassume that Q + R is true, wecannot have, simultaneously, Qtrue andR false; thuqwemay disregard the sixthline of the table.In allof the remaininglines,Pv Q+ PvR takesthevalue 'ii) The proof that P rQ + P R is analogousto theabove. IWe agree that P e Q is to be an abbreviation f9r (P+ Q\ x(Q + P\'1.8Theorem For all sentencesP, Q andR, thefollowing are true:i) Pv Q+ Qv P,ii) Pv(0vR)+(PvQ)vR,iii) Pn(0v R) + (P n Q)v(P nR),iv) PvP+P,i)'PnQ+Qt'P,ii)'Pn(QnR)e(PnQ)nR,iii)' Pv(g nR) +. (Pv Q)n(PvR),iv)'PnP+P.The proof of thistheoremis leftasanexercisefor the reader.Pla lRl PvQPvQ+PvRIn thisand thesubsequent chapters,+ will beusedas anabbreviation forimplies, e will be used as anabbrwiation for if and.onry rtwe will sometimeswrite "iff" insteadof +), n will be usedas an abbreuiuiio' f", ;;;';;i";will be usedas an abbreviation for or. rf p, Q, R, ... are any statements, anexpression of theform p - e => R => ... should be understood to mean thatP - Q, Q = R,andso on;analogously, p o eoR e... should be under_stood to meanthat p o e, e o R,and-so on.- As is customary, r is to be readthereexists, y is to be read for au, and = isto be readsuchthat.EXERCTSES 1.1Building Classes25ProveTheorem L8.Prove that the following sentexcesaretrue for all p and g ioevrorgan,sLaws)a) -(PvQ)+-p-e.b) -(PnQ)+-pv-e.3. Prove thatthefollowing sentences are true,for everysentence p.a) --P-P.b) P+ --P4.Provethatthe following sentences are truefor all p and e.a)(P+ Q)o(-e + -p).c) (P+ Q) o -(P -e).e) [(PvQ)n -p]> e.5. Provethe following sentences aretruefor a1l p, e and R.a) [(P + 0@ - R)] =, (p - R).b) !{l - 0)n(R + 0)l + [(pvR ) - e]c) [(p>e)(p-R)]e[p-(0^Rf.6. Prove that,forall sentences p, e and R, if e +R is true,then thefollowing aretrue.a) pv e pvR. b) p xe+ pR.c) (P+ 0)o (p*i).7. Provethatfor all sentences p, e, R and S,if p + e andR> S.thena)Pv,( + QvS,2BUILDING CTASSESb)PnR+ Qn,S.b) (P+ Q)+(pve).d) [Pn(P - Q)] - e.We will now beginour development ofaxiomatic set theory." E"1y axiomatic system, aswehaveseen,must start with a certain numberof undefined notions. For examplgin geomeiry, the words ..poi.rt,;-urro ..line,,and the relation of "incidenc"l' *"g"enerally taken to be undefined. whilewe arefree in ourown minds to attacha "meaning,,, in the formof a mental26Classesand Setspicture, to eachof these notions,mathematically wemustproceed ..as if ,, wedidnotknow whatthey meant. .Nnrv en "unrtefinert"notinirhasno rrolerfie3are explicitto it: taxiomsall I elewhichweexto have.In our systemof axiomatic set theorywechoosetwo undefinednotions:theword classand themembershprelation e. All theobjectsof our theoryarecalledclasses.certain classes,to becalled sers,will bedefined in section7.Everyset is a class,but not conversely;a classwhichis not a set is calledaproper class.Let us comment briefly on the "meaning" we intendto attachto thesenotions. In the intended interpretation ofour axiomaticsysterq theword c/assis understood to referto any collectionof objects.However,as wenoted insection2 of chapter0, certain"excessively large"collectionscanbeformedinintuitive settheory(for exampre,the collction of all x suchthat x I x), and ifwe do not exercisespecialcaution they leadto contradictions such as Russell,sparadox. The"termproper c/ass is understood to referto these ..excessivelylargie" collections; all other collections aresers.If x and ,4 are classegthe expression x e ,4 is read ..x is anelementof A,,,or "x belongsto .4," or "x is in A." It is convenientto writex $ A fot ,,x isnot an elementof A." Let x bea class;if thereexistsa class,4 such thatx e A,then x is called an element.Fromhereon we willusethe following notational'conventlon: rower-caseletters eb, c, x, !,...will be qsed only to designateelements.Thus, a "upi;;llet{er, suchasA,may denoteeitheran elementor a class which is not an element,but a lower-case letteq suchasx, maydenoteonly artelement.1.9 DefinitionLet A andBbeclasses;wedefineA:B to mean that everyelementof ,4 is an elementof B and vice versa. In symbols,A:BitrxeA+.xeB and xeB+xeA.wehavedefined two,classesto beequar if andonly if they havethe sameelements. Equal classeshavcanotherproperty: irx andy arequaland x isan element of A, wecertainly expecty to Lan element of . Thispropertyis statedas our first axiom:.A1. Ifx : yand x e A,then y e A.Thisaxiomis sometimes calledthe axiomof extent.1.I0 Definition Let Aand B be classes; wedefine A c B to meanthateveryelementof ,4 is anelementof B. In symbols,Building Classes 27' AeBitrxeA+xeB.lf A e B,thenwe say that A is a subclassof B.Wedefine A e B to mean that A c B andA t' B; in thiscasgwesaythatA is a strict subclassof B.If A is a subclassof B,and.4 is a set,we will call