Mathematical Proofs. A Transition to Advanced Mathematics

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