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DM:010222A six-set inclusion order has two peaks and a missing join

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DM Hasse Cover ExtractionDM Maximal Meet Join Lattice Audit

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Sets, functions, relations, and equivalence classesDiscrete Mathematics · Mixed review

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DM:010326Formation trees keep one more atom than binary connective

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DM Structural Induction FormulasDM Formation Tree Leaf Invariant

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Induction, recursion, and invariantsDiscrete Mathematics · Mixed review

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DM:010423Three coprime clocks determine a unique residue modulo 504

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DM Constructive Crt BezoutDM Crt Uniqueness Modulus

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Divisibility, modular arithmetic, and elementary number theoryDiscrete Mathematics · Mixed review

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DM:010424RSA decryption needs Euler, and Euler needs a coprime message

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DM Euler Totient Units PermutationDM Rsa Coprime Message Correctness

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Divisibility, modular arithmetic, and elementary number theoryDiscrete Mathematics · Mixed review

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DM:010517Six patches onto four queues: the surjection count survives a type audit

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DM Surjection Inclusion ExclusionDM Stirling Second Kind Type Audit

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Combinatorics and inclusion-exclusionDiscrete Mathematics · Mixed review

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DM:010518Four labs sharing ten hours cannot exceed a per-lab cap

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DM Bounded Stars And BarsDM Inclusion Exclusion Generating Function Audit

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Combinatorics and inclusion-exclusionDiscrete Mathematics · Mixed review

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DM:010519Six-bead two-color bracelets split one chiral necklace pair

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DM Burnside Cycle Index DihedralDM Bracelet Listing Chiral Audit

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Combinatorics and inclusion-exclusionDiscrete Mathematics · Mixed review

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DM:010520The sixth derangement number is 265 by two independent recurrences

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DM Derangement Inclusion ExclusionDM Derangement Recurrence Consistency

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Combinatorics and inclusion-exclusionDiscrete Mathematics · Mixed review

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DM:010521A first-touch reflection subtracts the lattice paths that cross the diagonal

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DM Reflection Principle BarrierDM Catalan Two Formula Audit

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Combinatorics and inclusion-exclusionDiscrete Mathematics · Mixed review

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DM:010525Prefix residues force a consecutive block divisible by n

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DM Prefix Residue PigeonholeDM Consecutive Block Divisibility

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Combinatorics and inclusion-exclusionDiscrete Mathematics · Mixed review

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DM:010727A Prüfer word rebuilds its labeled tree and counts all of them

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DM Prufer Encode Decode AuditDM Cayley Count From Word Bijection

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Graphs, trees, connectivity, and matchingsDiscrete Mathematics · Mixed review

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DM:010728Hierholzer certifies an Euler trail that Hamiltonicity cannot replace

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DM Euler Trail Exact CriterionDM Hierholzer Splice Certificate

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Graphs, trees, connectivity, and matchingsDiscrete Mathematics · Mixed review

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DM:010729Four-sided faces make Euler's formula refuse a specified K_{3,3}

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DM Euler Formula Plane InductionDM Bipartite Face Bound Nonplanarity

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Graphs, trees, connectivity, and matchingsDiscrete Mathematics · Mixed review

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DM:010730BFS two-coloring is the certificate that no odd cycle exists

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DM Bipartite Odd Cycle CharacterizationDM Bfs Layer 2color Certificate

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Graphs, trees, connectivity, and matchingsDiscrete Mathematics · Mixed review

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DM:010731A Laplacian cofactor enumerates the weighted spanning trees

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DM Kirchhoff Cofactor ArithmeticDM Weighted Spanning Tree Audit

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Graphs, trees, connectivity, and matchingsDiscrete Mathematics · Mixed review

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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Logic ProofDiscrete Mathematics Problem Solving

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Logic ProofDiscrete Mathematics Problem Solving

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Logic ProofDiscrete Mathematics Problem Solving

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Logic ProofDiscrete Mathematics Problem Solving

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Logic ProofDiscrete Mathematics Problem Solving

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919011Prove that for every integer n, if n^2 is even then n is even, by proving the contrapositive.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919012Prove that there is no integer n such that n^2\equiv 2\pmod{4}.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919013Let a, b, and c be integers.

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919014Prove that \sqrt{2} is irrational.

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919015Prove by contradiction that if A and B are sets with A\cap B=\emptyset and A\cup B=A, then B=\emptyset.

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919016Prove that for every positive integer n, by mathematical induction.

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919017Prove by induction on n that for every integer n\ge 0 and every real number x\neq 1,

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919018Let P(n) be the statement “n^2+n is even”.

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919019(Honors) Prove that for every integer n\ge 1, by induction.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919020Let A=\{1,2,3,4\} and B=\{3,4,5,6\}.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919021Let U=\{1,2,3,4,5,6,7,8\} be the universe, A=\{1,2,3,4\}, and B=\{3,4,5,6\}.

