Mathematics is a science that concerns theorems that must be proved within a system of axioms and definitions. With this book, the mathematical novice will learn how to prove theorems and explore the universe of abstract mathematics. The introductory chapters familiarise the reader with some fundamental ideas, including the axiomatic method, symbolic logic and mathematical language. This leads to a discussion of the nature of proof, along with various methods for proving statements. The subsequent chapters present some foundational topics in pure mathematics, including detailed introductions to set theory, number systems and calculus. Through these fascinating topics, supplemented by plenty of examples and exercises, the reader will hone their proof skills. This complete guide to proof is ideal preparation for a university course in pure mathematics, and a valuable resource for educators.
• A complete guide to constructing proofs • Introduces students to the world of abstract mathematics • Prepares students for further study in linear algebra, calculus and topology
Contents
Some notes on notation; To the students; For the professors; Part I. The Axiomatic Method: 1. Introduction; 2. Statements in mathematics; 3. Proofs in mathematics; Part II. Set Theory: 4. Basic set operations; 5. Functions; 6. Relations on a set; 7. Cardinality; Part III. Number Systems: 8. Algebra of number systems; 9. The natural numbers; 10. The integers; 11. The rational numbers; 12. The real numbers; 13. Cantor's reals; 14. The complex numbers; Part IV. Time Scales: 15. Time scales; 16. The Delta Derivative; Part V. Hints: 17. Hints for (and comments on) the exercises; Index.
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