**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**

**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**

**2. Logic**

**2.1. Statements**

**2.2. The Negation of a Statement**

**2.3. The Disjunction and Conjunction of Statements**

**2.4. The Implication**

**2.5. More On Implications**

**2.6. The Biconditional**

**2.7. Tautologies and Contradictions**

**2.8. Logical Equivalence**

**2.9. Some Fundamental Properties of Logical Equivalence**

**2.10. Quantified Statements**

**2.11. Characterizations of Statements**

**Chapter 2 Supplemental Exercises**

**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**

**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**

**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**

**6. Mathematical Induction**

**6.1 The Principle of Mathematical Induction**

**6.2 A More General Principle of Mathematical Induction**

**6.3 Proof By Minimum Counterexample**

**6.4 The Strong Principle of Mathematical Induction**

**Chapter 6 Supplemental Exercises**

**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**

**Chapter 7 Supplemental Exercises**

**8. Prove or Disprove**

**8.1 Conjectures in Mathematics**

**8.2 Revisiting Quantified Statements**

**8.3 Testing Statements**

**Chapter 8 Supplemental Exercises**

**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**

**10. Functions**

**10.1 The Definition of Function**

**10.2 The Set of All Functions From A to B**

**10.3 One-to-one and Onto Functions**

**10.4 Bijective Functions**

**10.5 Composition of Functions**

**10.6 Inverse Functions**

**10.7 Permutations**

**Chapter 10 Supplemental Exercises**

**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 Schroeder - Bernstein Theorem**

**Chapter 11 Supplemental Exercises**

**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**

**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**

**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**

**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**

**16. Proofs in Ring Theory (Online)**

**16.1 Rings**

**16.2 Elementary Properties of Rings**

**16.3 Subrings**

**16.4 Integral Domains**

**16.5 Fields**

**Chapter 16 Supplemental Exercises**

**17. Proofs in Linear Algebra (Online)**

**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**

**Chapter 17 Supplemental Exercises**

**18. Proofs with Real and Complex Numbers (Online)**

**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**

**Chapter 18 Supplemental Exercises**

**19. Proofs in Topology (Online)**

**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**

**Chapter 19 Supplemental Exercises**

**Answers and Hints to Odd-Numbered Section Exercises**

**References**

**Index of Symbols**

**Index**