A a subsetofB.A few simple properties of equalityandinclusionare given in thenexttheorem.1.11TheoremForall classes A, B and C, the followinghold:1) A:A.11)A:B+B:A..nDA:BandB:C>A:C.iv\ A c BandB c A> A:B.v)AcBandBeC+AcC.Proqfi) The statement xeA-xeAandxeA+xeAis obviouslytrue;thus,by Definition 1.9,A : A.ii) SupposeA:B; thenx eA+ xeBand xB+xeA; henceby 1.8(i)'x e B + x e A and x e A > x e B;thuqby Definition I.9, B : A.iii) Suppose A : B andB : C; then wehavethefollowing:xeA+,fi,xB+,E1,:Z',liZ';From the first and thirdof these statements we conclude(by 1.6)thatx e A + x e C.Fromthe secondandfourthof thesestatementsweconcludethatxe C+xe,4. ThuqbyDefinition1.9,A: C.We leavetheproofsof(iv) and(v) as an exercisefor thereader. IWe have seenthatthe intuitivewayof makingclassesis to name a propertyof objectsand form the class of all the objectswhichhavethat property. Oursecondaxiomallowsus to make classes in thismanner.!4. LetP(x) designatea statement about x whichcan be expressedentirelyin termsof the symbols, v, A, -, *,3,V, brackets, and variablesx, !, z,A, B, ... Then thereexists a classC whichconsistsof all theelements x whichsatisfyP(x).2A ClassesandSetsAxiom42 is calledthe axiom qf classconstruction.Thereadershouldnotethataxiom A2 permitsus to formthe class of alltbe elementsx whichsatisff P(x),nottheclassof all the classes x whichsatisfyP(x);aswediscussedon page13, this distinctionis sufficientto eliminatethelogical paradoxes.Thesemantic paradoxes have beenavoidedby admittingin axiom A2 only those statementsP(x) whichcan be writtenentirelyin termsofthe symbolse, v, A, r, *, 3, V, bracketsand variables.TheclassC whose existenceis assertedby Axiom A,2will be designatedby the symbolc : {x lrlx}.l.l2 Remurk Theuseof a small x in theexpression {x I f(x)} is not accidental,but quiteessntial.Indeed,wehave ugt..dthat lower-casl"tters x, y, etc.,will beusedonly to designateelements.Thusc : {*lp(r)}asserts that C is the class of all theelementsx whichsatisfy p(x).Wewill now usetheaxiomof classconstructionto build,or. ,r"* classesfrom givenclasses.l.l3 DefinitionLetA andBbe classes ; theunionof A andBis defined to bethe classof all the elementswhichbelongeither to A, or to B, or to both,4and B. In symbols.AvB: {rl" eAorxeB}.Thus,x e A v B if and only if x e A or x e B.l.l4 Definition Let andB beclasses;rheintersectionol Aand B is definedto be the class of all the elementswhich belong to bthA and B. In symbols,AB: {xl xeAandxeB\.Thus, x e A B if and onlyif xe AandxeB.:1.15 Defnition By the unitersalclass ll we mean theclassof all elements.Theexistenceof the universal classis a consequenceof the axiom of classconstruction,forif we take P(x) to be the statementx : x,then M guaranteestheexistenceof a class whichconsistsof all the elementswhich satisff x : x;by 1.11(i),everyelementis in thisclass.tiIBuildingClasses1.16 DefinitionBy iheemptyc/asswemean the class @ whichhas no elementsat all. The existenceof the emptyclassis a consequenceof the axiomof classconstruction;indeed, M guarantees theexistenceof a class which consistsofall the elements which satisfyx * x; by Theorem 1.11(i),this classhas noelements.1.17Theorem For everyclass-4,thefollowing hold:i)=..Ii)Acolt.Proofi) In orderto prove that @ c A,wemustshow thatx e@+xeA.Itsufficestoprovethe contrapositiveof thisstatement,that is, x ( A + x O.Well,supposex Q A; then certainly x 4 , for @ hasno elements;thusx(A+x4.ii) If x e A, thenx is an element; hencex e all. f1.18 DefinitionIf two classeshave no elementsin commortheyare said tobe disjoint. ln symbols,A and B are disjoint iffAnB:@.1.19Definition Thecomplementof a class,4 is theclassof all the elementswhichdo notbelong to A. In symbols,A' : {xl*d t\.Thus,x e A' if and only if x ( A.Relationsamongclasses canbe represented graphicallyby meansof ausefuldeviceknownas the Venndagram. A classis representedby a simpleplanearea(circular or ovalin shape);if it is desiredto show the complementof a class,then the circleor ovalis drawnwithin a rectangle which representsthe universalclass. Thug.4u B is renderedby the shaded areaof Fig.1,A B by the shadedareaof Fig.2, andA'bytheshaded areaof Fig. 3. Thereaderwillfind'that Venndiagrams are helpfulin guidinghis reasoning aboutclasses,and thattheygivemoremeaning to set-theoretic formulasby makingthemmoreconcrete.For example,in Section3 of this chapterwe will provetheformulaAo(BuC):(AoB)v(AtC).This formula is illustratedin Fig. 4, where the shadedarea representsClasses and SetsA o (B u C); one immediately noticesthatthissameshadedarea represents(AnB)v(,4nC).EXERCISES 1.21. Supposethat A c B and C c D; provethata)(AvC)c(BuD),b)(.anOg(BnD).lHnt t IJse theresultofExercise7, ExerciseSet 1.1.]2.SupposeA:BandC: D;Provethata)AvC:BvD,b).4nC:BD.