Definition, Symbol, Proper and Improper Subset | Power Set

What is a Subset in Maths? ... Set A is said to be a subset of Set B if all the elements of Set A are also present in Set B. In other words, set A ... MathsMathArticleSubsets Subsets SubsetsareapartofoneofthemathematicalconceptscalledSets.Asetisacollectionofobjectsorelements,groupedinthecurlybraces,suchas{a,b,c,d}.IfasetAisacollectionofevennumberandsetBconsistsof{2,4,6},thenBissaidtobeasubsetofA,denotedbyB⊆AandAisthesupersetofB.Learn SetsSubsetAndSuperset tounderstandthedifference. Theelementsofsetscouldbeanythingsuchasagroupofrealnumbers,variables,constants,wholenumbers,etc.Itconsistsofanullsetaswell.Letusdiscusssubsetsherewithitstypesandexamples. Tableofcontents: Definition Symbol Allsubsets Types ProperSubset ProperSubsetSymbol Formula SubsetsandProperSubsets ImproperSubsets Powerset Properties SolvedExamples FAQs WhatisaSubsetinMaths? SetAissaidtobeasubsetofSetBifalltheelementsofSetAarealsopresentinSetB.Inotherwords,setAiscontainedinsideSetB. Example:IfsetAhas{X,Y}andsetBhas{X,Y,Z},thenAisthesubsetofBbecauseelementsofAarealsopresentinsetB. SubsetSymbol Insettheory,asubsetisdenotedbythesymbol⊆andreadas‘isasubsetof’. Usingthissymbolwecanexpresssubsetsasfollows: A⊆B;whichmeansSetAisasubsetofSetB. Note:Asubsetcanbeequaltotheset.Thatis,asubsetcancontainalltheelementsthatarepresentintheset. AllSubsetsofaSet Thesubsetsofanysetconsistsofallpossiblesetsincludingitselementsandthenullset.Letusunderstandwiththehelpofanexample. Example:FindallthesubsetsofsetA={1,2,3,4} Solution:Given,A={1,2,3,4} Subsets= {} {1},{2},{3},{4}, {1,2},{1,3},{1,4},{2,3},{2,4},{3,4}, {1,2,3},{2,3,4},{1,3,4},{1,2,4} {1,2,3,4} TypesofSubsets Subsetsareclassifiedas ProperSubset ImproperSubsets Apropersubsetisonethatcontainsafewelementsoftheoriginalsetwhereasanimpropersubset,containseveryelementoftheoriginalsetalongwiththenullset. Forexample,ifsetA={2,4,6},then, Numberofsubsets:{2},{4},{6},{2,4},{4,6},{2,6},{2,4,6}andΦor{}. ProperSubsets:{},{2},{4},{6},{2,4},{4,6},{2,6} ImproperSubset:{2,4,6} Thereisnoparticularformulatofindthesubsets,instead,wehavetolistthemall,todifferentiatebetweenproperandimproperone.Thesettheorysymbolsweredevelopedbymathematicianstodescribethecollectionsofobjects. WhatareProperSubsets? SetAisconsideredtobeapropersubsetofSetBifSetBcontainsatleastoneelementthatisnotpresentinSetA. Example:IfsetAhaselementsas{12,24}andsetBhaselementsas{12,24,36},thensetAisthepropersubsetofBbecause36isnotpresentinthesetA. ProperSubsetSymbol Apropersubsetisdenotedby⊂andisreadas‘isapropersubsetof’.Usingthissymbol,wecanexpressapropersubsetforsetAandsetBas; A⊂B ProperSubsetFormula IfwehavetopicknnumberofelementsfromasetcontainingNnumberofelements,itcanbedoneinNCn numberofways. Therefore,thenumberofpossiblesubsetscontainingnnumberofelementsfromasetcontainingNnumberofelementsisequaltoNCn. Howmanysubsetsandpropersubsetsdoesasethave? Ifasethas“n”elements,thenthenumberofsubsetofthegivensetis2nandthenumberofpropersubsetsofthegivensubsetisgivenby2n-1.  Consideranexample,IfsetAhastheelements,A={a,b},thenthepropersubsetofthegivensubsetare{},{a},and{b}. Here,thenumberofelementsinthesetis2.  Weknowthattheformulatocalculatethenumberofpropersubsetsis2n–1.  =22–1 =4–1 =3 Thus,thenumberofpropersubsetforthegivensetis3({},{a},{b}). WhatisImproperSubset? Asubsetwhichcontainsalltheelementsoftheoriginalsetiscalledanimpropersubset.Itisdenotedby ⊆. Forexample:SetP={2,4,6}Then,thesubsetsofPare; {},{2},{4},{6},{2,4},{4,6},{2,6}and{2,4,6}. Where,{},{2},{4},{6},{2,4},{4,6},{2,6}arethepropersubsetsand{2,4,6}istheimpropersubsets.Therefore,wecanwrite{2,4,6}⊆P. Note: Theemptyset isanimpropersubsetofitself(sinceitisequaltoitself)butitisapropersubsetofanyotherset. PowerSet Thepowersetissaidtobethecollectionofallthesubsets.ItisrepresentedbyP(A). IfAissethavingelements{a,b}.ThenthepowersetofAwillbe; P(A)= {∅, {a},{b},{a,b}} Tolearnmoreinbrief,clickonthearticlelinkofpowerset. PropertiesofSubsets Someoftheimportantpropertiesofsubsetsare: Everysetisconsideredasasubsetofthegivensetitself.ItmeansthatX⊂XorY⊂Y,etc Wecansay,anemptysetisconsideredasasubsetofeveryset.  XisasubsetofY.ItmeansthatXiscontainedinY IfasetXisasubsetofsetY,wecansaythatYisasupersetofX Also,read: SetsForClass11 SetsSubsetAndSuperset UnionOfSets UniversalSet SubsetsExampleProblems Example1:Howmanynumberofsubsetscontainingthreeelementscanbeformedfromtheset? S={1,2,3,4,5,6,7,8,9,10} Solution:Numberofelementsintheset=10 Numberofelementsinthesubset=3 Therefore,thenumberofpossiblesubsetscontaining3elements=10C3 Therefore,thenumberofpossiblesubsetscontaining3elementsfromthesetS={1,2,3,4,5,6,7,8,9,10}is120. Example2:Givenanytworeal-lifeexamplesonthesubset. Solution:Wecanfindavarietyofexamplesofsubsetsineverydaylifesuchas: Ifweconsiderallthebooksinalibraryasoneset,thenbookspertainingtoMathsisasubset. Ifalltheitemsinagroceryshopformaset,thencerealsformasubset. Example3:FindthenumberofsubsetsandthenumberofpropersubsetsforthegivensetA={5,6,7,8}. Solution: Given:A={5,6,7,8} Thenumberofelementsinthesetis4 Weknowthat, Theformulatocalculatethenumberofsubsetsofagivensetis2n =24=16 Numberofsubsetsis16 Theformulatocalculatethenumberofpropersubsetsofagivensetis2n–1 =24–1 =16–1=15 Thenumberofpropersubsetsis15. FrequentlyAskedQuestionsonSubsetsDefinesubset Insettheory,asetXisdefinedasasubsetoftheothersetY,ifalltheelementsofsetXshouldbepresentinthesetY.ThiscanbesymbolicallyrepresentedbyX⊂Y Whatarethetwoclassificationsofsubset? Theclassificationsofsubsetsare: Propersubset Impropersubset Defineproperandimpropersubsets. Animpropersubsetisdefinedasasubsetwhichcontainsalltheelementspresentintheothersubset.Butinpropersubsets,ifXisasubsetofY,ifandonlyifeveryelementofsetXshouldbepresentinsetY,butthereisoneormorethanelementsofsetYisnotpresentinsetX. Giveanexampleofproperandimpropersubsets. Propersubset: X={2,5,6}andY={2,3,5,6} ImproperSubset: X={A,B,C,D}andY={A,B,C,D} Whatistheformulatocalculatethenumberofsubsetsandpropersubsetforanygivenset? If“n”isthenumberofelementsofagivenset,thentheformulastocalculatethenumberofsubsetsandapropersubsetisgivenby: Numberofsubsets=2n Numberofpropersubsets=2n–1 Learnmoreaboutsettheorysymbolsandotherrelatedtopics.RegisterwiththeBYJU’S–TheLearningApptoday. TestyourknowledgeonSubsets Q5 PutyourunderstandingofthisconcepttotestbyansweringafewMCQs.Click‘StartQuiz’tobegin! Selectthecorrectanswerandclickonthe“Finish”buttonCheckyourscoreandanswersattheendofthequiz StartQuiz Congrats! VisitBYJU’SforallMathsrelatedqueriesandstudymaterials Yourresultisasbelow 0outof0arewrong 0outof0arecorrect 0outof0areUnattempted ViewQuizAnswersandAnalysis MATHSRelatedLinks TotalSurfaceAreaOfHemisphere CircleTheorems TriangleSimilarityTheorems SumOfNNaturalNumbers Discriminant ProbabilityMathDefinition IntegrationRules ArithmeticProgression StokesTheorem ParabolicCurve 3Comments Ramakant January25,2020at4:55am Ifanynonemptysethave2impropersubsetthanhowyouwrotethattotalnumberofpropersubset=2raisedtothepowern_1 Reply OBENUTETTEYDANIEL January22,2021at11:37am pleaseIneedproofofformulaforDeMorgan’slaw Reply lavanya January25,2021at10:15am Pleasecheck:https://byjus.com/maths/de-morgans-first-law/ Reply LeaveaCommentCancelreply YourMobilenumberandEmailidwillnotbepublished.Requiredfieldsaremarked* * SendOTP DidnotreceiveOTP? 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