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RA:001Negating alternating quantifiers

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Proof FoundationsQuantifier NegationArchimedean LanguageLogical Equivalence

Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

RA:002Contrapositive versus converse

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Proof FoundationsContrapositiveTruth ValuesImplication

Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

RA:003Rational plus irrational

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Proof FoundationsRational ClosureIrrational NumbersAdditive Inverses

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:004Even square forces even integer

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Proof FoundationsParityDivisibilityContrapositive

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:005Irrationality of square root two

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Proof FoundationsSqrt TwoLowest TermsParity Descent

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:006De Morgan by element chasing

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Proof FoundationsSet ExtensionalityDE Morgan LawComplements

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:007Power-set inclusion

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Proof FoundationsPower SetSubset RelationSingleton Witness

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:008Preimage of indexed intersections

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Proof FoundationsPreimageIndexed IntersectionFunction Membership

Proof language, sets, functions, and quantifiersReal Analysis I · Calculation

RA:009Image of an intersection

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Proof FoundationsSet ImageInjective FunctionsIntersection Counterexample

Proof language, sets, functions, and quantifiersReal Analysis I · Calculation

RA:010Injective compositions

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Proof FoundationsFunction CompositionInjectivityRange Restriction

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:011Surjective compositions

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Proof FoundationsSurjective FunctionsComposition RangeFinite Counterexample

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:012Congruence modulo five

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Proof FoundationsEquivalence RelationModular CongruenceInteger Divisibility

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:013Equality of equivalence classes

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Proof FoundationsEquivalence ClassesSet ExtensionalityTransitive Relations

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:014Sum of the first odd integers

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Proof FoundationsMathematical InductionFinite SumsOdd Numbers

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:015Exponential versus linear growth

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Proof FoundationsInduction InequalityExponential GrowthEquality Cases

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:016Existence of a prime divisor

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Proof FoundationsStrong InductionPrime DivisorsFactorization

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:017A recursive affine sequence

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Proof FoundationsRecursive SequenceClosed FormInduction

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:018Finite geometric sum

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Proof FoundationsGeometric SeriesFinite SumInduction Algebra

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:019Enumerating the integers

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Proof FoundationsCountable SetsInteger EnumerationSurjective Map

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:020Diagonal enumeration of pairs

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Proof FoundationsCartesian ProductDiagonal EnumerationCountability

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:021Countable union theorem

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Proof FoundationsCountable UnionEnumerationsPairing Function

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:022Countability of the rationals

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Proof FoundationsRational NumbersQuotient RepresentationCountable Image

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:023Cantor diagonal binary sequence

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Proof FoundationsCantor DiagonalBinary SequencesUncountability

Proof language, sets, functions, and quantifiersReal Analysis I · Proof

RA:024Negation of sequence convergence

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Proof FoundationsSequence ConvergenceEpsilon NegationQuantifier Order

Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

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