[Hnr: Usetheresultof the precedingexercise.]3. Provethat If A c B,thenB' c A'.lHint,rJsethe resultofExercise 4(a), ExerciseSet1.1.]4.Proverhatif A : B, thenA' : B'.5. Prove thatifA : B and B c C,then A c C-6. Provethatlf A c B andB c C,then A c C.7.ProveTheorem1.11,parts(iv)and(v).8. Let S : {" | , I x}; useRussell'sargumentto provethatS is not an element'9.DoesAxiomA2 allowus to formthe"classofall classes"?Explain'10.ExplainwhyRussell'sparadoxand Berry'sparadoxcannotbe producedby usingAxiom,A2.3THE ALGEBRAOFCLASSESOneof the mostinteresting andusefulfactsaboutclassesis thatundertheoperationsof union,intersection,andcomplementation they satisScertainTheAlgebraof Classes31algebraiclawsfrom which we can.develop an algebra of classes. Weshallseelater (Chapter 4)that the algebraofclassesis merely oneexampleof a dtructureknown asa Booleanalgebra;anotherexampleis the "algebraof logic,"wherev, A, r Lre regardedas operationson sentences.Ourpurpose in thissection is to develop the basiclawsof the algebraofclasses.We remindthe readerthat the word clcssshould be understoodtomeanany collecton of objects; thug thelaws weareabout to present shouldbethoughtofas applyingtoevery collectionofobjects; in particular,theyapplyto all sets.1.20Theorem If A and B areany classes,theni\Ac.AvBandBcAvB,ii)AaBcAandAaBcB.Proofi) To prove that A c A'vB, we must showthat x e A > x e A v B:xeA+xeAvxeBby1.5(i)+xeAvBby1.13.Analogously,we canshowthat B c A v B.ii) To prove thatAn B c,4, we mustshowthat x e A B - x e A.xeAaB +xeA nxeB- ,r-{ I1.21TheoremIf A and B are classes, then1) A c. Bifand only if.4 v B : B,ii) A c Bifand onlyifA o B : A.Proofi) LetusfirstassumethatA c B;thatis,xel+xeB. ThenxeAvB-xeAorxeB+xeBorxeB+xeBby l.I4by 1.s(ii).by 1.13by 1.7(i)by 1.8(iv).Thus,4u B c B;but B c A u Bby 1.20(i);consequently,AvB:B.Conversely,let us assumethatAvB:B. By 1.20(i), Ac AwB;thus.4 c B.ii) The proof is leftasanexercisefor thereader.I32 Clsses and Sets1.22Theorem (Absorption Laws)'For all classes A and B'i) Av(AaB): ' ii) '4 n (AwB)- A'Proofi) By1.20(ii), AB c '4;therefore'bv1'21(i)'Av(AoB): 'ii) By1.20(i),A = Au B;therefore'by 1'21(ii\'A (Av B):A'l1.23Theorem For every classA, (A')' : 'Proof xe(A',)',+x(A',+xeAxeA+xQA'+xe(A')'1.24Theore m (DeMorgan's Laws)' For all classes A and B'i) (,a u B)' : A' r B'' ii) ('n B)' : A' v B''Proo.fi)First, xe(AuB)'+x(AwB bY1'19+ xQ Aandx( B(becauseif either x e A otx e B' then x e A w B)by 1.19,by 1.19.1by 1.19by r.14.by l.r4by 1.19by 1.19.+ xe A'andxeB'+xe (A'aB')Next, x e (A' n B) i'xe A' andxe B'- x(AandxfB+x(AvB>xe(AvB)'ii) The proof is leftas an exercise for theeader' I1.25 Theorem For all classesA, B and C' thefollowing aretrue'commutatiue Laws: i) A v B : B w Aii)AaB:BrAiil\ Av A:AIdempotent Laws:^ lv)AoA':AAssocatiae r,aws: ,ll i: [l : :] ::,"^:;"^zTheAlgebraof Classesvii) A n(B u C) : (A a B)v (A a C)viii) Au (Bn C) :(A v B) n (A w C)DistributiueLaws:Proqf1)xeAvB+xeAorxeB+xeBorxeA+xeBwAv) xe Av(B "': l=t1ti:"i!.::l>x(AwB)vCvii) xe Aa(BuC) ixeAn xeBvCii.',ool 1: 1r",.,!'o ^: lZl\u,,",i;:;,'by 1.13by 1.8(i)by 1.13.by 1.13by 1.13by 1.8(ii)by 1.13by 1.13.by t.r4by 1.13x e C)by 1.8(iii)by l.I4by 1.13.The proofs of (ii), (iii), (iv), (vi), and(viii)are exercisesfor the reader.ITheempty classand theuniversalclassareidentityelementsfor unionand intersectionrespectively;they satisfithefollowingsimple rules:1.26TheoremFor everyclass.4,'l)Aw@:A.ii)Aa:@.iii) A v 4,/ * Q/.iv) ,4n 4t : A.v) '?l' :' g.vi) Q' : a.vii) Iu A' : 4t..viii),4n A' : @.Proofi) Byt.I7, g A, and therefore by 1.21(i),A w @ + A.iii) By 1.17(ii)A e al/, andtherefore by 1.21(i),A v Ql : 4.Theproofs of the remaining parts of thistheoremareleft as an exerciseforthe reader.IBy using thelaws ofclassalgebrawhichwe have developedabove,we canproveall the elementaryproperties of classeswithout referringtothe definitionsofthesymbols u, n, ', and -c. The following is an exampleofhow such proofsare carriedout.Classesand SetsExampleProve that A a (A' u B) : A r B.ProofA n (A' u B) : (A n A,) v (A a B): Ow(AnB):ABA-B:AaB':B'nA: B' n (A,),:B'-A'Thefollowingdefinitionis f'requently useful:The dffirenceof twoclassesA and B is theclassof all elementswhichbelong to A, butdo not belong to B.In symbols,A-B:AaB'.Example provethat A - B:B'-A".Proofby 1.25(vii)by 1.26(viii)by L26(i\.Definitionby 1.2s(ii)by t.23Definitionof B' - A'.It is useful to note thatwith theaid of rheorem1.21,relationsinvolvinginclusion (c), not merely equalitcan beprovedusingclassalgebra.EXERCTSES1.31. Prove Theoreml.2l(1i).2. Prove Theoremt.24(ii).3.ProveTheorem1.25,parrs(ii),(iii), (iv), (vi) and(vi).4.ProveTheorem1.26,parts(ii), (ii,), {v) through(viii).5. Use classalgebrato prove thefollowing.a)(A B) v C : (A v C)n (B v C), b) (,au B) n C : (A ^C) u (Bn e.6.Use classalgebrato provethefollowing.a\ If A ^ C : A,then,4 n (B v C) : A n B.b) lf A a B : 6,thenz4 - B -- A.