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Discrete Mathematics

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Discrete Mathematics

Logic, sets, combinatorics, graphs, and algorithm analysis

190Modules

11Units

Being rebuilt

This course is still in our original format. All the content is here and free to use, but it has not yet been through the rebuild that gives our newer courses their interactive animations and cleaner visuals — so you may hit some rough edges. Spot an error? Send us a note and we will fix it.

Recommended preparation

Algebra 2 and some exposure to proofs (Intro to Proofs) make the best preparation.

Algebra 2 Intro to Proofs

The Language of LogicINTRO

Detective's DeductionINTRO

Propositions and ConnectivesLESSON

Logical Equivalence PracticePRACTICE

Truth Table ConstructorACTIVITY

Valid ArgumentsLESSON

Proving an Argument ValidEXAMPLE

Inference Rule MatcherACTIVITY

Building Mathematical ArgumentsLESSON

Proving √2 is IrrationalEXAMPLE

Proof WorkshopACTIVITY

Mathematical InductionLESSON

Digital Circuit DesignerREAL-WORLD

Evaluating (p → q) ∧ (¬q → ¬p)EXAMPLE

Logic and Proofs SummarySUMMARY

Logic and Proofs ReferenceFORMULA SHEET

Propositional Logic AssessmentASSESSMENT

Organizing CollectionsINTRO

Set FundamentalsLESSON

Set Operations VisualizerACTIVITY

Set Operations PracticePRACTICE

Algebraic Properties of SetsLESSON

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

Set Proof BuilderACTIVITY

Power Sets and ProductsLESSON

Computing P({a, b, c})EXAMPLE

Database Queries as SetsREAL-WORLD

Finding A ∪ (B ∩ C')EXAMPLE

Relation ExplorerACTIVITY

Set Theory SummarySUMMARY

Set Theory ReferenceFORMULA SHEET

Set Theory AssessmentASSESSMENT

Connecting ElementsINTRO

Relations as SetsLESSON

Analyzing a Divisibility RelationEXAMPLE

Relation Property CheckerACTIVITY

Equivalence RelationsLESSON

Functions as Special RelationsLESSON

Testing Injectivity and SurjectivityEXAMPLE

Function MapperACTIVITY

Function CompositionLESSON

Finding Inverse FunctionsEXAMPLE

Composition and Inverse LabPRACTICE

Cryptography and BijectionsREAL-WORLD

Functions SummarySUMMARY

Functions and Relations ReferenceFORMULA SHEET

Relations and Functions AssessmentASSESSMENT

The Art of CountingINTRO

Fundamental Counting PrinciplesLESSON

License Plate PossibilitiesEXAMPLE

Advanced Counting ProblemsPRACTICE

Counting ScenariosPRACTICE

Order Matters vs. Order Doesn't MatterLESSON

Selecting a CommitteeEXAMPLE

Arrangement VisualizerACTIVITY

Circular PermutationsLESSON

The Binomial TheoremLESSON

Expanding (2x - 3)⁵EXAMPLE

Pascal's Triangle ExplorerACTIVITY

Password Strength AnalyzerREAL-WORLD

Lottery Odds CalculatorREAL-WORLD

Probability and CountingLESSON

Tournament BracketsREAL-WORLD

Combinatorics SummarySUMMARY

Combinatorics ReferenceFORMULA SHEET

Combinatorics AssessmentASSESSMENT

Networks EverywhereINTRO

Graph FundamentalsLESSON

Analyzing the Petersen GraphEXAMPLE

Graph Properties PracticePRACTICE

Graph Builder LaboratoryACTIVITY

Social Network AnalysisREAL-WORLD

Spanning TreesLESSON

Binary Search TreesLESSON

Minimum Spanning TreesLESSON

Shortest Path AlgorithmsLESSON

Finding Shortest RouteEXAMPLE

Algorithm RaceACTIVITY

Internet Routing SimulatorREAL-WORLD

Network FlowLESSON

Decision Tree OptimizerREAL-WORLD

Hamiltonian CircuitsEXAMPLE

Depth-First vs. Breadth-FirstEXAMPLE

Graph ColoringACTIVITY