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919022Using only the definitions of union, intersection, and complement (element-chasing), prove that for all sets A, B,…

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919023Prove that for all sets A and B, where complements are taken relative to a fixed universe U containing both A and B.

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919024Let A, B, and C be sets.

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919025Let A and B be sets.

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919026Using set identities (associativity, commutativity, distributivity, De Morgan, identity, and complement laws), alge…

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919027(Honors) Let A, B, and C be sets.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919028Define a binary relation R on the power set \mathcal{P}(\{1,2,3\}) by Prove that R is an equivalence relation, and…

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919029(Honors) Let A and B be nonempty finite sets, and let f:A\to B be a function.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919030Let f \colon \mathbb{Z} \to \mathbb{Z} be defined by f(n) = 3n+2 .

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919031Let g \colon \mathbb{R} \to \mathbb{R} be defined by g(x) = x^3 - x .

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919032Let A = \{1,2,3,4\} and B = \{a,b,c\} .

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919033Let f \colon \mathbb{N} \to \mathbb{N} be defined by f(n) = n+1 , and let g \colon \mathbb{N} \to \mathbb{N} be def…

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919034Let f \colon X \to Y and g \colon Y \to Z be functions.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919035Let f \colon X \to Y and g \colon Y \to Z be functions.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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Problem library

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Topic choices for Discrete Mathematics are shortened. Search still covers every problem.

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Problem results

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DM:010222A six-set inclusion order has two peaks and a missing join

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DM Hasse Cover ExtractionDM Maximal Meet Join Lattice Audit

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Sets, functions, relations, and equivalence classesDiscrete Mathematics · Mixed review

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DM:010326Formation trees keep one more atom than binary connective

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DM Structural Induction FormulasDM Formation Tree Leaf Invariant

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Induction, recursion, and invariantsDiscrete Mathematics · Mixed review

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DM:010423Three coprime clocks determine a unique residue modulo 504

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DM Constructive Crt BezoutDM Crt Uniqueness Modulus

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Divisibility, modular arithmetic, and elementary number theoryDiscrete Mathematics · Mixed review

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DM:010424RSA decryption needs Euler, and Euler needs a coprime message

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DM Euler Totient Units PermutationDM Rsa Coprime Message Correctness

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Divisibility, modular arithmetic, and elementary number theoryDiscrete Mathematics · Mixed review

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DM:010517Six patches onto four queues: the surjection count survives a type audit

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DM Surjection Inclusion ExclusionDM Stirling Second Kind Type Audit

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Combinatorics and inclusion-exclusionDiscrete Mathematics · Mixed review

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DM:010518Four labs sharing ten hours cannot exceed a per-lab cap

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DM Bounded Stars And BarsDM Inclusion Exclusion Generating Function Audit

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Combinatorics and inclusion-exclusionDiscrete Mathematics · Mixed review

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DM:010519Six-bead two-color bracelets split one chiral necklace pair

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DM Burnside Cycle Index DihedralDM Bracelet Listing Chiral Audit

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Combinatorics and inclusion-exclusionDiscrete Mathematics · Mixed review

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DM:010520The sixth derangement number is 265 by two independent recurrences

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DM Derangement Inclusion ExclusionDM Derangement Recurrence Consistency

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Combinatorics and inclusion-exclusionDiscrete Mathematics · Mixed review

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DM:010521A first-touch reflection subtracts the lattice paths that cross the diagonal

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DM Reflection Principle BarrierDM Catalan Two Formula Audit

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Combinatorics and inclusion-exclusionDiscrete Mathematics · Mixed review

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DM:010525Prefix residues force a consecutive block divisible by n

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DM Prefix Residue PigeonholeDM Consecutive Block Divisibility

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Combinatorics and inclusion-exclusionDiscrete Mathematics · Mixed review

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DM:010727A Prüfer word rebuilds its labeled tree and counts all of them

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DM Prufer Encode Decode AuditDM Cayley Count From Word Bijection

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Graphs, trees, connectivity, and matchingsDiscrete Mathematics · Mixed review

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DM:010728Hierholzer certifies an Euler trail that Hamiltonicity cannot replace

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DM Euler Trail Exact CriterionDM Hierholzer Splice Certificate

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Graphs, trees, connectivity, and matchingsDiscrete Mathematics · Mixed review

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DM:010729Four-sided faces make Euler's formula refuse a specified K_{3,3}

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DM Euler Formula Plane InductionDM Bipartite Face Bound Nonplanarity

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Graphs, trees, connectivity, and matchingsDiscrete Mathematics · Mixed review

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DM:010730BFS two-coloring is the certificate that no odd cycle exists