-c) If A n B : and, Av B : C,thenA : C - B.7. Using class algebra,prove echof thefollowing.a) Aa(B-C):(A^B)-C.b) (.a v q - C : (A - c)u(B - C).8.9.Ordered PairsCartesianProducts 35c) A - (B u C) : (A - B) a(A - C).d) A - (B n c) : (A - B) v (A - C\.Wedefinetheoperation+ on classesasfollows:lf A andB are classes,thenA+B:(A-B)v(B-A\.Proveeachof the following.a\A+B:B+A, , b\A+(B+C):(A+B)+c,c) Aa(B-+C):(AaB)+(AnC), d) A+ A: @, e) A+ @: A.Prove eachof thefollowing.ti)AvB:g+A:6andB:9.b) A a B' : ifandonly tf A c B.c) A + B -- gifandonly if A : B.10.Proveeachof the following.a)AwC:BvCifandonlyif,4+BcC.b) (,a u Q + (nu Q : @.+ B) - C.11. Use classalgebrato prove thatif ,4 c B and C : B-r4,thenA : B - C.4ORDEREDPAIRSCARTESIANPRODUCTSIf a is anelement,wemay use theaxiomofclassconstruction to form theclass{a} : {x l* : o).It is easyto seethat {a} contains only oneelemen!namely theelementa.A classcontaininga singleelementis calleda singleton.lf a and b are elementqwemayusethe axiomof classconstructiontoform theclass{o,b} : {, lx: aorx:b\.clearly {a, b} containstwo elementgnamelythe elementsa and b. A classcontainingexactlytwo elementsis calledan unorderedpair,r, more simply,a doubleton.In like fashion,we can formthe classes {a, b, cl, {a, b, c,d), and so on.Frequently,in mathematics, weneedto form ciasieswhose elements aredoubletons.In order to be able to do tiris legitimately,weneed a newaxiomwhichwillguaranteethat if a and,b areelements,then the doubleton {a, b} isan element.Thismotivatesournextaxiom, which is often calledtheAxiomof Pairing:A3. If a and b are elements , then {a, } is anelement.It is clear fhat {a, a}': la}; thus, settinga : b in AxiomA3, we immedi.ately getif is anelement,then the singleton {a} is anelement..il36Classesand Sets1.27Theorem If {x, y} : {u, u}, thenlx: uand y:ufor lx: uand y:u].Theproof is left asan exercisefor thereader. lHint: considerthecases x : y,x * !, separately.Use AxiomAl.]An importantnotionin mathematics is that ofan ordereilpcrr ofelements.Intuitively,an orderedpairis a class consistingof twoelementsin a specifiedorder.In factheorderis not reallyessential;whatis essential is thatorderedpp{rs havethefollowing property.t1.28 Let(a,b) and (c,d) b orderedpairs. If {a,b): (c,d\then a : c andb:d.we wouldlike to define orderedpain in sucha wayas to. avoid introducinga new undfinednotionof "order." It is an interesting factthat this canlindeed,beaccomplished;weproceed asfollows.129 DefinitionLeta and,bbeelements; theorderedpair(a,) is definedtobe theclass. (a,b) : {{a},{a,b}}.By AxiomA3,(a,)can be legitimately formed,and is tserf an erement.It is worthnoting that(b, a\ : {{b},{b, o}} : {{b},{a,b}}.Hencethere is a clear distinctionbetweenthe two possible ..orders,, (a, b) and(b, a): they are distinctclasses.It remains to provi thatorderedpairg aswehave just defined therrhave property 1.2g.1.30TheoremIf (a, b):(c,d),thena: candb: d.Proof. Supposethat (a,bl :(c, d);that is,{{o},{o, b} } : {{c}, {c, d}}.By Theorem1.27. either[{r} : {"} and {a,b} : {r,it}],Ordered PairsCartesianProductsorl{a} : {c,d} and {a,b} : {c}l;we willconsider thesetwo casesseparately.Casel. {o} : {"} and{a,b} : {",d}. From {o\ : {r\,itfollows thata:c.From {a, b\ : {",d} andTheorem1.21,itfollows that either a:c andb : d,or a : d and b : c; :r'the first casgwe aredone; in thesecondcase,we haveb : c : a : d, so again we are done.Case 2. {o):{t,d} and {a,b} : {c}. Here c e{c,d\and {c,d} : {a}, soc e {a}; thusc :q; analogously,d = a. Also,b e {a,b} and {o,b} : {c}, sob e {c\; hence b : c. Thus : b : c : d,andwearedone.I1.31 DefinitionTheCartesianproiluct of two classesA and B is the classofall ordered pairs (x, y) wherex e A andy e B. ln symbols,' A x.B: {(*.illxe land yeBI.The followin g zrea fewsimple propertiesof Cartesian products.1.32 TheoremFor all classes A, B.and, C, :i\ A x (Bn C) :(A x B\ a(A x C).1l)A x (B v C) :(A x B)u(,4 x C).1i1)(A x B)n (C x D) : (A o C) x (Bn D).Proqf1)(x,y)eAx (BnC)+xeAand yeBnC+xeAand yeBand yeCo(x,y)eAxBand (x,y)eAxCo(x,y)e(AxB)n(AxC).B)o(C xD)e(x,y)eAxBand (x,y)eCxD+xeAand yeB and xeC and yeDxeArCand yeBaD.