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This book introduces students to the process of doing mathematics and prepares them to succeed in higher-level mathematics courses. By discussing proof techniques, problem solving methods, and the understanding of mathematical ideas, the book provides a solid foundation for students majoring in mathematics, science, and engineering. Students will learn to grasp the underlying concepts of a subject and how to apply these concepts to solving problems. While being able to understand and reproduce proofs of theorems, they will also gain the ability to comprehend the connections among the important concepts and techniques of each subject. This book is intended for a course on proofs and mathematical reasoning, and could also be used as a supplemental text in courses such as algebra, analysis, and linear algebra.
- A new chapter on cardinality includes material on finite, denumerable, uncountable sets, and cardinal numbers.
- Refined explanations of important concepts.
- A new chapter on discrete math covers basic counting principles, binomial coefficients, and recurrence relations.
- Problem solving methods are thoroughly integrated into the text.
- A brief but lucid development of the basic number systems of mathematics: the integers from the natural numbers, the rational from the integers, and the complex numbers from the reals; an axiomatic presentation of the reals is given.
- Many more problems appear than in the previous edition. New and updated examples and exercises appear throughout.
- Chapter II deals with logic and sets, two of the fundamental building blocks of mathematics. It covers both propositional and predicate calculus, and also introduces informally the concept of a set.
- Chapter III covers methods of proving theorems. The principle proof techniques presented are direct proof, proof by contradiction, proof by contraposition, mathematical induction, and case analysis.
- Based on the work of mathematician George Polya, Section 5 of Chapter III looks at the different methods one can use to attack a problem.
- Written in an entirely nontechnical fashion, Chapter VII explores some aspects of an individual's behavior and attitude that affect the way he or she actually learns and does mathematics. Following a path blazed by Alan Schoenfeld, the book examines the beliefs that many people bring to the study of mathematics and the way people behave when they work on mathematical problems.
1. How to Solve It.
2. Understanding the Problem.
Chapter II. THINKING LOGICALLY.
3. Propositional Calculus.
5. Predicates and Quantifiers.
Chapter III. PROVING THEOREMS.
6. Direct Proof.
7. Indirect Proof.
8. Mathematical Induction.
9. Case Analysis.
10. Attacking the Problem/Proof.
11. Looking Back.
Chapter IV. SETS AND RELATIONS.
12. Sets and Set Operations.
15. Equivalence Relations.
16. Number Systems and Well-Defined Operations.
17. The Real and Complex Numbers.
Chapter V. CARDINALITY.
18. Equinumerous Sets.
19. Finite Sets.
20. Denumerable Sets.
21. Uncountable Sets.
22. Cardinality and the Cantor-Bernstein Theorem.
Chapter VI. DISCRETE STRUCTURES.
23. Fundamental Combinatorial Principles.
24. Permutations and Combinations.
25. Binomial Coefficients and the Binomial Theorem.
26. Recurrence Relations.
27. Algebraic Properties of the Integers.
Chapter VII. DOING MATHEMATICS.
28. Controlling Your Thinking.
29. Attitudes and Beliefs.