Introduction to Sets - Math is Fun

Subsets. When we define a set, if we take pieces of that set, we can form what is called a subset. Example: the ... Advanced ShowAds HideAds AboutAds IntroductiontoSets Forgeteverythingyouknowaboutnumbers. Infact,forgetyouevenknowwhatanumberis. Thisiswheremathematicsstarts. Insteadofmathwithnumbers,wewillnowthinkaboutmathwith"things". Definition Whatisaset?Well,simplyput,it'sacollection. Firstwespecifyacommonpropertyamong"things"(wedefinethiswordlater)andthenwegatherupallthe"things"thathavethiscommonproperty. Forexample,theitemsyouwear:hat,shirt,jacket,pants,andsoon. I'msureyoucouldcomeupwithatleastahundred. Thisisknownasaset. Oranotherexampleistypesoffingers. Thissetincludesindex,middle,ring,andpinky.   Soitisjustthingsgroupedtogetherwithacertainpropertyincommon. Notation Thereisafairlysimplenotationforsets.Wesimplylisteachelement(or"member")separatedbyacomma,andthenputsomecurlybracketsaroundthewholething: Thecurlybrackets{}aresometimescalled"setbrackets"or"braces". Thisisthenotationforthetwopreviousexamples: {socks,shoes,watches,shirts,...} {index,middle,ring,pinky} Noticehowthefirstexamplehasthe"..."(threedotstogether). Thethreedots...arecalledanellipsis,andmean"continueon". Sothatmeansthefirstexamplecontinueson...forinfinity. (OK,thereisn'treallyaninfiniteamountofthingsyoucouldwear,butI'mnotentirelysureaboutthat!Afteranhourofthinkingofdifferentthings,I'mstillnotsure.Solet'sjustsayitisinfiniteforthisexample.) So: Thefirstset{socks,shoes,watches,shirts,...}wecallaninfiniteset, thesecondset{index,middle,ring,pinky}wecallafiniteset. Butsometimesthe"..."canbeusedinthemiddletosavewritinglonglists: Example:thesetofletters: {a,b,c,...,x,y,z} Inthiscaseitisafiniteset(thereareonly26letters,right?) NumericalSets Sowhatdoesthishavetodowithmathematics?Whenwedefineaset,allwehavetospecifyisacommoncharacteristic.Whosayswecan'tdosowithnumbers? Setofevennumbers:{...,−4,−2,0,2,4,...} Setofoddnumbers:{...,−3,−1,1,3,...} Setofprimenumbers:{2,3,5,7,11,13,17,...} Positivemultiplesof3thatarelessthan10:{3,6,9} Andsoon.Wecancomeupwithalldifferenttypesofsets. Wecanalsodefineasetbyitsproperties,suchas{x|x>0}whichmeans"thesetofallx's,suchthatxisgreaterthan0",seeSet-BuilderNotationtolearnmore. Andwecanhavesetsofnumbersthathavenocommonproperty,theyarejustdefinedthatway.Forexample: {2,3,6,828,3839,8827} {4,5,6,10,21} {2,949,48282,42882959,119484203} AreallsetsthatIjustrandomlybangedonmykeyboardtoproduce. WhyareSetsImportant? Setsarethefundamentalpropertyofmathematics.Nowasawordofwarning,sets,bythemselves,seemprettypointless.Butit'sonlywhenweapplysetsindifferentsituationsdotheybecomethepowerfulbuildingblockofmathematicsthattheyare. Mathcangetamazinglycomplicatedquitefast.GraphTheory,AbstractAlgebra,RealAnalysis,ComplexAnalysis,LinearAlgebra,NumberTheory,andthelistgoeson.Butthereisonethingthatalloftheseshareincommon:Sets. UniversalSet   Atthestartweusedtheword"things"inquotes. Wecallthistheuniversalset.It'sasetthatcontainseverything.Well,notexactlyeverything.Everythingthatisrelevanttoourquestion.         InNumberTheorytheuniversalsetisalltheintegers,asNumberTheoryissimplythestudyofintegers.   ButinCalculus(alsoknownasrealanalysis),theuniversalsetisalmostalwaystherealnumbers.   Andincomplexanalysis,youguessedit,theuniversalsetisthecomplexnumbers. SomeMoreNotation Whentalkingaboutsets,itisfairlystandardtouseCapitalLetterstorepresenttheset,andlowercaseletterstorepresentanelementinthatset. Soforexample,Aisaset,andaisanelementinA.SamewithBandb,andCandc. Nowyoudon'thavetolistentothestandard,youcanusesomethinglikemtorepresentasetwithoutbreakinganymathematicallaws(watchout,youcangetπyearsinmathjailfordividingby0),butthisnotationisprettyniceandeasytofollow,sowhynot? Also,whenwesayanelementaisinasetA,weusethesymboltoshowit. Andifsomethingisnotinasetuse. Example:SetAis{1,2,3}.Wecanseethat1A,but5A Equality Twosetsareequaliftheyhavepreciselythesamemembers.Now,atfirstglancetheymaynotseemequal,sowemayhavetoexaminethemclosely! Example:AreAandBequalwhere: Aisthesetwhosemembersarethefirstfourpositivewholenumbers