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Contents
1Logic1
1.1Statements,Sets,Subsets,andImplication......1
1.2StatementsandTheirTruthValues..........2
1.2.aAnds/Ors/NotsasIntersections/Unions/Complements.......2
1.2.bImplies/EquivalenceasSubset/Equality...3
1.2.cTheEmptySetandVacuouslyTrueStatements...........4
1.2.dIndicatorsandAnds/Ors/Nots........4
1.3Proofs,aFirstLook......4
1.4LogicalQuantiers......7
1.5TaxonomyofProofs......8
2SetTheory11
2.1SomeSimpleQuestions....11
2.2NotationandOtherBasics..12
2.3Products,Relations,Correspondences,andFunctions14
2.4EquivalenceRelations.....17
2.5OptimalChoiceforFiniteSets19
2.5.aTheBasics......19
2.5.bRepresentingPreferences...........20
2.5.cRevealedPreference.21
2.6DirectandInverseImages,Compositions......23
2.6.aDirectImages.....23
2.6.bInverseRelations...23
2.6.cInjections,Surjections,andBijections....24
2.6.dCompositionsofFunctions..........25
2.7WeakandPartialOrders,Lattices...........26
2.8MonotonicChangesinOptima:Supermodularity,andLattices.......28
2.8.aTheImplicitFunctionApproach.......29
2.8.bTheSimpleSupermodularityApproach...30
2.8.cMonotoneComparativeStatics........31
2.8.dQuasi-Supermodularity32
2.9Tarski’sLatticeFixedPointTheoremandStableMatchings.........33
2.9.aMatchingProblemsandStability.......34
2.9.bStableMatchingsasFixedPoints......34
2.9.cTarski’sFixedPointTheorem........35
2.10FiniteandInniteSets....37
2.10.aComparingtheSizeofFiniteSets......38
2.10.bComparingtheSizeofInniteSets.....38
2.11TheAxiomofChoiceandSomeEquivalentResults.42
2.12RevealedPreferenceandRationalizability......43
2.13Superstructures........45
2.14EndofChapterProblems....46
2.15BibliographyforChapter2..47
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CONTENTS
3TheSpaceofRealNumbers49
3.1WhyWeWantMorethantheRationals.......49
3.2BasicPropertiesofRationals.50
3.3Distance,CauchySequences,andtheRealNumbers.51
3.3.aDistanceandSequences...........51
3.3.bCauchySequences..52
3.3.cTheRealNumbers:DenitionandBasicProperties.........53
3.4TheCompletenessoftheRealNumbers.......56
3.5Newton’sBisectionMethod..59
3.6SupremumandInmum...60
3.7SummabilityandaGrowthModel..........62
3.7.aTheIdealizedFisheryModel.........62
3.7.bSummability.....63
3.7.cSummabilityintheFishery..........65
3.7.dOptimality,EulerEquations,andSteadyStatesintheFishery....65
3.7.eMonotoneSequences.66
3.7.fOptimalityAwayFromSteadyStates....66
3.8Patience,Liminf,andLimsup.67
3.8.aLiminfandLimsup..67
3.8.bPatience........68
3.8.cDiscountFactorsClosetoOne........69
3.9BibliographyforChapter3..70
4TheMetricSpacesR,=1,2,...71
4.1TheBasicDenitionsforMetricSpaces.......71
4.1.aMetrics........71
4.1.bSequencesandCompleteness........72
4.1.cSeparability......73
4.1.dOpenandClosedSets73
4.1.eNeighborhoods,Topologies,andtheTopologicalEquivalenceofMetrics.........73
4.1.fOpenCovers,CompactnessandConnectedness...........75
4.1.gContinuousFunctions75
4.1.hHomeomorphismsandIsometries......75
4.1.iAWordtotheReader76
4.2DiscreteSpaces........76
4.3RasaNormedVectorSpace.77
4.3.aThreeNormsonR..77
4.3.bInner(orDot)ProductsandtheCauchy-SchwarzInequality....79
4.3.cThep-Norms,p∈[1,∞)...........80
4.4Completeness.........80
4.4.aComparableMetrics.80
4.4.bTheCompletenessof(R,d1),(R,d2),and(R,d∞).......81
4.4ompletenessandContractionMappings..81
4.5Closure,Convergence,andCompleteness......83
4.5.aCharacterizingClosedSets..........83
4.5.bClosureandCompleteness..........85
4.6Separability..........86
4.7CompactnessinR......86
4.7.aTheCompactnessof[0,1]..........86
4.7.bContextMatters....88
4.7.cTheDenitionRedux.88
4.7.dBoundednessandTotalBoundedness....88
4.7.eTheFiniteIntersectionProperty.......89
4.7.fCharacterizationsofCompactness......89。