Introduction to Mathematical Proofs

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