Mathematical Proofs: A Transition to Advanced Mathematics

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Oktober 2007



"Mathematical Proofs: A Transition to Advanced Mathematics, 2/e, " prepares students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets. KEY TOPICS Communicating Mathematics, Sets, Logic, Direct Proof and Proof by Contrapositive, More on Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, Functions, Cardinalities of Sets, Proofs in Number Theory, Proofs in Calculus, Proofs in Group Theory. MARKET For all readers interested in advanced mathematics and logic.


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 7.4 A Quiz of "Prove or Disprove" Problems 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 Answers and Hints to Selected Odd-Numbered Exercises References Index of Symbols Index of Mathematical Terms
EAN: 9780321390530
ISBN: 0321390539
Untertitel: Sprache: Englisch.
Erscheinungsdatum: Oktober 2007
Seitenanzahl: 365 Seiten
Format: gebunden
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