Proper Subset -- from Wolfram MathWorld

Proper Subset. A proper subset S^' of a set S , denoted S^' subset S , is a subset that is strictly contained in S and so necessarily excludes at least one ... Algebra AppliedMathematics CalculusandAnalysis DiscreteMathematics FoundationsofMathematics Geometry HistoryandTerminology NumberTheory ProbabilityandStatistics RecreationalMathematics Topology AlphabeticalIndex InteractiveEntries RandomEntry NewinMathWorld MathWorldClassroom AboutMathWorld ContributetoMathWorld SendaMessagetotheTeam MathWorldBook WolframWebResources » 13,781entries Lastupdated:FriDec172021 Created,developed,andnurtured by Eric WeissteinatWolfram Research FoundationsofMathematics > SetTheory > Sets > ProperSubset Apropersubsetofaset,denoted,is asubsetthatisstrictlycontainedinandsonecessarily excludesatleastonememberof.Theempty setisthereforeapropersubsetofanynonempty set. Forexample,consideraset.Then andarepropersubsets, whileandare not. SEEALSO:ImproperSubset,Set, Subset REFERENCES: Kamke,E.Theory ofSets.NewYork:Dover,p. 6,1950. ReferencedonWolfram|Alpha:ProperSubset CITETHISAS: Weisstein,EricW."ProperSubset."From MathWorld--AWolframWebResource.https://mathworld.wolfram.com/ProperSubset.html WolframWebResources Mathematica » The#1toolforcreatingDemonstrationsandanythingtechnical. Wolfram|Alpha » Exploreanythingwiththefirstcomputationalknowledgeengine. WolframDemonstrationsProject » Explorethousandsoffreeapplicationsacrossscience,mathematics,engineering,technology,business,art,finance,socialsciences,andmore. Computerbasedmath.org » Jointheinitiativeformodernizingmatheducation. OnlineIntegralCalculator » SolveintegralswithWolfram|Alpha. Step-by-stepSolutions » Walkthroughhomeworkproblemsstep-by-stepfrombeginningtoend.Hintshelpyoutrythenextsteponyourown. WolframProblemGenerator » Unlimitedrandompracticeproblemsandanswerswithbuilt-inStep-by-stepsolutions.Practiceonlineormakeaprintablestudysheet. WolframEducationPortal » CollectionofteachingandlearningtoolsbuiltbyWolframeducationexperts:dynamictextbook,lessonplans,widgets,interactiveDemonstrations,andmore. WolframLanguage » Knowledge-basedprogrammingforeveryone. ContacttheMathWorldTeam ©1999-2021WolframResearch,Inc.|TermsofUse THINGSTOTRY: 343toRomannumerals Dynamic Is1/(1+sqrt(5))analgebraicinteger?