Introduction to Mathematical Proofs

Häftad, Engelska, 2023

AvNicholas A. Loehr,USA) Loehr, Nicholas A. (Virginia Tech, Virginia,Nicholas A Loehr

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An Introduction to Mathematical Proofs presents fundamental material on logic, proof methods, set theory, number theory, relations, functions, cardinality, and the real number system. The text uses a methodical, detailed, and highly structured approach to proof techniques and related topics. No prerequisites are needed beyond high-school algebra. New material is presented in small chunks that are easy for beginners to digest. The author offers a friendly style without sacrificing mathematical rigor. Ideas are developed through motivating examples, precise definitions, carefully stated theorems, clear proofs, and a continual review of preceding topics. Features Study aids including section summaries and over 1100 exercises Careful coverage of individual proof-writing skills Proof annotations and structural outlines clarify tricky steps in proofs Thorough treatment of multiple quantifiers and their role in proofs Unified explanation of recursive definitions and induction proofs, with applications to greatest common divisors and prime factorizations About the Author:Nicholas A. Loehr is an associate professor of mathematics at Virginia Technical University. He has taught at College of William and Mary, United States Naval Academy, and University of Pennsylvania. He has won many teaching awards at three different schools. He has published over 50 journal articles. He also authored three other books for CRC Press, including Combinatorics, Second Edition, and Advanced Linear Algebra.

Produktinformation

Utgivningsdatum2023-01-21
Mått178 x 254 x 25 mm
Vikt1 220 g
FormatHäftad
SpråkEngelska
SerieTextbooks in Mathematics
Antal sidor412
FörlagTaylor & Francis Ltd
ISBN9781032475226

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Nicholas A. Loehr is an associate professor of mathematics at Virginia Technical University. He has taught at College of William and Mary, United States Naval Academy, and University of Pennsylvania. He has won many teaching awards at three different schools. He has published over 50 journal articles. He also authored three other books for CRC Press, including Combinatorics, Second Edition, and Advanced Linear Algebra.

LogicPropositions; Logical Connectives; Truth TablesLogical Equivalence; IF-StatementsIF, IFF, Tautologies, and ContradictionsTautologies; Quantifiers; UniversesProperties of Quantifiers: Useful DenialsDenial Practice; UniquenessProofsDefinitions, Axioms, Theorems, and ProofsProving Existence Statements and IF StatementsContrapositive Proofs; IFF ProofsProofs by Contradiction; OR ProofsProof by Cases; DisproofsProving Universal Statements; Multiple QuantifiersMore Quantifier Properties and Proofs (Optional)SetsSet Operations; Subset ProofsMore Subset Proofs; Set Equality ProofsMore Set Quality Proofs; Circle Proofs; Chain ProofsSmall Sets; Power Sets; Contrasting ∈ and ⊆Ordered Pairs; Product SetsGeneral Unions and IntersectionsAxiomatic Set Theory (Optional)IntegersRecursive Definitions; Proofs by InductionInduction Starting Anywhere: Backwards InductionStrong InductionPrime Numbers; Division with RemainderGreatest Common Divisors; Euclid’s GCD AlgorithmMore on GCDs; Uniqueness of Prime FactorizationsConsequences of Prime Factorization (Optional)Relations and FunctionsRelations; Images of Sets under RelationsInverses, Identity, and Composition of RelationsProperties of RelationsDefinition of FunctionsExamples of Functions; Proving Equality of FunctionsComposition, Restriction, and GluingDirect Images and PreimagesInjective, Surjective, and Bijective FunctionsInverse FunctionsEquivalence Relations and Partial OrdersReflexive, Symmetric, and Transitive RelationsEquivalence RelationsEquivalence ClassesSet PartitionsPartially Ordered SetsEquivalence Relations and Algebraic Structures (Optional)CardinalityFinite SetsCountably Infinite SetsCountable SetsUncountable SetsReal Numbers (Optional)Axioms for R; Properties of AdditionAlgebraic Properties of Real NumbersNatural Numbers, Integers, and Rational NumbersOrdering, Absolute Value, and DistanceGreatest Elements, Least Upper Bounds, and CompletenessSuggestions for Further Reading