Mathematical Proofs. A Transition to Advanced Mathematics

✍ Scribed by Gary Chartrand, Albert D. Polimeni, Ping Zhang

• Publisher: Pearson
• Year: 2018
• Tongue: English
• Leaves: 622
• Edition: 4
• Category: Library

✦ Table of Contents

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