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Discrete mathematics

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All knowledge/Mathematics/Discrete mathematics

foundationsopen problemsAdder circuitsAddition and difference rulesAdjacency matrixAlgorithmAlgorithm correctnessAntisymmetric relationArithmetic sequenceAugmenting pathSimplifying finite-state automataAxiom of choiceBayes’ theoremBiconditionalBig-O notationBig-Omega notationBig-Theta notationOne-to-one correspondenceBinary notationBinary searchBinary search treeBinary treeBinomial theoremBipartite graphBoolean algebraBoolean expressionBoolean functionBreadth-first searchCantor diagonalizationCantor's theoremCardinalityCartesian productCharacteristic functionChernoff boundChinese remainder theoremClassical logicClique and independent setCombination (binomial coefficient)Combinational circuitCombinatorial proofComplete graphConditional probabilityConditional statementCongruence modulo nConnected graphContrapositiveConverse and inverseCountable setDisproof by counterexampleCovarianceCut edgeDe Morgan’s lawsDerangementDijkstra’s shortest path algorithmDirichlet's approximation theoremDirect proofDirected graphDisjunctive normal formDivide and conquerDivisibilityDivision algorithmDual graphEdge coloringElement method of proofEqually likely probabilityEquivalence classEquivalence relationEuclidean algorithmEuclid’s lemmaEuler circuits and the degree criterionEuler's formula for planar graphsProof of existenceExistential quantifierExponential and logarithmic functionsExpected valueExtended Euclidean algorithmLogical fallaciesFermat’s little theoremFibonacci numbersFinite differencesFinite-state automatonFirst-order logicTarski's fixed-point theoremFloor and ceilingFormal languageFour-color theoremFunctionComposition of functionsFundamental theorem of arithmeticGreatest common divisorGenerating functionGeometric sequence and seriesGraphGraph coloringGraph isomorphismGroupGroup isomorphismHall's marriage theoremHalting problemHamiltonian circuitHandshake theoremHash functionHasse diagramHeapHeyting algebraIdealImage and inverse imageIncidence matrixInclusion-exclusion principleIndependent eventsIndexed familyInfinitude of the primesInsertion sortIntuitionistic logicInverse functionIrrationality of the square root of 2Kleene’s theoremKruskal’s algorithmLattice (order theory)Lattice pathLine graphLinear congruences and Diophantine equationsSecond-order linear homogeneous recurrencesLogarithmic and exponential ordersLogic gateLogical connectivesLogical equivalenceLoop invariantMatchingMathematical inductionMathematical proofMax-flow min-cut theoremMeet and joinMerge sortMersenne primeSolving recurrences by iterationMinimum spanning treeMöbius inversionModular arithmeticInverse modulo nMultinomial coefficientMultiplication ruleCombinations with repetitionNAND and NOR completenessNatural deductionNecessary and sufficient conditionsNegation of quantified statementsMultiple (nested) quantifiersNetwork flowExistence of noncomputable functionsNonregular languagesOne-to-one functionOnto functionDimension of a partial orderEven and odd integersParse treePartial functionPartial orderPartition of a setRefinement of a partitionPascal’s formulaPeano arithmeticPermutationPERT and critical path methodPigeonhole principlePlanar graphThe five regular polyhedraPower setPredicatePrime numberPrim’s algorithmProbability axiomsLogic programming (Prolog)Proof by division into casesProof by contradictionProof by contrapositionQuotient-remainder theoremRabin cryptosystemRandom variableRational and irrational numbersRecurrence relationRecursively defined setReflexive relationRegular expressionRelationRelational databaseRooted treeRSA cryptographyRules of inferenceRussell’s paradoxSample space and eventSequenceSetSet identitiesSet operationsSimple graphSimply typed lambda calculusProof by smallest counterexampleSpanning treeStatementStringStrong mathematical inductionStrongly connected componentStructural inductionSubgraphSubgroups and cosetsSubsetSummation and product notationSymmetric relationSymmetry groupTautology and contradictionTime efficiency of an algorithmTopological sortingTotal orderTower of HanoiTractable and intractable problemsTransitive relationTraveling salesman problemTreeTriangular numbersTruth tableTruth-value semanticsTwo’s complementUniversal conditional statementUniversal modus ponens and tollensUniversal quantifierVacuous truthValid argumentVarianceVertex coverDegree of a vertexWalks, paths and circuitsWell-founded inductionWell-ordering principleErdős conjecture on arithmetic progressionsErdős–Faber–Lovász conjectureErdős–Gyárfás conjectureErdős–Hajnal conjectureErdős–Ko–Rado theoremGreen–Tao theoremHadwiger–Nelson problemOff-diagonal Ramsey R(3,k)Growth rate of diagonal Ramsey numbersRoth's theorem (3-term APs)Optimal algorithm for the girth of a sparse graphČerný conjecture (synchronizing automata)Amenability of Thompson's group FNontrivial acyclic finite groupComputable counterexample to Sarnak's Möbius conjectureFrankl's union-closed sets conjectureP versus NPLinear programming in strongly polynomial timeSzemerédi's theoremTriangle removal lemma

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  • Logic
  • Proof and number theory
  • Induction and recursion
  • Sets, functions and relations
  • Counting and probability
  • Graphs and trees
  • Algorithms and automata
  • Open problem (dashed: since solved).

How we made this graph

We built it from the structure of 4 textbooks: Susanna S. Epp's Discrete Mathematics with Applications (5th edition, 2020), Oscar Levin's Discrete Mathematics: An Open Introduction (4th edition, 2025, free online), Jean Gallier and Jocelyn Quaintance's Mathematical Foundations and Aspects of Discrete Mathematics (1st edition, 2024) and Edward A. Scheinerman's Mathematics: A Discrete Introduction (3rd edition, 2013). Every idea and most links cite the section they come from. When more than one book teaches an idea, it's still one dot, and it points you to each book. Links marked "our reading" are judgement calls with no page behind them. None of the books' text is here.

A small model running on our own machine read the extra books a chapter at a time and suggested ideas, links and sections. We checked every suggestion before it went in and wrote all the wording ourselves.

Left to right is roughly "what you'd want to know first". It's just a suggestion; you can start anywhere. The problems on the right edge come from our Atlas of open problems.

Everything is a draft until a Coragora contributor checks it. (Coming soon)