Mathematical Proofs: A Transition to Advanced Mathematics

For courses in Transition to Advanced Mathematics or Introduction to Proof. Meticulously crafted, student-friendly text that helps build mathematical maturity Mathematical Proofs: A Transition to Advanced Mathematics, 4th Edition introduces students to proof techniques, analyzing proofs, and writing proofs of their own that are not only mathematically correct but clearly written. Written in a student-friendly manner, it provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as optional excursions into fields such as number theory, combinatorics, and calculus. The exercises receive consistent praise from users for their thoughtfulness and creativity.  They help students progress from understanding and analyzing proofs and techniques to producing well-constructed proofs independently. This book is also an excellent reference for students to use in future courses when writing or reading proofs. 0134746759 / 9780134746753 Chartrand/Polimeni/Zhang, Mathematical Proofs: A Transition to Advanced Mathematics, 4/e

Table of Contents:
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 Schröder - 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