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Full Description
Mathematical Proofsprepares 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.
Contents
0. Communicating MathematicsLearning MathematicsWhat Others Have Said About WritingMathematical WritingUsing SymbolsWriting Mathematical ExpressionsCommon Words and Phrases in MathematicsSome Closing Comments About Writing1. Sets1.1. Describing a Set1.2. Subsets1.3. Set Operations1.4. Indexed Collections of Sets1.5. Partitions of Sets1.6. Cartesian Products of SetsExercises for Chapter 12. Logic2.1. Statements2.2. The Negation of a Statement2.3. The Disjunction and Conjunction of Statements2.4. The Implication2.5. More On Implications2.6. The Biconditional2.7. Tautologies and Contradictions2.8. Logical Equivalence2.9. Some Fundamental Properties of Logical Equivalence2.10. Quantified Statements2.11. Characterizations of StatementsExercises for Chapter 23. Direct Proof and Proof by Contrapositive3.1. Trivial and Vacuous Proofs3.2. Direct Proofs3.3. Proof by Contrapositive3.4. Proof by Cases3.5. Proof EvaluationsExercises for Chapter 34. More on Direct Proof and Proof by Contrapositive4.1. Proofs Involving Divisibility of Integers4.2. Proofs Involving Congruence of Integers4.3. Proofs Involving Real Numbers4.4. Proofs Involving Sets4.5. Fundamental Properties of Set Operations4.6. Proofs Involving Cartesian Products of SetsExercises for Chapter 45. Existence and Proof by Contradiction5.1. Counterexamples5.2. Proof by Contradiction5.3. A Review of Three Proof Techniques5.4. Existence Proofs5.5. Disproving Existence StatementsExercises for Chapter 56. Mathematical Induction6.1 The Principle of Mathematical Induction6.2 A More General Principle of Mathematical Induction6.3 Proof By Minimum Counterexample6.4 The Strong Principle of Mathematical InductionExercises for Chapter 67. Prove or Disprove7.1 Conjectures in Mathematics7.2 Revisiting Quantified Statements7.3 Testing StatementsExercises for Chapter 78. Equivalence Relations8.1 Relations8.2 Properties of Relations8.3 Equivalence Relations8.4 Properties of Equivalence Classes8.5 Congruence Modulo n8.6 The Integers Modulo nExercises for Chapter 89. Functions9.1 The Definition of Function9.2 The Set of All Functions from A to B9.3 One-to-one and Onto Functions9.4 Bijective Functions9.5 Composition of Functions9.6 Inverse Functions9.7 PermutationsExercises for Chapter 910. Cardinalities of Sets10.1 Numerically Equivalent Sets10.2 Denumerable Sets10.3 Uncountable Sets10.4 Comparing Cardinalities of Sets10.5 The Schroeder-Bernstein TheoremExercises for Chapter 1011. Proofs in Number Theory11.1 Divisibility Properties of Integers11.2 The Division Algorithm11.3 Greatest Common Divisors11.4 The Euclidean Algorithm11.5 Relatively Prime Integers11.6 The Fundamental Theorem of Arithmetic11.7 Concepts Involving Sums of DivisorsExercises for Chapter 1112. Proofs in Calculus12.1 Limits of Sequences12.2 Infinite Series12.3 Limits of Functions12.4 Fundamental Properties of Limits of Functions12.5 Continuity12.6 DifferentiabilityExercises for Chapter 1213. Proofs in Group Theory13.1 Binary Operations13.2 Groups13.3 Permutation Groups13.4 Fundamental Properties of Groups13.5 Subgroups13.6 Isomorphic GroupsExercises for Chapter 1314. Proofs in Ring Theory (Online)14.1 Rings14.2 Elementary Properties of Rings14.3 Subrings14.4 Integral Domains14.5 FieldsExercises for Chapter 1415. Proofs in Linear Algebra (Online)15.1 Properties of Vectors in 3-Space15.2 Vector Spaces15.3 Matrices15.4 Some Properties of Vector Spaces15.5 Subspaces15.6 Spans of Vectors15.7 Linear Dependence and Independence15.8 Linear Transformations15.9 Properties of Linear TransformationsExercises for Chapter 1516. Proofs in Topology (Online)16.1 Metric Spaces16.2 Open Sets in Metric Spaces16.3 Continuity in Metric Spaces16.4 Topological Spaces16.5 Continuity in Topological SpacesExercises for Chapter 16Answers and Hints to Odd-Numbered Section ExercisesReferencesIndex of SymbolsIndex of Mathematical Terms
