Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition, prepares students for the more abstract mathematics courses that follow calculus. Appropriate for self-study or for use in the classroom, this text introduces students to proof techniques, analyzing proofs, and writing proofs of their own. Written in a clear, conversational style, this book provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as the theoretical aspects of fields such as number theory, abstract algebra, and group theory. It is also a great reference text that students can look back to when writing or reading proofs in their more advanced courses.
0. Communicating Mathematics Learning Mathematics What Others Have Said About Writing Mathematical Writing Using Symbols Writing Mathematical Expressions Common Words and Phrases in Mathematics 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 Exercises for Chapter 1 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 Exercises for Chapter 2 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 Exercises for Chapter 3 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 Exercises for Chapter 4 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 Exercises for Chapter 5 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 Exercises for Chapter 6 7. Prove or Disprove 7.1 Conjectures in Mathematics 7.2 Revisiting Quantified Statements 7.3 Testing Statements Exercises for Chapter 7 8. Equivalence Relations 8.1 Relations 8.2 Properties of Relations 8.3 Equivalence Relations 8.4 Properties of Equivalence Classes 8.5 Congruence Modulo n 8.6 The Integers Modulo n Exercises for Chapter 8 9. Functions 9.1 The Definition of Function 9.2 The Set of All Functions from A to B 9.3 One-to-one and Onto Functions 9.4 Bijective Functions 9.5 Composition of Functions 9.6 Inverse Functions 9.7 Permutations Exercises for Chapter 9 10. Cardinalities of Sets 10.1 Numerically Equivalent Sets 10.2 Denumerable Sets 10.3 Uncountable Sets 10.4 Comparing Cardinalities of Sets 10.5 The Schroder-Bernstein Theorem Exercises for Chapter 10 11. Proofs in Number Theory 11.1 Divisibility Properties of Integers 11.2 The Division Algorithm 11.3 Greatest Common Divisors 11.4 The Euclidean Algorithm 11.5 Relatively Prime Integers 11.6 The Fundamental Theorem of Arithmetic 11.7 Concepts Involving Sums of Divisors Exercises for Chapter 11 12. Proofs in Calculus 12.1 Limits of Sequences 12.2 Infinite Series 12.3 Limits of Functions 12.4 Fundamental Properties of Limits of Functions 12.5 Continuity 12.6 Differentiability Exercises for Chapter 12 13. Proofs in Group Theory 13.1 Binary Operations 13.2 Groups 13.3 Permutation Groups 13.4 Fundamental Properties of Groups 13.5 Subgroups 13.6 Isomorphic Groups Exercises for Chapter 13 14. Proofs in Ring Theory (Online) 14.1 Rings 14.2 Elementary Properties of Rings 14.3 Subrings 14.4 Integral Domains 14.5 Fields Exercises for Chapter 14 15. Proofs in Linear Algebra (Online) 15.1 Properties of Vectors in 3-Space 15.2 Vector Spaces 15.3 Matrices 15.4 Some Properties of Vector Spaces 15.5 Subspaces 15.6 Spans of Vectors 15.7 Linear Dependence and Independence 15.8 Linear Transformations 15.9 Properties of Linear Transformations Exercises for Chapter 15 16. Proofs in Topology (Online) 16.1 Metric Spaces 16.2 Open Sets in Metric Spaces 16.3 Continuity in Metric Spaces 16.4 Topological Spaces 16.5 Continuity in Topological Spaces Exercises for Chapter 16 Answers and Hints to Odd-Numbered Section Exercises References Index of Symbols Index of Mathematical Terms
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Pearson New International Edition
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