This book introduces students to the art and craft of writing proofs, beginning with the basics of writing proofs and logic, and continuing on with more in-depth issues and examples of creating proofs in different parts of mathematics, as well as introducing proofs-of-correctness for algorithms. The creation of proofs is covered for theorems in both discrete and continuous mathematics, and in difficulty ranging from elementary to beginning graduate level.
Just beyond the standard introductory courses on calculus, theorems and proofs become central to mathematics. Students often find this emphasis difficult and new. This book is a guide to understanding and creating proofs. It explains the standard “moves” in mathematical proofs: direct computation, expanding definitions, proof by contradiction, proof by induction, as well as choosing notation and strategies.
Contents:Getting Started:A First ExampleThe Starting Line: Definitions and AxiomsMatching and Dummy VariablesProof by Contradiction“If and Only If”Drawing PicturesNotationMore Examples of Proofs*Exercises Logic and Other Formalities:Propositional CalculusExpressions, Predicates, and QuantifiersRules of InferenceAxioms of Equality and InequalityDealing with SetsProof by InductionProofs and AlgorithmsExercisesDiscrete and Continuous:InequalitiesSome Proofs in Number TheoryCalculate the Same Thing in Two Different WaysAbstraction and AlgebraSwapping Sums, Swapping IntegralsEmphasizing the ImportantGraphs and NetworksReal Numbers and ConvergenceApproximating or Building “Bad” Things with “Nice” ThingsExercisesMore Advanced Proof-Making:Counterexamples and ProofsDealing with the InfiniteBootstrappingImpredicative DefinitionsDiagonal ProofsUsing DualityOptimizingGenerating FunctionsExercisesBuilding Theories:Choosing DefinitionsWhat Am I Modeling?Converting One Kind of Mathematics into AnotherWhat is an Interesting Question?ExercisesReadership: Undergraduates and graduates who study mathematical proofs, teachers, and high school students and general readers interested in mathematical proofs.Key Features:This book does not require students to master discrete structures or set theory before starting to understand proofs or how to write themConsistent with being a practical guide, the book starts with a proof, and explains how it worksWriting proofs is discussed for both discrete and continuous mathematics, including linear algebra, calculus, graph (or network) theory, number theory, and analysisStrategies for basic and more advanced proof writing are explained: when to use proof by contradiction, proof by induction, unpacking definitions, and so onLater chapters discuss more advanced issues that can be useful for more advanced undergraduate students and beginning graduate studentsThe book can be used as a textbook for a course of writing proofs, or as a supplement for courses involving proof writing, or as a self-study guide