B={4,2,1,3} Let'scheck.Theybothcontain1.Theybothcontain2.And3,And4.Andwehavecheckedeveryelementofbothsets,so:Yes,theyareequal! Andtheequalssign(=)isusedtoshowequality,sowewrite: A=B Example:Arethesesetsequal? Ais{1,2,3} Bis{3,1,2} Yes,theyareequal! Theybothcontainexactlythemembers1,2and3. Itdoesn'tmatterwhereeachmemberappears,solongasitisthere. Subsets Whenwedefineaset,ifwetakepiecesofthatset,wecanformwhatiscalledasubset. Example:theset{1,2,3,4,5} Asubsetofthisis{1,2,3}.Anothersubsetis{3,4}orevenanotheris{1},etc. But{1,6}isnotasubset,sinceithasanelement(6)whichisnotintheparentset. Ingeneral: AisasubsetofBifandonlyifeveryelementofAisinB. Solet'susethisdefinitioninsomeexamples. Example:IsAasubsetofB,whereA={1,3,4}andB={1,4,3,2}? 1isinA,and1isinBaswell.Sofarsogood. 3isinAand3isalsoinB. 4isinA,and4isinB. That'salltheelementsofA,andeverysingleoneisinB,sowe'redone. Yes,AisasubsetofB Notethat2isinB,but2isnotinA.Butremember,thatdoesn'tmatter,weonlylookattheelementsinA. Let'stryaharderexample. Example:LetAbeallmultiplesof4andBbeallmultiplesof2. IsAasubsetofB?AndisBasubsetofA? Well,wecan'tcheckeveryelementinthesesets,becausetheyhaveaninfinitenumberofelements.Soweneedtogetanideaofwhattheelementslooklikeineach,andthencomparethem. Thesetsare: A={...,−8,−4,0,4,8,...} B={...,−8,−6,−4,−2,0,2,4,6,8,...} Bypairingoffmembersofthetwosets,wecanseethateverymemberofAisalsoamemberofB,butnoteverymemberofBisamemberofA:   So: AisasubsetofB,butBisnotasubsetofA ProperSubsets Ifwelookatthedefintionofsubsetsandletourmindwanderabit,wecometoaweirdconclusion. LetAbeaset.IseveryelementofAinA? Well,umm,yesofcourse,right? SothatmeansthatAisasubsetofA.Itisasubsetofitself! Thisdoesn'tseemveryproper,doesit?Ifwewantoursubsetstobeproperweintroduce(whatelsebut)propersubsets: AisapropersubsetofBifandonlyifeveryelementofAisalsoinB,andthereexistsatleastoneelementinBthatisnotinA. ThislittlepieceattheendistheretomakesurethatAisnotapropersubsetofitself:wesaythatBmusthaveatleastoneextraelement. Example: {1,2,3}isasubsetof{1,2,3},butisnotapropersubsetof{1,2,3}. Example: {1,2,3}isapropersubsetof{1,2,3,4}becausetheelement4isnotinthefirstset. NoticethatwhenAisapropersubsetofBthenitisalsoasubsetofB. EvenMoreNotation WhenwesaythatAisasubsetofB,wewriteAB. OrwecansaythatAisnotasubsetofBbyAB("AisnotasubsetofB") Whenwetalkaboutpropersubsets,wetakeoutthelineunderneathandsoitbecomesABorifwewanttosaytheopposite,AB. Empty(orNull)Set Thisisprobablytheweirdestthingaboutsets. Asanexample,thinkofthesetofpianokeysonaguitar. "Butwait!"yousay,"Therearenopianokeysonaguitar!" Andrightyouare.Itisasetwithnoelements. ThisisknownastheEmptySet(orNullSet).Therearen'tanyelementsinit.Notone.Zero. Itisrepresentedby Orby{}(asetwithnoelements) Someotherexamplesoftheemptysetarethesetofcountriessouthofthesouthpole. Sowhat'ssoweirdabouttheemptyset?Well,thatpartcomesnext. EmptySetandSubsets Solet'sgobacktoourdefinitionofsubsets.WehaveasetA.Wewon'tdefineitanymorethanthat,itcouldbeanyset.IstheemptysetasubsetofA? Goingbacktoourdefinitionofsubsets,ifeveryelementintheemptysetisalsoinA,thentheemptysetisasubsetofA.Butwhatifwehavenoelements? Ittakesanintroductiontologictounderstandthis,butthisstatementisonethatis"vacuously"or"trivially"true. Agoodwaytothinkaboutitis:wecan'tfindanyelementsintheemptysetthataren'tinA,soitmustbethatallelementsintheemptysetareinA. Sotheanswertotheposedquestionisaresoundingyes. Theemptysetisasubsetofeveryset,includingtheemptysetitself. Order No,nottheorderoftheelements.Insetsitdoesnotmatterwhatordertheelementsarein. Example:{1,2,3,4}isthesamesetas{3,1,4,2} Whenwesayorderinsetswemeanthesizeoftheset. Another(better)nameforthisiscardinality. Afinitesethasfiniteorder(orcardinality).Aninfinitesethasinfiniteorder(orcardinality). Forfinitesetstheorder(orcardinality)isthenumberofelements. Example:{10,20,30,40}hasanorderof4. Forinfinitesets,allwecansayisthattheorderisinfinite.Oddlyenough,wecansaywithsetsthatsomeinfinitiesarelargerthanothers,butthisisamoreadvancedtopicinsets. Arg!Notmorenotation! Nah,justkidding.Nomorenotation. by RickyShadrachand RodPierce   Activity:Subsets VennDiagrams SetCalculator IntroductiontoGroups SetsIndex Copyright©2020MathsIsFun.com