+ (x, y) e (A a C) x (B n D).Iiii) (x, y) e (A xJustas wefoundit instructiveto represent relations between classesbymeansof Venndiagramg it is oftenconvenientto illustraterelationsbetweenproductsof classesby using a graphicdeviceknownas a coordinatediagram.A coordinatediagramis analogousto thefamiliarCartesian coordinateplane;thereare two axes-averticalone anda horizontalone-butwe consideronlyClassesandSetsone "quadrant."If we wishto representa classA x B,then a segment of thehorizontalaxisis markedoff to representA and a segmentof thevertical axisis marked off to representB; A x B is the rectaigledeterminedby these twosegments (Fig.5). As an exampleof the useof coordinatediagrams, Theorem1.32(iit)is illustrated in Fig. 6..Fig.5EXERCISES1.43r!t.LetA:{",b,c,d},B:{r,2,3},C:{r,y,"}.Find,4 xB, BxA, Cx(BxA),(A v B) x C, (Ax C)ur (B x C),(Aw B) x (B u C).2.ProveTheorem1.32(ii).3. Provethat'.4x (B - Dl : (A x Bl - (A x D).4.Prove tbat(A x B) n(C x D) :(A x D) n (C x B).5.lf A, B and,C are classes, provethefollowing.a)(A x A) a(B x C) :(A a B\ x (A C\.b) (,a x D - (c x C) : l@ - C) x Bl u la x (n - Ql.c) (A x A) - (B x C): f(A - B) x AlwlA x (A - C)f.6. ProvethatA and B are disjoint if and onlyif, for anynonemptyclassC, A x C andB x Caredisjoint.7.lf AandCarenonemptyolasses,provethatA c BandC = D ifandonlyif,4x C cBxD.8. LetA.B,C,Dbenonemptyclasses.Provethat A x B:C x Difandonlyif A:CandB:D.9.lf A, B, and C are anyplasses,prove?) , *BandA'xCaredisjoint, b) B x AandC x A'aredisjoint.10.Prove that A x B : ifandonly if A : A or B : @."JrIt"ItA.ir.BDIIncl11,12.Graphs 39Prove eachof thefollowing.a) lf a: {b},tbenb e a.b) x:yif andonlyif {x}:{y}.c) x e a if andonly1f {x\ c q.d) {a,b} : {a} ifand onlyif a : b.We give the followingalternativedefinitionof orderedpairs:(x,y):{{,,@},{t,{}}}uu"nr,}Tl";rl,T;!,il_7."5GRAPHSA class of orderedpairsis calleda graph.In other words,a graphis anarbitrarysubclassof all xi?/ .Theimportanceof graphswillbecomeapparentto the readerin Chapters2 and 3. It may be shown,for instance,thata function from A to B is a graphG c A x B with certain .special properties. Specifically,G consistsof all thepairs(x,y) such thaty : /(x). This examplemayhelptomotivatethe followingdefinitions.1.33 .Definition If G is a grapt\thenG-l is thegraph defined byG-t : {(r, y) I $, x) e G}. {$r:1.34 Definition If G ,and H arc grapfus, then G ".EI is the graph definedasfollows:G. H : {(r,y) I 1z=(x,z)eHand (z,y)e G\.The followin g are a fewbasic propertiesof graphs.1.35Theorem If G, 1,and Ji) (G. H)"J:G.(II.J):ii) 1C-t-t * O.iii) (G " l)-t : H-L " G-r.Proqfaregraphs,thenthe followingstatementshold:i) (x' v) e (G ' r{) " t :i1= *u''=:: r"::i. ,o;i]^'ri,l. H and (w, y) e G' +lw=(x,w)eH.Jand (w,y)eG(x,y)e6"(H"J).ItjIIq ClassesandSetsii) (x, y) e (c-1)-t o(y,x) e G-L+(x,y)eG.iii) (x'Y)e (G 'H)-':T1'#1i1;:ru "',izu '1.36DefinitionLetG be a graph.By the ilomqinof G wemeanthe classdomG: {xl3ya(x,y)eG},and by the range of G we meantheclassranG+{yll*=(x,y)eG}.ln other words,the domain of G is the class of all "first components"ofelementsof G, andthe range of G is the class of all "second components" ofelementsof G.1.37'TheoremIfG and Il are graphs,theni) dom G : ranG-t,ii) ran G : dom G-1,iii) dom(G " H) c dom 1,iv) ran (G " H) c ranG.tProqfi) xedomG+1y=(x,y)eGe3y=(y,x)eG-1xcranG-iii) xedom(G "H)- 3y:(x. y)e(G"H\+12=(x.z)eHand (z,y)e G+_\domFI. I1.38 CorollaryLet G and Il be graphs.If ran.EI c domG thendom G o H :domH.The proof of thistheorem is leftasanexercisefor thereader.EXERCTSES1.5l. LetG = {(b, b),(b, c),(c" c), (c,d)}GraphsandH : {A.a\(c.bl,(d.c)}.Find G-1,H-',G " H,H " G,(c " H)-t,(Gv H)-1,H-t " G.2. ProveTheorem1.37,parts(ii) and(iv).3. ProveTheorem1.38.4. If G, U, andJare graphs, proveeachofthe following.a) (Hu J)"G:(H.G)u(J"G),b)(C - Hl-r:G-l - H-t,c) G"(Hn,r) c (c"H)n(G""/),d) (c" H) -(c"J)=G"(H- J).5. lf G and H are graphs.proveeachof the following.a)(Gn Hl-t : G-1 ^ H'1, b) (G u H)-' : G-t v H*1.6.lf G, H, J , and K are graphs, provea)ifG c H andJcK,thenG " J c H " K,b) C c Hlf andonlyifG-1 s H-t.7.IfA,B,andCareclasses,proveeachofthefollowing.a)(AxB)-':BxA.b) If,4 ^B + O,then(,4 x B)"(A x B):A x B.c) ll,4 and B aredisjoint,then(,4x B) " (Ax q: A.d) If B /@,then(B x C) "(A x B\ -- A x C.8. LetGandH begraphs; proveeachofthe following.a)IfG c A x B,thenG-1 c B x A.b)IfG-AxBandIIcBxC,thenIoGcAxC. t&9. If Gand H are graphs, prove eachof the following.a)dom (G v H):(dom G) u (domH).b) ran (G u H) : (ran G) u (ran H).c) domG - domH c dom(G - H).d) ranG - ran H c ran (GIl).10. LetG be a graph, andlet B be a subclass ol thedomainof G. Byfhe restrictionof 'GtoB we meanthe graphGror: {(x, y)l@,y)eGandxeB}.Proveeachof thefollowing.a) Gn:G o(B x ranG),s) Ga^cl : G.