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Intro to Proofs

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Intro to Proofs

Logic, direct proofs, induction, sets, and advanced proof techniques

197Modules

11Units

1

Foundations of Logic

Master foundations of logic

20 modules

The Language of MathematicsINTRO

The Proof ToolkitINTRO

From Statements to PredicatesINTRO

Structure of Mathematical ProofsLESSON

Propositions and Logical ConnectivesLESSON

Universal and Existential QuantifiersLESSON

Database Query LogicLESSON

Digital Circuit VerificationLESSON

Computer-Assisted Proof VerificationLESSON

Foundations SummaryLESSON

Translating Mathematical StatementsEXAMPLE

Evaluating (p → q) ↔EXAMPLE

Analyzing a Simple ProofEXAMPLE

Truth Table BuilderPRACTICE

Predicate Logic MasteryPRACTICE

Quantifier Negation GamePRACTICE

Logical Equivalence PracticePRACTICE

Proof Strategy PracticePRACTICE

Logic and Proof Foundations ReferenceFORMULA SHEET

Section 1 AssessmentQUIZ

2

Direct Proofs

Master direct proofs

18 modules

Algebraic Manipulation MasteryINTRO

Building Bridges of LogicINTRO

Complete Coverage StrategyINTRO

Direct Proof SummaryLESSON

The Direct Proof MethodLESSON

The Method of CasesLESSON

Properties and AxiomsLESSON

Proving Sum of Odds is EvenEXAMPLE

Proving (a+b)² ≥ 4abEXAMPLE

Proving |ab| = |a||b|EXAMPLE

Algebraic Proof TechniquesPRACTICE

Divisibility ProofsPRACTICE

Direct Proof ConstructorPRACTICE

Modular Arithmetic ProofsPRACTICE

Inequality Proof LabPRACTICE

Case Analysis OrganizerPRACTICE

Direct Proof ReferenceFORMULA SHEET

Section 2 AssessmentQUIZ

3

Indirect Methods

Master indirect methods

19 modules

Choosing Your WeaponINTRO

The Impossible ScenarioINTRO

Reversing the FlowINTRO

Indirect Methods SummaryLESSON

Contrapositive vs ContradictionLESSON

The Contradiction MethodLESSON

The Contrapositive MethodLESSON

Common PitfallsEXAMPLE

Infinity of PrimesEXAMPLE

Proving √2 is IrrationalEXAMPLE

Proving n² odd implies n oddEXAMPLE

Indirect Proof PracticePRACTICE

Method Selection WorkshopPRACTICE

Contradiction FinderPRACTICE

Rational Root Theorem ApplicationPRACTICE

When to Use ContrapositivePRACTICE

Contrapositive ConverterPRACTICE

Indirect Proof ReferenceFORMULA SHEET

Section 3 AssessmentQUIZ

4

Mathematical Induction

Master mathematical induction

19 modules

Beyond Natural NumbersINTRO

Multiple DominoesINTRO

The Domino EffectINTRO

Induction SummaryLESSON

Algorithm CorrectnessLESSON

Inequality InductionLESSON

Strong vs Weak InductionLESSON

Mathematical Induction ExplainedLESSON

Proving 2ⁿ > n² for n ≥ 5EXAMPLE

Proving Every Integer > 1 Has Prime FactorizationEXAMPLE

Proving Sum Formula 1+2+...+n = n(n+1)/2EXAMPLE

Advanced InductionPRACTICE

Induction Challenge LabPRACTICE

Strong Induction ApplicationsPRACTICE

Recursive Sequence ProverPRACTICE

Sum and Product FormulasPRACTICE

Induction Template BuilderPRACTICE

Induction ReferenceFORMULA SHEET

Section 4 AssessmentQUIZ

5

Sets and Relations

Master sets and relations

20 modules

Function DetectiveINTRO

Relationship MapperINTRO

Venn Diagram LogicINTRO

Sets and Relations SummaryLESSON

Properties of FunctionsLESSON

Equivalence ClassesLESSON

Properties of RelationsLESSON

Set Operations and IdentitiesLESSON

Proving a Function is BijectiveEXAMPLE

Proving Relation PropertiesEXAMPLE

Proving A ∩ (B ∪ C) = (A ∩ B) ∪EXAMPLE

Proof Technique SorterPRACTICE

Function Proof TechniquesPRACTICE

Composition PropertiesPRACTICE

Partition Proof WorkshopPRACTICE

Relation Property CheckerPRACTICE

Power Set PropertiesPRACTICE

Set Identity ProverPRACTICE

Set Theory ReferenceFORMULA SHEET