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DM Bipartite Odd Cycle CharacterizationDM Bfs Layer 2color Certificate

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Graphs, trees, connectivity, and matchingsDiscrete Mathematics · Mixed review

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DM:010731A Laplacian cofactor enumerates the weighted spanning trees

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DM Kirchhoff Cofactor ArithmeticDM Weighted Spanning Tree Audit

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Graphs, trees, connectivity, and matchingsDiscrete Mathematics · Mixed review

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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Logic ProofDiscrete Mathematics Problem Solving

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Logic ProofDiscrete Mathematics Problem Solving

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Logic ProofDiscrete Mathematics Problem Solving

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Logic ProofDiscrete Mathematics Problem Solving

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919011Prove that for every integer n, if n^2 is even then n is even, by proving the contrapositive.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919012Prove that there is no integer n such that n^2\equiv 2\pmod{4}.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919013Let a, b, and c be integers.

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919014Prove that \sqrt{2} is irrational.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919015Prove by contradiction that if A and B are sets with A\cap B=\emptyset and A\cup B=A, then B=\emptyset.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919016Prove that for every positive integer n, by mathematical induction.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919017Prove by induction on n that for every integer n\ge 0 and every real number x\neq 1,

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919018Let P(n) be the statement “n^2+n is even”.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919019(Honors) Prove that for every integer n\ge 1, by induction.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919020Let A=\{1,2,3,4\} and B=\{3,4,5,6\}.

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919021Let U=\{1,2,3,4,5,6,7,8\} be the universe, A=\{1,2,3,4\}, and B=\{3,4,5,6\}.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919022Using only the definitions of union, intersection, and complement (element-chasing), prove that for all sets A, B,…

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919023Prove that for all sets A and B, where complements are taken relative to a fixed universe U containing both A and B.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919024Let A, B, and C be sets.

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Logic ProofDiscrete Mathematics Problem Solving

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DM:9919025Let A and B be sets.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919026Using set identities (associativity, commutativity, distributivity, De Morgan, identity, and complement laws), alge…

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919027(Honors) Let A, B, and C be sets.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919028Define a binary relation R on the power set \mathcal{P}(\{1,2,3\}) by Prove that R is an equivalence relation, and…

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919029(Honors) Let A and B be nonempty finite sets, and let f:A\to B be a function.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919030Let f \colon \mathbb{Z} \to \mathbb{Z} be defined by f(n) = 3n+2 .

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919031Let g \colon \mathbb{R} \to \mathbb{R} be defined by g(x) = x^3 - x .

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919032Let A = \{1,2,3,4\} and B = \{a,b,c\} .

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919033Let f \colon \mathbb{N} \to \mathbb{N} be defined by f(n) = n+1 , and let g \colon \mathbb{N} \to \mathbb{N} be def…

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919034Let f \colon X \to Y and g \colon Y \to Z be functions.

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Logic ProofDiscrete Mathematics Problem Solving

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Propositional and predicate logic with proof methodsDiscrete Mathematics · Calculation

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DM:9919035Let f \colon X \to Y and g \colon Y \to Z be functions.

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Logic ProofDiscrete Mathematics Problem Solving

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Problem results

DM:9919001Let p, q, and r be propositions.

DM:9919002Let p, q, and r be propositions.

DM:9919003Negate the compound proposition and simplify the result so that negation appears only immediately before atomic pro…

DM:9919004Determine, with a truth table, whether the argument is valid.

DM:9919005Express the English sentence “A student passes the course if and only if the student attends every lecture and subm…

DM:9919006Let P(x) denote “x is prime” and E(x) denote “x is even”, where the domain of discourse is the set of positive inte…

DM:9919007Let the domain of discourse be the set of all real numbers.

DM:9919008Let the domain be the set of all people, and let L(x,y) mean “x likes y”.

DM:9919009Let the domain of discourse be the integers.

DM:9919010Consider the open sentences P(x,y): “x divides y” and Q(x): “x is odd”, with domain the set of integers.

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Problem results

DM:9919001Let p, q, and r be propositions.

DM:9919002Let p, q, and r be propositions.

DM:9919003Negate the compound proposition and simplify the result so that negation appears only immediately before atomic pro…

DM:9919004Determine, with a truth table, whether the argument is valid.

DM:9919005Express the English sentence “A student passes the course if and only if the student attends every lecture and subm…

DM:9919006Let P(x) denote “x is prime” and E(x) denote “x is even”, where the domain of discourse is the set of positive inte…

DM:9919007Let the domain of discourse be the set of all real numbers.

DM:9919008Let the domain be the set of all people, and let L(x,y) mean “x likes y”.

DM:9919009Let the domain of discourse be the integers.

DM:9919010Consider the open sentences P(x,y): “x divides y” and Q(x): “x is odd”, with domain the set of integers.