\ | Grq,11.LetG be a graphandletB be a subclassof the domainofG. WeusethesymbolG(B)to designate the classG(B) : {y I l, e B = (x, y)e G\.Proveeachofthefollowing..a) G(B) : ran Gr,c) G(Bn C) : G(B) a G(C),b) Ga,cr : Ga1v Gc1,d) (G. H)s:G " Htut.b) G(Bu C) : c(B)u G(C), "d) If B c C,thenc(B) s c(C).Classesand Sets6GENERALIZEDUNIONANDINTERSECTIONConsidertheclass {Ap A2,...,A,\; its elementsare indexedby thenumbers1,2, ..., n. Sucha class il oftencalld an indexedfamilyof classes;the numbers1,2,...,n arecalled indices andthe class {1,2,...,n} is calledthe indexclass.Moregenerally, wearefrequentlyled to think ofa classI whoseelementsi,j,k,...serveas indicesto designatetheelementsof a class {A,Aj,Au,...}.The class {Ar, A,Ao,...} is calledat indexed family of classes,1 is calleditsinilexclass, andihe elementsof 1 are called indices.A compactnotation whichis often usedto designatetheclass {A, A,,40,...} is{At}or'Thus,speaking informally, {A,\n, is the class of all the classesA,as i rangesover/.Remark.Thedefinition of an indexedfamily of classeswhichwe have justgiven is,admittedlanintuitiveone;itrelies on the intuitive notion of indexing.ihis intuitivedefinition is adequate at the present time; however,frfuturereference, we nowgivea.formal definitionof the sameconcept:By anindexedfamily ofclasseg {,A,},.r, we meana graphG whosedomainis 1; for each e / we define,4, bYA,: {x l1;,xec}.For*grample,consider {A}n where.I -{1,2\, A,.: {a,b}, andAr: {c,d\-Theff, formally,{Atli.r is thegraphO : {(t, a), (1, b),(2, c),(2, d)}.If {Ai}rd is an indexedfamilyof classessuch thatforeachiel,A,is'anelement, thenwelet {A,l ; e f} designatethe classwhoseelementsare all theA,, that.is, {,4, | ; e f} : {t l, : .4, forsome i e I}.However,weshall followcurrentmathernaticalusage anduse the twoexpressions, {,4,},.. and {Arl ; e f},interchangeably.1.39 DefinitionLet {A,},., be an indexed familyof classes.The union of theclasses, , consistsof all theelementswhich belongto at least oneclass .4, ofthefamily.In symbols,UA,: {xll.e I=xeA1.Theintersection of the classes,4.,consistsof all the elementswhichbelongtoeveryclassA, of thefamily.In symbols,)A,:{xlv;e I,xeA,\.ieIThe followingaresomebasicpropertiesofindexed familiesof classes.ho - t vv a0 s- u"c t''A e"Lo tato \".tt'-t-./vo ob:k a k\l- n,\uch:f ut\ff\tr!, s Q'^- vYrvc\. h.*vl,r"od o,, totr,oo,GeneralizedUnionand lntersection 431.40TheoremLet {A,},., bean indexed family ofclasses.i) If At c B for everYi e 1,then fl'1, = n.ii) IfB c Aforeveryi el, thenU - )O,.Proqfi) Supposethat A,c B for everyiel;nowif xe[J A,, thanxeA, forsome j e 1;but Aj - B, so x e B. Thus [J A, = B. ielieIThe proof of (ii) is left asanexercisefor thereader. I1.41Theorem(GeneralizeddeMorgan'sLaws). Let {A,\n, be an indexfamilyofclasses.Then,i) (U /,)' : ),q;.ielielii) (l^).4,)' :0 ,q;:ielieIProo.fi) x e (lJ A,)' o xdU A,ieliel+VjeI,x(A,YieI,xeAr'+xe ,]ot''The proof of (ii) is left as anexercisefor thereader. I1.42 Theorem(Generalized DistributiueLaws). Let lA,j,., and {Br}r., beindexed families ofclasses.Theni) (U ,4,)n ([J B.) : U (A, n B),i.4jeJ(i,j\elxJii) (n ,4,)u (l B;) : 0 (A, v B). :ieljeJ(i,j)elx JProofi) xe([J,4,)n(U B,)+x.UA, and xel) n,ieI x ,to,o. ,o-" n.llx e Boforsome k e Jx Ao o Boforsome(h, k) e Ix Jx U (A,aB)..(i,i)eIxJProqfi)4ClassesandSets.Theproof of (ii) is left as an exerciseforthe reader. IA theoremconcerningtheunionof graphswill be usefulto us in thenextchapter.1.43TheoremLet {G,},.t bea family of graphs. Theni) dom(U c,) : ! (domGr).ieI ielii) ran(U C,) : [J (ranGr).ieliIxedom(U G,)e3ya(",y)e U G bY 1.36, t" + 3y : (x, y) e ]ro, ,o.n.;. rby 1.39+ x e domG, for some j e fbY 1.36+ x lJ (domG') bY 1'39-ieIThe proof of (ii) is leftasanexercisefor thereader. IA variantnotationfor the unionandintersectionof a familyof classesissometimesuseful. If . is a class (its elements arenecessarilyclasses).wedefinetheunion o.f., or unonof the elementso.f, to be the union of all the classeswhichare elementsof,. In sYmbols,1.4l) A: {xlxe ,4.forsomeAe.\.In other words,*.^"oif andonly if thereis a class1 suchthatx e A andAe'&Ae,.Analogouslwe define the intersectionof d,ot intetsectionof theelementsof ., to be the intersectionof all the classeswhichare elementsof..Insymbols,1.451.46 .Example Letu: {d,e}. ThenU A: {a,b,c,d,e\ and ) A: {d\.'t. Ae"t1.47RemarkIt is frequentpractice, in theliterature of settheory,to writeUd for l) at/) e: {xlxe.4forevery Ae.