Lesson on Subsets | Math Goodies

Another way to define a subset is: A is a subset of B if every element of A is contained in B. Both definitions are demonstrated in the Venn diagram above. Skiptomaincontent Searchform Search Example1: Given A ={1,2,4}and B ={1,2,3,4,5},whatistherelationshipbetweenthesesets? Wesaythat A isasubsetof B,sinceeveryelementof A isalsoin B.Thisisdenotedby: A Venndiagram fortherelationshipbetweenthesesetsisshowntotheright.   Answer: A isasubsetof B. Anotherwaytodefineasubsetis: A isasubsetof B if everyelementof A iscontainedin B. BothdefinitionsaredemonstratedintheVenndiagramabove. Example2: Given X ={a,r,e}and Y ={r,e,a,d},whatistherelationshipbetweenthesesets? Wesaythat X isasubsetof Y,sinceeveryelementof X isalsoin Y. Thisisdenotedby: AVenndiagramfortherelationshipbetweenthesesetsisshowntotheright.   Answer: X isasubsetof Y. Example3: Given P ={1,3,4}and Q ={2,3,4,5,6},whatistherelationshipbetweenthesesets? Wesaythat P isnotasubsetof Q sincenoteveryelementof P isnotcontainedin Q. Forexample,wecanseethat1 Q.Thestatement "PisnotasubsetofQ" isdenotedby: Notethatthesesetsdohavesomeelementsincommon.TheintersectionofthesesetsisshownintheVenndiagrambelow. Answer: P isnotasubsetof Q. Thenotationforsubsetsisshownbelow. Symbol Meaning isasubsetof isnotasubsetof Example4: Given A ={1,2,3,4,5}and B ={3,1,2,5,4},whatistherelationshipbetween A and B? Analysis: Recallthattheorderinwhichtheelementsappearinasetisnotimportant.Lookingattheelementsofthesesets,itisclearthat: Answer: A and B areequivalent. Definition: Foranytwosets,if A  B  and  B  A, then A=B. Thus A and B areequivalent. Example5: Listallsubsetsof theset C ={1,2,3}. Answer:    Subset Listallpossiblecombinationsofelements... D ={1} oneatatime E ={2} oneatatime F ={3} oneatatime G ={1,2} twoatatime M ={1,3} twoatatime N ={2,3} twoatatime P ={1,2,3} threeatatime Ø Thenullsethasnoelements. Lookingatexample5,youmaybewonderingwhythenullsetislistedasasubsetof C. Therearenoelementsinanullset,sotherecanbenoelementsinthenullsetthataren'tcontainedinthecompleteset.Therefore, thenullsetisasubsetofeveryset. Youmayalsobewondering: Isasetasubsetofitself? Theanswerisyes:Anysetcontainsitselfasasubset.Thisisdenotedby: A  A. Asubsetthatissmallerthanthecompletesetisreferredtoasa propersubset.Sotheset{1,2}isapropersubsetoftheset {1,2,3}becausetheelement3isnotinthefirstset.Inexample5,youcanseethat G isapropersubsetof C,Infact,everysubsetlistedinexample5isapropersubsetof C, except P.Thisisbecause P and C areequivalentsets(P = C).Somemathematiciansusethesymbol  todenoteasubsetandthesymbol  todenoteapropersubset,withthedefinitionforpropersubsetsasfollows: If A  B,and A ≠ B,then A issaidtobea propersubset of B anditisdenotedby A  B. Whileitisimportanttopointouttheinformationabove,itcangetabitconfusing,Solet'sthinkofsubsetsandpropersubsetsthisway: SubsetsandProperSubsets Theset{1,2}isapropersubsetoftheset{1,2,3}. Theset{1,2,3}isanotapropersubsetoftheset {1,2,3}. Doyouseeapatternintheexamplesbelow? Example6: Listallsubsetsof theset R ={x,y,z}.Howmanyarethere? Subsets D ={x} E ={y} F ={z} G ={x,y} H ={x,z} J ={y,z} K ={x,y,z} Ø Answer: Thereareeightsubsetsofthe set R ={x,y,z}. Example7: Listallsubsetsof theset C ={1,2,3,4}.Howmanyarethere? Subsets D ={1} M ={2,4} E ={2} N ={3,4} F ={3} O ={1,2,3} G ={4} P ={1,2,4} H ={1,2} Q ={1,3,4} J ={1,3} R ={2,3,4} K ={1,4} S ={1,2,3,4} L={2,3} Ø Answer: Thereare16subsetsofthe set C ={1,2,3,4}. Inexample6,set R hasthree(3)elementsandeight(8)subsets.Inexample7,setChasfour(4)elementsand16subsets.Tofindthenumberofsubsetsofasetwithnelements,raise2tothenthpower:Thatis: ThenumberofsubsetsinsetAis2n ,wherenisthenumberofelementsinset A. Lesson  Summary Subset: A isasubsetof B: if everyelementof A iscontainedin B.Thisis denotedby A  B. EquivalentSets: Foranytwosets,if A  B  and  B  A, then A=B. Nullset: Thenullsetisasubsetofeveryset. Setsandsubsets: Anysetcontainsitselfasasubset.Thisisdenotedby A  A. ProperSubsets: If A  B,and A ≠ B,then A issaidtobeapropersubsetof B anditisdenotedby A  B. NumberofSubsets: ThenumberofsubsetsinsetAis2n ,wherenisthenumberofelementsinset A. Exercises Directions:Readeachquestionbelow.Selectyouranswerbyclickingonitsbutton.FeedbacktoyouranswerisprovidedintheRESULTSBOX.Ifyoumakeamistake,rethinkyouranswer,thenchooseadifferentbutton. 1. Whichofthefollowingisasubsetofset G?  G ={d,a,r,e}      X ={e,a,r} Y= {e,r,a} Z ={r,e,d} Alloftheabove. RESULTSBOX:  2. Whichofthefollowingstatementsistrue?      {vowels}  {consonants} {consonants}  {vowels} {vowels}  {alphabet} Noneoftheabove. RESULTSBOX:  3. WhichofthefollowingisNOTasubsetofset A? A ={2,3,5,7,11}      B ={3,5,2,7} C ={2,3,7,9} D ={7,2,3,11} Alloftheabove. RESULTSBOX:  4. Howmanysubsetswillthesetbelowhave?  T ={Monday,Tuesday,Wednesday,Thursday,Friday}      5 10 32 Noneoftheabove. RESULTSBOX:  5. If R ={wholenumbers<5} and S ={4,2,0,3,1},thenwhichofthefollowingstatementsistrue?      R = S R hasmoreelementsthan S. S isnull. Noneoftheabove. RESULTSBOX:  IntroductiontoSets IntroductiontoSets BasicSetNotation TypesofSets SetEquality VennDiagrams Subsets UniversalSet Set-BuilderNotation Complement Intersection Union PracticeExercises ChallengeExercises Solutions SignUpForOurFREENewsletter! * Bysigningup,youagreetoreceiveusefulinformationandtoourprivacypolicy SignUpForOurFREENewsletter! E-MailAddress* FeaturedSites: EducationWorld TeacherPlanet StudentAwardCertificates ShopMathGames