Section 5 AssessmentQUIZ

6

Number Theory

Master number theory

21 modules

Clock ArithmeticINTRO

Prime Sieve ExplorerINTRO

Factor Finding MissionINTRO

Number Theory SummaryLESSON

Calendar CalculationsLESSON

Congruences and Residue ClassesLESSON

RSA Cryptography FoundationLESSON

Fundamental Theorem of ArithmeticLESSON

Linear Diophantine EquationsLESSON

Divisibility TheoryLESSON

Proving Fermat's Little TheoremEXAMPLE

Proving Infinitude of PrimesEXAMPLE

Proving GCD PropertiesEXAMPLE

Modular Arithmetic ProofsPRACTICE

Modular Equation SolverPRACTICE

Prime Number ProofsPRACTICE

Prime Testing LabPRACTICE

Divisibility Proof ProblemsPRACTICE

Euclidean Algorithm VisualizerPRACTICE

Number Theory ReferenceFORMULA SHEET

Section 6 AssessmentQUIZ

7

Combinatorial Proofs

Master combinatorial proofs

19 modules

Sequence DetectiveINTRO

Overcrowded CagesINTRO

Two Ways to CountINTRO

Combinatorial Proofs SummaryLESSON

Recurrence Relation ProofsLESSON

Party Problem ApplicationsLESSON

Pigeonhole Principle and GeneralizationsLESSON

Combinatorial Proof TechniqueLESSON

Fibonacci Formula ProofEXAMPLE

Proving Repetition in SequencesEXAMPLE

Proving C(n,k) = CEXAMPLE

Catalan Number ApplicationsPRACTICE

Generating Function LabPRACTICE

Existence ProofsPRACTICE

Pigeonhole Problem SolverPRACTICE

Committee Selection ProofsPRACTICE

Identity Proof BuilderPRACTICE

Combinatorial Proof ReferenceFORMULA SHEET

Section 7 AssessmentQUIZ

8

Advanced Methods

Master advanced methods

19 modules

The Impossible StaircaseINTRO

The Unchanging TruthINTRO

The Smallest CounterexampleINTRO

Advanced Methods SummaryLESSON

Method of Infinite DescentLESSON

Algorithm Termination ProofsLESSON

Invariant MethodLESSON

Well-Ordering PrincipleLESSON

Proving x⁴ + y⁴ = z² Has No Integer SolutionsEXAMPLE

15-Puzzle ImpossibilityEXAMPLE

Proving √2 Irrational via Well-OrderingEXAMPLE

Constructive vs Non-constructivePRACTICE

Descent Construction WorkshopPRACTICE

Game Strategy ProofsPRACTICE

Invariant FinderPRACTICE

Extremal Principle ExplorerPRACTICE

Minimal Criminal AlgorithmPRACTICE

Advanced Techniques ReferenceFORMULA SHEET

Section 8 AssessmentQUIZ

9

Abstract Algebra

Master abstract algebra

19 modules

Universal Proof PatternsINTRO

Number System BuilderINTRO

Symmetry DetectiveINTRO

Abstract Algebra SummaryLESSON

Proving by Universal PropertiesLESSON

Ring Theory FoundationsLESSON

Lagrange's TheoremLESSON

Groups and SubgroupsLESSON

Uniqueness via Universal PropertyEXAMPLE

Proving ℤ[√2] is a RingEXAMPLE

Proving Cancellation LawsEXAMPLE

Abstract Algebra ApplicationsPRACTICE

Isomorphism Class ExplorerPRACTICE

Field Extension ProofsPRACTICE

Polynomial Ring ExplorerPRACTICE

Homomorphism HunterPRACTICE

Group Property VerifierPRACTICE

Abstract Algebra ReferenceFORMULA SHEET

Section 9 AssessmentQUIZ

10

Analysis and Topology

Master analysis and topology

19 modules

Open Set ExplorerINTRO

Convergence DetectiveINTRO

The Precision GameINTRO

Analysis SummaryLESSON

Fixed Point TheoremsLESSON

Open and Closed SetsLESSON

Sequence ConvergenceLESSON

Uniform ContinuityLESSON

Limits and ContinuityLESSON

Proving aₙ = n/(n+1) → 1EXAMPLE

Proving lim(x→2) x² = 4EXAMPLE

Topological PropertiesPRACTICE

Topology Proof BuilderPRACTICE

Power Series ProofsPRACTICE

Series Convergence TestsPRACTICE

Epsilon-Delta ChallengesPRACTICE

Delta FinderPRACTICE

Analysis ReferenceFORMULA SHEET

Section 10 AssessmentQUIZ

11

Course Conclusion

Master course conclusion

4 modules

The Art of ProofLESSON

Proof Portfolio BuilderPRACTICE

Master Proof ReferenceFORMULA SHEET

Comprehensive Proof ExamQUIZ