}.Ae.r'/,d : {K,L, M}, whereK : {s,b,d\, f : {a,c.it\, andGeneralizedUnion and lntersection45and)d for ),.Ae,ry'We shall occasionallyfollowthat practice in thisbook.EXERCISES1.61. ProveTheorem1.40(ii).2.ProveTheorem 1.41(ii).3. ProveTheorem1.42(11\.4. ProveTheorem1.43(ii). ,5. Let {A,},,, and {8,},., betwofamiliesofclasseswith thesameindexclassI. SupposethatVi e I, A, c Br;provethata)UA,cU4,eIeIb)0.a,sn4.eIiel6.Let {A,}r., and {^B,}n, be indexedfamiliesofclasses.Provethefollowing.a)(n.4,)x (l B;) : n (A, x B1,ie/jeJ(i,j\el x Jb) (U,4,) x ([J B.) : U (A, x B).eIjeJ(,j\elx J7.Let {A,}r., and {B}r be indexedfamiliesofclasses.SupposethatVeI,)jeJ=B, = 4,. Provethat)a, s ).a,.8. Let {A,}r.,and{B};..r beindexedfamiliesofclasses.Provethata)(U 1,) - (U ) : U (n lA,- Bl),ieljeJieljelb) (n.4,) - (llBj) : n (u lA, - Bf).ieljeJ eI je,I9. Wesaythat an indexedfamily {8,},., isacouerngof Aif A = !J 4. Supposethat{.B,}". and {C,}ru aretwodistinctcoverings of ,4.Provethatthe family{(4 ^ c)}rt,.r*,isa coveringof.4.10.Let a: {u,u,w},b:{r,r}, c: {w,y},r:{a,b\,s--{b,"},andp: {r,s}. Findtheclassesu(up), n(np),u(np)n(up).11.Provethata(.v g) :(a.) n (n0).12. Proveeachofthefollowing.a\ If A e 9,thenA c v Q anda0 c A.b) U c 0ifandonlyifw. c vQ.c) lf @ e,,then a, : .16Classes andSets7SETSundoubtedly, everythingwe have said in the preceding pagesis fairlyfamiliarto thereadei. nu*tftorrgh wesaid,classwherethereaderis more accustomedio.hearing sef, it is obvius that'the "union" and"intersection" definedinthischapterare prcisely the familiarunion andintersectionof sets, the..Cartesn product"is eiactly the usual Cartesiannproduct of sets,andsimi-larlyfortheother conceptsintroducedin this chapter. 'At this poing it appearsastrougheverything*" ur" accustomed to doing withsetscan bedone withclasses.Thus,the reader may very wellaslg"why bother distinguishingbetween classesandsets? Whynoi developall of mathematics in terms ofclasses?,,Since,as we havesaid,a class means"anf collection of elements,"why not simplycall a classa se! and be donewith it? Theanswer to thisl,rstiol is oi lreat importance;thechief purpose of this section is to explain*hy t"" do'wantto distinguish betweenclasses and sets'First, we notethatthe axiom of classconstruction (Axiom A2)permitsusto form theclassof all elemens whichsatisfya givenproperty; it does notallow us to form the class of all classeswhichsatis$a givenproperty'Thereason forthislimitationis obvious:if we were enabled to formthe classofall classes which satisff anygivan property,then we could_form "Russell'sclass"of all classeswhich aienot elementsof themselveg andthiswould giveusRussell's paradox.Next,we note that in mathematicsweoftenneedto form particularsesof sets.A fewexampleswhichcometo mind are thefollowing:The set of all closedintervals la, b] of real numbers'Theset of all convergent sequencesof real numbers'Thesetof all the lines in the plane(where eachlineis regardedas a setof points)'Letus look more closelyat the first example;in a discussion in elementarycalculus,we wouldfeel perfectlyfreeto say "let .-ql consistof all the setswhichareclosld intervals la,-b] of realnumbers." Nowif "sets"were no differentfrom "classes"then, by the precedingparagraph,we wouldnot be allowedtoform..This wouldban iniolerablerestrictionuponour freedomof operatingwith setsin mathematics.Let us recapitulate: Thenotionof class is appealing becauseofits irituitivesimplicity and lenerality; however,thereis a seriousdrawbackto dealing with"lur."r, nu-.t/ttrut it is not permissible [for an arbitrary propertyP(X)]toform tie "classof all classes X whichsatisff P(X)."Thiswould be an intoler-able restrictionon our mathematical freedom of action if we wereto basemathematics uponclasses.Insteadwebasemathematics uponses; theconcept ofa sef is somewhat narrowerthanthat ofa class;setswillbedefin"dSets 47i"\,svch;a JfLy; h-o_wg.u.el,that .for any propertvp(X\, it is legitimate to formtheclass o.l atl sets x which satisy p(x). Thus, thefreedom we require is resioFdI withsetsratherthanthe broadernotionof classes.we are morethan willing to do so, because,aswe will beable toshow,all theclasses we dealwith in mathematics aresets.tsamethin n element.Thenit us toformtheclassof all sefs v whichsatisfvanv prooertyp(X).This simpleansweris, in fact,the onewhichhas beenadoptedby mostmathematicians.We willuse it here; thus,1.48DefinitionBy a setis.meantanyclasswhich is an elementof a class.ihir d.finitionis supportedby ourintuitive perception of what a set shouldbe.For if A, B, c, ... are'sets, it is perfectlyreasonable thatwe should beableto form the class .{A, B, C, ...} whoseelementsare A, B, C, ....In otherwords, we wouldquite certainlyexpecteverysetto be anelement.Theconverseis equally reasonable:for i[.] !s nota set, then A is a proper class.andwe haveatreadyseenthatln orderto avoidcontradictions,proper classes should not beelements of anything. Thus, if Iis not a set, then Iis notanelement.In theremainder of this section,we will state thebasic axiomsdealingwithsets. The main purpose of these axiomsis to guarantee thatwhenthe usualset-theoreticoperations areperform'ed on sets,the result, each time,is a set.First, we note that theAxiomof Pairing,our Axiom43, may be re-statedthus:43.. If a and b aresets, then {a, } is a set.Now,if Iis a set and A = ,8,onewould reasonablyexpectIto be a set.Thisis thecontentof our next axiom,called theAxiomof sibsets.44. Every subclassof a set is a set.By Theorem1.20(ii), A n B c l;thus,by AxiomA4,if A is a set, thenA B is a set. In particular, theintersectionof any twosetsis a set.Theunion of "not toomany"sets shouldbe a set.This is guaranteed byour next axiom,calledthe Axiomof Unions:A5. If . is a setof sets,thenUAedIis a set.Ig\!