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.
Table of Contents
Chapter I. SOLVING 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.