Graph Algorithms LabPRACTICE

Traversal AnimatorACTIVITY

Planar GraphsLESSON

Graph Theory SummarySUMMARY

Graph Theory ReferenceFORMULA SHEET

Graph Theory AssessmentASSESSMENT

Factors and MultiplesINTRO

Clock ArithmeticINTRO

Division AlgorithmLESSON

The Euclidean AlgorithmEXAMPLE

Divisibility PatternsPRACTICE

GCD Calculator WorkshopACTIVITY

The Building BlocksLESSON

Testing PrimalityEXAMPLE

Prime ExplorerACTIVITY

Congruences and ResiduesLESSON

Solving 7x ≡ 3 (mod 11)EXAMPLE

Modular Arithmetic LabACTIVITY

Fermat's Little TheoremLESSON

Calendar CalculationsREAL-WORLD

RSA EncryptionREAL-WORLD

Number Theory ChallengesPRACTICE

Number Theory SummarySUMMARY

Number Theory ReferenceFORMULA SHEET

Number Theory AssessmentASSESSMENT

Switches and SignalsINTRO

Boolean Operations and LawsLESSON

Simplifying Boolean ExpressionsEXAMPLE

Logic Gate FundamentalsLESSON

Karnaugh MapsLESSON

Minimizing a 4-Variable FunctionEXAMPLE

Circuit MinimizerACTIVITY

Combinational Circuit DesignLESSON

Seven-Segment Display DriverREAL-WORLD

Memory ElementsLESSON

Designing a CounterEXAMPLE

State Machine DesignerACTIVITY

Memory and RegistersLESSON

CPU Control UnitREAL-WORLD

Digital Logic PracticePRACTICE

Boolean Algebra SummarySUMMARY

Digital Logic ReferenceFORMULA SHEET

Boolean Algebra AssessmentASSESSMENT

Updating BeliefsINTRO

Games of ChanceINTRO

Probability SpacesLESSON

Rolling Two DiceEXAMPLE

Basic Probability PracticePRACTICE

Probability CalculatorACTIVITY

Conditional Probability and IndependenceLESSON

The Monty Hall ProblemEXAMPLE

Conditional Probability LabACTIVITY

Bayes' TheoremLESSON

Disease Testing AnalysisREAL-WORLD

From Outcomes to NumbersLESSON

Expected Value in GamesREAL-WORLD

Analyzing Binomial DistributionEXAMPLE

Probability Distributions PracticePRACTICE

Distribution ExplorerACTIVITY

Indicator Random VariablesLESSON

Monte Carlo MethodsREAL-WORLD

Discrete Probability SummarySUMMARY

Probability ReferenceFORMULA SHEET

Discrete Probability AssessmentASSESSMENT

Patterns in SequencesINTRO

Recurrence Relations FundamentalsLESSON

Modeling Compound InterestEXAMPLE

Recurrence Modeling PracticePRACTICE

Sequence GeneratorACTIVITY

Characteristic Equation MethodLESSON

Solving Fibonacci RecurrenceEXAMPLE

Recurrence SolverACTIVITY

Non-Homogeneous RecurrencesLESSON

Generating FunctionsLESSON

Catalan NumbersEXAMPLE

Generating Function LabACTIVITY

Divide and Conquer RecurrencesLESSON

Merge Sort AnalysisEXAMPLE

Tower of HanoiEXAMPLE

Dynamic ProgrammingREAL-WORLD

Advanced Recurrences PracticePRACTICE

Recurrences SummarySUMMARY

Recurrences ReferenceFORMULA SHEET

Recurrence Relations AssessmentASSESSMENT

Racing AlgorithmsINTRO

Big-O NotationLESSON

Analyzing Nested LoopsEXAMPLE

Complexity Analysis PracticePRACTICE

Search Algorithm ShowdownLESSON

Sorting Algorithm VisualizerACTIVITY

Algorithm Efficiency ComparisonREAL-WORLD

Lower BoundsLESSON

Complexity Theory PracticePRACTICE

Graph Algorithm ComplexityLESSON

Cache-Efficient AlgorithmsREAL-WORLD

The P vs NP ProblemLESSON

Traveling Salesman AnalysisEXAMPLE

Reduction DemonstratorACTIVITY

Algorithm Selection GuideREAL-WORLD

QuickSort AnalysisEXAMPLE

Algorithm Analysis SummarySUMMARY

Algorithm Analysis ReferenceFORMULA SHEET

Algorithm Analysis AssessmentASSESSMENT

Bridge to Advanced TopicsSUMMARY

Master ReferenceFORMULA SHEET

Comprehensive Final ExamQUIZ