-b.ow Shouldsetsbe defined?In order to answ"' this-que"ti.rn,it ise_ssentialto remindourselves, once again,fhtwe are seekinga way of definingrf A and -Bare sets, then, by AxiomA3, {A, B} is a set; it follows immediately48ClassesandStsfrom Definition1.44that l-l X : A u .B;thus, by Axiom A5,Au B is aset. Thisshows that ,n, ,fi'] twosetsis a set.1.49 Remark. ByAxiom A3,. every doubletonis a set. Furthermore, lettinga * b in A3, everysingletonis a set. Sincethe unionof two setsis a set,follows that everyclass of three elementsis a set, everyclass of four elementsis a set,and soon. Thus, in an intuitivesense,every finiteclassis a set.Next, we willestablishthat if Iis a set, then theclassof all the subsetsofIis a set. We beginwitha definition.1.50 DefinitionLetAbeaset; bythepowersetof Awemeantheclassof allthe subsetsaf A. In symbols,thepowerset of Iis theclassg(A): {BlB q A}.Notethatby Artiom A4, g(A) is the classof all the ses,B whichsatisfyB c A.By Definition1.48 andAxiomA2, it is legitimateto form this class.The followingis calledtheAxiom of power Sets:A6. If Iis a set, then g(A) is a set.1.51 Example If A : ,o,], then g(A) : {, {a}, {b}, {a, b}}.Fromall thatwe have said so far, it doesnot yetfollowthatthere exisrany sets at all. To fill thisvacuum, we state a temporaryaxiom,whichwill besuperseded by AxiomA9:T The empty classis a set.Henceforth,we willrefer to @ as theempty sel.FromAxiom T, togetherwith 43 and45, wemayinferthe existenceofa great many sets. We have the emptyset, @; we have singletons such as{}, {{A}}, etc.;we havedoubletons suches {, {@}}, forriedby anytwoof the above. Similarly, taking unionsof the above'repeatedly, we may formsets with any fnite numberof elements.1.52 Remark An important consequence of Axiom 46 is the following. lf ,4is a set, thenclearlyB:{Xlxet and p(x)}is theclassof all the subsetsof rwhichsatisfythe property p; by Axiom44, Axiom A2 may legitimately be usedto formttre class .8. Now if x e B,Sets 49thenXis a subsetof l,so X e 9(A); thus.B c 9(A), But by Axiom A6, g(A)is a set, henceby Axiom A4, .B is a set, Wemaysummarizeas follows:if Iis asetandP(X) is a property of X, thentheclass of all the subsetsof A whichsatisfyP(Xl is a set.1.53Theoremlf A andBaresetgthen Ix B is a set.Proof.LetA andB be sets. By Axiom A5,AvBis a set; by AxiomA6,9(A v B) is a set; finally,by AxiomA6again, glg(A u B)] is a set. We willprovethatA x B = glg(A u B)],andit willfollow,by Axiom,A.4, thatA x Bisaset.Let(x, y)e A x B. By1.29,(x,y): {{r}, {r,y}}. NowxeAv B,hence{*} = 'u B, so {x} eA(AwB). Similarl xe AvB andye A w B, so{*, y} s A v B, hence {x, y} e 9(A w B). We have just shownthat {x} and{x, y} ae elementsof 9(Au B), hence{ {"}, {r, y} } s 9(A v B);{ {r}, {r, y}} e ele(A u B)1,(x, "y) e qlgl u B)1.AxBcglg(AuB)1.Iit followsthatthat is,ThusIt follows from Theorem1.53arrdAxiomA4 that if A and B aresets, thenanygraphG c A x Bisaset.It is easyto show that if G is a set, thendomG and ranG are sets(seeExercise5, ExerciseSet1.7).Usingthisfact,one can easilyshowthat if G andEl aresets,then G " H and G-r aresets(see Exercise6, ExerciseSet1.7).EXERCTSES1.71. rf u4 and B aresets, provethat A - B ado * ui,firP,i?;llse8, ExerciseSet 1.3.)2. lf A is a properclassandA c B, provethat B is a properclass. Concludethat theunionof twoproper classes isa proper class. ^.r tla*3.Prove that the "Russell class" and theunivrsalclassare properclasses. [Ainr$setheresultof Exercise8, ExerciseSet 1.2.]4. Let {Ail.r be an indexedfamilyof sets.Provethat (-) ,4,isa set.5. I,et G beagraph. Provethat if G isa set,theno-'rrrran G aresets. [Ifinr: ShowthatbothdomG and ran Gare subsetsof u(uG).]50Classs andSets6. ].t G andII begraphs. Provethatif Gandf[ are sets,then G- 1 and G o -EIare sets'7. Let r : {o, b}, s : {b, c}, p : {r, s}. Findthe sets 0(r\,9(9(r))' and 9(vp)'8. Let AandB besets;provethefollowing'' a) A = B if andonlY1f9(A) = E(B)'b) A:B if and olY1f9(A) = 9(B\'c) E(A) 9(B) = e(A B)'d) E(A)w 9(B) c 0(Av B)'"\ t' n' n :gJ 1f arrd ov tf 9(A) 9(B) : {q}'9.lf A and Iarcsets,prove the following'a\ v(e(6\): fi.0 n(s(s\):'c) lf 9(A) e A(0\thenA e A'10.Exhibit thesetss(s()) and Els(E(@\\l'