Mathematical Proofs. A Transition to Advanced Mathematics

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Cover Title Page Copyright Page Dedication Contents Chapter 0: Communicating Mathematics 0.1 Learning Mathematics 0.2 What Others Have Said About Writing 0.3 Mathematical Writing 0.4 Using Symbols 0.5 Writing Mathematical Expressions 0.6 Common Words and Phrases in Mathematics 0.7 Some Closing Comments About Writing Chapter 1: Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets 1.6 Cartesian Products of Sets Chapter 1 Supplemental Exercises Chapter 2: Logic 2.1 Statements 2.2 Negations 2.3 Disjunctions and Conjunctions 2.4 Implications 2.5 More on Implications 2.6 Biconditionals 2.7 Tautologies and Contradictions 2.8 Logical Equivalence 2.9 Some Fundamental Properties of Logical Equivalence 2.10 Quantified Statements 2.11 Characterizations Chapter 2 Supplemental Exercises Chapter 3: Direct Proof and Proof by Contrapositive 3.1 Trivial and Vacuous Proofs 3.2 Direct Proofs 3.3 Proof by Contrapositive 3.4 Proof by Cases 3.5 Proof Evaluations Chapter 3 Supplemental Exercises Chapter 4: More on Direct Proof and Proof by Contrapositive 4.1 Proofs Involving Divisibility of Integers 4.2 Proofs Involving Congruence of Integers 4.3 Proofs Involving Real Numbers 4.4 Proofs Involving Sets 4.5 Fundamental Properties of Set Operations 4.6 Proofs Involving Cartesian Products of Sets Chapter 4 Supplemental Exercises Chapter 5: Existence and Proof by Contradiction 5.1 Counterexamples 5.2 Proof by Contradiction 5.3 A Review of Three Proof Techniques 5.4 Existence Proofs 5.5 Disproving Existence Statements Chapter 5 Supplemental Exercises Chapter 6: Mathematical Induction 6.1 The Principle of Mathematical Induction 6.2 A More General Principle of Mathematical Induction 6.3 The Strong Principle of Mathematical Induction 6.4 Proof by Minimum Counterexample Chapter 6 Supplemental Exercises Chapter 7: Reviewing Proof Techniques 7.1 Reviewing Direct Proof and Proof by Contrapositive 7.2 Reviewing Proof by Contradiction and Existence Proofs 7.3 Reviewing Induction Proofs 7.4 Reviewing Evaluations of Proposed Proofs Exercises for Chapter 7 Chapter 8: Prove or Disprove 8.1 Conjectures in Mathematics 8.2 Revisiting Quantified Statements 8.3 Testing Statements Chapter 8 Supplemental Exercises Chapter 9: Equivalence Relations 9.1 Relations 9.2 Properties of Relations 9.3 Equivalence Relations 9.4 Properties of Equivalence Classes 9.5 Congruence Modulo n 9.6 The Integers Modulo n Chapter 9 Supplemental Exercises Chapter 10: Functions 10.1 The Definition of Function 10.2 One-to-one and Onto Functions 10.3 Bijective Functions 10.4 Composition of Functions 10.5 Inverse Functions Chapter 10 Supplemental Exercises Chapter 11: Cardinalities of Sets 11.1 Numerically Equivalent Sets 11.2 Denumerable Sets 11.3 Uncountable Sets 11.4 Comparing Cardinalities of Sets 11.5 The Schr¨oder-Bernstein Theorem Chapter 11 Supplemental Exercises Chapter 12: Proofs in Number Theory 12.1 Divisibility Properties of Integers 12.2 The Division Algorithm 12.3 Greatest Common Divisors 12.4 The Euclidean Algorithm 12.5 Relatively Prime Integers 12.6 The Fundamental Theorem of Arithmetic 12.7 Concepts Involving Sums of Divisors Chapter 12 Supplemental Exercises Chapter 13: Proofs in Combinatorics 13.1 The Multiplication and Addition Principles 13.2 The Principle of Inclusion-Exclusion 13.3 The Pigeonhole Principle 13.4 Permutations and Combinations 13.5 The Pascal Triangle 13.6 The Binomial Theorem 13.7 Permutations and Combinations with Repetition Chapter 13 Supplemental Exercises Chapter 14: Proofs in Calculus 14.1 Limits of Sequences 14.2 Infinite Series 14.3 Limits of Functions 14.4 Fundamental Properties of Limits of Functions 14.5 Continuity 14.6 Differentiability Chapter 14 Supplemental Exercises Chapter 15: Proofs in Group Theory 15.1 Binary Operations 15.2 Groups 15.3 Permutation Groups 15.4 Fundamental Properties of Groups 15.5 Subgroups 15.6 Isomorphic Groups Chapter 15 Supplemental Exercises Chapter 16: Proofs in Ring Theory 16.1 Rings 16.2 Elementary Properties of Rings 16.3 Subrings 16.4 Integral Domains 16.5 Fields Exercises for Chapter 16 Chapter 17: Proofs in Linear Algebra 17.1 Properties of Vectors in 3-Space 17.2 Vector Spaces 17.3 Matrices 17.4 Some Properties of Vector Spaces 17.5 Subspaces 17.6 Spans of Vectors 17.7 Linear Dependence and Independence 17.8 Linear Transformations 17.9 Properties of Linear Transformations Exercises for Chapter 17 Chapter 18: Proofs with Real and Complex Numbers 18.1 The Real Numbers as an Ordered Field 18.2 The Real Numbers and the Completeness Axiom 18.3 Open and Closed Sets of Real Numbers 18.4 Compact Sets of Real Numbers 18.5 Complex Numbers 18.6 De Moivre’s Theorem and Euler’s Formula Exercises for Chapter 18 Chapter 19: Proofs in Topology 19.1 Metric Spaces 19.2 Open Sets in Metric Spaces 19.3 Continuity in Metric Spaces 19.4 Topological Spaces 19.5 Continuity in Topological Spaces Exercises for Chapter 19 Answers and Hints to Selected Odd-NumberedExercises in Chapters 16–19 Answers to Odd-Numbered Section Exercises References Credits Index of Symbols Index